NOISE R EDUCTION A NALYSIS Uno Ingard
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NOISE REDUCTION ANALYSIS
NOISE REDUCTION ANALYSIS Uno Ingard
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[email protected]. Copyright © 2010 by Jones and Bartlett Publishers, LLC All rights reserved. No part of the material protected by this copyright may be reproduced or utilized in any form, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without written permission from the copyright owner. The publisher recognizes and respects all marks used by companies, manufacturers, and developers as a means to distinguish their products. All brand names and product names mentioned in this book are trademarked or service marks of their respective companies. Any omission or misuse (of any kind) of service marks or trademarks, etc., is not an attempt to infringe on the property of others. ISBN: 978-1-934015-31-5 ISBN: 978-0-7637-8289-4 (e) Cover Design: Tyler Creative Library of Congress Cataloging-in-Publication Data Ingard, K. Uno. Noise reduction analysis / Uno Ingard. p. cm. ISBN 978-1-934015-31-5 (hardcover) 1. Noise control. I. Title. TD892.I56 2010 620.2’3–dc22 2009000229
6048 8875 Printed in the United States of America 13 12 11 10 09 10 9 8 7 6 5 4 3 2 1
Preface This book is based on notes resulting from my sporadic involvement in a variety of projects on noise and vibration and from lectures given in the department of Aeronautics and Astronautics at M. I. T. In principle, the most obvious and effective way of noise reduction is of course to eliminate the sources of the noise, providing that they can be identified and that their removal does not adversely affect the function of the machine or facility involved. The sources are of many different kinds and the noise reducing measures may be quite different. An example comes to mind from direct personal involvement. A power plant was forced to shut down because it emitted a very powerful howling tone which was deemed environmentally unacceptable. The source of the noise was traced to a number of solid bars which had been installed between opposing walls of the exaust stack to stabilize the stack. It so happened that the Aeolian tone from the wakes of the bars happened to coincide with the first acoustic cross mode of the duct and that led to a feedback instability and a very powerful excitation of this mode. Actually, the process was a little more complicated as the howling was periodic with a period of a few seconds. The cross mode became so intense that the related flow instability apparently increased the overall flow resistance of the duct, which in turn reduced the flow speed. Because of the flow dependence of the frequency, this led to a reduction of the frequency of the Aeolian tone from the rods and thus a removal of the coincidence with the acoustic cross mode frequency of the duct. This broke the feedback process and reduced the overall flow resistance of the rods resulting in an increase of the duct flow. In this manner the process was repeated periodically with a corresponding very intense periodic ‘howling.’ The problem to eliminate the noise in this case simply involved cutting down the rods. The stabilizing of the walls, if necessary, could be done externally. Often, as in this case, the mechanism of noise generation is unique to each particular facility. On the other hand, noise reduction by means of absorbers and attenuators is less dependent on the sources involved and lends itself to a treatment as given here. Discussions with students and colleagues at M. I. T. and engineers in industry have been very helpful, and they are gratefully acknowledged. Special thanks go to my former students at M. I. T., Dr. William Patrick, United Technologies, and Dr. George Maling, Du Pont, and to Terry Dear, Du Pont, for many helpful discussions and
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vi comments on a variety of interesting problems in noise control and for providing many of the references assembled in the lists at the end of the book. A grant from the Du Pont Company to the M. I. T. Aeronautics Department to support work in acoustics is also gratefully acknowledged. Uno Ingard, Professor Emeritus of Physics and of Aeronautics, M. I. T., October 2008.
Contents Preface
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I Absorbers 1 Introduction 1.1 General Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Terminology and Notation . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Chapter Organization . . . . . . . . . . . . . . . . . . . . . 2 Sound Absorption Mechanisms 2.1 Brief Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Steady Flow Through a (Narrow) Channel . . . . . . . . . . . . . . 2.2.1 Flow Resistance . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Acoustic Boundary Layers . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 The Viscous Boundary Layer . . . . . . . . . . . . . . . . . 2.3.2 The Thermal Boundary Layer . . . . . . . . . . . . . . . . . 2.3.3 Power Dissipation, Visco-Thermal Boundary Layer . . . . . 2.4 Sound Propagation in a Narrow Channel . . . . . . . . . . . . . . . 2.4.1 Propagation Constant . . . . . . . . . . . . . . . . . . . . . 2.4.2 Velocity and Temperature Profiles . . . . . . . . . . . . . . 2.4.3 Effect of Internal Damping of Flexible Wall . . . . . . . . . 2.4.4 Relaxation Times and a Note on Complex Compressibility in a Channel . . . . . . . . . . . . . . . . . 2.5 Impedances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Impedance Per Unit Length . . . . . . . . . . . . . . . . . . 2.5.2 Complex Density and Wave Impedance . . . . . . . . . . . 2.5.3 Perforated Plate . . . . . . . . . . . . . . . . . . . . . . . . 2.5.4 Wire Mesh Screen . . . . . . . . . . . . . . . . . . . . . . . 2.5.5 Perforated Plate-Screen Combination, Laminates . . . . . . 2.5.6 Effect of Acoustically Induced Motion . . . . . . . . . . . . 2.5.7 A Note on the Interpretation of Steady Flow Resistance Data 2.6 Visco-Thermal Admittance and Absorption Coefficient of a Rigid Wall 2.6.1 Equivalent Admittance . . . . . . . . . . . . . . . . . . . . 2.6.2 Absorption Coefficient . . . . . . . . . . . . . . . . . . . . . vii
3 3 4 6 7 7 9 9 10 10 12 13 14 15 18 19 19 21 22 23 24 28 29 30 30 31 31 33
viii 2.7
Mathematical Supplement . . . . . . . . . . . . . . . 2.7.1 Steady Flow Through a Narrow Channel . . . 2.7.2 Oscillatory Flow and Viscous Boundary Layer 2.7.3 The Thermal Boundary Layer . . . . . . . . . 2.7.4 Power Dissipation in the Boundary Layer . . 2.7.5 Sound Propagation in a Narrow Channel . . . 2.7.6 Impedances . . . . . . . . . . . . . . . . . .
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53 53 53 55 55 56 56 57 59 62 62 63 66 66 68 69 70 77 77 80 82 83 83 85 87 87 90 94 98 99
4 Resonators 4.1 Introduction and Summary . . . . . . . . . . . . . . 4.2 Absorption and Scattering . . . . . . . . . . . . . . . 4.2.1 Q-Value . . . . . . . . . . . . . . . . . . . . . 4.2.2 Helmholtz Resonator . . . . . . . . . . . . . 4.2.3 Resonator Absorber in a Diffuse Sound Field 4.2.4 Two-Dimensional Arrays of Resonators . . . .
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105 105 106 109 109 110 112
3 Sheet Absorbers 3.1 Introduction and Brief Summary . . . . . . . . . 3.1.1 Single Sheet Surface Absorber . . . . . . 3.1.2 Multisheet Absorber . . . . . . . . . . . . 3.1.3 Single Sheet as a ‘Volume’ Absorber . . . 3.2 Rigid Single Sheet with Cavity Backing . . . . . . 3.2.1 Flow Resistance and Impedances . . . . . 3.2.2 Resonances and Anti-Resonances . . . . . 3.2.3 Absorption Spectra . . . . . . . . . . . . . 3.2.4 Wire Screens . . . . . . . . . . . . . . . . 3.2.5 Effect of Honeycomb Cell Size . . . . . . 3.2.6 Examples and Comments . . . . . . . . . 3.3 Flexible Porous Sheet with Cavity Backing . . . . 3.3.1 The ‘Equivalent’ Impedance . . . . . . . 3.3.2 A Low Frequency Resonance . . . . . . . 3.3.3 Absorption Spectra . . . . . . . . . . . . . 3.3.4 Examples and Comments . . . . . . . . . 3.4 Lattice Absorbers . . . . . . . . . . . . . . . . . . 3.4.1 Periodic Lattice . . . . . . . . . . . . . . 3.4.2 Nonperiodic Lattice . . . . . . . . . . . . 3.5 ‘Volume’ Absorbers . . . . . . . . . . . . . . . . . 3.5.1 Reflection, Transmission, and Absorption . 3.5.2 Absorption Spectra, Infinite Sheet . . . . 3.5.3 Finite Sheet, Effect of Diffraction . . . . 3.6 Mathematical Supplement . . . . . . . . . . . . . 3.6.1 Rigid Single Sheet Cavity Absorber . . . . 3.6.2 Flexible Sheet Cavity Absorber . . . . . . 3.6.3 Uniform (Periodic) Lattice . . . . . . . . 3.6.4 Nonuniform Lattice . . . . . . . . . . . . 3.6.5 Sheet as a Volume Absorber . . . . . . . .
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4.2.5 Three-Dimensional Lattice of Resonators . . . . . . . 4.2.6 Transient Response and Reverberation . . . . . . . . . Acoustic Nonlinearity . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Perforated Plate With (Porous) Cavity Backing . . . . . 4.3.2 Nonlinear Absorption Characteristics . . . . . . . . . . Effects of Flow . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Flow Induced Acoustic Resistance . . . . . . . . . . . 4.4.2 Flow Excitation of Pipes and Orifices . . . . . . . . . . 4.4.3 Resonator in Free Field With Grazing Flow . . . . . . 4.4.4 Flow Excitation of a Side-Branch Resonator in a Duct . Mathematical Supplement . . . . . . . . . . . . . . . . . . . 4.5.1 Impedance of a Tube Resonator . . . . . . . . . . . . . 4.5.2 Absorption and Scattering Cross Sections . . . . . . . 4.5.3 Helmholtz Resonator . . . . . . . . . . . . . . . . . . 4.5.4 Three-Dimensional Array of Resonators . . . . . . . . 4.5.5 Acoustic Nonlinearity, Perforated Plate . . . . . . . . .
5 Rigid Porous Materials 5.1 Introduction and Summary . . . . . . . . . . . . . . 5.2 The Slot Absorber . . . . . . . . . . . . . . . . . . . 5.2.1 Input Impedance, Absorption Spectra . . . . 5.3 Isotropic Porous Layer, Physical Parameters . . . . . 5.3.1 Porosity . . . . . . . . . . . . . . . . . . . . . 5.3.2 Flow Resistance and Impedance . . . . . . . 5.3.3 Structure Factor . . . . . . . . . . . . . . . . 5.3.4 Mass Density of a Porous Material . . . . . . 5.3.5 Compressibility . . . . . . . . . . . . . . . . . 5.3.6 Discussion . . . . . . . . . . . . . . . . . . . 5.4 Wave Motion . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Propagation Constant . . . . . . . . . . . . . 5.4.2 Penetration Depth . . . . . . . . . . . . . . . 5.5 Absorption Spectra . . . . . . . . . . . . . . . . . . . 5.5.1 Infinite Layer . . . . . . . . . . . . . . . . . . 5.5.2 Finite Layer . . . . . . . . . . . . . . . . . . 5.5.3 Examples and Comments . . . . . . . . . . . 5.5.4 Effect of a Perforated Facing, Its Nonlinearity and Induced Motion . . . . . . . . . . . . . . 5.5.5 Effect of a Screen on a Porous Layer . . . . . 5.5.6 Nonuniform Porous Absorbers . . . . . . . . 5.5.7 Sheet Absorbers vs Uniform Porous Layer . . 5.6 Effect of Refraction in Grazing Flow . . . . . . . . . 5.6.1 View Angle vs Emission Angle . . . . . . . . . 5.6.2 The Boundary Layer . . . . . . . . . . . . . . 5.6.3 Effect on Absorption . . . . . . . . . . . . . . 5.6.4 Region of Total Reflection . . . . . . . . . . . 5.7 Mathematical Supplement . . . . . . . . . . . . . . .
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113 115 116 117 120 122 122 123 127 128 132 132 133 136 137 139
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143 143 146 148 154 154 155 156 156 157 157 157 157 159 161 161 162 164
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166 167 169 171 173 174 175 178 179 181
x 5.7.1 5.7.2 5.7.3
Slot Absorber . . . . . . . . . . . . . . . . . . . . . . . . Isotropic Porous Layer . . . . . . . . . . . . . . . . . . . Interaction Impedance, Impedance Per Unit Length, and Complex Density . . . . . . . . . . . . . . . . . . . 5.7.4 Propagation Constant and Wave Impedance . . . . . . . 5.7.5 Angle of Refraction . . . . . . . . . . . . . . . . . . . . 5.7.6 Input Impedance and Admittance, Absorption Coefficient . . . . . . . . . . . . . . . . . . . 5.7.7 Perforated Facing, Its Nonlinearity and Induced Motion 5.7.8 Anisotropic Layer . . . . . . . . . . . . . . . . . . . . . 5.7.9 Effect of Grazing Flow . . . . . . . . . . . . . . . . . . 5.7.10 Computational Considerations . . . . . . . . . . . . . . 6 Flexible Porous Materials 6.1 Introduction and Summary . . . . . . . . . . . 6.2 Coupled Waves . . . . . . . . . . . . . . . . . . 6.3 Dispersion Relation . . . . . . . . . . . . . . . 6.4 Field Distributions . . . . . . . . . . . . . . . . 6.4.1 Pressure and Velocity Fields . . . . . . . 6.4.2 Dissipation Function . . . . . . . . . . . 6.4.3 Examples and Comments . . . . . . . . 6.5 Absorption Spectra . . . . . . . . . . . . . . . . 6.5.1 General Comments . . . . . . . . . . . 6.5.2 Absorption Peaks But Not at Resonances 6.5.3 Effect of a Bonded Perforated Facing . 6.5.4 Examples and Comments . . . . . . . . 6.5.5 Porous Material with Closed Cells . . . 6.6 Nonlinear Effects and Shock Wave Reflection . 6.6.1 Apparatus . . . . . . . . . . . . . . . . . 6.6.2 Amplitude Dependence of Wave Speed 6.6.3 Reflection From a Flexible Porous layer 6.7 Measurement of Complex Elastic Modulus . . . 6.7.1 Apparatus . . . . . . . . . . . . . . . . . 6.7.2 Data Analysis . . . . . . . . . . . . . . . 6.8 Mathematical Supplement . . . . . . . . . . . . 6.8.1 Limp Material . . . . . . . . . . . . . . 6.8.2 Equations for Coupled Waves . . . . . . 6.8.3 Pressure and Velocity Fields . . . . . . . 6.8.4 Absorption Coefficients . . . . . . . . .
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197 197 198 199 201 201 203 204 204 204 205 206 208 210 211 212 214 214 217 217 218 219 219 221 225 228
II Duct Attenuators 7 Duct Acoustics 235 7.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 7.2 Wave Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
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7.4 7.5 7.6 7.7
7.2.1 Simple Illustration . . . . . . . . . . . . . . . . Measures of Silencer Performance . . . . . . . . . . . 7.3.1 Attenuation . . . . . . . . . . . . . . . . . . . . 7.3.2 Transmission Loss, TL and TL0 . . . . . . . . . 7.3.3 Insertion Loss, IL . . . . . . . . . . . . . . . . 7.3.4 Noise Reduction, NR . . . . . . . . . . . . . . 7.3.5 Numerical Examples . . . . . . . . . . . . . . . 7.3.6 Pressure Drop and Flow Noise (Self-Noise, SN) Lined Ducts . . . . . . . . . . . . . . . . . . . . . . . . ‘Reactive’ Silencers . . . . . . . . . . . . . . . . . . . . Acoustically Equivalent Silencers . . . . . . . . . . . . Additional Comments on Silencer Testing . . . . . . .
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8 Lined Ducts 8.1 Attenuation Mechanisms . . . . . . . . . . . . . . . . . . 8.1.1 Dissipation in Duct Liners . . . . . . . . . . . . . 8.1.2 Interference . . . . . . . . . . . . . . . . . . . . 8.2 Rectangular Ducts . . . . . . . . . . . . . . . . . . . . . 8.2.1 Locally Reacting Liner . . . . . . . . . . . . . . . 8.2.2 Nonlocally Reacting Liner . . . . . . . . . . . . . 8.2.3 Locally vs Nonlocally Reacting Liner, An Example 8.2.4 Attenuation vs Flow Resistance of Liner . . . . . 8.2.5 Example: A Proposed Air Intake Silencer for an Automobile . . . . . . . . . . . . . . . . . 8.3 Additional Duct Shapes . . . . . . . . . . . . . . . . . . 8.3.1 Rectangular Duct With All Sides Lined . . . . . . 8.3.2 Circular Duct . . . . . . . . . . . . . . . . . . . . 8.3.3 A Comparison, Circular vs Square Lined Duct . . 8.3.4 Annular Duct . . . . . . . . . . . . . . . . . . . . 8.4 Ducts in Series and in Parallel . . . . . . . . . . . . . . . 8.4.1 Ducts in Series . . . . . . . . . . . . . . . . . . . 8.4.2 Parallel Ducts, Interference Filter . . . . . . . . 8.5 Duct Liner Configurations . . . . . . . . . . . . . . . . . 8.5.1 Effects of a Perforated Facing . . . . . . . . . . . 8.5.2 Effect of Duct Liner Flexibility . . . . . . . . . . 8.5.3 Multilayer Liners . . . . . . . . . . . . . . . . . . 8.5.4 Slotted Liner . . . . . . . . . . . . . . . . . . . . 8.5.5 Effect of Partition Spacing . . . . . . . . . . . . . 8.6 Effects of Higher Modes and Flow . . . . . . . . . . . . 8.6.1 Higher Modes . . . . . . . . . . . . . . . . . . . 8.6.2 Convection . . . . . . . . . . . . . . . . . . . . . 8.6.3 Refraction . . . . . . . . . . . . . . . . . . . . . 8.6.4 Scaling Laws . . . . . . . . . . . . . . . . . . . . 8.6.5 Static Pressure Drop in Ducts . . . . . . . . . . . 8.7 Liquid Pipe lines, Elementary Aspects . . . . . . . . . . 8.7.1 Liquid Pipe Line with Slightly Compliant Walls .
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267 268 268 270 272 272 274 274 274 275 276 278 278 279 280 282 282 284 285 287 290 293 293
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Liquid Pipe Line With Air Layer Wall Treatment . . . . . . 295
9 Reactive Duct Elements 9.1 Uniform Duct Section . . . . . . . . . . . . . . . . . . . 9.1.1 Role of Source Impedance . . . . . . . . . . . . 9.2 Expansion Chamber . . . . . . . . . . . . . . . . . . . . 9.2.1 Transmission Loss . . . . . . . . . . . . . . . . . 9.2.2 Insertion Loss . . . . . . . . . . . . . . . . . . . 9.3 ‘Contraction’ Chamber . . . . . . . . . . . . . . . . . . . 9.3.1 Transmission Loss . . . . . . . . . . . . . . . . . 9.3.2 Insertion Loss . . . . . . . . . . . . . . . . . . . 9.4 Side-Branch Resonator in a Duct . . . . . . . . . . . . . 9.4.1 Transmission Loss . . . . . . . . . . . . . . . . . 9.4.2 Insertion Loss . . . . . . . . . . . . . . . . . . . 9.5 Perforated Plate . . . . . . . . . . . . . . . . . . . . . . 9.5.1 Effect of Mean Flow On the Acoustic Resistance 9.5.2 Shock Wave Interaction With an Orifice Plate . . 9.6 Attenuation in Turbulent Flow in Ducts . . . . . . . . . . 9.6.1 Static Pressure Drop . . . . . . . . . . . . . . . . 9.6.2 Sound Attenuation . . . . . . . . . . . . . . . . . 9.6.3 A Proposed AeroAcoustic Instability . . . . . . . 9.7 Nonlinear Attenuation . . . . . . . . . . . . . . . . . . . 9.8 On Air Induction Acoustics . . . . . . . . . . . . . . . . 9.8.1 Sound Pressure and Radiated Power . . . . . . . 9.8.2 Pipe Impedance . . . . . . . . . . . . . . . . . . 9.8.3 Radiated Power . . . . . . . . . . . . . . . . . . 9.8.4 Acoustic ‘Supercharge’ . . . . . . . . . . . . . . . 9.8.5 Numerical Example . . . . . . . . . . . . . . . .
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10 Mathematical Supplements and Comments 10.1 Supplement to Section 8.1 . . . . . . . . . . . . . . . . . . . . . 10.1.1 High Frequency Attenuation of Fundamental Mode in Lined Duct, Average Compressibility . . . . . . . . . 10.2 Supplement to Section 8.2 . . . . . . . . . . . . . . . . . . . . . 10.2.1 Locally Reacting Liners . . . . . . . . . . . . . . . . . . 10.2.2 Nonlocally Reacting Liner . . . . . . . . . . . . . . . . . 10.3 Supplement to Section 8.3, Other Duct Types . . . . . . . . . . 10.3.1 Rectangular Duct Lined On All Sides . . . . . . . . . . . 10.3.2 Circular Duct . . . . . . . . . . . . . . . . . . . . . . . . 10.3.3 Annular Duct . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Supplement to Section 8.6, Higher Modes and Flow . . . . . . . 10.5 Supplement to Section 8.7, Liquid Pipe Lines . . . . . . . . . . 10.5.1 Liquid Pipe Line With Slightly Compliant Walls . . . . . 10.5.2 Liquid Pipe Line With Air Layer Wall Treatment . . . . 10.6 Supplement to Section 9.1, Uniform Duct . . . . . . . . . . . . 10.7 Supplement to Section 9.6, Attenuation in Turbulent Duct Flow
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299 299 300 303 304 304 305 305 306 307 308 309 312 312 316 317 318 318 319 319 320 321 327 328 328 329
331 . . 331 . . . . . . . . . . . . . .
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331 332 332 337 341 341 342 344 346 349 349 352 356 357
xiii 10.7.1 Friction Factor in Turbulent Duct Flow . . . . . . . . . . . 357 10.7.2 Acoustic Perturbations and Dispersion Relation . . . . . . . 358 10.7.3 A Comparison With Visco-Thermal Attenuation . . . . . . . 359 A Transmission Matrices A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . A.1.1 Choice of Variables . . . . . . . . . . . . . . . . A.2 Application of Matrices . . . . . . . . . . . . . . . . . . A.2.1 Impedance . . . . . . . . . . . . . . . . . . . . A.2.2 Reflection and Absorption Coefficients . . . . . A.2.3 Transmission Coefficient and Transmission Loss A.2.4 Insertion Loss . . . . . . . . . . . . . . . . . . A.2.5 Noise Reduction . . . . . . . . . . . . . . . . . A.3 Commonly Used Matrices . . . . . . . . . . . . . . . . A.3.1 Porous Screen . . . . . . . . . . . . . . . . . . A.3.2 Area Discontinuities . . . . . . . . . . . . . . . A.3.3 Duct Element . . . . . . . . . . . . . . . . . . A.3.4 Contracted Duct Section, Perforated Plate . . . A.3.5 ‘Expansion Chamber’ and Elbow . . . . . . . . A.3.6 Lined Duct . . . . . . . . . . . . . . . . . . . . A.3.7 Side-Branch Tube . . . . . . . . . . . . . . . . A.3.8 Side-Branch Helmholtz Resonator . . . . . . . A.3.9 Parallel Channels . . . . . . . . . . . . . . . . . A.3.10 Rigid Porous Layers . . . . . . . . . . . . . . . A.3.11 Flexible Layer . . . . . . . . . . . . . . . . . . A.3.12 Thin Porous Plate . . . . . . . . . . . . . . . .
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361 361 362 362 362 363 364 367 369 370 370 372 375 376 377 378 379 380 381 383 383 387
B Flow Resistance Measurements B.1 Simple Method for Steady Flow . . . . . B.1.1 Equations of Motion . . . . . . . B.1.2 Nonlinearity of Flow Resistance . B.2 Simple Method for Oscillatory Flow . . . B.2.1 Some Experimental Results . . . B.2.2 Other Materials . . . . . . . . . B.2.3 A Supplementary Note . . . . .
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389 389 392 393 395 398 400 401
C Historical Notes and References, Absorbers C.1 ‘The Absorption Coefficient Problem’ . . . . . . . C.1.1 Sound Absorption in Porous Materials . . C.1.2 Regarding the Lists of Publications . . . . C.2 Lists of References . . . . . . . . . . . . . . . . . C.2.1 Sound Absorption, Concepts and Analysis C.2.2 Measurements, Methods, and Data . . . . C.2.3 Anechoic Wedges and Rooms . . . . . . . C.2.4 Resonators and Related Matters . . . . . C.2.5 ‘Functional’ or ‘Volume’ Absorbers . . . .
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403 403 404 406 407 407 412 420 424 426
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xiv D Historical Notes and References, Ducts 429 D.1 Brief Historical Note . . . . . . . . . . . . . . . . . . . . . . . . . . 429 D.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430 Index
439
List of Figures 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15 3.16 3.17 3.18 3.19 3.20 3.21 3.22
The Normalized Propagation Constant for Waves in a (Narrow) Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Velocity and Temperature Distribution in a Narrow Channel . . . Complex Compressibility . . . . . . . . . . . . . . . . . . . . . . . Impedance Per Unit Length in a Channel . . . . . . . . . . . . . . Example of ‘Micro-Perforate’ Characteristics . . . . . . . . . . . . Acoustic Impedances of Perforated Plate and Screen . . . . . . . . Absorption Characteristics, Perforated Plate-Screen Combination Absorption Coefficient of A Rigid Wall Due to Visco-Thermal Losses Function F Used in Propagation in Narrow Channels . . . . . . . Single Porous Sheet Cavity Absorber . . . . . . . . . . . . . . . . ‘Universal’ Rigid Sheet Cavity Absorption Spectra . . . . . . . . . 1/3 and 1/1 Octave Band Absorption Spectra . . . . . . . . . . . . Absorption Spectra, Discussion of Partition Spacing . . . . . . . . Rigid Sheet/Cavity Absorber, Optimum NRC . . . . . . . . . . . . Nonlinearity of Cavity Absorber . . . . . . . . . . . . . . . . . . . Equivalent Impedance of a Limp, Porous Sheet . . . . . . . . . . Limp Porous Sheet Cavity Normal Incidence Absorption Spectra . Limp Porous Sheet Cavity Diffuse Field Absorption Spectra . . . Absorption Spectra for a Perforated Plate/Screen Cavity Absorber Low Frequency Limp Sheet Cavity Resonance . . . . . . . . . . . Sheet Absorber, Alpha vs Resistance . . . . . . . . . . . . . . . . . Periodic Lattice Absorber . . . . . . . . . . . . . . . . . . . . . . Absorption Spectra of a Uniform Sheet Lattice . . . . . . . . . . . Absorption Spectra of Nonuniform Lattice of Limp Sheets . . . . Absorption Spectra of Nonuniform Lattice of Sheet Absorbers . . ‘Surface’ and ‘Volume’ Absorbers . . . . . . . . . . . . . . . . . . Reflection, Transmission, and Absorption: Plane Wave Incident on a Thin Porous Sheet . . . . . . . . . . . . . . . . . . . . . . . . . Coefficients of Reflection, Transmission, and Absorption . . . . . . Absorption Area or Cross Section of a Sheet in a Diffuse Field . . Effect of Diffraction on Sound Absorption by a Sheet . . . . . . . Two Different Unit Cells in Periodic Lattice . . . . . . . . . . . . xv
16 18 21 22 26 27 30 33 45 54 55 61 62 64 65 67 70 71 72 75 76 78 79 80 81 82 82 84 85 87 94
xvi 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 4.15
Radiation Impedance of a Resonator in a Wall . . . . . . . . . Absorption and Scattering Cross Sections . . . . . . . . . . . . Helmholtz Resonator . . . . . . . . . . . . . . . . . . . . . . . Propagation Constant in Three-Dimensional Resonator Array . Experimental Demonstration of Transient Resonator Response Acoustically Driven Vortex Rings . . . . . . . . . . . . . . . . Nonlinear Orifice Resistance . . . . . . . . . . . . . . . . . . . Nonlinear Absorption, Perforated Plate Resonator . . . . . . . Nonlinear Absorption Characteristics of Resonator . . . . . . . Orifice Whistle . . . . . . . . . . . . . . . . . . . . . . . . . . Tone Generation in Industrial Driers . . . . . . . . . . . . . . Stability Diagram of a Flow Excited Resonator . . . . . . . . . Flow Excited Resonances of a Side-Branch Cavity in a Duct . . Mode Coupling in Flow Excited Resonators . . . . . . . . . . Flow Excitation of a Slanted Resonator in a Duct . . . . . . . .
5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14
Maximum Possible Absorption Coefficient of a Rigid Porous Layer The ‘Slot Absorber’ . . . . . . . . . . . . . . . . . . . . . . . . . . Input Impedance of a Slot Absorber . . . . . . . . . . . . . . . . . Normal Incidence Absorption Spectra of a Slot Absorbers . . . . . Influence of Porosity on Sound Absorption of a Slot Absorber . . . Effect of Heat Conduction on the Absorption in a Slot Absorber . Angle of Refraction in a Slot Absorber . . . . . . . . . . . . . . . . Diffuse Field Absorption Characteristics of a Slot Absorber . . . . Propagation Constant in a Rigid Porous Material . . . . . . . . . . Penetration Depth of Sound in a Porous Layer . . . . . . . . . . . Absorption Spectra of a Locally Reacting Rigid Porous Layer . . . Angular Dependence of Absorption Coefficient . . . . . . . . . . Absorption Characteristics of a Locally Reacting Rigid Porous Layer Absorption Characteristics of a Nonlocally Reacting Rigid Porous Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Absorption Coefficient of Porous Layer vs Flow Resistance . . . . Effect of a Perforated Facing on Absorption Characteristics . . . . Effect of Resistive Screen on a Porous Layer . . . . . . . . . . . . Absorption Characteristics, Three Porous Layers . . . . . . . . . . Comparison, Uniform Porous Layer with Nonuniform . . . . . . . Porous Layer with Air Backing, a Comparison of Absorption Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Comparison of Sheet Absorbers and Rigid Porous Layers . . . . View Angle vs Emission Angle in a Wind Tunnel . . . . . . . . . . Refraction at a Flow Boundary . . . . . . . . . . . . . . . . . . . . Vortex Sheet Model of Boundary Layer . . . . . . . . . . . . . . . Effect of Boundary Layer Flow on Sound Absorption . . . . . . . Effect of Refraction on Sound Absorption for Stiffness Reactive Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Region of Total Reflection in a Wind Tunnel . . . . . . . . . . . .
5.15 5.16 5.17 5.18 5.19 5.20 5.21 5.22 5.23 5.24 5.25 5.26 5.27
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106 108 110 114 115 117 119 120 121 124 126 128 129 130 131 145 146 149 150 151 152 153 154 160 160 161 162 163 164 165 167 168 169 170 171 172 175 176 177 178 180 181
xvii 5.28 5.29
Polar and Azimuthal Angles, Slot Absorber . . . . . . . . . . . . . 184 Angle of Refraction in a Porous Material . . . . . . . . . . . . . . 189
6.1 6.2 6.3 6.4 6.5 6.6 6.7
Dispersion Relation, Flexible Porous Material . . . . . . . . . Velocity Distributions in Flexible Porous Layer . . . . . . . . . Pressure Amplitude Distributions in a Flexible Porous Layer . Dissipation Functions in a Flexible Porous Layer . . . . . . . . Absorption Spectra of a Flexible Porous Layer . . . . . . . . . Effect of Perforated Plate Bonded to a Flexible Porous Layer . Absorption Spectra of a Flexible Porous Layer with Perforated Facing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flexible Porous Layer with Perforated/Screen Facing . . . . . Shock Tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . Amplitude Dependence on Wave Speed . . . . . . . . . . . . Shock Wave Reflection Coefficients . . . . . . . . . . . . . . . Shock Wave Reflections from Flexible Porous Layers . . . . . Compression of Flexible Layer by a Shock Wave . . . . . . . . Apparatus for Measuring Compliance of a Porous Material . .
6.8 6.9 6.10 6.11 6.12 6.13 6.14 7.1 7.2 7.3 7.4 7.5 7.6 7.7 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10 8.11 8.12 8.13 8.14 8.15
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200 202 203 204 205 207
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208 210 212 214 215 216 216 218
Generation of a Higher Acoustic Mode in a Duct . . . . . . . . . . Measurement of Insertion Loss . . . . . . . . . . . . . . . . . . . Comparion of TL, TL0, IL, and NR of Lined Duct . . . . . . . . . Attenuation Spectrum . . . . . . . . . . . . . . . . . . . . . . . . Comparison of Insertion Loss and Transmission Loss . . . . . . . Acoustically Equivalent Duct Configurations for the Fundamental Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Comparison of TL, TL0, and NR . . . . . . . . . . . . . . . . .
238 242 243 247 247
Attenuation Spectra of a Rectangular Duct, Local Reaction . . . . Optimum Design of Lined Duct . . . . . . . . . . . . . . . . . . . Attenuation Characteristics of Rectangular Duct, Nonlocally Reacting Liner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Porous Plug Characteristics . . . . . . . . . . . . . . . . . . . . . A High Frequency Muffler for Special Application . . . . . . . . . Locally vs Nonlocally Reacting Duct Liner, an Example . . . . . . Attenuation in a Rectangular Duct vs Flow Resistance/Inch of Liner Proposed Air Induction Muffler for an Automobile . . . . . . . . . Rectangular Duct Lined on all Four Sides . . . . . . . . . . . . . Attenuation in a Rectangular Duct with Four Sides Lined . . . . . Attenuation Spectra of a Square Duct Lined with a Locally Reacting Porous Liner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Circular and Annular Lined Ducts . . . . . . . . . . . . . . . . . . Attenuation Spectra of a Circular Duct with a Locally Reacting Porous Liner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Attenuation in a Circular Lined Duct . . . . . . . . . . . . . . . . Attenuation in an Annular Duct . . . . . . . . . . . . . . . . . . .
250 253 258 260 262 263 264 265 266 267 268 268 269 270 271 272 273
xviii 8.16 8.17 8.18 8.19 8.20 8.21 8.22 8.23 8.24 8.25 8.26 8.27 8.28 8.29 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 9.10
Nonuniform Duct . . . . . . . . . . . . . . . . . . . . . . . . . . Effect of Open Area of a Perforated Facing on Attenuation . . . . Effect of Nonlinearity and Induced Motion of a Perforated Facing on Attenuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . Transmission Loss, Duct with Multilayer Liner . . . . . . . . . . . Attenuation in Duct with Slotted Duct Liner . . . . . . . . . . . . Concerning Optimum Partition Spacing in a Duct Liner . . . . . . Geometrical Acoustics and Duct Attenuation . . . . . . . . . . . . Effect of Refraction in a Duct . . . . . . . . . . . . . . . . . . . . Example of the Effect of Flow on Insertion Loss . . . . . . . . . . Effect of Temperature on Attenuation in a Lined Duct . . . . . . Friction Factor in Pipe Flow . . . . . . . . . . . . . . . . . . . . . Propagation in a Liquid Pipe Line . . . . . . . . . . . . . . . . . . Air Layer Attenuator in a Liquid Pipe Line . . . . . . . . . . . . . Transmission Characteristics in a Water Line with Air Layers . . .
277 279 280 281 283 286 286 289 291 295 296 296
9.13 9.14 9.15 9.16 9.17
Insertion Loss and Input Impedance of a Straight Pipe Section . Transmission Loss of an Expansion Chamber . . . . . . . . . . . Insertion Loss of an Expansion Chamber . . . . . . . . . . . . . Transmission Loss of a Contraction Chamber . . . . . . . . . . . Insertion Loss of the Contraction Chamber . . . . . . . . . . . . Side-Branch Resonator in a Duct . . . . . . . . . . . . . . . . . Transmission Loss of a Side-Branch Resonator . . . . . . . . . . Insertion Loss of Side-Branch Tube, High Source Impedance . . Automotive Labyrinth Resonator for Air Induction . . . . . . . . Insertion Loss of a Side-Branch Resonator, Low Source Impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Absorption, Reflection, and Transmission Coefficients of a Perforated Plate with Flow . . . . . . . . . . . . . . . . . . Nonlinear Reflection, Transmission, and Absorption Coefficient of Perforated Plate . . . . . . . . . . . . . . . . . . . . . . . . . Shock Wave Reflection from an Orifice Plate . . . . . . . . . . . Nonlinear Attenuation . . . . . . . . . . . . . . . . . . . . . . . Air Induction into a Cylinder of a Combustion Engine . . . . . . Regarding Acoustic Supercharging . . . . . . . . . . . . . . . . Regarding Acoustic Supercharging . . . . . . . . . . . . . . . .
10.1 10.2
Rectangular Duct Lined on Two Sides with Different Liners . . . 332 Liquid Pipe Line with Slightly Compliant Walls . . . . . . . . . . 351
A.1 A.2 A.3 A.4 A.5 A.6
Four-Pole Network . . . . . Acoustic ‘Barrier’ . . . . . . Area Expansion in a Duct . Area Contraction in a Duct . Side-Branch Tube in a Duct Parallel Duct Attenuator . .
9.11 9.12
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302 303 304 306 307 308 309 310 312
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361 363 372 374 379 381
xix B.1 B.2 B.3 B.4 B.5
Simple Apparatus for Steady Flow Resistance Measurement Flow Speed Dependence on Flow Resistance . . . . . . . . Flow Resistance Apparatus for Oscillatory Flow . . . . . . . Measured Frequency Dependence of Resistance . . . . . . . Flow Resistance and Reactance of Screens . . . . . . . . . .
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390 394 396 399 399
Part I
Absorbers
Chapter 1
Introduction 1.1 GENERAL COMMENTS Normally, noise is not only a nuisance, but it can have more serious adverse effects on both human beings and machines. It is frequently referred to as a form of ‘pollution’ and has been the object of environmental regulations. Noisy facilities, in violation of such regulations, in some cases have been forced to close until compliance is achieved.1 Furthermore, high level noise and vibration often give rise to structural failures as occasionally occur in control valves as a result of flow induced instability (through feedback) and resulting violent oscillations. In practice the efforts to reduce noise often involve a ‘quick fix’ attempt based on engineering ‘common sense’ and on trial and error approaches. If this turns out to be unsatisfactory, however, a ‘basic’ study of the problem is often called for. This involves identifying and trying to understand the physical mechanisms involved in the generation and transmission of sound and vibration and then finding feasible means of noise reduction. This can become a tedious and lengthy effort often because of the constraints that the measures undertaken must not adversely affect the function of the machine or process involved. As the subtitle of this book indicates, it is divided into two parts, noise reduction by means of 1. absorbers and 2. duct attenuators.2 The First Part, Absorbers, starts out with a discussion of basic mechanisms of absorption by which acoustic energy is converted into heat in viscous and thermal boundary layers in a sound field. Analysis of sound propagation through narrow channels is then used as a basis for applications that follow. Thus, extensive treatments are given of the absorption characteristics of porous sheets, both rigid and flexible, of acoustic resonators, and of porous material in bulk, both rigid and flexible. Multilayer absorbers are also analyzed to seek means of improving and ‘shaping’ the absorption characteristics. 1 Occasionally, noise is generated intentionally for the purpose of masking irritating sounds; the masking noise often is referred to as acoustic ‘perfume.’ 2 The term ‘Noise Control’ has not been used as the title of the book since it implies a broader scope including such topics as noise sources, atmospheric acoustics, human response to noise, and considerable emphasis on noise regulations.
3
4
NOISE REDUCTION ANALYSIS
The bulk of these studies deal with ‘ordinary’ conditions, where linear acoustic theory is applicable, but a section is devoted to nonlinear effects including a discussion of some experimental results with shock waves. The Second Part of the book, Duct Attenuators, starts out with a description of how their performance is described, calculated, and measured. We distinguish between ‘dissipative’ and ‘reactive’ ducts. The dissipative ducts are lined with one or more layers of porous material, and the chapters that follow present extensive analyses of the attenuation characteristics of different duct configurations. Of particular interest is the comparison of ducts with locally and nonlocally reacting liners. In the ‘reactive’ duct, reflection plays a major role. In addition to the chapters dealing directly with absorbers and ducts, there are four appendices. The first of these is a fairly substantial discussion of transmission matrices as they are of considerable value in facilitating the analysis of both absorber and duct problems. The second appendix describes simple methods (and apparatus) for measuring the flow resistance of porous materials for both steady and oscillatory flow. The two remaining appendices present brief historical reviews of absorbers and duct attenuators and categorized lists with a fairly large number of references. The present lists could provide a start for anyone who is intent on producing a database on everything published in the fields considered here. No attempts have been made to assign credit for contributions to various aspects; it would have required a considerable additional effort, which was not deemed essential in this context. The main part of each chapter is essentially descriptive and contains numerical results, which should be of direct interest for design work. The mathematical analysis is generally placed in an analytical supplement at the end of each chapter and is included for readers who want to go deeper into the subject matter. To strictly follow this scheme of separation turns out to be awkward at times, and some deviations are to be found here and there. Actually, important final equations are presented in the descriptive part and are framed for emphasis. Usually they are the basis for the numerical results shown in numerous graphs, but can be skipped in a first reading without lack of continuity.
1.2 TERMINOLOGY AND NOTATION This section deals with definitions of acoustically relevant physical quantities and a few commonly used terms. Harmonic motion refers to a time dependence expressed by a periodic function, A cos(ωt − φ), where A is the amplitude, ω the angular frequency, and φ, the phase angle. The angular frequency is ω = 2πf and the frequency is f = 1/T , where T is the period. Frequently, both of these quantities will be referred to simply as frequency when there is no risk for confusion. In defining a complex amplitude, we have used cos(ωt) = {exp(−iωt)} with the time factor exp(−iωt) rather than exp(iωt), so that in going from the time domain to the frequency domain, ∂/∂t → −iω. Sound pressure p(t) is the time dependent variation in pressure caused by compressions and rarefactions of a gas (or whatever material is involved) and the corresponding time dependent velocity of the fluid will be denoted by u(t). Harmonic time
INTRODUCTION
5
dependence is generally assumed and the complex amplitudes of these quantities are p(ω) and u(ω). Unless there is risk for confusion, the argument ω will be omitted. The fluid velocity u should not be confused with the speed of propagation of a sound wave, which will be referred to as the speed of sound, c, or sound speed. The fluid velocity u is sometimes referred to as particle velocity. The material used for sound absorption is usually porous. The difference between a porous sheet and a porous layer in bulk in general is that the sheet has to be very thin in comparison with a wavelength, typically a small fraction of an inch, so that the compression of the material and the air in it can be neglected. The terms screen and sheet will be regarded as synonyms. By an acoustically rigid sheet or porous material in this book is meant that it does not move when it interacts with sound. Normally, this usage would be a bit of a misnomer since any object will be induced to move to some extent by a sound wave. By contrast, a flexible material is mobile (even though the term normally implies that the material can be bent or flexed, which is not necessary for it to be mobile). A limp porous sheet is one without bending stiffness and a limp porous layer in bulk has no stiffness. A rigid wall, which is also impervious to air, will sometimes be called acoustically hard. The impedance is the ratio of the complex amplitudes of pressure and fluid velocity in harmonic time dependence at a given frequency. In some presentations, but not here, the volume velocity (i.e., the product of velocity and the flow area involved) is used as a variable instead of velocity. To distinguish the impedance thus obtained and the impedance used here, our impedance is sometimes referred to as ‘specific’ to indicate that a unit area is involved for the volume flow. We shall use the term impedance only, without the qualifier ‘specific.’ (The distinction between the use of velocity and volume velocity as variables is slightly more than cosmetic when we come to the definition and use of transmission matrices, as discussed in Appendix A.) The notation used here for the impedance is normally a lowercase Roman letter, such as z, with the resistive and reactive (real and imaginary) parts typically denoted by r and x. The impedance in a plane traveling wave is ρc, the wave impedance (resistance),3 which is approximately 420 MKS units (42 CGS) for air at room temperature. In most of the equations in this book the normalized impedance, resistance, and reactance, will be used, normalized with respect to ρc and designated usually by Greek letters, such as ζ , θ , and χ , where ζ = z/ρc, etc. Thus, a quantity ζ , will be referred to simply as ‘impedance’ without the qualifier ‘normalized,’ since it is already implied by the use of Greek letters. The use of ‘rayl’ as a name for a unit of resistance is not used here, CGS and MKS are sufficient. Furthermore, since most of the time we deal with normalized quantities, the question of units is secondary. The absorption coefficient is denoted by α and the pressure reflection coefficient by R. In a diffuse field, where all directions of sound propagation are equally probable, approximately realized in a large reverberation room, it is often the average absorption
3 ρ, density, and c, sound speed.
6
NOISE REDUCTION ANALYSIS
coefficient over all angles of incidence which is of interest, and we have usually used the term diffuse field absorption coefficient rather than the frequently used ‘statistical average’ to designate this quantity. To differentiate between normal incidence and diffuse field values one may use α0 for normal incidence and α1 and α2 for the diffuse field values corresponding to a locally and nonlocally reacting absorber, respectively. In practice, an average absorption coefficient in a 1/3 or 1/1 octave band, is often used; no special notations are given to these quantities. The Noise Reduction Coefficient, NRC, is the arithmetic mean of the octave band absorption coefficients in the octave bands at 250, 500, 1000, and 2000 Hz, and the corresponding values for normal incidence and diffuse field have been denoted by NRC0, NRC1 (local reaction), and NRC2 (nonlocal reaction). The propagation constant of a plane wave in free field is k = ω/c = 2π/λ, where c is the sound speed. When dealing with other materials, we have generally used the notation q for the propagation constant with the normalized value Q = q/k. It is difficult to avoid conflicts in notation, and we have allowed that to happen several times in regard to the use of the symbol q or Q. The ‘quality’ of a resonator has been denoted by Q as have been the mass flow rate and heat flow rate; only the context saves us from confusion. The term acoustically compact refers to an object with a characteristic linear dimension small compared to a wavelength. In the discussion of local and nonlocal reaction of absorbers we have used the term ‘honeycombed’ to indicate that the (air) layer in an absorber is partitioned into acoustically compact cells. In most figure captions, English units have been used. However, whenever numerical computations are involved, conversion to CGS or MKS is usually made.
1.2.1 Chapter Organization Each chapter is divided into two parts; the first, the main part, is mainly descriptive and the second, the supplement, contains derivations and mathematical details. Results of numerical analysis and related graphs are shown and discussed in the main part, which should be sufficient for the reader who has no interest or time to devote to mathematical details. Actually, some important equations, which have been used for graphs, have been included in the main part and placed within boxes for emphasis, but they can be skipped if desired without lack of continuity. Although this scheme appears to be a good one at first sight, it becomes a bit awkward at times if it is to be followed strictly, and some flexibility in this matter has been allowed.
Chapter 2
Sound Absorption Mechanisms As indicated in Section 1.2.1 on chapter organization, we have attempted to place most of the mathematical details and derivations of each chapter in a separate section which can be skipped at the first reading or skipped altogether by the reader who is interested mainly in numerical results. In this chapter, most of this mathematical analysis is summarized in Section 2.7. Some of the most important results, often the basis for the numerical results presented in graphs, are duplicated (and often framed) in the main part of the chapter.
2.1 BRIEF SUMMARY The most important mechanism of sound absorption in porous materials is due to viscosity and, to some extent, heat conduction. Absorption materials generally are porous structures, rigid or flexible, in which the pore size is quite small, typically of the order of a few thousandths of an inch. This chapter deals in some detail with viscous and thermal effects at solid boundaries in a sound field, particularly in narrow channels. It is essential to the understanding of sound absorption in porous materials and a basis for many parts of this book. The macroscopic physical parameters of a porous material that can readily be measured include porosity, flow resistance, weight, and quantities that describe the flexibility of the material. On the microscopic level, the geometry and distribution of the pores or voids in the material or the size distribution of fibers and their arrangement need to be specified, but in general, such details are not available. In Chapter 2, on sheet absorbers, flow resistance is defined operationally and considered to be a known quantity in terms of which the absorption spectra of sheet-cavity absorbers can be determined. Now the flow resistance will be pursued a bit further. For a porous material in general with its complex structure, it appears intractable to calculate the flow resistance in terms of geometrical parameters that describe the micro-structure of the material, even if such a description were available. Unless specific simple models are used, the best one can do is to use dimensional considerations to express the dependence of the flow resistance on basic geometrical and physical parameters, such as pore size distribution, porosity, and shear viscosity. 7
8
NOISE REDUCTION ANALYSIS
The flow resistance per unit thickness of the porous material is then found to be proportional to the coefficient of shear viscosity of the fluid involved and inversely proportional to the square of the characteristic pore size of the material. For a fibrous material with a given porosity, this means that the flow resistance per unit thickness is inversely proportional to the square of the fiber diameter. To gain further insight, it is useful to study the behavior of a structure that can be described in simple geometrical terms and analyzed from first principles. Sound absorption by capillary tube absorbers was studied already by Rayleigh (what else is new!) in his classic Theory of Sound (§348–351), where he extended the also classic Kirchhoff analysis of the visco-thermal attenuation in a tube to include the case when the acoustic boundary layer is not necessarily thin compared to the width of the tube. In this chapter we extend the analysis of this problem, including studies of the temperature and axial velocity distributions, and express the results in such a manner as to make them readily applicable to porous materials in Chapter 5. Steady flow. The first part of the chapter is devoted to steady flow and expressions for the steady flow resistance per unit length in narrow channels are obtained. It is found, as expected from dimensional considerations, that the flow resistance per unit length is proportional to the shear viscosity and inversely proportional to the square of the channel width. The shear viscosity is temperature dependent, and this has to be accounted for in many applications. It is then important to realize that the sound absorption coefficient of a material depends on the ratio of the resistance and the wave impedance ρc of the fluid involved and both these quantities are temperature dependent. Oscillatory viscous flow. In steady flow through a channel, only viscous forces are involved in the interaction between the flow and the channel wall, and the drag force on the wall and the corresponding reaction force on the fluid are proportional to the mean velocity in the channel. In acoustically driven oscillatory flow, inertial forces also play a role, and the velocity profile in the channel is changed from the steady flow parabolic to a flatter profile with a larger velocity gradient at the walls. Because of the effect of inertia, the interaction force on the wall and the corresponding reaction force on the fluid no longer are proportional to the velocity only, but there is also a contribution proportional to the acceleration. The velocity profile in a channel for oscillatory flow is characterized by a velocity gradient at the wall, which increases with frequency. The change in tangential velocity from zero at the wall to the free stream value far away from the wall defines the (acoustic) viscous boundary layer, the thickness of which decreases with increasing frequency. It plays an important role throughout the discussion of porous materials. Thermal effects. A sound wave in free field, far away from boundaries, produces a temperature fluctuation due to the isentropic (adiabatic) compressions and rarefactions in the sound wave. Close to a solid boundary, however, the large heat conduction of a solid compared to the heat conduction of air, prevents such a fluctuation from occurring and the conditions will be isothermal. The transition from isothermal conditions at the boundary to isentropic away from the boundary defines a thermal boundary layer and a corresponding thickness. It is of the same order of magnitude as that of the viscous boundary layer, and, like the viscous boundary layer thickness,
SOUND ABSORPTION MECHANISMS
9
decreases with frequency. The layer is usually quite thin. For example, at 100 Hz, it is about 0.022 cm in air, and it varies as the inverse of the square root of frequency. At frequencies low enough so that the boundary layer fills the entire space in the pores of a porous material, the conditions are isothermal, and this has an important effect on the low frequency absorption characteristics, since the compressibility and the sound speed will be reduced, as will be shown shortly. The present chapter is devoted to a detailed study of these visco-thermal effects in the sound field within a capillary tube. It will enable us to compute the attenuation of sound in such tubes and gain insight into the process of sound absorption in a porous material.
2.2 STEADY FLOW THROUGH A (NARROW) CHANNEL 2.2.1 Flow Resistance The nature of the flow through a channel is characterized by the Reynolds number, which is the ratio of the inertial and the viscous forces in the fluid. The inertial forces are of the order of ρu2 , where u is the velocity and ρ the density, and the viscous forces of the order of μu/d, where d is the characteristic dimension of the cross section and μ the coefficient of shear viscosity (the velocity gradient is of the order of u/d). The Reynolds number is then R = ρu2 /(μu/d) = ud/ν, where ν = μ/ρ is the kinematic viscosity. It is known from experiments that the transition from laminar to turbulent flow in a channel occurs at a Reynolds number of approximately 2000. Since we are here interested in laminar flow, the Reynolds number which we deal with has to be considerably less than 2000. The kinematic viscosity of air is ≈0.15 CGS units and in a tube with a diameter of 1 cm, the flow velocity has to be smaller than 300 cm/sec in order for laminar flow to be expected. With a diameter of 1 mm, the velocity can be 10 times larger. This will give us some idea about the velocity range we deal with here. The velocity is not uniform in the channel. The profile is parabolic, with zero velocity at the walls and maximum in the center. The flow resistance is defined in terms of the average velocity uav in the channel as the ratio of the pressure drop and uav and the flow resistance per unit length, as derived in Section 2.7. Steadyflow channel resistance per unit length 3μ/a 2 = 12μ/d 2 , parallel plates r0c = 8μ/a 2 = 32μ/d 2 , circular tube
(2.1)
μ: Coeff. of shear viscosity ≈ 2 × 10−4 CGS (air). d = 2a: Channel width (diameter). See also Eq. 2.33. The subscript c in r0c refers to ‘channel’ and 0 to steady flow (zero frequency). Elementary √ kinetic theory of gases shows that the coefficient of viscosity is proportional to T , where T is the absolute temperature, but in reality the temperature √ dependence is a bit more complicated. However, it is a good approximation to use T dependence for our temperature scaling purposes. In this context we note that sound
10
NOISE REDUCTION ANALYSIS
absorption depends on the normalized flow resistance r0c /ρc. The sound speed is pro√ portional to T and, at a√ constant static pressure, the density is inversely proportional to T . Therefore, using a T -dependence for viscosity, the normalized flow resistance will be proportional to T . It is very important to keep this in mind in design applications. Sound absorbers and attenuators are frequently used at temperatures of about 1000◦ F, i.e., ≈ 811 K, (in gas turbine exhaust stacks, for example), and the normalized flow resistance of a porous material is then about 2.8 times larger than at room temperature. This can have a large effect on sound absorption and attenuation characteristics. It is sometimes of interest to get an idea of the pore size in a porous material directly from the measured flow resistance per unit length, which is often expressed in terms of ρc units per inch. Thus, if we express the channel width d in mil (1 mil = 0.001 inch = 0.00254 cm) and r0c /ρc in inch−1 , the following numerical relations between√normalized flow resistance per inch and the channel width in mil results, d ≈ 4.6/ r0c /ρc. Thus, a flow resistance of 1 ρc per inch corresponds to an average channel width of 4.6 mil (≈ 0.012 cm).
2.3 ACOUSTIC BOUNDARY LAYERS Sound involves oscillatory flow, and, as for steady viscous flow, there is a viscous boundary layer in which the tangential velocity amplitude goes to zero at the boundary. We shall find that the viscous interaction with a solid boundary not only involves a force component proportional to the velocity, as for steady flow, but also a component proportional to the fluid acceleration. This must be accounted for in a study of sound propagation in narrow channels. There is also a thermal boundary layer in which there is a transition from the temperature fluctuation in the sound in free field, away from the boundary, to zero at the boundary. It will be shown that the general character of sound propagation in a narrow channel depends intimately on the ratio of the channel width and the boundary layer thickness. As a preliminary to such sound propagation studies, we proceed with a discussion of the viscous and thermal acoustic boundary layers at a plane solid wall.
2.3.1 The Viscous Boundary Layer First, let us consider the shear flow generated by a flat infinite plate, which oscillates in harmonic motion in the plane of the plate. Due to friction, this induces a harmonic motion also in the surrounding fluid in which the velocity is the same as that of the plate at the plate contact surface. However, the velocity is found to decrease exponentially with the distance y from the plate (Eqs. 2.36 and 2.37). The distance from the plate at which the amplitude is 1/e ≈ .37 times the amplitude of the plate is called the viscous boundary layer thickness, dv . As shown in Eq. 2.2, it decreases with frequency and density and increases with shear viscosity. Associated with the decay in amplitude, there is also a time delay in the motion corresponding to a phase (lag) angle y/dv . The ‘transmission’ of the motion from the plate out into the fluid is governed by a diffusion process and not a wave motion. The quantity that ‘diffuses’ is the vorticity in the fluid.
SOUND ABSORPTION MECHANISMS Viscous boundary layer thickness √ √ dv = 2ν/ω ≈ 0.22/ f cm. (normal air)
11
(2.2)
ν = μ/ρ: kinematic viscosity (≈ 0.15, standard air). μ: Coeff. of shear viscosity. ρ: Density. f : Frequency, in Hz. ω = 2πf (see also Section 2.7: Eq. 2.37). With f = 100 Hz, the boundary layer thickness is ≈ 0.022 cm in air at room temperature. Surface Impedance For Shear Flow From the velocity field in the fluid, we can determine the (shear) force per unit area of the plate that is required to drive the oscillating flow. The ratio of the complex amplitudes of this stress and the velocity outside the boundary layer is defined as a surface impedance per unit area. The resistive and reactive parts of the impedance turn out to be equal and the magnitude can be expressed as (kdv )ρc/2, where k = ω/c. From the frequency dependence of dv (Eq. 2.2), it follows that the surface impedance is proportional to the square root of frequency. Surface impedance (shear flow) per unit area √ Zs ≡ Rs + iXs = F /u0 = (1 − i) μρω/2 = (1/2)(1 − i)(kdv )ρc
(2.3)
dv : Boundary layer thickness (Eq. 2.2). k = ω/c. μ: Coeff. of shear viscosity. ρ: Density. (See also Section 2.7: Eq. 2.38.) We can interpret the mass reactance in terms of the total kinetic energy of the oscillatory flow in the boundary layer. Integrating the kinetic energy density from 0 to ∞, and expressing the result as (1/2)m|u0 |2 , we find that the corresponding normalized mass reactance ωm/ρc agrees with the expression given in Eq. 2.3. The reverse situation, when the plate is stationary and the velocity of the fluid far away from the plate has harmonic time dependence, the fluid velocity goes to zero at the plate. The transition from the ‘free stream’ velocity to zero occurs in a boundary layer, which has the same form as above. Obviously, it is only the relative motion of the fluid and the plate that matters. There will be an oscillatory force on the plate and a corresponding surface impedance Zs per unit area for the flow with resistive and mass reactive parts, the same as before. We can use this impedance as a good approximation also for a curved surface as long as the radius of the curvature of the surface is much larger than the acoustic boundary layer thickness in which case the surface can be treated locally as a plane. Using this approximation, we can determine the total surface impedance for oscillatory flow in a channel of arbitrary cross section as long as the transverse dimensions are large compared to the boundary layer thickness. Thus, if the perimeter of the channel is S and its area A, the total
12
NOISE REDUCTION ANALYSIS
surface impedance per unit length of the channel will be (S/A)Zs . In addition, there is the mass reactance ωρ of the air itself. When combined with the reactive part of the surface impedance, the total reactance can be expressed as ωρe , where ρe is an equivalent mass density and ρe /ρ a viscous contribution to a ‘structure factor,’ which will be discussed later.
2.3.2 The Thermal Boundary Layer By analogy with the discussion of the viscous boundary layer, consider next the temperature field produced by a plane boundary with a temperature which has a harmonic time dependence about its mean value. The surrounding fluid will be heated periodically. Temperature rather than vorticity is now diffused into the fluid, and the temperature field takes the place of the velocity field in the shear motion discussed above. Thus, the temperature amplitude decreases exponentially with distance from the plate and the thermal boundary layer thickness dh is defined in an analogous manner. It is determined by the coefficient of heat conduction rather than the coefficient of shear viscosity and is slightly larger than the viscous boundary layer thickness (by about 10 percent). The expression for it can be obtained from the viscous boundary layer thickness given above by replacing the coefficient of shear viscosity μ by K/Cp , where K is the coefficient of heat conduction and Cp the specific heat per unit mass at constant pressure. Thermal boundary layer thickness K √ cm. (normal air) dh = 2K/ρCp ω = μC dv ≈ 0.25 f p
(2.4)
f : Frequency, Hz. dv : Viscous boundary layer thickness (Eq. 2.2.). K: Heat conduction coeff. Cp : Spec. heat, unit mass. (See also Section 2.7: Eq. 2.43.) The reverse situation when a temperature fluctuation far from the plate varies harmonically and the temperature fluctuation at the plate is zero is analogous to the case of an oscillating fluid above a stationary plate. The temperature fluctuation goes to zero at the plate through the thermal boundary layer. The reason why the temperature fluctuation at the solid boundary can be considered to be zero is that the thermal conduction (and heat capacity) of a solid is much greater than for air. In the free field, far from the boundary, the temperature fluctuations are caused by the periodic compressions and rarefactions in the sound field since the conditions there are very nearly isentropic (adiabatic). An important influence of heat conduction on the sound field is the variation of the compressibility from the isentropic value in free field, 1/γ P , to the isothermal value, 1/P , at the boundary (γ is the specific heat ratio, ≈ 1.4 for air, and P the static pressure). In both these regions, a compression of a fluid element is in phase with the pressure increase. This means that the velocity of the surface of a volume element will be 90 degrees out of phase so that there will be no net work done on the element in one period of harmonic motion.
SOUND ABSORPTION MECHANISMS
13
The situation is different within the boundary layer where the conditions are neither isothermal, nor isentropic. A compression leads to a delayed leakage of heat as a result of the diffusion in the air into the boundary and the pressure and velocity will not be 90 degrees out of phase. Thus, within the boundary layer, a net energy transfer will take place from the sound field into the gas and then via conduction into the boundary. The maximum transfer per unit volume of the gas occurs at a distance from the boundary approximately equal to the boundary layer thickness. This is the nature of the acoustic losses caused by heat conduction. Formally, it can be accounted for by means of a complex compressibility κ˜ in the thermal boundary layer which goes from the isothermal value at the boundary to the isentropic value outside the boundary layer. The loss rate per unit volume is proportional to the imaginary part of κ. ˜ There is some heat conduction also in the free field, far away from the plate which leads to a slight deviation from purely isentropic conditions. However, the heat flow is now a result of a gradient in which the characteristic length is the wavelength λ of the sound wave rather than the boundary layer thickness dh , and with λ >> dh , this effect can be neglected in the present discussion.
2.3.3 Power Dissipation, Visco-Thermal Boundary Layer With reference to the discussion in Section 2.7, the acoustic power dissipation at a boundary is the sum of two contributions. The first is due to the shear stresses in the viscous boundary layer and is proportional to the squared velocity amplitude tangential to the boundary. The second is due to the heat conduction in the thermal boundary layer and is proportional to the squared sound pressure amplitude at the boundary. They are obtained by integrating the viscous and thermal losses per unit volume over the boundary layers. Visco-thermal acoustic losses per unit area of a boundary Ls = Lv + Lh = (k/2)[dv |u|2 ρc + (γ − 1)dh |p|2 /ρc] √ ≈ 2 × 10−5 f [|u|2 ρc + 0.45|p|2 /ρc]
(2.5)
dv : Viscous boundary layer thickness, Eq. 2.2. dh : Thermal boundary layer thickness, Eq. 2.4. f : Frequency in Hz. |p|: Sound pressure amplitude (rms). |u|: Tangential velocity amplitude just outside the boundary layer. (See also Section 2.7: Eq. 2.53.) Since the velocity and pressure amplitudes are simply related, the total viscothermal power dissipation per unit area at the boundary can be expressed in terms of either the pressure amplitude or the velocity amplitude. The result obtained for a plane boundary can be used also for a curved boundary, if the local radius of curvature is much larger than the thermal boundary layer thickness.
14
NOISE REDUCTION ANALYSIS
Example: The Q-Value of a Cavity Resonator For a simple mass-spring oscillator with relatively small damping, the sharpness of its resonance is usually expressed as 1/(2π) times the ratio of the total energy of oscillation (twice the kinetic energy) and the power dissipated in one period. This relation is valid also for an acoustic cavity resonator. The total energy of oscillation is now obtained from the known pressure and velocity fields in the resonator and by dividing it with the total visco-thermal losses at the boundaries, the Q-value can be determined since both quantities are proportional to the maximum pressure amplitude in the resonator. The constant of proportionality for the total losses contains a visco-thermal boundary layer thickness dvh = dv + (γ − 1)dh , where dv and dh are the viscous and thermal boundary layer thicknesses and γ = Cp /Cv ≈ 1.4 (for air) is the specific heat ratio. If this scheme is used for a circular tube (quarter wavelength resonator), the Q-value turns out to be simply ≈a/dvh , where a is the radius of the tube. By introducing the frequency dependence of the boundary layer thickness, this can be expressed as √ ≈3.11 a f , where a is expressed in cm and f is the frequency in Hz. (The expression for a parallel plate cavity is the same if a stands for the separation of the plates.) Thus, a circular resonator with a radius of 1 cm and a resonance frequency of 100 Hz has a Q-value of 31.1. In this context we should be aware of the fact that the boundary layer thickness depends on the kinematic viscosity ν = μ/ρ and that it will decrease with increasing static pressure at a given temperature (μ is essentially independent of density). Thus, if a very high Q-value is desired in an experiment, a high pressure and a high density gas should be used. In a typical steam turbine in a nuclear power plant, the static pressure typically is of the order of 1000 atmospheres and the Q-value of acoustic resonances typically will be very high and the damping very low. This has a bearing on the problem of acoustically induced flow instabilities and their impact on key components in such planes; for example, control valves and related structures.
2.4 SOUND PROPAGATION IN A NARROW CHANNEL In the idealized case of a gas with no viscosity and heat conduction, the analysis of propagation of sound in a channel with acoustically hard walls is relatively simple, since the boundary condition which is needed for a solution is simply that the normal velocity at the boundary be zero. The governing mathematical description of the field is the ordinary scalar wave equation. The solution which is then obtained will have a tangential velocity and a temperature fluctuation at the boundary, the latter corresponding to the isentropic pressure fluctuation. In a real fluid, however, such a solution cannot be valid since the tangential velocity must be zero at the boundary as imposed by viscosity (on the basis of the standard assumption of no ‘slip’). Furthermore, since the heat conductivity and the heat capacity of a solid is much greater than for a gas, the temperature fluctuation at the boundary can be assumed to be zero.
SOUND ABSORPTION MECHANISMS
15
In order to meet these conditions, it is necessary to start from the acoustic equations for a real fluid. In the linear approximation, it turns out that the solution can be composed of a linear superposition of the solution of an ordinary wave equation and two diffusion equations, one for vorticity and one for temperature. For a discussion of this problem, see, for example, Morse and Ingard, Theoretical Acoustics, Section 6.4, in particular, Problem 6.13, where propagation in the channel between two parallel walls is considered in detail. We shall extend this analysis here and add some observations and numerical results. In the channel under consideration, the two walls are placed at y = ±a and the x-axis is chosen to be the direction of propagation. The sound field is assumed to be independent of z. As mentioned, the wave field in the channel can be shown to be a linear combination of three wave modes, a ‘propagational,’ a ‘thermal,’ and a ‘viscous’ mode, using the terminology in the reference above. The pressure in the propagational mode satisfies an ordinary wave equation and the temperature and velocity in the thermal and viscous modes satisfy diffusion equations. These equations have to be solved subject to the appropriate boundary conditions for velocity and temperature in the total field.
2.4.1 Propagation Constant The temperature fluctuation in the viscous mode and the axial velocity component in the thermal mode turn out to be negligible. Thus, the boundary condition for temperature requires the temperature fluctuations in the propagational and thermal mode to cancel each other. On the basis of this condition, the amplitude constant in the thermal mode can be expressed in terms of the propagational mode amplitude. Similarly, the axial velocity components of the propagational and viscous modes have to cancel each other at the boundary in order to get zero total velocity, and this determines the amplitude of the viscous mode in terms of the propagational mode amplitude. All the modes have velocity contributions in the transverse direction and the boundary conditions of zero normal total velocity at the walls establish a relation from which the transverse propagation constant is obtained, which in turn determines the axial propagation constant q = qr + iqi . Axial propagation constant q −1)Fh Q ≡ q/k ≡ (Qr + iQi )/k = 1+(γ 1−Fv √ 3γ /4 (1 + i)a/dv for a/dv , a/dh << 1 ≈ 1 + i[dv + (γ − 1)dh ]/4a for a/dv , a/dh >> 1
(2.6)
dv , dh : Viscous and thermal boundary layer thicknesses (Eq. 2.2, Eq. 2.4.). a: Half-width of channel (Eq. 2.79). k = ω/c = 2π/λ. Functions Fv and Fh are defined in Eqs. 2.43 and 2.79. The real part, Qr , is the ratio of the free field sound speed, and the phase velocity in the channel and the imaginary part, Qi , yields the attenuation per wavelength.
16
NOISE REDUCTION ANALYSIS
These quantities are plotted vs a/dv in Figure 2.1. The value a/dv = 1 can be considered to represent the transition between the friction dominated and the inertia dominated wave propagation regimes. For values less than 1, the real and imaginary parts of Q approach the same value, which is inversely proportional to a/dv , and hence inversely proportional to the frequency. Since the frequency dependence is expressed in terms of a dimensionless parameter a/dv , it is not immediately clear what the plot means in terms of numerical values. To get a reference, let us consider a separation of the channel wall of 0.1√cm, so that a = 0.05 cm. The boundary layer thickness (Eq. 2.37) is dv ≈ 0.22/ f cm, so that at 100 Hz it is dv ≈ 0.022 cm, i.e., a/dv ≈ 2.3. At this value the real part of Q is only slightly larger than 1 so that the phase velocity is slightly smaller than the free field value. The imaginary part is ≈ 0.2, which means that the attenuation is 54.8 × 0.2 ≈ 11 dB per wavelength, which at 100 Hz is 340 cm (11.2 ft). Thus, at this channel separation and frequency, the propagation is inertia dominated, and the channel surfaces do not have much effect on the wave speed. In order to get into the friction dominated regime at this frequency, the channel width has to be reduced at least by a factor of 2.3, i.e., to a value less than 0.043 cm. The attenuation per wavelength, 11 dB at 100 Hz (about 1 dB/ft) in the example √ above, increases with decreasing frequency approximately as √ 1/ f . This means that the attenuation per unit length will increase approximately as ω. With a wavelength of 340 cm in our example, the attenuation √ per cm is 0.032 dB, and this value will increase with frequency approximately as f/100. The results obtained apply to the fundamental acoustic mode; there is no restriction on the value of λ/a. It is left for the reader to show that for the wavelength to be of the order of the boundary layer thickness in normal air, the frequency must be of the order of 10 mega Hz. In this context it should be pointed out that at sufficiently high frequencies, the visco-thermal and molecular relaxation effects in the bulk of the fluid (away from boundaries) must be accounted for.
Figure 2.1: The real and imaginary parts of the normalized propagation constant in a channel between two parallel plates with a separation d = 2a. The real part is the ratio of the free field sound speed and the wave speed (phase velocity) in the channel and the imaginary part, when multiplied by 54.8, gives the attenuation in dB per wavelength. The result is approximately valid for a circular tube with a diameter equal to twice the channel width (see Eq. 2.80).
SOUND ABSORPTION MECHANISMS
17
Kirchhoff Attenuation These results are approximately valid for a circular tube with a diameter which is twice the channel separation so that in the high frequency regime a/dv >> 1 and √ Qi ≈ dvh /4a, where dvh = dv + (γ − 1)dh ≈ 0.31/ f cm, with f in Hz. As before, dv and dh are the viscous and thermal boundary layer thicknesses (Eqs. 2.37 and 2.43), and γ ≈ 1.4 is the specific heat ratio. For the circular tube this expression for Qi is often referred to as the ‘classical’ or Kirchhoff attenuation. The attenuation in dB per unit length is then given by 20 log(e)Qi k ≈ 8.7Qi k, where k = ω/c = 2π/λ, the sound speed c and the wavelength λ referring to free field. In this high frequency region where the boundary layer thickness is small compared to the transverse dimensions of a channel, the attenuation in a channel with an arbitrary cross section of area A and perimeter S can be expressed in the same manner, i.e., Qi = dvh /4D, where D = A/S is the ‘hydraulic’ diameter of the channel (A, area, and S, perimeter). For a circular tube of diameter d, D = d/4, and for the channel between two parallel plates, D = d/2, where d is the width of the channel. It should be mentioned that the attenuation can be obtained to a good approximation by starting from the loss-free wave field in a duct and then using the power dissipation in Section 2.3.3 to express the loss per unit length in a duct. The loss, βw, is proportional to the acoustic power w in the duct and the exponential spatial decay is then obtained from dw/dx = −βW . This approach can be used for both the fundamental and a higher mode. Penetration Depth The amplitude reduction in a distance x is exp(−Qi kx). The distance at which the reduction is 1/e is then Qi /k, which is defined as the penetration depth dp , i.e., dp = Qi λ/2π of a wave in the channel, where λ is the free field wavelength. In the low frequency friction controlled or capillary regime, a/dv << 1, the real and imaginary parts of Q are about the same, Qi ≈ Qr , so that Qi λ ≈ Qr λ. Then, the penetration depth simply becomes the wavelength in the channel (recall that Qr is the ratio of the free field sound speed and wave speed in the channel). As a numerical example, consider a frequency of 50 Hz (dv ≈ 0.03 cm) and a channel width of 0.006 cm (a/dv ≈ 0.1). The corresponding wave speed in the channel then is 10 times smaller than in free space so that the wavelength in the channel is ≈ 68 cm and the penetration depth 68/(2π ) ≈ 10.8 cm. Wave Propagation vs Diffusion Wave propagation in a narrow channel involves a competition between inertial and viscous forces. At low frequencies, where the resistive force far exceeds the inertial, wave propagation is altered drastically. The situation is analogous to what happens when a harmonic oscillator approaches the overdamped state in which the frequency of free oscillations decreases until the motion no longer is oscillatory. In an analogous fashion, the phase velocity of the sound will decrease with decreasing frequency and ‘sound propagation’ becomes a process similar to the diffusion of heat from a source with harmonic time dependence (compare the penetration of heat into the earth as
18
NOISE REDUCTION ANALYSIS
a result of the local periodic heating of the earth’s surface by the sun). The acoustic diffusive regime corresponds to the left portion in Figure 2.1 in which the real and imaginary parts of the propagation constant become equal, increasing with decreasing frequency. This corresponds to decreasing wave speed and increasing attenuation per wavelength; the attenuation per unit length, however, will decrease.
2.4.2 Velocity and Temperature Profiles The total axial velocity distribution across our channel is obtained as the sum of the contributions from the propagational and viscous modes, as mentioned above. The axial velocity profile or distribution across the channel thus obtained is shown in Figure 2.2 as a function of y/a, where y is the transverse coordinate, y = 0 in the center of the channel, and a is the half-width of the channel. Plots for several values of a/dv are shown, where dv is the viscous boundary layer thickness (see Eq. 2.37). For large values of frequency and a/dv , characterizing inertial mass dominance over friction, the profile is almost flat as in the ideal gas case. However, as the frequency decreases, so that friction becomes more important, the profile approaches the well-known parabolic shape of steady viscous flow and for a/dv < 1, is indistinguishable from it. Profiles of axial velocity in a channel ux (y) cos(kv y)−cos(kv a) ux (0) = 1−cos(kv a)
(2.7)
kv = (1 + i)/dv . dv : Viscous boundary layer thickness (Eq. 2.2.). a: Half-width of channel (Eq. 2.75.). Each of these wave profiles travels with a frequency dependent speed and attenuation along the channel. As expected, the almost flat profile, which occurs at high frequencies (large values of a/dv ), turns out to have almost the same speed as the
Figure 2.2: Axial velocity amplitude profile in a sound wave transmitted in a channel between two parallel walls vs y/a, where y is the distance from the center and a the half-width of the channel. The parameter values a/dv range from 1 to 32. The same distribution applies also to the temperature amplitude with dv replaced by dh (see Eq. 2.75).
SOUND ABSORPTION MECHANISMS
19
wave in free field and has very small attenuation. However, as the frequency decreases (a/dv decreases) the wave speed (phase velocity) decreases and the attenuation per wavelength increases, as already demonstrated in Figure 2.1. The parameter a/dv = a 2 ω/2ν, where ν = μ/ρ is the kinematic viscosity, can be thought of as the ratio of the inertial and viscous forces in the viscous mode. This follows from Eq. 2.71 by expressing the orders of magnitude of the left- and righthand sides of the equation, ρωuvx and (μ/a 2 )uvx , respectively. It also follows that the characteristic time of diffusion of vorticity in the shear mode is τv ≈ a 2 /ν so that √ a/dv = ωτv . If the period of the sound wave is long compared to the characteristic time of diffusion (to be discussed further below) so that a/dv is small, there will be ample time for the vorticity to diffuse from the boundary into the channel and significantly influence the velocity profile and the oscillatory motion in the channel. In an analogous manner the total temperature profile in the channel is the sum of the contributions from the propagational and thermal waves. The shape is the same as for the axial velocity distributions (Figure 2.2, Eq. 2.7) with the viscous boundary layer thickness dv replaced by the thermal, dh .
2.4.3 Effect of Internal Damping of Flexible Wall In discussing the visco-thermal attenuation in a tube, we have tacitly assumed that the tube wall is rigid so that only visco-thermal losses are present. This is a good assumption for a steel tube, for example, except at the radial resonance frequency of the tube. For a rubber or plastic tube, on the other hand, the radial displacement of the tube wall has to be accounted for since the internal losses in the material result in attenuation. We find then that for a typical rubber or plastic hose, the attenuation due to the internal damping in the material generally far exceeds the visco-thermal attenuation, at least at low frequencies. In practice the question sometimes arises about the design of a simple ρc (reflection-free) termination to be used in laboratory experiments to prevent reflection of sound from the end of a duct. One possibility is the use of a porous wedge (to be discussed later) or, alternatively, a long rubber or plastic hose which, if necessary, can be wound up in the form of a coil. An attenuation of 10 dB over the length of the hose will be sufficient since that means a round trip attenuation of 20 dB. This corresponds to an equivalent pressure reflection coefficient of 0.1 and an absorption coefficient of 0.99, a typical criterion for a termination to be considered ‘anechoic.’
2.4.4 Relaxation Times and a Note on Complex Compressibility in a Channel In connection with the discussion of the thermal boundary layer above a plane boundary, we introduced a complex compressibility and studied its variation with the distance from the boundary. The complex compressibility in a channel is obtained in a similar manner. It is isothermal at the boundaries but does not reach the isentropic (free field) value in the middle of the channel unless the channel width is much larger than the thermal boundary layer thickness, or, which amounts to the same thing,
20
NOISE REDUCTION ANALYSIS
the acoustic period is short enough (frequency high) so that there is no time for the heat to flow across the channel. The time required for the heat flow is referred to as the thermal ‘relaxation time.’ Its relation to the channel width can be seen directly from the diffusion equation for temperature fluctuation ∂T /∂t = (K/ρCp )∂ 2 T /∂y 2 , where K is the heat conduction coefficient, ρ, the density, and Cp , the specific heat at constant pressure and unit mass, and y the transverse coordinate of the channel. It follows from this equation that the thermal relaxation time τh will be proportional to a 2 (K/ρCp ). The diffusion of the shear flow in the channel is described by the diffusion equation ∂ux /∂t = ν∂ 2 ux /∂y 2 , where ν is the kinematic viscosity from which follows that the viscous relaxation time is τv ≈ a 2 /ν with a corresponding relaxation frequency fv = 1τv . The ratio of ν and K/(ρCp ) is the Prandtl number, which for standard air is ≈ 0.77. This means that τh ≈ 0.77τv with the relation between the corresponding relaxation frequencies being fh ≈ 1.3fv . The flow resistance per unit length of a channel in Eq. 2.1 is 3μ/a 2 = 3ρν/a 2 ∝ fv ρ, i.e., intimately related to the viscous relaxation frequency. We shall use this relation also for a rigid porous material in general and define the viscous relation frequency as (2.8) ωv = 2πfv = r/ρ. The corresponding thermal relaxation frequency is defined by the Prandtl number so that fh ≈ 1.3fv , as given above. In carrying out numerical studies of the porous absorber, several approaches can be used. One is to define an equivalent channel width from the known flow resistance of the material and then proceed to calculate wave impedance and propagation constant from the corresponding equivalent slot absorber from first principles. In this manner, the frequency dependence of the flow resistance and of the complex compressibility is automatically included. The only thing that has to be added is an empirical structure factor s . This approach has been used in the calculation of absorption spectra of porous layers in this book. Another approach is to assume the flow resistance to be equal to the steady flow value and use a semi-empirical expression for the frequency dependence of the complex compressibility. The frequency dependence of this latter compressibility is compared in Figure 2.3 with that computed from first principles in a narrow channel; the curve labeled (1) corresponds to the average compressibility in the channel of the porous material model, and curve (2) is the semi-empirical expression with an explicit frequency dependence. Derivations of these expressions are given in Section 2.7. From a practical standpoint, the difference between the two is not significant.
κ˜ av2
Complex compressibility h a) κ˜ av1 = κ 1 + (γ − 1) tan(k kh a γ −1 i(γ −1)(ω/ωh ) dρ = ρ1 dP = κ 1 + 1+(ω/ω + 1+(ω/ω )2 )2 h
h
kh = (1 + i)/dh . dh : Thermal boundary layer thickness (Eq. 2.4.). κ = 1/ρc2 (see Figure 2.3).
(2.9)
SOUND ABSORPTION MECHANISMS
21
Figure 2.3: Real and imaginary parts of the normalized average complex compressibility κ/κ ˜ in the channel between two parallel plates, where κ = 1/ρc2 is the isentropic compressibility. The abscissa is a/dh , where a is the half width of the channel and dh the thermal boundary layer thickness. (1) From the true expression in Eq. 2.93. (2) From the approximate formula in Eq. 2.94.
γ = Cp /Cv : Specific heat ratio. ≈ 1.4 for air. ωh : Eq. 2.94. (See also Section 2.7: Eqs. 2.93 and 2.98.). The ordinate in the figure is the compressibility normalized with respect to the isentropic (free field) value κ and the abscissa is a/dh , where dh is the thermal boundary layer thickness and a the half-width of the channel. The isothermal compressibility is larger than the isentropic by the factor γ , the specific heat ratio, which is ≈ 1.4 for air. We note that for small channel widths compared to the boundary layer thickness, a/dh << 1, this normalized average compressibility indeed becomes 1.4 and the imaginary part is zero (no losses). For large values of a/dh (a/dh >> 1), on the other hand, the compressibility becomes isentropic, and again, there are no losses. The losses are zero in both the low and high limits of a/dh (low and high frequencies); there is a maximum at a/dh = 1.12, i.e., close to 1, where the imaginary part of the normalized complex compressibility is 0.16. We shall take this conditionto represent the transition from isothermal to isentropic conditions. Since dh = 2K/(ρCp ω), this means that for a given channel width, the maximum occurs at a characteristic thermal transition frequency ωh ≈ 1.25(2K/ρCp a 2 ) and a corresponding relaxation time, the same as the relaxation frequency defined in Section 2.7. With 2K/Cp ≈ 1.3ν, as given earlier, it follows that ωh ≈ 3.25ν/a 2 = 1.08ωv . In other words, this thermal relaxation frequency is about the same as the viscous relaxation frequency in Eq. 2.8 and it can then be obtained from the known steady flow resistance per unit length of a porous material.
2.5 IMPEDANCES We can treat wave propagation in the channel approximately as a one-dimensional transmission line by introducing the average values of the amplitude of sound pressure
22
NOISE REDUCTION ANALYSIS
and the axial velocity over the channel. In the discussion of the acoustical characteristics of such a channel, it is convenient to introduce some impedances.
2.5.1 Impedance Per Unit Length The first is the channel impedance per unit length, i.e., the ratio of the drop in the (complex) sound pressure amplitude per unit length and the average velocity amplitude. Both the resistive and reactive parts of this impedance will be frequency dependent. As the frequency goes to zero, the resistance is found to go to the steady flow value 3μ/a 2 , where μ is the shear velocity and a the half-width of the channel, and does not change noticeably until a/dv becomes greater than 1, as shown in Figure 2.4. Channelimpedance per unit length 3μ/a 2 − i1.2ωρ for ξv << 1 −iωρ zc ≡ rc + ixc = 1−F = v ωρ/2ξv − iωρ(1 + 1/2ξv ) for ξv >> 1
(2.10)
Fv = tan(kv a)/kv a: kv = a/dv , dv : Viscous boundary layer thickness (Eq. 2.2.), a: half-width of channel. (See also Section 2.7: Eq. 2.83.) The other curve shows the frequency dependence of the reactance, normalized with respect to the reactance ωρ per unit length in free field. This curve approaches 1.2 as the frequency goes to zero. In other words, the air in the channel appears to have an equivalent inertial mass density ρe , which is 1.2 times the actual density ρ. Thus, curve (1) in the figure shows the frequency dependence of ρe /ρ, and this ratio sometimes is referred to as a viscous contribution to a structure factor. There is another way to understand the apparent increase in the inertial mass, as follows. Direct calculation of the average of the kinetic energy density ρu2x /2
Figure 2.4: (1): The normalized reactive part of the impedance per unit length of a channel
between two parallel plates, ωρe /ωρ = ρe /ρ = c , where ρec = is the effective mass and c a ‘structure factor.’ The abscissa is the ratio of the half-width of the channel and the viscous boundary layer thickness. (2): The normalized resistance rc /r0c per unit length, where r1c , is given in Eq. 2.83 and r0c is the resistance for steady flow in Eq. 2.32.
SOUND ABSORPTION MECHANISMS
23
by integration over the channel using the known parabolic velocity distribution and 2 /2 indeed yields the value ρ = 1.2ρ. Actually, it is expressing the result as Aρe Uxav e sometimes convenient to use this point of view in computing the equivalent inertial mass density for other channels. For example, using the velocity distribution for the steady viscous flow in a circular duct (see Eq. 2.31) leads to ρe ≈ 1.33ρ at low frequencies. √ The flow resistance in the channel increases with a/dv , and hence as ω for a/dv >> 1. The normalized resistance per unit length in the channel approaches kdv /d for a channel of width d between parallel plates and the value 2k/dv for a circular tube of diameter d. The physical reason for the increase in resistance is simply that the velocity gradient at the boundary increases with frequency, as shown in Figure 2.2. In the same high frequency regime, the equivalent mass reactance decreases from the low frequency value 1.2ωρ to ωρ but the reactance ωρ increases faster with frequency than the resistance so that the reactance/resistance ratio will be proportional to the square root of frequency. This is the reason for the increase with frequency of the Q-value of a tube resonator, as derived earlier and which will be discussed later. In the low frequency regime, the resistive and reactive parts of the impedance per unit length are equal at a frequency given by ωv = r0c /1.2ρ, where r0c is the steady flow resistance per unit length. This frequency is not much different from that which makes the viscous boundary layer thickness equal to the half-width of the channel. Both these frequencies can be expressed in terms of r0c /ρ, which has been called the viscous relaxation frequency in the channel.
2.5.2 Complex Density and Wave Impedance In the description of wave motion in a channel or a porous material in which viscous interaction with a boundary is involved, it is convenient to introduce a complex quantity for the equivalent density. In this manner, the friction interaction can be included in the density. It is analogous to the complex mass in an ordinary mass-spring oscillator with friction. The equation of motion is Mdu/dt − Ru = F , where M is the mass, u the velocity, R the friction coefficient, and F the driving force. In harmonic motion with angular frequency ω, the corresponding equation for the complex amplitudes ˜ is (−iωM − R)u(ω) = F (ω). If we introduce a complex mass M(ω) = M − iR/ω ˜ the equation can be written −i M(ω)u(ω) = F (ω), which has the same form as the ˜ friction free oscillator with M replaced by M. Thus, in terms of a complex density ρ, ˜ the impedance per unit length is expressed as −iωρ, ˜ i.e., ρ˜ = zc /(−iω), where zc is given in Eq. 2.10. In this manner, we can readily carry over well-known expressions from wave propagation in a loss-free channel to the corresponding expressions for a lossy channel merely by replacing the ordinary density by the complex density and the ordinary compressibility by the complex compressibility κ. ˜ As examples of the use of the complex density and √ compressibility, consider the well-known–expression for the speed of sound, c = 1/(ρκ), where κ is the isentropic compressibility 1/γ P , γ the specific heat ratio, and P the static pres√ sure. The corresponding propagation constant for a plane wave is k = ω/c = ω ρκ. These expressions are valid also in a lossy channel if we replace ρ with a complex
24
NOISE REDUCTION ANALYSIS
density and κ with a complex compressibility κ. ˜ The propagation constant now becomes complex, the real part determines the phase velocity, and the imaginary √ part the attenuation of the wave in the channel. Similarly, the wave impedance ρ/κ becomes complex and it equals the input impedance of an infinitely long channel including both the resistive and reactive parts. The impedance per unit length, of course, is −i ρω. ˜ For the expression for the complex density we refer to Section 2.7.
2.5.3 Perforated Plate In the analysis in Chapter 2, it was implied that the impedances of the porous sheets or screens could be regarded as experimentally known quantities. It is of interest in design work, however, to be able to calculate the impedance, at least approximately, in terms of geometrical parameters. Thus, guided by the results obtained earlier in this chapter, we shall develop semi-empirical expressions for the impedances of a perforated plate and a wire mesh screen, two elements of considerable practical interest in acoustical applications. In this subsection we consider the first of these elements. We have shown (Eq. 2.33) that the steady flow resistance per unit length is 12μ/d 2 in a channel between two parallel plates and 32μ/d 2 in a circular tube, where μ is the shear viscosity and d the channel width or tube diameter. These expressions are approximately valid for oscillatory flow at frequencies √ below the value at which the acoustic viscous boundary layer thickness dv = 2μ/ω is about half the channel width. At high frequencies, with a boundary layer thickness substantially smaller than the channel width d, the resistance per unit length approaches ρckdv /d for parallel plates and 2ρckdv /d for the circular tube (Eq. 2.83). Actually, this high frequency resistance can be expressed in a form which is valid for any shape of the channel cross section if we use as a characteristic dimension the hydraulic diameter D, which is the ratio of the area and the perimeter of the channel. For two parallel plates we have D = d/2 and for the circular tube, D = d/4. The general expression for the specific resistance per unit length is then ρckdv /2D. Furthermore, we have found that the reactive part of the channel impedance per unit length can be expressed in terms of an effective mass density, which at low frequencies (dv >> d) is (6/5)ρ for a channel between two parallel plates and (4/3)ρ for a circular tube. At high frequencies, dv << d, the value approaches ρ asymptotically in both cases. The complete frequency dependence of the resistance and the reactance we have expressed in terms of the function F = tan(X)/X, where X = (1 + i)d/2dv (see Eq. 2.83) for the parallel plate channel. For a circular tube, there is an analogous expression in terms of Bessel functions. For the present purpose, however, we shall construct a semi-empirical expression for the normalized impedance in order to be able to explicitly express the frequency dependence. The low frequency (approximately the steady flow) value of the resistance in a 2 2 2 circular tube per unit length is ρcθ √1 = 32μ/d , i.e., θ1 = 32μ/ρcd = 16k(dv /d) , where we have introduced dv = 2μ/ωρ and k = ω/c. The high frequency value is 2kdv /d. For the complete frequency dependence, we shall use the semi-empirical
SOUND ABSORPTION MECHANISMS
25
expression in Eq. 2.11. At low frequencies, x >> 1, the second term within the bracket can be neglected, and θ1 goes to the value for steady flow. For dv /d << 1, we obtain the high frequency value 2kx. Similarly, for the reactive part of the impedance per unit length we shall use the expression for χp in the same equation. It goes to the correct values (4/3)k and k in the low and high frequency limits. (With our choice of time factor exp(−iωt) in defining complex amplitudes, a negative sign of χp corresponds to a mass reactance.) Perforated facing, Normalized impedance, Semi-empirical ζp = θp + iχp θp ≈ 16kx 2[1 + 1/[8x(1 + 4x2 )] /s
(2.11)
10 χp ≈ −k 4/3 − (1/3) x+10 /s
x = dv /d. dv : Viscous boundary layer thickness (Eq. 2.2.). d: Orifice diam. = + 0.85(1 − s)d. : Plate thickness. s: Open area fraction. We have here used an effective or equivalent length for the orifice including an end correction accounting for the induced mass in the vicinity of the orifice. For a single orifice in an infinitely extended rigid plate it is approximately the end correction of a piston, which is (8/3π)d, accounting for both sides of the piston. When there is a uniform distribution of orifices in a plate so that the open area fraction is s, we need to modify the end correction in such a manner that it will vanish when s = 1. We have done that empirically by including a factor (1 − s) so that the equivalent length of the orifice channel will be ≈ + (1 − s)(8/3π )d ≈ + 0.85(1 − s)d. Furthermore, we note that the average velocity amplitude over the perforated plate is s times the velocity amplitude in one orifice, and if we use this average velocity in defining the specific impedance ζp of the perforated plate, we have to multiply the single orifice impedance by the factor 1/s, as indicated in Eq. 2.11. Actually, in a more detailed analysis one should have different end corrections for the resistive and the reactive components. The resistive end correction accounts for the friction losses on the plate due to the tangential flow over the plate outside the orifice. The reactive end correction accounts for kinetic energy increase of the flow in the vicinity of the orifice as it is distorted in going through the orifice. As a good approximation, we have here assumed the viscous and mass end corrections to be the same (Eq. 2.11). An example of the computed frequency dependence of the resistive and reactive parts of the impedance is shown in the first graph in Figure 2.6 for a perforated plate with thickness 0.1 inch, orifice diameter 0.1 inch, and open area 50 percent. In this case the reactance dominates over practically the entire frequency range. Micro-Perforates The situation can be drastically changed, however, by making the hole size sufficiently small, of the order of the viscous boundary layer thickness. The resistive component of the perforated plate impedance then can be made larger than the reactive over a wide
26
NOISE REDUCTION ANALYSIS
Figure 2.5: Left: The normalized resistance and reactance of a micro-perforated plate. Right: The corresponding absorption coefficient. N: Normal incidence, DL: Diffuse field, local reaction, DNL: Diffuse field, nonlocal reaction. Air layer backing: 8 inches. Facing: Hole diameter = plate thickness: 0.01 inches. Open area: 1 percent. These values were chosen to illustrate a case for which the normalized steady flow resistance is equal to 1.
frequency range. The material thus obtained is often referred to as a micro-perforate plate or simply a micro-perforate. As was found in Chapter 2, the absorption coefficient of a resistive sheet-cavity absorber has its first maximum at normal incidence at a frequency for which the depth of the air cavity is one quarter wavelength. In order for the peak absorption to be 100 percent, the normalized sheet resistance should be unity. From the resistance data for a typical perforated plate with the hole diameter much larger than the viscous boundary layer thickness, it is possible to achieve a resistance at resonance close to unity by making the open area fraction very small. Although a high peak absorption can be obtained in this way, the width of the absorption curve will be small and an absorber built in this manner normally is of little practical use. √ √ The viscous boundary layer thickness dv (Eq. 2.2) is dv = 2ν/ω ≈ 0.21/ f cm (ν ≈ 0.15 for air), where f is the frequency in Hz. Thus, at a frequency of 100 Hz, dv ≈ 0.021 ≈ 0.0083 inch. A typical orifice diameter of 1/8 to 1/4 inch is much larger than the boundary layer thickness. We call such a plate ‘inertia dominated.’ The normalized acoustic resistance and reactance of the orifice plate are given in Eq. 2.11 and exemplified in Figure 2.6. On the other hand, with an orifice diameter of the same order of magnitude as the viscous boundary layer thickness, the resistance is θp ≈ (1/s)16kx 2 and the reactance, χp ≈ (1/s)(4/3)k , is no longer dominant. On the contrary, as shown in the example in Figure 2.5, the resistance exceeds the reactance over a substantial portion of the frequency range. Actually, the resistance and reactance are equal when x 2 = 1/12, i.e., when 2ν/(ωd 2 ) = 1/12, i.e, f ≈ 0.57/d 2 , with d expressed in cm. The steady flow resistance of the perforated plate is θp = (1/s)16kx 2 . If we consider the special (but common) case when the plate thickness is equal to the hole diameter, ≈ 1.85d (if s << 1), the steady flow resistance can be written (1/s)59.2ν/cd ≈ 2.6 × 10−4 /sd, with d expressed in cm. If s is expressed in percent, the condition for having a normalized flow resistance of unity is sd ≈ 0.026. For
SOUND ABSORPTION MECHANISMS
27
Figure 2.6: Example of computed frequency dependence of normalized resistance and reactance, as indicated. (a) Perforated plate, thickness, 0.1 inch, hole diameter, 0.1 inch, open area, 50 percent. (b) Wire mesh screen, 100 wires per inch, open area, 30 percent, (c) Combination of the perforated plate and wire mesh screen in firm contact.
example, with d = 0.025 cm (≈ 0.01 inch), s should be ≈ 1 percent to yield a steady normalized flow resistance of unity. We should keep in mind that this expression applies only in the viscous regime. Figure 2.5 shows the frequency dependence of the resistance and the reactance for this particular choice of parameters. We note that the resistance indeed is ≈1 at low frequencies and varies only weakly with a frequency below 1000 Hz. The reactance becomes equal to the resistance at a frequency a little above 1000 Hz, not far from the approximate result f ≈ 0.57/d 2 (where d is in cm) given above, which is based on the assumption that the resistance is frequency independent and equal to the steady flow value. Thus, when this sheet is used in a resonator absorber, the frequency dependence of the absorption (also shown in the figure) below 1000 Hz is determined largely by the impedance of the air backing and can be obtained from the results given in Chapter 2. It should be emphasized that these results refer to a stationary plate. If acoustically induced motion of the plate is substantial, the results may differ considerably as shown in Chapter 2, and the same holds true in regard to nonlinear effects, discussed in the next chapter. As a rough rule of thumb, a plate-cavity absorber with an open area of the plate less than, say, 3 percent, should be of practical interest as an absorber. Steady flow through the plate and grazing flow as well as nonlinear effect will influence this conclusion as discussed in Chapters 4 and 8.
28
NOISE REDUCTION ANALYSIS
2.5.4 Wire Mesh Screen Next, we consider a wire mesh screen with a square wire pattern, with n wires per unit length. If the wire diameter is (also the thickness of the screen) and the air space between adjacent wires is d, we have n = 1/( + d) and √ the open area fraction s = d 2 /(d + )2 = d 2 n2 . In terms of s and n, we have d = s/n and = 1/n − d. Because of the cylindrical form of the wires, the open square air cell between wires does not have a uniform area, as in the case of a circular orifice in a plate but, qualitatively, the friction losses in the oscillatory flow through a screen and the corresponding acoustic resistance are similar to those in a perforated plate and the expression for the acoustic screen impedance will have the same general form as for the perforated plate. Without going into a detailed analysis of the problem, we use the same expression for the acoustic screen impedance as for the perforated plate by using equivalent (‘acoustic’) lengths of the channel (‘orifice’) between adjacent wires by replacing length of the air channel (thickness of the screen) with 0.8 in the expression for the perforated plate resistance θp in Eq. 2.11. For the screen reactance χs , we use the same expression as for the perforated plate, as given above. Wire mesh screen impedance, Semi-empirical θs ≈ (16kx 2 + 1/[8x(1 + 4x 2 )])0.8/s
(2.12)
|χs | ≈ k[4/3 − (1/3)10/(10 + x)] /s n: number of wires per unit length. s: open √ area fraction. d = s/n: air gap between adjacent wires. = 1/n − d: wire thickness. x = dv /d. dv : Eq. 2.2. = + 0.85(1 − s)d: ‘Acoustic’ screen thickness, including end correction. An example of the computed frequency dependence of the impedance of a wire mesh screen is shown in the second graph in Figure 2.6. It should be noted that although one frequently thinks of a screen as a purely resistance acoustical element, the resistance dominates only at sufficiently low frequencies, in this example below approximately 2000 Hz. It is important to note that for a constant open area the resistance increases and the reactance decreases with n. The basic reason for this behavior is that the channel resistance per unit length is inversely proportional to d 2 , where d is the width of the air channel between adjacent wires, and the reactance is proportional to the thickness of the screen (which decreases with n). To see this explicitly, we keep only the first terms in the expressions for θs and χs in Eq. 2.6 and make use of s = d 2 /(d + )2 = d 2 n2 and the expressions for the channel resistance and reactance per unit length derived earlier to get θs √≈ 32(ν/c)(n2 /s))0.8(/s) = and χs ≈ (ω/c)( /s). Then, with = (1/n(1 − (s)) we see that for a given open area fraction s, θs becomes proportional to n and χs inversely proportional to n. Still ignoring the frequency dependent corrections to θs and χs (the second terms in Eq. 2.12, the resistance and reactance become equal at a frequency given by ωc = 2πf ≈ 32ν(n2 /s)0.8(/ ). For example, with n = 100 per inch (39.4 per cm),
SOUND ABSORPTION MECHANISMS
29
ν = (μ/ρ) ≈ 0.15 CGS (air), and s = 0.3, we find f ≈ 1835 Hz, consistent with the result in Figure 2.6. Thus, a wire mesh screen should not be treated as predominantly resistive except at frequencies below the critical frequency ωc given above. For a screen with 30 percent open area, the critical frequencies are approximately 120, 450, 1800, and 7500 Hz for n = 25, 50, 100, and 200 wires per inch. Accurate measurement of small screen impedances such as those in the figure cannot normally be done with conventional apparatus. One method is to place the screen at the entrance of an acoustic cavity resonator and measure the frequency response of the resonator with and without the screen. From the measured change of the Q-value (or damping parameter) and the shift in resonance frequency, the resistance and reactance can be determined. With other methods (see Appendix B), stacks of several screens have to be used. One problem with using stacks of screens is that the total resistance of the stack is only approximately equal to the sum of the individual screen resistances. The flow field through a stack is complex and the total resistance is expected to be larger than the sum. The same holds true for the reactance. In subsequent chapters, dealing with porous materials, a ‘structure factor’ is introduced to account (empirically) for the effects of the tortuous path which the oscillatory flow is forced to follow within the material.
2.5.5 Perforated Plate-Screen Combination, Laminates A perforated plate or a wire screen often has too small a flow resistance to be of practical interest in the design of a sound absorber. However, when combined into a laminate, the average resistance of the unit can be increased substantially, provided that the plate and the screen are in ‘firm’ or ‘hard’ with each other so that the flow velocity through the screen at a perforation is the same as the flow velocity in a perforation, i.e., equal to 1/s times the average velocity over the surface of the plate. As a result, the average impedance of the combination will be zps = zp + (1/s)zs . On the other hand, if the contact is ‘loose,’ i.e., if the separation between the plate and the screen is larger than approximately a hole diameter in the plate, the flow will be almost uniform as it enters the screen (rather than concentrated at the perforations), and the total impedance of the combination will be the sum zp + zs . As an example, consider a combination in which the flow resistance of the perforated plate is negligible compared to that of the screen. The combination in close contact will have an average flow resistance, which is approximately 1/s times greater than for a combination in loose contact. As a result, the absorption coefficient can be quite different for the two types. Another example is given in Figure 2.7 with a comparison of the absorption spectra of perforated plate-screen combinations with loose and hard contact and with an 8 inch thick air cavity backing. The flow resistance of the screen in this case is only θ = 0.25, but for the hard contact case, the total resistance of the laminate, with a 23 percent open area, will be almost 4.3 times higher, i.e., close to unity, with a correspondingly
30
NOISE REDUCTION ANALYSIS
Figure 2.7: The absorption characteristics of a perforated plate-screen combination backed by an 8 inch cavity. N: Normal incidence, DL: Diffuse field, local reaction, DNL: Diffuse field, nonlocal reaction, Screen: Normalized flow resistance = .25, weight = 0.2 lb/ft2 . Plate: Thickness = 0.1 inch, Hole diameter = 0.1 inch. Open area = 23 percent. Left: Loose contact. Right: Hard contact (laminate).
higher resonance absorption (close to 100 percent) and a better overall absorption as well.
2.5.6 Effect of Acoustically Induced Motion With reference to the discussion of a flexible, resistive screen in Chapter 3, we can account for the induced motion of the perforated plate-screen combination (laminate) by introducing the ‘structural’ impedance of the laminate to obtain the equivalent impedance. Thus, if the structural impedance is ζst , the equivalent impedance
= ζ ζ /(ζ ζ ), which takes the place of ζ . If the structural (fundabecomes ζps ps st ps st ps mental) resonance frequency is considerably below the frequency range of interest, it is a good approximation to assume the laminate to be limp so that zst = −iωm, where m is the mass per unit area of the laminate.
2.5.7 A Note on the Interpretation of Steady Flow Resistance Data Frequently, data on the flow through a porous material is presented in the form of a graph showing the pressure drop P vs flow velocity U through the material. In lieu of a numerical curve fitting procedure we shall here merely assume that this relation is of the form P = r1 U + r2 U 2 . (2.13) The quantity of interest for acoustical purposes is r1 , the linear flow resistance of the material. If the values of P at the velocities U1 and U2 are P1 and P2 , it follows that P1 1 − (U1 /U2 )2 (P2 /P1 ) . (2.14) r1 = U1 1 − (U1 /U2 )
SOUND ABSORPTION MECHANISMS
31
As an example we mention that for a particular porous metal it was found that for U1 = 400 ft/min and U2 = 1000 ft/min, P1 = 2 and P2 = 10 inches of water. Then, from Eq. 2.14 it follows r1 = (1/3)(P1 /U1 ), where P1 is in inches of water and U1 in ft/min. Converting to CGS and dividing by ρc ≈ 41, we obtain for the normalized linear resistance the value θ1 ≈ 0.20. It should be emphasized that although the low frequency limit of the acoustic resistance can be obtained in this manner, it should be remembered that the acoustic resistance is frequency dependent, increasing approximately as the square root of the frequency, as demonstrated above. Even more important is the realization that although a porous material is thought of as ‘resistive,’ it also has a reactance. Thus, if the thickness of a porous sheet material is , the reactance will be at least that of an air layer of thickness , i.e., with the magnitude ωρ or, normalized, ωmρ/(ρc) = k = 2π /λ. Thus, as an example, if the thickness is 0.5 cm, the normalized reactance at 1000 Hz will be at least 0.09. Actually, accounting for the structure factor of the material (see Chapter 5: this value, in reality, can be increased by a factor of 2 or more).
2.6 VISCO-THERMAL ADMITTANCE AND ABSORPTION COEFFICIENT OF A RIGID WALL 2.6.1 Equivalent Admittance With reference to the discussion of visco-thermal effects, in particular, to wave propagation in a channel, we shall show here the normalized equivalent admittance of a rigid wall due to the visco-thermal boundary layer is η=
1−i (kdv sin2 φ + (γ − 1)kdh ), 2
(2.15)
where dv and dh are the viscous and thermal boundary layer thicknesses (see Eqs. 2.2 and 2.4), k = ω/c and γ , the specific heat ratio (≈ 1.4 for air). As in the section referenced above, the total wave field is composed of a linear superposition of a ‘propagational,’ a ‘thermal,’ and a ‘viscous’ mode. The propagation wave field is made up of an incident and a reflected wave and this wave field generates a viscous and thermal wave in the boundary layer at the wall. The strategy in solving the problem is to express the amplitude of the thermal and viscous waves in terms of the amplitude of the incident propagational wave by requiring that the total temperature amplitude and the tangential velocity amplitudes at the wall be zero. The x-components of the three waves can then be expressed in terms of the amplitude of the incident wave and the (unknown) reflection coefficient. By putting the sum of these components equal to zero at the wall, we get an equation for the reflection coefficient and the equivalent wall admittance. The x-axis is normal to the plane, which is placed at x = 0. Let the amplitude of the incoming wave be 1 and the pressure reflection coefficient R. The wave is incident from the left in the xy-plane at an angle φ with respect to the x-axis.
32
NOISE REDUCTION ANALYSIS
Let the amplitude of the incoming wave be 1 and the pressure reflection coefficient R. The complex amplitudes of the sound pressure, the x- and y-components ux , uy of the velocity, and the temperature fluctuation θp in this ‘propagational mode’ to the left of the plane are then
uxp
p = (eikx x + Re−ikx x )eiky y = (cos φ/ρc)(eikx x − Re−ikx x )eiky y
uyp = (sin φ/ρc)(eikx x + Re−ikx x )eiky y θp = [(γ − 1)/γ ](pT /P ),
(2.16)
where kx = k cos φ, ky = k sin φ, k = ω/c, P , is the static pressure, and T the ambient temperature. The complex amplitude of the temperature in the ‘thermal mode’ is of the form θt = Ae − ikh xeiky y ,
(2.17)
where kh = (1 + i)/dh , as in Chapter 3. Since the heat capacity and heat conduction coefficient of the solid wall is much greater than for the gas, the total temperature amplitude at the wall will be to zero, for all practical purposes, i.e., θt + θp = 0. This means that the constant A must be such that θT will cancel θp , i.e., A = −[(γ − 1)/γ ](p/P )T .
(2.18)
Similarly, with the y-component of the viscous mode being of the form uyv = Be−ikv xeiky y ,
(2.19)
we get from the boundary condition uyp + uuv = 0, B = −(sin φ/ρc)(1 + R).
(2.20)
Next, we express the three contributions to the x-component of the velocity. The first is already in Eq. 2.16. The contribution from the thermal wave is known to be uxt = (γ P h /T ρc)∂θt /∂x, where h = iω/(ckh2 ) (see discussion in Chapter 3 in connection with Eqs. 2.69 and 2.70) and with the value of A, given above, we get uxt = −(γ − 1)[(1 + R)/ρc]kdh /(1 + i).
(2.21)
Finally, the x-component of the velocity contribution from the viscous mode is obtained from ∂uxv /∂x + ∂uyv /∂y = 0, expressing the fact that there is no compression in the vorticity field. Then, from Eqs. 2.19 and 2.20, we get uxv = −[(1 + R)/ρc] sin2 φkdv /(1 + i).
(2.22)
33
SOUND ABSORPTION MECHANISMS
The boundary condition uxp +yxt +uxv = 0 at x = 0 yields R, and the corresponding admittance of the boundary is obtained from the well-known relation R=
cos φ − η cos φ + η
η ≡ ηr + iηi = (1 − i)[kdv sin2 φ + (γ − 1)kdh ]/2.
(2.23) (2.24)
2.6.2 Absorption Coefficient The absorption coefficient follows from Eqs. 2.23 and 2.24, α(φ) = 1 − |R|2 =
4ηr cos φ . (ηr + cos φ)2 + (ηi )2
(2.25)
As for any absorber, the absorption coefficient at grazing incidence is zero for both the heat conduction and viscosity. The contribution from heat conduction is a maximum at normal incidence but for viscosity it is zero. The angular dependence of the total absorption coefficient is shown for two frequencies, 1 kHz and 10 kHz, on the left in Figure 2.8. It shows the typical behavior for a low admittance (high impedance) boundary with a maximum at an angle close to 90 degrees at which the wave admittance in the normal direction is well matched with the normal admittance of the boundary. The maximum value of the absorption coefficient is about 0.82 and it occurs at an angle of 89.9 degrees, the same for both 1 kHz and 10 kHz. The fact that a rigid, impervious wall can have such a high absorption coefficient is a curiosity and an eye-opener. It contributes to making the average value in a diffuse field considerably higher than for normal incidence. This average absorption coefficient is obtained from π/2 αst = 2 α(φ) cos φ sin φ dφ (2.26) 0
Figure 2.8: Left: Angular dependence of the absorption coefficient of a rigid impervious wall due to the visco-thermal boundary layer. Frequencies: 1 kHz and 10 kHz. The maximum absorption coefficient is 0.82. Right: The diffuse field average absorption coefficient of the same wall. Air at 20◦ C and 1 atm. Notice the difference in scale on the absorption axes.
34
NOISE REDUCTION ANALYSIS
as shown in the mathematical supplement in the previous chapter. The computed frequency dependence is shown on the right in Figure 2.8. It is less than 0.01 at frequencies below approximately 3000 Hz and it can be expressed approximately as (2.27) α ≈ 1.7 × 10−4 f , where f is the frequency in Hz. Comparing the corresponding power loss with the power transmitted through a wall, we note that a one percent loss corresponds to a transmission loss of 20 dB of the wall. Thus, for a wall with a transmission loss above 20 dB, the visco-thermal losses dominate.
2.7 MATHEMATICAL SUPPLEMENT 2.7.1 Steady Flow Through a Narrow Channel Let us consider first the flow between two parallel plates, located at y = −a and y = a. The equation of motion for a mass element of unit width between −y and y is −
∂u ∂P 2y = −2μ , ∂x ∂y
(2.28)
which expresses the fact that the gradient of the pressure force 2yP is balanced by the friction force from the two walls in the channel. The solution to the equation is u(y) = u(0)[1 − (y/a)2 ] u(0) = − ∂P ∂x
a2 2μ ,
(2.29)
where μ is the coefficient of shear viscosity and P the pressure. For a circular tube of radius a, the corresponding equation of motion is −
∂u ∂P πr 2 = −2π rμ ∂x ∂r
(2.30)
with u(r) = u(0)[1 − (r/a)2 ] u(0) = − ∂P ∂x
a2 4μ .
(2.31)
This is the velocity distribution in the well-known Poiseuille flow, included in many elementary texts in physics. Steady Flow Resistance The average values of the velocities in the channels are (2/3)u(0) parallel plates uav = (1/2)u(0) circular tube
(2.32)
35
SOUND ABSORPTION MECHANISMS and the flow resistances per unit length are 1 ∂P 3μ/a 2 = 12μ/d 2 = r0c = − 8μ/a 2 = 32μ/d 2 uav ∂x
parallel plates circular tube
(2.33)
where μ: Coeff. of shear visc. ≈ 2×10−4 CGS (air), d = 2a: Channel width (diameter). The subscript c refers to ‘channel’ and 0 to steady flow (zero frequency) used here to distinguish it from the resistance for oscillatory flow.
2.7.2 Oscillatory Flow and Viscous Boundary Layer The Acoustic Viscous Boundary Layer A flat plate oscillates in harmonic motion with the velocity u0 cos(ωt) in the x-direction (in the plane of the plate). With the y-direction chosen normal to the plate, the rate momentum flux (shear stress) in the y-direction is τ (y) = −μ∂u/∂y per unit area so that the net force (in the x-direction) per unit area on a fluid element of thickness dy is τ (y) − τ (y + dy) = −(∂τ/∂y)dy. The equation for the x-component of the fluid velocity is then ∂ 2u ∂u (2.34) = μ 2. ρ ∂t ∂y The corresponding equation for the complex velocity amplitude u(ω) is1 ∂ 2u + (iρω)/μ = 0 ∂y 2
(2.35)
with the solution u = u0 eikv y = u0 e−y/dv eiy/dv √ kv = (1 + i) ρω/2μ ≡ (1 + i)/dv .
(2.36)
The velocity amplitude decreases exponentially with y and is reduced by a factor e at the distance dv above the plate which defines the boundary layer thickness, dv =
0.22 2μ/ρω ≈ √ cm. f
(normal air),
(2.37)
where f : Frequency, Hz. Surface Impedance For Shear Flow The complex amplitude of the shear stress μ∂u/∂y on the plate (y = 0) is F = −μikv u0 = u0 (1 − i)μ/dv , and the corresponding shear impedance per unit area is (2.38) Zs ≡ Rs + iXs = F /u0 = (1 − i) μρω/2 = (1/2)(1 − i)(kdv )ρc, where k = ω/c. 1 The complex velocity amplitude is defined by u(t) = {u(ω) exp(−iωt)}.
36
NOISE REDUCTION ANALYSIS
In the reverse situation, when the plate is stationary and the velocity of the fluid in the free stream far away from the plate is u0 cos(ωt), the corresponding complex amplitude equation of motion in the free stream is −iρωu0 = −∂p/∂x, where the right-hand side is the pressure gradient required to maintain the oscillatory flow. If we assume that this pressure gradient is independent of y, the equation of motion in the boundary layer will be −iωρu(y) = μ∂ 2 u/∂y 2 − iωρu0 , where we have replaced −∂p/∂x by −iωρu0 , as given above. The solution is u = u0 [1 − exp(ikv y)]. Thus, the velocity increases exponentially with y from 0 to the free stream value u0 , and we can use the same definition for the boundary layer thickness as in Eq. 2.37. The viscous stress on the plate will be the same as before and the real part represents the resistive friction force per unit area of the plate and is responsible for the viscous boundary losses in the interaction of sound with the boundary. The time average power dissipation per unit area in the shear flow at the boundary is then simply Rs |u0 |2 , i.e., Lv = Rs |u0 |2 = (1/2)kdv ρc |u0 |2 √ Rs = (1/2)kdv ρc = ρc νω/2c2 ≈ 2 × 10−5 f ρc
(normal air).
(2.39)
(k = ω/c. dv : Eq. 2.37. u0 : Tangential velocity amplitude, rms, outside the boundary layer, where |u0 | is the rms magnitude of the tangential velocity outside the boundary layer and Rs the surface resistance in the numerical approximation for normal air, f is the frequency in Hz.) This function, i.e., the viscous losses per unit volume in the shear flow, can be obtained also by direct integration of the viscous dissipation function over the boundary layer, as follows. Consider an element of thickness y. In a frame of reference moving with the fluid with its origin at the center of y, the velocity at the top surface is (∂ux /∂y) y/2. The shear stress is −μ∂ux /∂y (in the positive y-direction) and the power transfer to the element through the top surface (which is in the negative y-direction) is then μ∂ux /∂y (y/2)(∂ux /∂y). There is a similar transfer from the bottom surface, so that the total power transfer per unit volume will be μ(∂ux /∂y)(∂ux /∂y). In harmonic time dependence, the time average of this quantity will be Lv = μ{(∂ux /∂y) (∂u∗x /∂y)}, where ux is the rms value and u∗x the complex conjugate of ux . With ux = u0 exp(ikv y), integration from y = 0 to y = ∞ yields the result in Eq. 2.39. F = −μ∂u/∂y is the viscous stress on the surface, which can be used to obtain an approximate value for the impedance per unit length of a channel of arbitrary cross section as long as its transverse dimensions are large compared to the boundary layer thickness. The flow velocity in the center of the channel can then be considered to be the free stream velocity. With the perimeter of the channel denoted by S and the area by A, the total viscous stress per unit length of the channel is SF = SZs u0 , where F = −μ∂u/∂y and Zs = F /u0 . The reaction force on the fluid will be the same but with the opposite sign and the equation of motion for a fluid element of unit length is −iAωρu0 = −A∂p/∂x −SZs u0 . With Zs = Rs +iXs , the corresponding impedance per unit length of the channel is then
∂u (2.40) = (S/A)[Rs − iωρ + iXs ]. z1c = r1c + ix1c = (1/u0 ) − ∂x
37
SOUND ABSORPTION MECHANISMS
Since Xs represents a mass reactance (i.e., it is negative), as explained earlier, the total reactance can be written −iωρe , where ρe = ρ + |X|/ω is an equivalent mass density. The impedance per unit length is then z1c = (S/A)[Rs − iωρe ]. The quantity c = ρe /ρ can be regarded as a viscous contribution to the structure factor, which will discussed in more detail later.
2.7.3 The Thermal Boundary Layer By analogy with the discussion of the viscous boundary layer, we consider now the temperature field produced by a plane boundary with a temperature, which varies harmonically with time about its mean value, the variation being τ (t) = τ0 cos(ωt). The temperature away from the boundary is obtained from the diffusion equation ∂τ ∂ 2τ = (K/Cp ρ) 2 , ∂t ∂y
(2.41)
where K, Cp , and ρ are the heat conduction coefficient, the specific heat at constant pressure and unit mass, and the density, respectively. For harmonic time dependence (∂/∂t → −iω) and with the y-dependence expressed as τ (y, ω) = τ0 exp(ikh y), (2.42) it follows from Eq. 2.41 kh2 = i(ωρCp )/K, i.e., kh = (1 + i)/dh K √ cm. dh = 2K/ρCp ω = μC dv ≈ 0.25 f p
(normal air),
(2.43)
(f : Frequency, Hz. dv : Eq. 2.37.) where dh is the thermal boundary layer thickness. In the numerical approximation, f is the frequency in Hz. From Eq. 2.42, the complex amplitude of the temperature is τ (y, ω) = τ0 e−y/dh eiy/dh
(2.44)
so that at a distance from the plate equal to the thermal boundary layer thickness, y = dh , the magnitude of the temperature is 1/e of the value at the plate at y = 0. The ratio of the viscous and thermal boundary layer thicknesses is √ dv /dh = μCp /K = Pr Pr = μCp /K, (2.45) where Pr is the Prandtl number. For air at 1 atm and 20 degrees centigrade, μ ≈ 1.83 × 10−4 CGS (poise), Cp ≈ 0.24 cal/gram/degree, and K ≈ 5.68 × 10−5 cal cm/degree, so that Pr ≈ 0.77, dv /dh ≈ 0.88 and kh ≈ 0.88kv .
38
NOISE REDUCTION ANALYSIS
The reverse situation, when the temperature fluctuation in a sound wave far from the plate is equal to τ0 and the temperature fluctuation at the plate is zero (due to a heat conduction coefficient and the heat capacity of a solid is much larger than for air) the appropriate solution is τ (y, ω) = τ0 [1 − eikh y ].
(2.46)
This solution is applicable to the case when a harmonic sound wave is incident on the plate. Far away from the plate, y >> dh , the conditions in the fluid are isentropic and the compressions and rarefactions in the sound wave produce a harmonic temperature fluctuation with the amplitude τ0 =
γ −1 p T. γ P
(2.47)
Quantity p is the sound pressure amplitude, γ = Cp /Cv , the specific heat ratio, P the ambient pressure, and T the absolute temperature. The acoustic wavelength of interest is large compared to the boundary layer thickness so that we need not be concerned about any change of the sound pressure with position across the boundary layer. However, the compressibility varies, going from the isentropic value 1/γ P to the isothermal, 1/P , as the boundary is approached. These values refer to an ideal gas. Complex Compressibility To determine the complex compressibility throughout the boundary layer, we start with the density ρ(P , T ) being a function of both pressure P and temperature T (not only of pressure alone) so that
dρ =
∂ρ ∂P
dP + T
∂ρ ∂T
dT .
(2.48)
P
From the gas law, P = rρT , we have (∂ρ/∂P )T = ρ/P and (∂ρ/∂T )P = −ρ/T . Then, the quantities dP = p, dρ and dT = τ (y, ω) are treated as complex amplitudes, where τ is given in Eqs. 2.46 and 2.47 in terms of the sound pressure amplitude p, and the compressibility κ˜ = (1/ρ)(dρ/dP ) =
1 [1 + (γ − 1) e−kh y eikh y ]. γP
(2.49)
The tilde symbol is used to indicate that the compressibility is complex and different from the normal isentropic compressibility κ = 1/γ P = 1/ρc2 . For y = 0, κ˜ = 1/P equals the isothermal value, and for y = ∞, κ˜ = 1/γ P , the isentropic value; in the transition region, κ˜ is complex. The imaginary part can be written (2.50) κi = κ(γ − 1)e−y/dh sin(y/dh ). It has a maximum 0.321κ at y/dh = π/4.
SOUND ABSORPTION MECHANISMS
39
2.7.4 Power Dissipation in the Boundary Layer The power dissipation per unit area due to viscosity in the acoustically driven oscillatory shear flow over a solid wall has already been expressed in Eq. 2.39. To determine the dissipation due to heat conduction, we start from the conservation of mass equation for the fluid ∂ρ/∂t +ρdiv u = 0. For harmonic time dependence and with the relation between the complex amplitudes of density and pressure (δ and p) expressed as δ = ρ κp ˜ in terms of a complex compressibility κ, ˜ this equation becomes −iωκp ˜ + div u = 0. After integration of this equation over a small volume V with surface area A, and replacing the volume integral of div u by a surface integral over A, we can express the time average power {un p∗}A transmitted through A into the volume element as {(−iω)κ|p| ˜ 2 }V , where un is the inward normal velocity component of the velocity at the surface, |un | and |p| being rms values to avoid an additional factor of 1/2. Thus, the corresponding power dissipation per unit volume becomes Dh = ωκi |p|2 .
(2.51)
The integral of this expression over the boundary layer yields the corresponding acoustic power loss per unit area of the wall. The integration can be taken from 0 clear out to infinity. The contribution to the integral comes mainly from y-values less than a couple of boundary layer thicknesses and quickly goes to zero with increasing y outside the boundary layer. The pressure amplitude |p| can be taken to be constant throughout the layer since the wavelength of interest is much larger than the boundary layer thickness. After insertion of the expression for the compressibility in Eq. 2.49, the loss due to heat conduction per unit area of the wall can be expressed as Lh = (1/2)(γ − 1)kdh |p|2 ,
(2.52)
which is the counterpart of the expression for the viscous power dissipation Lv in Eq. 2.39. The total visco-thermal power dissipation per unit area of the wall then becomes Ls = Lv + Lh = (k/2)[dv |u|2 ρc + (γ − 1)dh |p|2 /ρc] √ ≈ 2 × 10−5 f [|u|2 ρc + 0.45|p|2 /ρc]
(2.53)
where dv : Eq. 2.37. dh : Eq. 2.43. f : Frequency in Hz. |u|: Tangential velocity outside the boundary layer. |p|: Pressure amplitude at the wall, both rms magnitudes. Q-Value of a Cavity Resonator For a simple harmonic oscillator (spring constant K, mass M, and resistance constant R) driven by a harmonic force with frequency independent amplitude, the frequency dependence (response) of the velocity by the familiar amplitude u(ω) is characterized √ 2 2 resonance at the frequency ωr = ω0 − γ , where ω0 = K/M and γ = R/2M. For small damping, ωr ≈ ω0 and the sharpness of the resonance curve is often
40
NOISE REDUCTION ANALYSIS
expressed in terms of the Q-value, Q = ω0 M/R. It can be interpreted as the ratio of the resonance frequency and the total width of the response curve at the ‘half-power point,’ defined by |u(ω)/u(ω0 )|2 = 1/2. Q = ω0 M/R = ω0 M|u|2 /R|u|2 can be interpreted also as ω0 times the ratio of the time average of the energy of oscillation (being twice the kinetic energy average) and the dissipation rate or, apart from a factor of 2π, as the ratio of the energy of oscillation and the dissipation in one period. This relation is valid also for an acoustic cavity resonator in the vicinity of a resonance. The cavity under consideration is a straight tube of length L, area A, perimeter S, open at one end, and terminated by a rigid wall at the other (at x = L). With the pressure amplitude at the wall being p(L), the amplitude at a distance x from the wall is p(x) = p(L) cos(kx), where k = ω/c. Then, if the ‘driving pressure’ at the open end (x = 0) is p0 , we get p(x) = p0 cos[k(L − x)]/ cos(kL). Similarly, the velocity amplitude distribution is |u(x)| = |u0 | sin[k(L − x)]/ cos(kL), where |u0 | = |p0 |/ρc. Integrating the kinetic and potential energy densities ρ|u|2 /2 and |p|2 /2ρc2 over the volume of the tube gives the total energy E = (AL/2)|p0 |2 /ρc2 .
(2.54)
Using these expressions for |u| and |p| in Eq. 2.53 and integrating over the tube walls and accounting for the thermal losses at the wall, we get for the total loss rate W = (SL/2)[|u0 |2 (kdv ρc/2) + (|p0 |2 /ρc)(γ − 1)kdh /2] + A(|p0 |2 /ρc)(γ − 1)kdh /2 = ω(SL/2)(|p0 |2 /ρc2 )(dvh /2)[1 + (A/SL)(γ − 1)dh /dvh ] ≈ ω(SL/2)(|p0 |2 /ρc2 )(dvh /2), where
dvh = dv + (γ − 1)dh = dv [1 + (γ − 1)/ Pr ] ≈ 1.46dv
(2.55) (2.56)
is the ‘visco-thermal’ boundary layer thickness and Pr = μCp /K ≈ 0.77 (air), the Prandtl number, defined in Eq. 2.45. The term in Eq. 2.55, which contains the area A expresses the heat conduction loss at the rigid wall termination, which is usually small compared to the rest.2 Without this contribution and with ω = ω0 , the Q-value of the resonator becomes 2A Q = ω0 E/W ≈ , (2.57) Sdvh which can be interpreted as twice the ratio of the volume of the tube and the volume occupied by the visco-thermal boundary layer. As an example, consider a circular tube of radius a. With A = π a 2 and S = 2π a we get Q-value of a circular tube resonator (2.58) √ Q = a/dvh ≈ 3.11a f 2 There is no tangential velocity at the end wall and no viscous losses.
41
SOUND ABSORPTION MECHANISMS
where a: Tube radius, cm. f : Frequency, Hz. dvh : Visco-thermal boundary layer thickness (Eq. 2.56). Thus, Q is simply the ratio of the radius and the boundary layer thickness. The approximate numerical expression was obtained by using dvh ≈ 1.46dv and dv ≈ √ 0.22/ f . Thus, a 100 Hz quarter wavelength circular tube resonator with a one-inch diameter will have a Q-value of ≈39.5. For the channel between two parallel plates, separation d and width w, the area dw and S = 2(w + d) ≈ 2w. Therefore, Q ≈ d/dvh
(parallel plates).
(2.59)
In Chapter 4, we reconsider this problem without making the assumption that the cross sectional dimensions of the tube are large compared to the boundary layer thickness. The Q-value thus obtained is found to be consistent with that obtained here.
2.7.5 Sound Propagation in a Narrow Channel The two walls bounding the channel under consideration are placed at y = ±a, and the x-axis is chosen to be the direction of propagation. The sound field is assumed to be independent of z. The wave field in the channel can be shown to be a linear combination of three wave modes, a ‘propagational,’ a ‘thermal,’ and a ‘viscous’ mode, using the terminology in the reference above. The pressure in the propagational mode satisfies an ordinary wave equation and the temperature and velocity in the thermal and viscous modes satisfy diffusion equations. These equations have to be solved subject to the appropriate boundary conditions for velocity and temperature. The Propagational Mode The pressure field is contributed almost exclusively by the propagational mode, which is a solution to the ordinary wave equation ∇ 2 p + (ω/c)2 p = 0,
(2.60)
where c is the isentropic (free field) sound speed. Thus, the complex amplitude of the sound pressure p(x, y, z, ω) is a superposition of waves of the form exp(iqx + iqy y + iqz z), where q 2 + qy2 + qz2 = (ω/c)2 . With the sound field being independent of z, i.e., qz = 0, so that q 2 + qy2 = (ω/c)2 .
(2.61)
The fundamental acoustic mode (wavelength much longer than the width of the channel) is symmetrical with respect to the center of the channel. Thus, since y = 0 is placed at the center of the channel (with the boundaries at y = ±a, as already mentioned), it follows that pp (x, y, ω) = A cos(qy y)eiqx ,
(2.62)
where qy , yet to be determined, is the propagation constant in the transverse direction of the channel. In the absence of visco-thermal effects, qy = 0 but in general it is not,
42
NOISE REDUCTION ANALYSIS
and after having determined it, the propagation constant q, which is the quantity of primary interest, is obtained from Eq. 2.61. The velocity amplitude components corresponding to the pressure field in the propagational mode are Velocity amplitude, Propagational mode upx = A(q/ωρ)eiqx cos(qy y) upy = A(iqy /ωρ) sin(qy
(2.63)
y)eiqx
where qy : Eq. 2.77. q: Eq. 2.61. The subscript p indicates ‘propagational mode.’ The temperature field associated with the propagational pressure mode is due to the isentropic compression-rarefaction involved. The corresponding complex temperature amplitude is τp = (γ − 1)pp T /γ P ,
(2.64)
where γ is the specific heat ratio Cp /Cv , T the absolute temperature, and P the ambient pressure. The Thermal Mode With the boundary assumed to have a heat conduction coefficient and heat capacity much larger than the fluid, the temperature fluctuation can be assumed to be zero at the boundary. To satisfy this boundary condition, there must be a contribution from a heat conduction mode, which cancels the temperature fluctuation in the propagational mode at the boundary (the temperature fluctuation in the viscous mode is negligible). The temperature τh in the thermal mode is a solution to the diffusion equation ∂ 2 τh ∂τh = (K/ρCp )∇ 2 τ ≈ (K/ρCp ) 2 ∂t ∂y
(2.65)
subject to the condition that at the boundaries, the temperature τh cancels the temperature fluctuation τp in the propagational mode given in Eq. 2.68. For harmonic time dependence and with the y-dependence of the complex temperature amplitude of the form exp(ikh y), it follows that kh = (1 + i)/dh , (2.66) where dh = 2K/(ρCp ) is the thermal boundary layer thickness. The temperature fluctuation caused by the propagational (pressure) mode can be written γ − 1 pp γ −1A τp = (2.67) T = T cos(qy y)eiqx . γ P γ P The appropriate solution to the diffusion equation for the temperature amplitude in the thermal mode must be such that it cancels the contribution from the pressure mode at the boundaries. This means that τh = −τp
cos(kh y) , cos(kh a)
(2.68)
SOUND ABSORPTION MECHANISMS
43
which becomes −τp at the boundaries, y = ±a, where it will cancel τp in the propagational mode. The y-dependence of the total temperature amplitude is then
cos(kh y) τ (ω) = τp 1 − . (2.69) cos(kh a) Although the sound pressure in the thermal mode is negligible, its gradient and the corresponding fluid velocity component in the y-direction are not. The velocity component in the thermal mode can be shown to be uh = (γ P /ρc)(1/T )h grad τ , where h = K/ρCp c = iω/ckh2 , where h is of the order of the mean free path. From kinetic theory, h ≈ K/ρCp ct , where ct is the average molecular thermal speed, which is approximately the sound speed. The x-component of this velocity is negligible, since the acoustic wavelength is much greater than the boundary layer thickness, and the significant velocity is the y-component, uhy . From Eqs. 2.62 and 2.67 and the expressions for h and τh , given above, it follows that Transverse velocity amplitude, Thermal mode iqx cos(q a) h y) uhy = i(A/ρc)(γ − 1)(ω/c) khsin(k y sin(kh a) e
(2.70)
where kh : Eq. 2.66, qy : Eq. 2.77, q: Eq. 2.61. The Viscous Mode and the Total Velocity Profile The total velocity components, both in the x- and y-directions, must be zero at the boundaries and we start with the construction of the axial velocity distribution, which satisfies this condition as well as the equations of motion. Thus, the x-component of the velocity in the propagational mode in Eq. 2.63 must be canceled at the boundary by the velocity component in viscous mode, which is a solution to ρ
∂ 2 uvx ∂uvx = μ∇ 2 uvx ≈ μ ∂t ∂y 2
(2.71)
encountered earlier. It is of the same form as the diffusion equation for temperature with μ/ρ corresponding to K/ρCp . Thus, with the y-dependence for the complex velocity being exp(±ikv y) (or cos(kv y) or sin(kv y)), it follows that kv = (1 + i)/dv ,
(2.72)
where dv = sqrt2ν/ω is the viscous boundary layer thickness. The x-component uvx of the velocity in the viscous mode must be such as to cancel the velocity amplitude qP /ωρ of the propagational mode at the boundaries y = ±a. By analogy with the solution for the temperature equation we realize that the appropriate solution is Axial velocity amplitude, Viscous mode v y) iqx uvx = −(Aq/ωρ) cos(k cos(qy a) cos(kv a) e where kv : Eq. 2.72, qy : Eq. 2.77, q: Eq. 2.61.
(2.73)
44
NOISE REDUCTION ANALYSIS
The total axial velocity is obtained by adding the contribution upx from the propagational mode in Eq. 2.63,
cos(kv y) cos(qy a) cos(kv y) ux = upx 1 − ≈ upx 1 − . cos(kv a) cos(qy y) cos(kv a)
(2.74)
It will be established shortly that qy a << 1 so that cos(qy a) = cos(qy y) ≈ 1, and the equation above can be expressed as Axial velocity in a channel distribution ux (y) cos(kv y)−cos(kv a) ux (0) = 1−cos(kv a)
(2.75)
where kv = (1 + i)a/dv : Eq. 2.60, a: Half-width of channel, ux (0): Velocity at the center of the channel. The profile is plotted in Figure 2.2 as a function of y/a for a/dv = 1, 2, 4, 8, 16, and 32. It is noteworthy also that for sufficiently large values of a/dv , the maximum value of the velocity amplitude is not at the center of the channel but moves toward the wall with increasing a/dv , i.e., increasing frequency. Eq. 2.69 shows that the variation of the temperature amplitude across a channel has the same form as the velocity with kv replaced by kh . For a circular tube an analogous derivation can be carried out with a similar result. The characteristic function F = tan(X)/X will be replaced by a similar one in terms of Bessel Functions, as discussed at the end of this chapter. As we shall see shortly, both cos(qy y) and cos(qy a) can be replaced by 1, since qy a is found to be << 1. The y-component uvy of the viscous mode is obtained from the fact that there is no compression in the shear flow so that div uv = 0, i.e., ∂uvy /∂y = −∂uvx /∂x. Hence, Transverse velocity component, Viscous mode iqx cos(q a) v y) uvy = i(A/ωρ)q 2 kv sin(k y sin(kv d/2) e
(2.76)
where kv : Eq. 2.72, qy : Eq. 2.77, q: Eq. 2.61. The pressure field associated with the viscous mode is negligible. Propagation Constant, Phase Velocity, and Attenuation Having obtained the y-components of the velocity contributions from the propagational, thermal, and viscous modes in Eqs. 2.63, 2.70, and 2.76, respectively, we now turn to the boundary condition that requires the sum of these components to vanish at the boundaries. This condition, as we shall see, determines the ‘transverse’ propagation constant qy . Since the wavelength of the sound wave is assumed to be much larger than the channel width, we expect the variation in pressure across the channel (expressed by cos(qy y) to be small and hence qy a << 1. This enables us to simplify the boundary condition equation by putting cos(qy a) ≈ 1 and sin(qy a) ≈ qy a. Then, with
45
SOUND ABSORPTION MECHANISMS
q 2 = (ω/c)2 − qy2 , the boundary condition equation for qy , upy + uhy + uvy = 0 (see Eqs. 2.63, 2.70, 2.76), can be reduced to Transverse propagation constant qy qy2 (ω/c)2
(wh )+F (wv ) = − (γ −1)F 1−F (wv )
(2.77)
F (w) = tan(w)/w w = (1 + i)ξ,
ξv = a/dv ,
ξh = a/dh
where dv : Eq. 2.37, dh : Eq. 2.43. We shall often use the notations Fv ≡ F (wv ) and Fh ≡ F (wh ) for short. The behavior of F for small and large arguments is tan[(1 + i)ξ ] = F [(1 + i)ξ ] ≡ (1 + i)ξ
1 + i2ξ 2 /3 − 8ξ 4 /15 (1 + i)/2ξ
for ξ << 1 . for ξ >> 1
(2.78)
The real and imaginary parts of F are shown in Figure 2.9. The maximum value of the imaginary part (≈ √ 0.4) occurs when ξ ≈ 1. For a fixed channel width, the variable ξ is proportional to ω and can be regarded as a frequency variable. low and high frequency regions, It follows from Eq. 2.77 for qy that in the√ √ corresponding to ξ << 1 and ξ >> 1, qy a ≈ 3γ /2(ωdv /c) and qy a ≈ ω adv /c, respectively, where ω/c = 2π/λ. For wavelengths much larger than a and dv , qy a becomes much less than 1 over the entire frequency range of interest, consistent with the assumption made earlier.
Figure 2.9: Real and imaginary parts of the function F = tan[(1 + i)ξ ]/(1 + i)ξ . In our case, ξ = a/dv , where a is the half-width of the channel and dv the viscous boundary layer thickness.
46
NOISE REDUCTION ANALYSIS
Under the conditions stated above, the propagation constant q in Eq. 2.60 becomes, with Fh ≡ F (kh a) and Fv ≡ F (kv a), Axial propagation constant q −1)Fh Q ≡ q/k ≡ (qr + iqi )/k = 1+(γ 1−Fv √ ≈ 3γ /4 (1 + i)/ξv for ξv , ξh << 1 ≈ 1 + (i/4)[1/ξv + (γ − 1)/ξh ] = 1 + idvh /4a for ξv , ξh >> 1
(2.79)
where√F , ξv , ξh : Eq. 2.77 and 2.78, dvh : Eq. 2.80, k = ω/c, ξv = a/dv , ξh = a/dh , dv = 2μ/ρω, dh = 2K/Cp ρω, dvh = dv + (γ − 1)dh . For a circular tube with a diameter d, the high frequency approximation of the propagation constant is Q ≈ 1+i[dv +(γ −1)dh ]/d = 1+idvh /d
(dvh = dv +(γ −1)dh << d). (2.80)
In Figure 2.1, we have plotted vs ξv the real and imaginary parts of the normalized propagation constant, Qr = qr /(ω/c) and Qi = qi /(ω/c), using dh ≈ dv , and hence ξh ≈ ξv as a good approximation.
2.7.6 Impedances The channel can be treated approximately as a one-dimensional transmission line by introducing the average values of the amplitude of sound pressure and the axial velocity in the channel. Under the assumption that qy a << 1, the average value of pressure and velocity in the propagational mode are simply A and (Aq/ωρ) (Eq. 2.63). Adding the average velocity in the viscous mode (the thermal mode contribution is negligible), we get for the total average axial velocity uav = upx [1 − Fv ],
(2.81)
where upx = (qA/ωρ) exp(iqx x). The channel impedance per unit length is defined as zc = −
∂p 1 −iωρ = ≡ −iωρ, ˜ ∂x uav 1 − Fv
(2.82)
where the complex density is ρ˜ = ρ/(1 − Fv ) and Fv is given in Eqs. 2.77 and 2.78. Thus, Channelimpedance per unit length for ξv << 1 3μ/a 2 − i1.2ωρ −iωρ zc ≡ rc + ixc = 1−F = v ωρ/2ξv − iωρ(1 + 1/2ξv ) for ξv >> 1
(2.83)
where Fv : Eqs. 2.77 and 2.78, a: half-width of channel, ξ : Eq. 2.80. The impedance is based on the force per unit area of a fluid element of unit length, not on the total force on the fluid element, which is proportional to the area of the
SOUND ABSORPTION MECHANISMS
47
channel. The subscript c refers to ‘channel’ to distinguish it from the impedance per unit length of a porous material in general, discussed in the next two chapters, rc /r0c = (2/3)(a/dv )2 {1/(1 − Fv )}. Figure 2.4 shows the normalized value of the resistance per unit length, and the equivalent mass density ratio c = ρec /ρ = {1/(1 − Fv )} vs ξv = a/dv . For ξv ≈ 5, the resistance is about twice the steady flow value. c decreases monotonically from the steady flow value of 1.2 to 1 as the frequency increases, the transition occurring at ξv ≈ 1. The low frequency approximations for the resistance, the equivalent mass density, and the corresponding reactance are often good approximations over most of the frequency range of interest. With increasing frequency, the structure factor approaches 1 asymptotically and the resistance becomes proportional to a/dv , i.e., proportional to the square root of frequency. It is reassuring that Eq. 2.83 at low frequencies, ξ << 1, yields the resistance r1c , which is the same as the flow resistance per unit length, r0c = 3μ/a 2 , derived for steady flow in Eq. 2.33, and that for ξv >> 1, the resistance agrees with that for oscillatory (incompressible) flow in Eq. 2.40. Wall Stress and Viscous Interaction Impedance The impedance in Eq. 2.83 contains a mass reactance, which for ξ << 1 is xc = −1.2ωρ. This means that in addition to the reactance −iωρ of the air in the channel there is an additional contribution 0.2ωρ. This ‘induced’ or ‘virtual’ mass contribution is related to the force transferred to the walls of the channel. To see explicitly that this is indeed the case, we compute the force per unit area on the wall at y = a and multiply by 2 to include the force on the other wall. Thus, we start with the axial velocity distribution and express the factor (qA/ωρ) exp(iqx) in terms of the average velocity in the channel. The total force per unit length and unit width of the channel is then, with d = 2a, fw = −2μ
Fv −iωρFv ∂utx = −μuav dkv2 = duav . ∂y 1 − Fv 1 − Fv
The viscous interaction impedance per unit length is then Viscous interaction impedance per unit length w v zvc = d1 ufav = −iωρF 1−Fv
(2.84)
where Fv : Eqs. 3.70, 3.71, and 2.83. For small values of ξv = a/dv , the approximation for F shows that zv ≈ 3μ/a 2 − i0.2ωρ. Thus, in addition to a friction component there is also an inertial mass component in the force, which corresponds to a mass density 0.2ρ, where 0.2 will be referred to as the virtual or induced mass factor. The impedance per unit length, already introduced in Eq. 2.83, is obtained by adding to zvc the mass reactance of air, −iωρ. It is often convenient to express this total impedance per unit length as −iωρ, ˜ where ρ˜ is the complex density, already introduced in Eq. 2.83.
48
NOISE REDUCTION ANALYSIS
Wave Impedance The normalized wave impedance is ζwc ρc ≡ zwc = p/uav , where p and uav refer to a traveling wave, and it follows from the expression for p, Fv , q, and uav in Eqs. 2.62, 2.78, 2.79, and 2.81 that Normalized wave impedance in a channel 1 ζwc ≡ θwc + iχwc = uavAρc = √[1−F ][1+(γ = (1−F1 v )Q −1)Fh ] v √ 3/4γ (1 + i)/ξv = Q/γ for ξv , ξh << 1 ≈ 1 for ξv , ξh >> 1
(2.85)
where F , ξ : Eqs. 2.77 and 2.78, Q: Eq. 2.79. Quantity Q ≡ Qr + iQi ≡ q/k is the normalized propagation constant. The qualitative difference between the wave impedance at low and high frequencies, both in terms of magnitude and phase, reflects the change in character of the interaction force from viscous (diffusive) to inertial (propagational) dominance as the frequency increases. As ξ increases, the conditions in the channel approach those in free field and zwc /ρc → 1, as expected. In the diffusion regime, ξ << 1, the real and imaginary parts have equal magnitude, which was the case also for the impedance zc per unit length. There is one important distinction, however. Unlike z1c , the wave impedance has a positive imaginary part, indicating a stiffness reactance ρcχwc = (1/γ )qi /k, where k = ω/c. In terms of the penetration depth dp = 1/qi it follows that ρcχcw = (1/γ )(ρc2 /ωdp ),
(2.86)
which we recognize as the (stiffness) reactance of an air cavity of depth dp backed by a rigid wall under isothermal conditions, the isothermal compressibility being γ /ρc2 . The real and imaginary parts of the normalized wave impedance zwc /ρc are shown vs ξ in Figure 2.4. Comparison With a One-Dimensional Transmission Line For harmonic time dependence, the one-dimensional equations for mass and momentum balance in a loss free one-dimensional air-wave guide are ∂u −iωδ = −ρ ∂x ∂p −iωρu = − ∂x ,
(2.87)
where δ, p, and u are the perturbations in density, pressure, and velocity. Introducing the compressibility κ = (1/ρ)δ/p = 1/ρc2 , the mass equation can be written −iκωp = −∂u/∂x. For a lossy transmission line, the compressional and friction losses can be accounted for by letting κ and ρ be complex, and the equations then take the form ∂u −iωκp ˜ = − ∂x ∂p −iωρu ˜ = − ∂x ,
(2.88)
49
SOUND ABSORPTION MECHANISMS
where κ˜ is the complex compressibility and ρ˜ the complex density, given in Eq. 2.83. The complex compressibility is analogous to the inverse of the frequently used complex spring constant, which expresses the combined effect of an ideal spring and a dashpot resistance in parallel. If the resistance is frequency independent, the irreversible stress component involved is proportional to the rate of change of compression. Similarly, the complex density is analogous to the complex mass, which can be used to account for the combined effect of inertia and a friction force proportional to its velocity. With the space dependence of p and u proportional to exp(iqx), it follows from these equations that the propagation constant, impedance per unit length, and the wave impedance are Propagation constant, impedance per unit length, wave impedance q = ω κ˜ ρ˜ zc = −iωρ˜ zwc = ρ/ ˜ κ˜ av
(2.89)
where ρ, ˜ κ˜ av : Eq. 2.90. By comparing these expressions with those for the channel wave under consideration, we see from the expressions for q and zc in Eqs. 2.79 and 2.83 that the equivalent complex density and compressibility of the fluid in the channel are ρ˜ =
ρ 1−Fv
κ˜ av = κ[1 + (γ − 1)Fh ],
(2.90)
where κ = 1/ρc2 = 1/γ P is the isentropic compressibility in free field. More About Complex Compressibility The derivation of the complex compressibility κ˜ in Eq. 2.90 was based on the general relations in Eq. 2.89 and Eq. 2.79. We can, of course, arrive at the complex compressibility by starting from the definition of compressibility κ = (1/ρ)∂ρ/∂P , with ρ = ρ(P , T ), and express the density perturbation in terms of the perturbations in pressure and temperature, dρ =
∂ρ ∂ρ dP + dT . ∂P ∂T
(2.91)
In the present case, the total temperature perturbation amplitude dT = τ is given by Eq. 2.69, which is proportional to the pressure amplitude (see Eq. 2.68). Thus, expressing the temperature perturbation dT = τ in terms of dP , dρ becomes proportional to dP and the compressibility is then obtained from (1/ρ)dρ/dP . This leads to the y-dependence of the compressibility in the channel
1 cos(kh y κ(y, ˜ ω) = 1 + (γ − 1) (2.92) γP cos(kh a)
50
NOISE REDUCTION ANALYSIS
with the average value The average complex compressibility ina channel h a) κ˜ av1 = κ 1 + (γ − 1) tan(k kh a
(2.93)
where kh = 1/dh , dh : Eq. 2.43, Curve (1) in Figure 2.3, κ = 1/γ P . With Fh = tan(kh a)/kh a, the result is the same as obtained in Eq. 2.90. The real and imaginary parts of the normalized complex compressibility κ/κ ˜ are plotted in Figure 2.3 as a function of a/dh . The compressibility goes from the isothermal to the isentropic value as a/dh (or frequency) increases. The imaginary part has a maximum of 0.16κ for a/dh = 1.12, and if we take this to represent the transition from isothermal to isentropic conditions, the corresponding transition frequency becomes ωh ≈ 1.25(2K/a 2 ρCp )
(2.94)
with a corresponding characteristic (relaxation) time th = 1/ωh . The frequency dependence of the compressibility is expressed implicitly by the function Fh . It is of interest in this context to derive an approximate expression for the compressibility in which the frequency dependence is expressed explicitly in a simple manner. The general approach is the same as before; the only difference is that we use an approximate expression for the average temperature fluctuation obtained from the diffusion equation by making an estimate of the temperature gradient at the wall. As before, we let the total temperature fluctuation τ be the sum of the contributions τp from the propagational (pressure) mode and τh = τ − τp from the thermal mode. The latter is a solution to the diffusion equation, and realizing that the spatial variation of τp can be neglected compared to that of τh (and hence of τ ) since the wavelength is much greater than the boundary layer thickness, we have ∂(τ − τp ) = (K/ρCp )∇ 2 τ. ∂t
(2.95)
To obtain the average temperature, we integrate this equation over a volume element of unit length and unit width of the channel and replace the integral over ∇ 2 τ = div grad τ by a surface integral of the normal component on grad τ in the outward direction. The variation of τ along the axis of the channel is neglected and the integral then reduces to (2K/ρCp )(∂τ/∂y)y=a , the factor of 2 accounting for the two wall surfaces of the channel. Next, we express the thermal gradient at the wall as (∂τ/∂y)y=a ≈ −τav /a1 , where the length a1 is of the order of the half-width a of the channel. The integrated righthand side of Eq. 2.95 then becomes −2(K/ρCp )τav /a1 , the factor of 2 accounting for the two walls of the channel. For harmonic time dependence, the integrated lefthand side of the equation becomes −iω(τav − τp )2a, where 2a is the volume of the element (unit width and length). The characteristic length a1 is chosen so that K/(ρCp a1 ) becomes equal to the characteristic frequency ωh in Eq. 2.94. This means that a1 = a/2.5 and Eq. 2.95 takes the form
51
SOUND ABSORPTION MECHANISMS −iω(τav − τp ) = −ωh τav so that τav (ω) =
−iωτp , ωh − iω
(2.96)
where τp = [(γ − 1)/γ ]T (p/P ) and p is the sound pressure amplitude. In the expansion of the density in terms of pressure and temperature, dρ = (∂ρ/∂P )dP + (∂ρ/∂T )dT , we insert the amplitudes p and τ for the perturbations dP and dT to obtain
γ −1 iω dρ = (ρ/P )dP 1 + . (2.97) γ ωh − iω The corresponding complex compressibility is then Approximate complex compressibility in a channel κ˜ av2 =
1 dρ ρ dP
≈κ 1+
γ −1 1+(ω/ωh )2
+
i(γ −1)(ω/ωh ) 1+(ω/ωh )2
(2.98)
where ωh : Eq. 2.94, κ = 1/γ P = 1/ρc2 , Curve (2) in Figure 2.3. In comparing this κ˜ av2 with κ˜ av1 (see Eq. 2.93 and Figure 2.3), it should be noted that ω/ωh = (a/dh )2 /1.25 (recall that dh2 = 2K/ρCp ω and ωh is given in Eq. 2.94). The complex compressibility κ˜ av2 has the correct limiting values for ω/ωh << 1 and ω/ωh >> 1, and it has a maximum value of the imaginary part of (γ − 1)/2 (equal to 0.2 for air) obtained for ω = ωh or a/dh ≈ 1.12. In Figure 2.3 we have plotted κ˜ av1 as a function of a/dh , and it can be seen to be in rather good agreement with κ˜ av1 given in Eq. 2.93. The expression for κav2 can be useful in the study of the rigid porous absorber since it is a good approximation to replace the thermal by the viscous relaxation time, which is determined by the known flow resistance of the material. For a channel we have ωv = roc /ρ and with r0c = 3μ/a 2 it follows that ωv = 3μ/a 2 ρ = 3Pr (K/ρCp a 2 ) ≈ 2.3(K/ρCp a 2 ) ≈ 0.92ωh ,
(2.99)
where we have used Pr = μCp /K ≈ 0.77 for the Prandtl number of air. In other words, the viscous and thermal ‘relaxation’ frequencies ωv and ωh are almost the same numerically so that ωv can also be used as an indicator of the transition between isothermal and isentropic conditions. For porous material with a normalized flow resistance θ = r/ρc per inch, we may use (see Eq. 2.99) Ratio of viscous and thermal relaxation frequencies fh ≈ fv = r/(2πρ) = cθ/2π ≈ 2140θ Hz
(2.100)
For example, a material with θ = 0.5 per inch yields fv ≈ 1070 Hz at room temperature, with c ≈ 1120 ft/sec. In other words, for most materials used in practice, the conditions within the porous material are isothermal over a substantial range of frequencies.
Chapter 3
Sheet Absorbers As indicated in Section 1.2.1 on chapter organization, we have attempted to gather most of the mathematical details and derivations in a separate section, which can be skipped at the first reading or skipped altogether by the reader who is interested mainly in results. In this chapter, most of this mathematical analysis is summarized in Section 3.6. Some of the most important results, often the basis for the numerical results presented in graphs, are duplicated in the main part of the chapter.
3.1 INTRODUCTION AND BRIEF SUMMARY Before proceeding with this (or any other) chapter, it is advisable that the reader become familiar with the discussion of terminology and notation in Chapter 7. This, chapter deals with a sound absorber consisting of a porous sheet or screen backed by an air layer and a rigid wall, which is parallel with the sheet; the air layer then forms an important part of the absorber. In another configuration, the sheet is simply hung from the ceiling in a room without any direct relation to any particular wall. These two configurations are sometimes referred to as ‘surface’ and ‘volume’ absorber, respectively (see Figure 3.17). When sound interacts with a sheet, some of the incident acoustic energy is lost to heat as a result of the friction drag between the sheet and the acoustic velocity field, i.e., sound is absorbed.
3.1.1 Single Sheet Surface Absorber Sheet absorbers are alternatives to conventional uniform porous layers in bulk and can have some advantages in regard to several nonacoustical factors having to do with erosion, water containment, ease of cleaning, hostile environments (high temperature and flow), contamination of the air by fibers, etc. When a sheet is used as a ‘volume’ or ‘functional’ absorber, ease of mounting and the possibility for creative architectural designs are other factors which may favor the use of sheets. When used as a surface absorber, the air backing or cavity between the sheet is largely responsible for the frequency dependence of the absorption. If the sheet is in direct contact with the wall, the absorption is essentially zero, and in order for the absorber to be effective, the depth of the air backing normally should not be less than 53
54
NOISE REDUCTION ANALYSIS
Locally reacting
Nonlocally reacting
Incident sound wave
L
L
Figure 3.1: Porous sheet cavity absorber. Left: Locally reacting. Right: Nonlocally reacting.
a quarter wavelength. With a quarter wavelength backing, the absorption coefficient can be made to be 100 percent at the peak and in excess of about 80 percent over a frequency band of approximately one octave. Narrower absorption bands with about the same peak value occur at higher frequencies corresponding to cavity depths of an odd number of quarter wavelengths. The cavity can be empty or ‘honeycombed’ (Figure 3.1) with acoustically compact honeycombs, which make the absorber locally reacting. If it is empty, the absorber is nonlocally reacting. A simple example of the nonlocally reacting kind is a sheet hung as a curtain or drape in front of a wall. For normal incident sound, the performance of the two configurations will be the same, of course, but in a diffuse sound field, the locally reacting absorber is better, except at the frequencies for which the backing depth is an integer number of half wavelengths. At these frequencies the absorption coefficient dips to zero for both normal incidence and diffuse field; this does not occur for the nonlocally reacting absorber. The dips are quite pronounced in a narrow band spectrum but less pronounced in broader bands, say 1/3 or 1/1 octaves, as will be demonstrated shortly. By absorption spectrum we mean the relationship between the absorption coefficient and frequency. The complete absorption spectra, for locally and nonlocally reacting sheet absorbers, are shown in Figure 3.2 for a rigid and in Figure 3.8 for a flexible sheet, which demonstrate the special characteristics mentioned above. For normal incidence, the absorption coefficient α0 becomes 1.0 at a quarter wavelength resonance if the flow resistance of the sheet is one ρc, but in a diffuse field, a flow resistance of ≈2 ρc yields the best overall performance. With proper choice of sheet parameters, the acoustically induced motion of the sheet can be utilized to lower the resonance frequency of the absorber substantially below the regular quarter wavelength resonance, and 100 percent absorption can be obtained at this new low frequency resonance if the normalized flow resistance of the sheet is chosen to equal the ratio mr of the sheet mass and the mass of the air
SHEET ABSORBERS
55
Figure 3.2: The absorption coefficient of a rigid sheet-cavity absorber vs L/λ, where L is cavity depth and λ, the wavelength. Normalized flow resistances of the sheet: 1, 2, 4. (a): Normal incidence. (b) Diffuse field, local reaction. (c): Diffuse field, nonlocal reaction. Bandwidth: ≈1/60 OB. For curves with explicit frequency dependence, see the next figure and Appendix C.
layer (assuming mr >> 1). The corresponding resonance frequency is then reduced √ by a factor, which is approximately equal to (π/2) mr . However, the reduction in resonance frequency is achieved at the expense of a decrease in the width of the resonance.
3.1.2 Multisheet Absorber An absorber consisting of several parallel porous sheets in the form of a lattice in front of and parallel with a rigid wall can yield an absorption which can come close to and even exceed that of a uniform porous layer as can be seen in Figure 5.21. Actually, at sufficiently low frequencies, typically below 200 Hz, a single sheet absorber can be better than a uniform layer of the same thickness and the same total flow resistance.
3.1.3 Single Sheet as a ‘Volume’ Absorber In this case, the appropriate measure of the performance is the absorption area or absorption cross section per unit area of sheet material. Under the idealized assumption of an immobile, infinite sheet, the absorption is frequency independent. However, due to the induced motion of the sheet, the low frequency performance is reduced as
56
NOISE REDUCTION ANALYSIS
shown in Figure 3.20. An additional, and generally even more important, reduction at low frequencies is caused by diffraction, which is considered in the last section (Figure 3.21). As for the surface absorber, there is an optimum flow resistance for each frequency. If the wavelength is smaller than the dimensions of the sheet, the best overall performance of the volume absorber in a diffuse field is obtained with a normalized flow resistance of 3.2. The corresponding high frequency limit of the absorption area is then almost equal to the physical area of the sheet material, corresponding to an absorption coefficient of almost 48 percent for the sound, which is incident on each of the two surfaces of the sheet. For normal incidence on a rigid infinite sheet under optimum conditions (flow resistance 2), 50 percent of the sound is absorbed by the sheet, 25 percent is reflected, and 25 percent transmitted through the sheet. It should be realized, however, that if the wavelength is large compared to the sheet dimensions, the optimum flow resistance and the absorption cross section per unit sheet area decrease with the size of the sheet because of diffraction, which tends to reduce the difference between the sound pressure amplitudes on two sides of the sheet and the corresponding flow velocity through the sheet. This effect is also included in the analysis and can be regarded as a low frequency approximation for the absorption, as shown in Figure 3.21.
3.2 RIGID SINGLE SHEET WITH CAVITY BACKING 3.2.1 Flow Resistance and Impedances In a harmonic sound wave, the drop in sound pressure amplitude across a sheet and the corresponding fluid velocity through the sheet are both oscillatory and in the idealized case of a purely resistive sheet, these oscillations are in phase. The ratio of the amplitudes of the pressure drop and the velocity is then a flow resistance, which is the same as for steady flow. However, in reality, the velocity lags behind the pressure by some phase angle due to an inertial reactance of the sheet. The phase angle goes to zero as the frequency goes to zero, but it can be substantial at sufficiently high frequencies. The amplitudes of the pressure drop and the velocity are now described by complex amplitudes which contain both the magnitudes and phase angles. Their ratio is also a complex quantity, a complex impedance, with a magnitude and a phase angle, which expresses the combined effects of the resistance and the reactance of the sheet. The measurements of the steady flow resistance as well as the impedance of the sheet are described in Appendix A, where some experimental data are shown and discussed. If the sheet is not moving, the velocity through the sheet, i.e., the relative velocity of the air with respect to the sheet, will be the same as the absolute velocity of the air just in front and just behind the sheet, and a measurement of the ratio of the amplitudes of the pressure drop and the air velocity then yields, by definition, the interaction impedance. If the sheet is not rigid but induced to oscillate as a result of the interaction with the sound, the velocity amplitude of the air outside the sheet will not be the same as the amplitude of the relative motion of the air and the sheet, and an impedance defined in terms of the absolute rather than the relative velocity amplitude will contain
SHEET ABSORBERS
57
an inertial contribution resulting from the mass of the screen, as discussed later in this chapter. The corresponding impedance, based on the absolute rather than the relative velocity, will be called the equivalent impedance. By making the mass of the screen large enough, the induced motion can be made negligible; the screen becomes immobile or ‘rigid.’ The presence of an inertial component in the equivalent impedance of the sheet is easy to understand because of the induced motion of the sheet, but an inertial component of the interaction impedance (the sheet is stationary) is a more subtle matter. A detailed analysis of an oscillatory flow (sound) in a porous material shows that there are actually two contributions to the inertial part of the impedance. One is due to the reaction force between the fluid and the structure as the fluid is forced to change direction and velocity along a tortuous path through the material. This force is proportional to the acceleration of the fluid, and thus acts like an inertial reactance and an apparent increase in the inertial mass density of the fluid. The flow resistive force is proportional to the velocity. Actually, there is a second contribution to the inertial reaction force on the fluid. It stems from the oscillatory friction force on the material, which is not quite in phase with the air velocity. The corresponding reaction force on the fluid from the boundary then contains a component proportional to the acceleration, i.e., an inertia component, which can be interpreted as a contribution to the induced mass of the fluid. Of course, there is also the inertia of the air itself within the sheet but in the present treatment of the sheet as a very thin layer, it will be neglected. For a rigid sheet, the assumption of being purely resistive is often quite good over the frequency range of most interest in noise control applications. One finds, for example, that for cloth-like sheets, the resistance typically exceeds the mass reactance at frequencies below 3000 Hz, but this frequency limit depends on the fiber diameter and the structure of the cloth. For a flexible sheet, however, the acoustically induced motion often contributes a significant inertial component to the impedance, particularly if the flow resistance is large.
3.2.2 Resonances and Anti-Resonances The resistance of a porous sheet varies only weakly with frequency, typically as the square root of frequency. The sheet reactance always represents an inertia. However, the reactance of the air layer between the sheet and the rigid backing wall can be either inertia- or stiffness-like. If the sheet is purely resistive, the reactance of the absorber is contributed only by the air layer and is zero when its thickness L is an odd number of quarter wavelengths yielding the ‘quarter wavelength resonances’ of the absorber. The sheet then finds itself in the plane, where the velocity amplitude is a maximum in the standing wave, which is established in the air layer. At resonance, the input impedance of the absorber is then the impedance of the sheet alone, and if the sheet resistance is chosen to equal the wave impedance ρc of a plane wave at normal incidence, impedance matching, 100 percent absorption results, and no sound will be reflected (at an angle of incidence φ, the corresponding
58
NOISE REDUCTION ANALYSIS
value is ρc/ cos φ). If the sheet itself contributes a reactive impedance component, the resonance occurs at a frequency slightly lower than the quarter wavelength value; the stiffness-like reactance of the air layer is then canceled by the inertial reactance of the sheet. On the other hand, at frequencies for which the layer thickness is an integer number of half wavelengths, the sheet is located in a plane where the velocity amplitude is zero and the pressure amplitude a maximum, and the layer impedance is infinite. No absorption occurs and we have an anti-resonance. For sound at oblique incidence, the impedance of the air backing depends on whether or not it is honeycombed, as indicated in the figure. By ‘honeycombed’ is meant that the air backing is divided into cells by partitions and that the cell size is much smaller than the wavelength (acoustically compact). With honeycombed backing, the air motion in the layer is forced to be perpendicular to the wall, and the input impedance will be the same as for normal incidence. The air motion in a cell depends only on the sound pressure at the surface of this cell and is not affected by the sound pressure at other cells (there is no acoustic communication between them). Therefore, the input impedance of the boundary, i.e., the ratio of the sound pressure amplitude at the boundary and the normal component of the velocity amplitude, will be independent of the angle of incidence. The surface is then said to be locally reacting. Sheet absorber, Input impedance, Local reaction ζi ≡ θi + iχi = ζ + i cot(kL)
(3.1)
k = ω/c = 2π/λ. ζ = θ + iχ : Normalized sheet impedance. If purely resistive: χ = 0. L: Air layer thickness (Figure 3.1). (See also Section 3.6, Eq. 3.19.) For an absorber which is not honeycombed, the motion of the air in the backing layer will be a superposition of waves traveling in the same directions as the incident and reflected wave outside the absorber, and it will depend not only on the local sound pressure amplitude but on the distribution over the entire absorber. The absorber is then called nonlocally reacting. It will perform like a locally reacting absorber at normal incidence, of course, but at oblique incidence important differences occur. As for the locally reacting absorber, there still will be a standing wave in the air backing, but the distance between adjacent pressure minima (or maxima), i.e., the spatial half period in the direction normal to the boundary, will be greater than the half wavelength of the incident sound. It will be half a wavelength for normal incidence, but it will go to infinity at grazing incidence when the wave fronts are perpendicular to the boundary so there is no periodic variation in a direction normal to the boundary, i.e., parallel with the wave fronts. At an arbitrary angle of incidence, the spatial period perpendicular to the boundary varies as λx = λ/ cos φ, where φ is the angle of incidence (the angle between the direction of propagation and the normal to the boundary, Figure 3.1) and λ is the wavelength of the incident sound. The corresponding x-component of the propagation constant is kx = k cos φ = 2π/λx . As far as the sound is concerned, it ‘measures’ the thickness of the air layer in units of the spatial period λx , and the thickness of the absorber thus appears to the incident
59
SHEET ABSORBERS
sound to decrease with increasing angle of incidence. The first quarter wavelength resonance now occurs when L = λx /4, the condition for maximum absorption, and if L = λx /2, we have an anti-resonance and no absorption. Note that the corresponding frequencies will be higher than for normal incidence. For a given L, the anti-resonance can only happen at one angle of incidence; the absorption at other angles will not be zero and this means that, unlike the locally reacting absorber, the nonlocal absorber cannot have a vanishing diffuse field absorption coefficient (which is an average over all angles of incidence) and the input impedance is dependent on the angle of incidence. Sheet absorber, Input impedance, Nonlocal reaction ζi ≡ θi + iχi = ζ + i(1/ cos φ) cot(kx L)
(3.2)
L: Layer thickness. φ: Angle of incidence, Figure 3.1. k = ω/c = 2π/λ, kx = k cos φ. ζ = θ + iχ : Normalized sheet impedance, Eq. 3.1. (See also Section 3.6, Eq. 3.20.) If the cell size in the honeycomb is not much smaller than the wavelength, the calculation of the input impedance becomes considerably more complicated and will not be considered here. The absorption spectrum then falls between those for the locally and nonlocally absorber, closer to the former at low frequencies and to the latter, at high.
3.2.3 Absorption Spectra Recall that the ratio of the amplitudes of sound pressure p and fluid velocity u in a plane traveling wave is ρc, where ρ is the density and c the sound speed, the wave impedance of the fluid. At an angle of incidence φ, the velocity normal to the boundary is ux = u cos φ, and the corresponding ‘component’ of the wave impedance is p/ux = ρc/ cos φ. If the sound encounters a boundary with an input impedance, which is different from this impedance, the boundary does not ‘feel’ like free field to the incoming sound and reflection will occur. The reflection coefficient R is the ratio of the reflected (complex) pressure amplitude at the boundary and the incident amplitude and, like the impedance, is a complex quantity. The corresponding absorption coefficient, the ratio of the reflected and incident intensities, is shown in Eq. 3.22, and is repeated here. Pressure reflection coefficient, Absorption coefficient R = (ζi cos φ − 1)/(ζi cos φ + 1) α(φ) = 1 − |R|2 =
(3.3)
4θi cos φ (1+θi cos φ)2 +(χi cos φ)2
ζi = θi + iχi : Normalized input impedance (Eqs. 3.1, 3.2). φ: Angle of incidence, Figure 3.1. see Eq. 3.22. (See also Section 3.6, Eq. 3.21.) As mentioned earlier, the input impedance of a sheet absorber at a quarter wavelength resonance is simply the sheet resistance, and if this is chosen to be ρc/ cos φ, 100 percent absorption will occur at resonance and an angle of incidence φ, as can be
60
NOISE REDUCTION ANALYSIS
seen for normal incidence (φ = 0) in Figure 3.2. This is the case of ‘impedance matching.’ At other frequencies and/or values of the sheet resistance, reflection occurs. Averaging the absorption coefficient over all angles of incidence in a diffuse field, we obtain the Diffuse field absorption coefficient π/2 α(φ) cos φ sin φ dφ αst = 0 π/2 = 2 0 α(φ) cos φ sin φ dφ π/2 0
(3.4)
cos φ sin φ dφ
α(φ): Absorption coefficient, Eq. 3.3. φ: Angle of incidence, Figure 3.1. (See Eq. 3.23.) The computed absorption spectra of a rigid, purely resistive sheet-cavity absorber with and without partitioned air backing are shown in Figure 3.2 for both normal incidence and diffuse field and for normalized sheet resistances θ of 1, 2, and 4. Note that the frequency parameter is expressed in normalized form as L/λ, where L is the layer thickness and λ the free field wavelength. We refer to this nondimensional presentation loosely as ‘universal,’ and it is chosen here so that the quarter wavelength resonances and the half-wavelength anti-resonances readily can be spotted. If instead an explicit frequency dependence is desired, with the frequency in Hz rather than the nondimensional parameter L/λ, one curve is needed for each layer thickness L, whereas here, one curve will do (hence, the designation ‘universal’). With a normalized sheet resistance θ = 1, the resonance absorption coefficient at normal incidence is 1, as we have seen, but it will be less than 1 if θ > 1. However, for oblique incidence, on the other hand, it will be 1 for waves with an angle of incidence given by cos φ = 1/θ . With θ < 1, the absorption coefficient is less than unity for all angles of incidence, and it follows that to optimize the diffuse field absorption coefficient, θ should be >1. If the cavity depth is an integer number of half wavelengths, the absorption coefficient becomes zero for the locally reacting absorber for both normal incidence and in diffuse field. In either case, the input impedance is infinite. The appearance of the spectra in the vicinity of these anti-resonances depends on the bandwidth used in the presentation, i.e., on the number of frequencies used in computing the graph. To get a ‘clean’ zero of the absorption coefficient in the graph requires a very narrow bandwidth. In this case, the bandwidth is approximately 1/60th of an octave (400 data points in L/λ range from 0.01 to 1, which covers 6.6 octaves). At a given frequency away from resonance so that the impedance is not purely resistive, there is an optimum resistance for maximum absorption. As shown in Section 3.6, this occurs when the resistance is approximately equal to the reactance. 1/3 and 1/1 Octave Band Average Absorption The ‘universal’ absorption spectra in Figure 3.2 are useful in general discussions of absorption since they are valid for all values of the thickness L of the air layer by the use of the frequency parameter L/λ. In practice, however, it is more convenient to have frequency in Hz as a variable.
SHEET ABSORBERS
61
Figure 3.3: Absorption spectra of a rigid resistive sheet backed by a 4 inch air layer and rigid wall. The sheet resistance is 1 ρc. The graphs refer to 1/12, 1/3, and 1/1 octave bands, respectively. In each graph, the three curves, starting from the top (at about 800 Hz), correspond to normal incidence, diffuse field local reaction, and diffuse field, nonlocal reaction. In the top graph in Figure 3.3, which concerns a rigid sheet-cavity absorber with a normalized flow resistance of 1 and a cavity depth of 4 inches, 121 values of the frequency have been used evenly distributed logarithmically over the frequency range 10 to 10000 Hz, approximately 10 octaves. This corresponds to an approximate bandwidth of 1/12th octave in the top graph, which is narrow but not as narrow is in Figure 3.2. In engineering applications, however, the average absorption in an even wider frequency band, typically 1/3 and 1/1 octaves, is generally of more interest, and the overall appearance of the absorption curves can then be markedly different, as can be seen in Figure 3.3. The average in a band was calculated from the absorption coefficients at 7 equally spaced (logarithmically) frequencies in a 1/3 octave band and at 13 frequencies in a full octave band (see Eq. 3.29). In the example in Figure 3.3, the graphs, starting from the top, refer to bandwidths 1/12, 1/3, and 1/1 octaves; the center frequencies of the bands are marked. Notice that only the narrow band curves in the top graph show clean zeroes at the anti-resonances. Incidentally, to check these, we note that at the first anti-resonance the wavelength is twice the air layer thickness, i.e., λ = 8 inches and the frequency 1680 Hz, which checks. The frequency at the quarter wavelength resonance is half of this value, i.e., 840 Hz, consistent with the location of the first absorption peak in the top graph. Notice that with a bandwidth of 1/3 octave, there are no clean zeroes at the antiresonance frequencies and for the 1/1 octave absorption spectrum, these frequencies cannot be identified.
62
NOISE REDUCTION ANALYSIS
3.2.4 Wire Screens In the analysis so far in this chapter, the absorption coefficient has been expressed in terms of the impedance of the sheet or screen involved, and it has been implied that this impedance is an experimentally known quantity. Furthermore, a purely resistive sheet has often been assumed and used in numerical examples. It is of interest to check this assumption by using the computed impedance of a wire mesh screen as given in Chapter 3, and it is of general interest in design work to be able to calculate the impedance, at least approximately, in terms of geometrical parameters of the screen or sheet.
3.2.5 Effect of Honeycomb Cell Size Two main kinds of absorbers have been considered in the text, locally and nonlocally reacting, i.e., honeycombed and not honeycombed. However, a liner can be nonlocally reacting even if honeycombed if the cell size in the honeycomb is not small compared to the wavelength. Measurements of the diffuse field absorption coefficient of sheet absorbers with air backing, partitioned into square cells, 2 ft by 2 ft, have shown that the absorption spectrum generally falls between those of locally (partition spacing small compared to a wavelength) and nonlocally reacting absorbers (no partitions), closer to the former at low frequencies and to the latter at high frequencies. An example is shown in the left graph in Figure 3.4. The absorber in this case consists of a resistive sheet in contact with a perforated facing of thickness 0.15 inches, hole diameter 3/16 inches, and hole spacing 1/2 inches (square pattern). The flow resistance of the combination was 1.7 ρc. The thickness of the air backing was 2 inches. The solid line is the measured and the dashed lines, the computed. The curve with the dots between the dashes refers to local reaction. The trend mentioned above is apparent; low frequency performance approaching that of local reaction and high frequency performance that of nonlocal reaction. As a reference, we note that the frequency at which the partition spacing is half a wavelength is 280 Hz, which normally is the frequency below which local reaction can be assumed. 1.0
1.0
0.8
0.8
0.6
0.6
f0
AV
0.4
0.4
f 0 = 870 CPS k L = 0.79 0 =1.7 MEASURED
0.2
1AV
f0 = 270 CPS k0 L = 1.26
0.2
= 1.5
f0
CALCULATED
2AV
0 100
2
3
4
5 6 7 8 9 1000 ν CPS
MEASURED 2 AV
2
3
4
5
6000
0 100
2
3
4
5
6
7
8 9
ν CPS
1000
2
CALCULATED 3
4
5
6000
Figure 3.4: Measured absorption spectra of a resistive sheet/perforated facing combination with air backing compared with computed values for local and nonlocal reactions. Left: Backing depth: 2 inches, Right: 10 inches.
63
SHEET ABSORBERS
The spectrum on the right in the figure refers to an absorber with a relatively low resonance frequency, with an air backing of 10 inches but with the same partition spacing as before. Now, the absorption is high at relatively low frequencies for which the wavelength is large enough (in comparison with the partition spacing) to qualify the absorber as nearly locally reacting. Indeed, the calculated absorption spectrum for a locally reacting absorber agrees well with the measured.
3.2.6 Examples and Comments 1. Effect of cell size in a partitioned air backing Consider first an absorber, such as a resistive sheet with air backing with partitions only in one direction, perpendicular to the yx-plane and with a spacing which is not necessarily small compared to a wavelength. In other words, except for the sheet, the boundary is like a diffraction grating or a slot absorber. Try to calculate the frequency dependence of the reflection and absorption coefficient of the boundary at a given angle of incidence using whatever approximations you feel reasonable. This is not a simple problem, and, as far as we know, has not been analyzed. For someone who has a mathematical inclination and wants to sharpen his or her mathematical tools and physical insights, this could be an interesting project. 2. Rigid sheet absorber for maximum NRC A rigid resistive sheet is backed by a 4 inch air layer and a rigid wall. By repeated computations (iterations), find the (normalized) flow resistance of the sheet, which gives the highest possible diffuse field noise reduction coefficient, NRC, when the absorber is locally reacting (partitioned air backing). For this optimum flow resistance, show the narrow band and octave band absorption curves in the frequency range from 10 to 10,000 Hz. SOLUTION The NRC is the average of the octave band absorption coefficients in the bands 250, 500, 1000, and 2000. We have to distinguish between 3 different values corresponding to normal incidence (NRC0), diffuse field for local reaction (NRC1), and diffuse field for nonlocal reaction (NRC2). The dependence of NRC on the resistance is not very strong in the vicinity of the optimum and, to within one percent, a maximum value of the diffuse field value, local reaction, is found to be NRC1 = 0.81 for values of the resistance between 1.6 and 2.3. The corresponding NRC values for normal incidence and diffuse field, nonlocal reaction, are 0.74 and 0.63, respectively. The narrow and octave band absorption spectra that correspond to a sheet resistance of 2 are shown in the Figure 3.5. 3. The noise reduction coefficient (a) Find the optimum resistance for maximum NRC0, NRC1, and NRC2. (b) In all cases make plots of the NRC-values vs the flow resistance.(c) Explain
64
NOISE REDUCTION ANALYSIS
Figure 3.5: Left: OB absorption spectra. Right: 1/12 OB spectra.
why at high frequencies the diffuse field octave band absorption coefficient is almost independent of frequency for the nonlocally reacting absorber, but not for the locally reacting. (d) What is the smallest weight of the sheet for which it can be regarded as acoustically rigid when the normalized flow resistance is 2? 4. Double sheet absorber (a) The outer screen in a double screen absorber is 8 inches from the rigid backing wall and the inner screen, 4 inches. The screens are identical, each with a normalized flow resistance of 1 and a weight of 0.2 lb/ft2 . Determine the noise reduction coefficients. Compare these values with those obtained if both screens are put together and placed 8 inches from the wall. (b) Is it possible to improve the NRC1 value by using another placement of the screens within the constraint of a total absorber thickness of 8 inches? If so, give these locations and the corresponding NRC1 value. SOLUTION (a) With the screens 4 inches apart, as indicated, the NRC values for normal incidence, diffuse field local reaction, and diffuse field, nonlocal reaction, are 0.90, 0.90, and 0.78, respectively. With the screens put together, 8 inches from the wall, the corresponding values are 0.75, 0.81, and 0.71. In the latter case they are treated as a single screen with twice the flow resistance and twice the weight. (b) Yes. By choosing the screen distances from the wall 8 and 6 inches, we get NRC1 = 0.91. The other values are then NRC0 = 0.88 and NRC2 = 0.78. 5. Optimization of the absorption spectrum (a) In some applications it is not always the NRC that is relevant. Often in noise control problems it is at low and mid frequencies where absorption is needed. Optimize the design for the 125 Hz octave band. (b) Carry out the analogous optimization for screen placement also for NRC0 and NRC2.
65
SHEET ABSORBERS
(c) Given the total thickness of a two screen (limp) absorber, determine the optimum resistances, weights, and placements to yield maximum NRC1. This is an ambitious undertaking and probably does not lead to a unique answer. However, it might be worth playing around with to get some design guidelines. (d) Extend the study to include 3 and 4 sheets with the same total thickness of the absorber as above and do some computer experiments to find optimum sheet parameters. 6. Effect of temperature and nonlinearity on sheet absorber Use the absorber described in the sample run, i.e., a perforated plate/sheet absorber. Cavity depth: 4 inch. Perforated plate: Open area: 5 percent. Thickness and hole diameter: 0.05 inch. Weight: 2 lb/ft2 . Repeat the determination of the octave band absorption spectra, now with a temperature of 1000◦ F. SOLUTION The results obtained are shown in Figure 3.6. (Compare the results obtained at 70◦ F.) At 1000◦ F, the flow resistance is larger than at 70◦ F, since both the shear viscosity and the wave impedance of air depends on temperature. As shown in Chapter 5, the normalized flow resistance increases approximately as T , where T is the absolute temperature (Kelvin). Thus, at 1000◦ F, T ≈ 811 K and at
Figure 3.6: Level dependence of absorption spectra at 1000◦ F.
66
NOISE REDUCTION ANALYSIS 70◦ F, ≈ 294, and the increase in the normalized flow resistance due to the temperature increase will be by a factor of 811/294 ≈ 2.76. The increase provides an improved impedance match of the absorber at 80 dB and the absorption is improved. At 1000◦ F, the resistance is already high enough at 80 dB for a good impedance match, and the increase in sound pressure does not alter the absorption substantially. The example serves to support what is already expressed in the text that temperature can play an important role in the design of absorbers.
3.3 FLEXIBLE POROUS SHEET WITH CAVITY BACKING The acoustically induced motion of a sheet can have a pronounced effect on absorption particularly when the flow resistance is larger than the mass reactance of the sheet. The effect will be studied for a limp sheet, but the results obtained can readily be extended to apply to a sheet with its own resonances.
3.3.1 The ‘Equivalent’ Impedance When the sheet is mobile, the interaction between a sound wave and the sheet results in an acoustically induced velocity u of the sheet so that the air velocity amplitude through the sheet (relative to the sheet) will be different from the absolute amplitude u of the air at the two sides of the sheet. The velocity through the sheet is the relative velocity ur = u − u . Based on these three velocities, ur , u , and u, we can define three different impedances of the sheet as the ratio of the pressure amplitude drop across the sheet and each of these velocities. The interaction impedance and the corresponding flow resistance, which were introduced in the previous section, refer to the relative velocity ur , the structural impedance refers to the velocity u of the sheet, and the equivalent impedance to the absolute air velocity amplitude u. The equivalent impedance will be denoted by z with the real and reactive parts being r and x . Even for a purely resistive sheet, the equivalent impedance will contain a mass reactive part due to the induced motion of the sheet. The prime, as used for the velocity of the sheet, serves as a reminder that z accounts for the induced motion of the sheet. In terms of an electrical circuit analogy, the equivalent impedance z can be thought of as the parallel combination of the interaction impedance z = r + ix and the structural impedance zs ≈ −iωm. Flexible sheet absorber, Equivalent sheet impedance z ≡ r + ix ≡ p/u = zzs /(z + zs )
(3.5)
z = ρcζ : Sheet impedance (Eq. 3.1). zs : ‘Structural’ impedance zs = −iωm for a limp sheet. m: Sheet mass per unit area. z : Accounts for induced motion. (See Section 3.6, Eq. 3.33.) The structural impedance depends on the elastic properties, tension, and mounting of the sheet but for a limp sheet without tension its magnitude is simply the mass
SHEET ABSORBERS
67
reactance ωm, where m is the mass of the sheet per unit area and ω the angular frequency. Most of our discussion will concern a limp sheet for which the degree of induced motion depends on the ratio r/ωm of the flow resistance r and the reactance ωm. In that case, the induced motion becomes most pronounced at low frequencies. At a sufficiently high frequency or sheet mass the sheet becomes essentially immobile and behaves as discussed in the previous section. The induced motion affects both the equivalent resistance r and reactance x of the sheet. Figure 3.7 shows the computed frequency dependence of r /r and |x |/r, where r is the flow resistance of the sheet. The frequency is normalized with respect to the characteristic angular frequency fm = (1/2π )r/m at which the flow resistance r (the interaction resistance) equals the mass reactance of the sheet. As the frequency goes to zero, the equivalent resistance goes to zero because the induced velocity of the sheet approaches the absolute velocity of the air and the entire pressure drop across the sheet is used up for the inertial reactance of the sheet, i.e., for the acceleration of the sheet. The air velocity is then 90 degrees out of phase with the pressure drop. The pressure drop caused by the in phase component of the velocity is essentially zero. As the frequency increases, the inertia of the sheet reduces the induced motion and the sheet response approaches that of the rigid sheet. In this (high frequency) limit, the equivalent flow resistance r approaches the flow resistance r of the sheet so that r /r approaches 1, as shown. At the characteristic frequency fm , the equivalent resistance r is half of the flow resistance r. The equivalent inertial reactance is represented by the bell-shaped curve in the figure. It is zero in the limit of both low and high frequencies. In the low frequency end this is because the frequency goes to zero and in the high end because the sheet is essentially immobile so that the velocity through the sheet is in phase with the pressure drop with no 90 degrees out of phase reactive component. The largest possible equivalent mass reactance of a limp porous sheet is half of its flow resistance, i.e., r/2, and it occurs at the characteristic angular frequency ωm = r/m.
Figure 3.7: The frequency dependence of the equivalent resistance r and reactance x of a limp sheet (accounting for the induced motion). These quantities are normalized with respect to the flow resistance r of the sheet. The frequency is normalized with respect to the characteristic frequency fm = (1/2π)r/m at which r = ωm m (see Eq. 3.33).
68
NOISE REDUCTION ANALYSIS
In the discussion above, it was implied that the interaction impedance z = r + ix is purely resistive, i.e., x = 0. In reality, there is also a mass reactive component (negative x), which is normally insignificant at low and middle range frequencies. If this reactance is accounted for, the equivalent impedance becomes a bit more complicated (see Eq. 3.33). Effect of Bending Stiffness and Structural Resonances of the Sheet For a limp sheet, the magnitude of the structural reactance is simply ωm. If the sheet is under tension or if it has bending stiffness, it will have structural resonances at which the structural impedance is very low so that the velocity amplitude of the sheet will be almost the same as that of the air. The equivalent resistance of the sheet then will be small and the absorption will have a (narrow) dip in the vicinity of a resonance. However, this is usually of little practical interest. As indicated in Section 3.6, the mathematical analysis becomes somewhat involved but the outcome generally can be said to be of relatively little importance.
3.3.2 A Low Frequency Resonance From a practical standpoint, the importance of the limpness of a porous sheet is that in a sheet-cavity absorber, the resonance absorption occurs at a frequency lower than the quarter wavelength resonance familiar from the discussion of the rigid sheet absorber in the previous section. Furthermore, the equivalent resistance of the sheet can be considerably lower than the flow resistance of the sheet, as illustrated in Figure 3.7, and, depending on the magnitude of the flow resistance, this can lead either to an increase or a decrease in sound absorption, as will be shown shortly. With a proper combination of flow resistance and mass of the sheet, 100 percent resonance absorption can be obtained at normal incidence. In many noise control applications, the absorption of low frequency sound is the most difficult to achieve, and the possibility of utilizing the flexibility of a sheet for the purpose of bringing the absorption maximum to a lower frequency can be of practical importance. For a qualitative understanding of this resonance, let us start with an impervious limp sheet. The air backing behind the sheet acts like a spring, and the resonance of the system is that of an ordinary mass-spring oscillator. Thus, the resonance frequency, in principle, can be made as low as we wish by making the mass sufficiently large. The problem is that if the sheet is impervious, there will be no sound absorption. However, with a sheet with a high but finite flow resistance r it is expected to have approximately the same resonance frequency as the impervious sheet. Due to the induced motion of the sheet, the equivalent resistance r will be lower than the flow resistance r (see Figure 3.7) and with a proper choice of parameters, it should be possible to make the equivalent flow resistance equal to ρc so that 100 percent resonance absorption will result. Rule of Thumb for 100% Absorption at the Lowest Resonance The analysis in Section 3.6 shows that for a sheet weight much larger than the weight of the air in the backing layer (at least by a factor of 5), an approximately 100 percent
SHEET ABSORBERS
69
normal incidence resonance absorption is obtained at the low frequency resonance if the normalized flow resistance of the sheet equals the ratio of the weight of the sheet and the weight of the air layer. The corresponding resonance frequency turns out to be lower than the ordinary quarter wavelength resonance by a factor approximately equal to π/2 times the square root of the weight ratio. Thus, with a weight ratio of 10, this factor is approximately 5. The penalty for the reduction in the resonance frequency is a narrower absorption peak. These observations are illustrated by the computed absorption spectra discussed in the following section (see Eq. 3.33).
3.3.3 Absorption Spectra For an absorber consisting of a rigid resistive sheet backed by an air layer and a rigid wall, the lowest resonance occurs when the thickness of the air layer backing is one quarter wavelength. For example, with a 4 inch cavity, this frequency is ≈ 840 Hz at room temperature, and 100 percent absorption at resonance is obtained if the flow resistance of the sheet is chosen to be ρc, i.e., 420 MKS. For the limp sheet, as indicated above, the first resonance can be made to occur at a frequency considerably lower than the quarter wavelength resonance frequency and if the parameters are chosen properly, the absorption coefficient at resonance can be 100 percent. As a start, the proper design parameter can be obtained from the ‘weight ratio’ rule of thumb, given above, but numerical parametric studies can easily be performed to establish the range of validity of this simple rule. The absorption coefficient is then computed in the same way as for the rigid sheet absorber (Eqs. 3.19, 3.20, and 3.23) but with the equivalent impedance of the sheet used instead of the interaction impedance, thus accounting for the induced motion of the sheet. Some computed absorption coefficients vs the layer thickness-to-wavelength ratio L/λ for a limp porous sheet absorber are shown in Figures 3.8 and 3.9. The graphs in each figure refer to sheet-to-air layer weight ratios of 4, 8, and 16, and the curves in each graph correspond to different flow resistances of the sheet, from 1 to 16. Figure 3.8 applies to sound at normal incidence and it is of particular interest to note the appearance of the low frequency resonance we discussed above. It occurs for relatively large weight ratios. These results are consistent with the rule of thumb which says that if the normalized flow resistance equals the weight ratio, 100 percent absorption is obtained at the resonance. For example, for the weight ratio mr = 8 and a normalized flow resistance of θ = mr = 8, the peak absorption is ≈ 100 percent and it occurs at L/λ ≈ 0.058, which is lower than the quarter wavelength resonance value 0.25 by a factor of about 4.3. According to the rule, this ratio is √ ≈ (π/2) mr ≈ 4.4. In a diffuse field, the locally reacting absorber (Figure 3.9) yields results, which are similar to those for normal incidence, exhibiting zero absorption when the thickness of the air backing is an integer number of half wavelengths. A notable feature is that for a sheet resistance of 1, the absorption is less than for a resistance of 2 at practically all frequencies and for this reason the absorption curve for unit resistance has been included only for the mass ratio 4. Generally, the best overall absorption is obtained when the normalized resistance lies between one quarter to one half of the mass ratio.
70
NOISE REDUCTION ANALYSIS
Figure 3.8: Normal incidence absorption spectra of a limp resistive sheet-cavity absorber. Weight ratios, sheet/air layer: 4, 8, and 16. Normalized sheet (flow) resistance values: 1 to 16, as indicated. (For explicit frequency dependence, see Appendix C.)
For example, with a mass ratio of 16, a resistance of 4 gives an almost flat absorption curve at about 80 percent between L/λ = 0.05 and 0.4. For the nonlocally reacting absorber, the diffuse field absorption peaks are markedly lower than for normal incidence and for a locally reacting absorber in a diffuse field but the absorption coefficient does not go to zero at any frequency (Figure 3.9). As mentioned earlier, this is due to the fact that the input impedance is angle dependent and an anti-resonance cannot occur at the same frequency for all angles of incidence. Even so, the performance of the nonlocally reacting absorber is generally inferior to that of the locally reacting absorber.
3.3.4 Examples and Comments 1. Limp sheet absorber for maximum NRC Consider the same situation as in Problem 2, in Section 3.2.6, but the sheet is now limp with a weight of 0.2 lb/ft2 . (a) Again, determine the maximum NRC1 that can be obtained with this absorber by varying the flow resistance of the sheet. (b) What would be the resonance frequency of the absorber if the sheet were impervious?
71
SHEET ABSORBERS
Figure 3.9: Diffuse field absorption spectra of a limp resistive screen-cavity absorber. Left: Locally reacting. Right: Nonlocally reacting. Weight ratios, screen/air layer: 4, 8, and 16. Normalized screen resistance values: 1 to 16, as indicated.
SOLUTION (a) We find from that for a flow resistance between 2.2 and 2.3, the NRC1 is 0.85, i.e., somewhat larger than for the rigid sheet in Problem 2. The corresponding values NRC0 and NRC2 for normal incidence and diffuse field, nonlocal reaction, are 0.78 and 0.67, respectively. (b) Resonance occurs when the total input reactance of the absorber is zero, i.e., when the mass reactance of the sheet is canceled by a stiffness reactance from the air layer. If the mass per unit area of the sheet is m, the depth of the air
72
NOISE REDUCTION ANALYSIS
Figure 3.10: Left: Hard contact (laminate). Right: Loose contact. layer, L, the angular frequency, ω = 2πf , and the sound speed, c. This leads to the equation for the resonance frequency ωm/ρc = cot(ωL/c).
(3.6)
However, rather than solve this equation numerically, we can use a computer program to simulate the impervious membrane by making the flow resistance large enough so that the sheet for practical purposes is impervious. Thus, running the program with a resistance of 200, say, we get a resonance at ≈ 190 Hz. We can confirm that this indeed satisfies Eq. 3.6. A weight of 0.2 lb/ft2 corresponds to a mass m = 0.098 g/cm2 . Then, with L = 10.2 cm, c = 34000 cm/sec and ρc = 42, the left-hand side of the equation becomes 2.78 and the righthand side 2.67. This means that the resonance frequency is somewhat below 190 Hz. It should be noted also that if we had assumed the wavelength at resonance to be much larger than the cavity depth, so that ωL/c << 1, we would have had cot(ωL/c) ≈ 1/(ωL/c) so that c ω ≈ f = 2π 2π
ρ , mL
(3.7)
which yields f ≈ 189.9 Hz. This is slightly larger than the true value since cot(x) < 1/x. The corresponding wavelength is 171 cm so that x = ωL/c = 2π L/λ ≈ 0.37 yielding 1/x ≈ 2.67 and cot(x) ≈ 2.54. 2. Collection of sheet data (a) From manufacturers’ catalogs and literature, obtain data on the flow resistance, weight, open area fraction, wire diameter, etc., of wire mesh screens, woven sheets, fabric, porous paper products, etc., and explore their potential as sound absorbers. If flow resistance data are not available, obtain samples and measure the flow resistance yourself (see Appendix A). (b) Find empirical relations between flow resistance, open area, and wire (fiber) diameter and compare the results with calculations (see Chapter 3).
73
SHEET ABSORBERS
3. Perforated plate-sheet combinations, loose vs hard contact (a) A resistive sheet is placed in hard contact with a perforated plate to form a laminate or ‘sandwich,’ which is backed by a 4 inch partitioned air layer and a rigid wall. The weight of the perforated plate is 4 and of the screen 0.2 lb/ft2 . The open area of the plate is 23 percent, thickness 0.1 inch, and hole diameter 0.1 inch. Determine the optimum flow resistance and the corresponding maximum NRC1. (b) Using the optimum resistance thus found, determine the octave band absorption curve for diffuse field, local reaction, when the elements are in loose contact. (c) Does a sheet resistance exist for which the normal incidence absorption coefficient at resonance is the same for loose and hard contact between the sheet and the perforated facing? If so, determine analytically this resistance and the corresponding resonance absorption coefficient. SOLUTION (a) In the program, it is assumed that when a screen and a perforated plate are adjacent elements, they are in loose contact. When they are in hard contact, the effective flow resistance of the combination will be larger than the flow resistance for the bare screen by a factor 1/s, where s is the open area fraction of the plate (0.23 in this case). Therefore, to simulate the hard contact case, the resistance that is entered as an input in the program is θs /s, where θs is the flow resistance of the bare screen. The optimum input resistance is found to be 2.2 and the corresponding NRC1 0.83. This means that the flow resistance of the bare screen is 0.51. It is not surprising that the maximum NRC1 will be the same as for the rigid bare screen in Problem 2 where the optimum flow resistance was found to be ≈ 2. There is one important difference, however. Due to the mass reactance of the perforated facing, the absorption coefficient at high frequencies will be reduced and there will be a slight increase at low frequencies, as can be seen by comparing the result with that in Problem 2. (b) The graph on the right in the figure refers to loose contact in which case the resistance of the absorber is about the same as the resistance of the bare sheet, i.e., 0.51 (normalized). Again, we notice how the perforated plate reduces the absorption coefficient at high frequencies. (c) As before, we denote the open area fraction of the perforated plate by s and the normalized flow resistance of the sheet by θ . The resistances of the perforated plate-sheet combinations are then θ/s and θ for hard and loose contact, respectively. These resistances are the input resistance θi of the absorber at resonance. The normal incidence absorption coefficient at resonance (see Chapters 1 and 3) is 4θi /(1 + θi )2 . Hence, equating the resonance absorption in the two cases, we get 4θ/s 4θ = (1 + θ/s)2 (1 + θ)2
(3.8)
74
NOISE REDUCTION ANALYSIS from which follows, solving for θ, √
θ=
s−s √ . 1− s
(3.9)
With s = 0.23, we get θ = 0.48, and the corresponding absorption coefficient at resonance is α0 = 0.88. It should be pointed out, though, that the absorption curve obtained with hard contact is considerably broader than with loose contact. 4. Can loose and hard contact yield the same result? (a) Is there a flow resistance value for which the NRC1 value of the plate-screen absorber is the same for loose and hard contact? If so, what is this resistance and the corresponding NRC1? (b) In the text we have considered the two extreme cases when a sheet is in hard contact with a perforated plate and when it is in loose contact. In the first case, the flow velocity through the screen is localized to each of the orifices in the plate and the flow velocity through the screen is the same as the velocity in the orifice. In the case of a bare perforated plate, the velocity becomes gradually less localized as the distance from the plate is increased until it becomes uniform and equal to s u0 , where u0 is the velocity amplitude in the orifice and s the open area fraction of the plate. The distance at which this occurs is of the order of an orifice diameter. The total flow resistance of the combination in the two limiting cases will be r/s and r, where r is the flow resistance of the sheet (neglecting the flow resistance of the orifice). Determine theoretically or experimentally, or both, the resistance of the combination as a function of the separation between the plate and the screen. Calculate the corresponding absorption coefficient using the data in the problem statement. 5. Low frequency resonance, limp sheet absorber A limp resistive sheet has a normalized flow resistance of 10. What should be its weight per unit area to produce a low frequency resonance with a normal incidence absorption coefficient of 1.0 or close to it? What is the corresponding value of the octave band absorption coefficient and the NRCs? SOLUTION With the present high flow resistance and relatively low weight, the lowest resonance is expected to involve the induced motion of the sheet and not the quarter wavelength resonance familiar from the rigid sheet absorber, which requires a normalized resistance of unity to get 100 percent absorption. It is the low frequency resonance referred to in Chapter 3, in which the induced motion of the sheet is large enough to make the equivalent resistance equal to unity even though the actual resistance is θ = 10. As a start, we make use of the ‘rule of thumb’ given in the text that approximately 100 percent absorption can be achieved if the ratio of the sheet weight Ws and the weight Wa of the air layer equals the normalized resistance θ . In this case, with an 8 inch air layer
75
SHEET ABSORBERS
the weight of it per square foot of surface area is approximately 0.054 lb/ft2 . Thus, from the thumb rule it follows that the weight of the screen should be Ws ≈ θWa ,
(3.10)
which in this case is ≈0.54 lb/ft2 . The rule also says that the resonance frequency should be f0 ≈
1 c/L √ , 2π θ
(3.11)
where c is the sound speed and L the cavity depth. In our case, with c ≈ 1120 ft/sec, L = 2/3 ft, and θ = 10, we get f0 ≈ 85 Hz. It is important to realize that for this rule of thumb to be valid, the normalized resistance of the sheet should be much larger than 1. Running the program with the inputs for θ and Ws , we obtain the curves shown in Figure 3.11, narrow band data (bandwidth about 1/12th octave) on the left and octave band data on the right. The normal incidence resonance absorption coefficient indeed is very close to unity and the frequency is close to the predicted value of 85 Hz, as can be seen in the figure. The bandwidth of the resonance is rather small, however, and the peak value of the normal incidence octave band absorption spectrum is considerably smaller. With the same rule of thumb for a screen with a resistance of only 5 and a weight reduced to 0.27 lb/ft2 , the predicted resonance frequency is ≈ 119 Hz, and the rule still works quite well. Even down to a resistance of 2.5 and a weight of 0.135 lb/ft2 with a predicted frequency of 170 Hz, the rule is satisfactory; the only flaw being that the predicted frequency is a bit too low. For still lower values of the resistance, the rule should not be used. 6. Bandwidth of low frequency resonance With reference to the low frequency resonance discussed in Problem 5, try to derive an expression for the bandwidth of the resonance. Define the bandwidth, for example, as the width of the absorption curve at which the absorption coefficient is half of the peak absorption.
Figure 3.11: Sheet resistance: 10 Weight: 0.54 lb/ft2 .
76
NOISE REDUCTION ANALYSIS 7. Absorption coefficient vs sheet resistance (a) Consider a rigid sheet-cavity absorber. Show that in the low frequency limit with the cavity depth L much smaller than the wavelength λ, the normal incidence absorption coefficient of a rigid sheet-cavity absorber is proportional to the sheet resistance and to the square of the frequency. (b) For a limp sheet, the angular frequency ωm at which the flow resistance r equals the mass reactance of the sheet is given by r = ωm m, where m is the mass per unit area of the sheet. In this case, show that in the low frequency limit where both λ >> L and f << fm , the normal incidence absorption coefficient is proportional to the 4th power of the frequency and inversely proportional to the sheet resistance (explain the last apparent peculiar dependence qualitatively). (c) It is useful for design purposes to have the complete explicit dependence of the absorption coefficients on the flow resistance of the sheet so that one can readily determine the optimum resistance at a given frequency. In such a calculation, the cavity depth and the frequency are input parameters. Thus, consider a depth of 8 inches and a limp resistive sheet with a weight 0.2 lb/ft2 . Determine the resistance dependence of the absorption coefficient at a frequency of 85 Hz and compare the result with that for a rigid sheet. SOLUTION (a) The normal incidence input impedance of a rigid sheet-cavity absorber is θ + i cot(kL), where θ = r/ρc and k = ω/c = 2π/λ (see Chapter 1). At long wavelengths, with kL << 1, this reduces to θ + i/kL. The corresponding normal incidence absorption coefficient is α=
(1 + θ)2
4θ ≈ 4θ(kL)2 , + (1/(kL)2 )
(3.12)
where kL = 2π L/λ. The approximation in the last step is based on the assumption that (1 + θ )kL << 1. The complete resistance dependence is shown in the right graph in Figure 3.12.
Figure 3.12: Left: Limp sheet, weight: 0.54 lb/ft2 . Right: Rigid. N: Normal incident. DL: Diffuse field, local reaction. DN: Diffuse field, nonlocal reaction.
77
SHEET ABSORBERS
(b) Accounting for the acoustically induced motion (see Chapter 1), and with ωm = r/ω, the equivalent impedance of the sheet is θ = θ/(1 + i(ωm /ω)2 ) ≈ θ (ω/ωm )2 , for ω << ωm . The low frequency input impedance of the sheetcavity absorber is then θ +i/(kL), neglecting the inertial reactance of the sheet and with cot(kL) ≈ 1/(kL). The corresponding normal incidence absorption coefficient is α≈
ω2 ωL 2 4θ
) , ≈ 4θ (kL)2 ≈ 4θ 2 ( 2 + (1/kl) ωm c
(1 + θ )2
(3.13)
where in the last step it is assumed that (1 + θ )(kL) << 1 and where we have used θ ≈ θ (ω/ωm )2 , ωm = r/m = θρc/m. Thus, we see that the low frequency approximation indeed is proportional to the 4th power of frequency and inversely proportional to the flow resistance. The reason for the latter dependence is that the induced motion of the sheet increases with increasing resistance and decreases the relative velocity of the sheet and the air, thus reducing the input resistance. (c) The complete resistance dependence of the absorption at 85 Hz is shown in the left graph in the figure. The reason for having chosen the weight of the sheet equal to 0.54 lb/ft2 was to see if this result is consistent with that in Figure 3.12. Recall that in that case we applied as a starting point the ‘rule of thumb,’ which says that the absorption coefficient is close to unity if the normalized sheet resistance, in this case 10, equals the ratio of the sheet √ mass and the air mass in the layer, and the resonance frequency is ≈ (c/L)/ θ, in this case ≈ 85 Hz. Indeed, we find from the graph that for θ = 10 the absorption coefficient is quite close to 1.
3.4 LATTICE ABSORBERS The studies of the single sheet-cavity absorbers of the last two sections will now be extended to include a multisheet absorber in the form of a lattice of an arbitrary number of sheets placed in front of and parallel with a rigid wall, as shown in Figure 3.13. First, a periodic or uniform lattice is considered and then a nonuniform lattice in which both the sheets and their separation are different.
3.4.1 Periodic Lattice Even if the analysis is now considerably more complicated than for the single sheet, the input impedance of the lattice can be expressed in closed form and the computation of the absorption coefficient turns out to be quick and simple. The sheets are located in front of a rigid wall a distance d apart as indicated in Figure 3.13. Each sheet is specified acoustically by the equivalent sheet impedance ζ , which is determined by the interaction impedance ζ and the structural impedance ζs of the sheet, as discussed earlier (Eq. 3.33). The input impedance of the lattice can be expressed in closed form in terms of these quantities, as shown in Section 3.6.
78
NOISE REDUCTION ANALYSIS
Inc. wave
d
Figure 3.13: Periodic lattice absorber consisting of N identical equidistant sheets placed in front of a rigid wall.
Input impedance of periodic lattice absorber ζi ≡ θi + iχi =
p(0) u(0)ρc
cos(qx d) =
=
ζ
2
x d) + i cossin(q φ sin(kx d) cot(N qx d),
φ cos(kx d) − i ζ cos 2
(3.14)
sin(kx d)
kx = k cos φ, k = ω/c. ζ : equivalent impedance, Eq. 3.33. d: Sheet separation (lattice constant) (see Figure 3.13). φ: Angle of incidence. N : Number of sheets. qx : Eq. 3.57 (see Eq. 3.61). Having obtained the input impedance, the absorption coefficients then follow from Eqs. 3.3 and 3.4. The separation of adjacent sheets is assumed to be larger than the acoustic thermal boundary layer (to be discussed later) so that the compressibility of the air can be assumed to be isentropic. This should be kept in mind when the absorption spectra of lattice absorbers are compared with those of uniform porous layers (to be treated later) in which heat conduction is accounted for, leading to isothermal compressibility of the air at low frequencies. The first example is shown in Figure 3.14. It refers to a lattice of limp, purely resistive sheets. In order to make it simple to compare the absorption curves for a different number of sheets in the lattice, we have shown the results for 2, 4, 8, and 16 sheets in each graph for sound at normal incidence. It turns out that for a locally reacting absorber the results are essentially the same as for a uniform porous layer with the same thickness and the same total flow resistance. For the nonlocally reacting absorber, however, the lattice absorber is not as good as expected. The absorption coefficient depends on the mass m per unit area of a sheet, the flow resistance of a sheet, r, the distance between the sheets, d, the number of sheets, N, and the frequency, f . We can reduce the number of variables by using as the frequency parameter the ratio d/λ and as a mass parameter mr = m/dρ, the ratio of the sheet mass and the mass of the air in one unit cell.
SHEET ABSORBERS
79
Figure 3.14: Normal incidence absorption spectra of a uniform lattice absorber (see Figure 3.13). Mass ratio, sheet/air layer in one cell: 16. Normalized sheet resistances: 0.2, 0.4, 0.6. N: number of sheets = 2, 4, 8, 16. (For examples of diffuse field characteristics, see Appendix C.)
As for the single limp sheet, the parameter r/ωm, i.e., the ratio of the flow resistance and the mass reactance of the sheet, determines to what extent acoustically induced motion of the sheet occurs. If r/ωm << 1, the sheet can be regarded as immobile. This condition can be expressed as d/λ >> 1/2π mr . In the numerical example in Figure 3.14, we have mr = 16, and this means that the sheet is essentially immobile if d/λ >> 0.01, i.e., over most of the frequency range shown in the figure. One significant feature is that the absorption coefficient is zero at frequencies for which the distance between the sheets is an integer number of half wavelengths, as is clearly evident from Figure 3.14. Actually, this is to be expected from the results for the single sheet absorber. Physically, this is understandable because in the absence of the sheet the standing wave in front of the rigid wall will have zero velocity at distances equal to an integer number of half wavelengths from the wall. Since the velocity is zero, we can introduce sheets at these locations without disturbing the sound field. With zero velocity, the friction loss in the sheets and the absorption coefficient of the absorbers will be zero. An incident sound wave then will be totally reflected and there is no absorption. In the physics of waves in lattices in general, this condition is often referred to as Bragg reflection stemming from Bragg’s work on x-ray diffraction in crystals.
80
NOISE REDUCTION ANALYSIS Sheet resistance = 0.1 0.2
0.4
0.8
1.6
3.2
Inc. wave 3’’ Normal inc. Diffuse, nonlocal.
1.0 0.9 0.8
Abs. coeff.
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0
10
100 1000 Frequency, Hz
10000
Figure 3.15: Normal incidence (upper curve) and diffuse field absorption coefficients (lower curve) of a nonuniform lattice absorber of limp sheets with unpartitioned air layers (nonlocal reaction). The sheets have different resistances but the same separation, 3 inches, as shown. The sheets are not purely resistive but have an interaction reactance χ = (f/5000)θ, where θ is the flow resistance. Weight: 10 oz/yd2 ≈ 0.068 lb/ft2 .
A Comparison with a Uniform Porous Layer It is of considerable practical interest to compare the performance of sheet absorbers and a uniform porous layer and to make such a comparison, we refer to the example in Figure 5.21.
3.4.2 Nonperiodic Lattice The problem with the uniform lattice absorber with equidistant identical porous screens or sheets is that the absorption will vanish at the frequencies for which the distance between the sheets is an integer number of half wavelengths. This can be remedied by using a varying sheet spacing. The input impedance no longer can be expressed in closed form, as was done for the periodic lattice, but is obtained by brute force multiplication of transmission matrices of the lattice cells involved. To illustrate results of a such computations, two lattices are considered, each with 6 sheets. In the first, Figure 3.15, the separations between the sheet are the same but the resistances are different, and in the second, the resistances are all the same but their separations are not. The sheets are limp with a weight of 10 ounces per square yard1 (≈ 0.033 g/cm2 ). The interaction impedance of the sheet has not only a resistive component θ but also a reactive component χ. The relation between the two depends on the microstructure of the sheet as discussed in earlier chapters. In the present example, we have used
1 A measure frequently used for woven sheet material in the textile industry.
81
SHEET ABSORBERS Distance from wall = 18”
13.5”
9.5”
6”
3” 1”
Inc. wave Resistance of each sheet = 0.4 Normal inc. Diffuse, nonlocal.
1.0 0.9 0.8
Abs. coeff.
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0
10
100
1000
Frequency, Hz
10000
Figure 3.16: Normal incidence (upper curve) and diffuse field (lower curve) absorption coefficients of a nonuniform, nonlocally reacting lattice absorber. The sheets all have the same normalized resistance of θ = 0.4 but their separations are different, as shown. The magnitude of the interaction reactance of each sheet is |χ | = (f/5000)θ. The sheets are limp with a weight of 10 ounces per square yard, ≈ 0.068lb/ft2 .
χ ≈ (f/5000)θ , fairly typical for a cloth, where f is the frequency in Hz, which makes the magnitude of the reactance equal the resistance at a frequency of 5000 Hz. In order to simulate the performance of rigid sheets at a given frequency, the mass must be large enough so that r << ωm. For example, with a flow resistance of 0.4 ρc ≈ 16.8 CGS and a frequency of 100 Hz, we must have m >> 0.027 g/cm2 . Thus, in the present example, with m ≈ 0.033, there will be some effect of the motion of the sheets at 100 Hz, but no significant effect at 1000 Hz and above. The absorption coefficient is close to 100 percent over substantial portions of the frequency range shown. However, due to the equal spacing, the absorber suffers from the same problem as the uniform lattice as it has zero normal incidence absorption at frequencies for which the sheet separation is an integer number of half wavelengths. This is true even if the sheets, as in the present example, are not identical. The diffuse field average absorption coefficient (nonlocal reaction), however, will not be zero at any frequency (different from zero), but it is markedly lower than for normal incidence. For a lattice with partitioned layers, the diffuse field absorption coefficient is close to that for normal incidence. In the second example (Figure 3.16), there are again six sheets but this time the flow resistance of the sheets are all the same (0.4 ρc) but their separations are not. The overall thickness (18 inches) of the lattice is the same as in the first example. The ‘half-wavelength’ dips in the absorption curves have now been eliminated although there are still some irregularities at high frequencies and with the resistance distribution used here, the absorption, on the average, is not quite as good as in the first example. The optimization of an absorber of this kind has to be done by continued experimentation on the computer, and these two examples most likely do not represent optimum configurations for a given total thickness of the lattice.
82
NOISE REDUCTION ANALYSIS
In regard to the design of the multisheet absorber, the flow resistance of the front sheet should be sufficiently small to prevent a substantial reflection from this sheet. To get an idea of this reflection, we use the pressure reflection coefficient ζ /(ζ + 2) of a rigid single sheet in free space, to be discussed in the next section. For an immobile, purely resistive sheet, ζ = θ (the normalized flow resistance), and if the reflection coefficient is to be kept below a certain value R, the flow resistance of the sheet must be smaller than 2R/(1 − R). According to this expression and with α = 1 − |R|2 , the reflection coefficient must be less than 0.1 and the normalized flow resistance of the first sheet less than 2/9 ≈ 0.22 to get an absorption coefficient of 0.99. This is merely a necessary requirement in the design, of course, and by no means guarantees that the reflection coefficient from the lattice absorber will be less than 0.1.
3.5 ‘VOLUME’ ABSORBERS In the present context, the sheet-cavity absorber might be called a ‘surface’ absorber since it is usually applied on an interior wall surface of a room. In the surface sheet absorber, the sound pressures on the two sides of the sheet are correlated since the pressure on the back of the sheet is a result of the reflection at the rigid backing of the wave transmitted through the sheet. It is this correlation which makes the resonances possible. This is in contrast to the use of a porous sheet as a
Volume abs.
Surface abs.
Figure 3.17: The designation of an absorber as a ‘surface’ or a ‘volume’ absorber refers to its placement in the room. In this figure, the surface absorber is located on the floor and the volume absorber is suspended from the ceiling away from the walls. y Reflected
Transmitted
x
Inc. wave
Sheet
Figure 3.18: Reflection, transmission, and absorption of a plane wave incident on a thin porous sheet.
83
SHEET ABSORBERS
volume absorber in which the sheet acts alone in the interior of a room without any combination with a particular or well-defined cavity backing. In this section, the absorption spectra of such a sheet are investigated under some simplifying assumptions. First, the linear dimensions of a sheet are assumed large compared to the acoustic wavelength, so that diffraction effects can be neglected. (Diffraction tends to reduce the absorption cross section at low frequencies.) Thus, the present analysis can be regarded as a high frequency approximation and this should be kept in mind when the results are examined. In the other extreme case, with the sheet dimensions small compared to a wavelength, diffraction plays an important role and will reduce the absorption, as shown at the end of this section. The sound field in the room is assumed to be diffuse, so that the sound incident on one side of the sheet is uncorrelated with the sound incident on the other side. This is an essential difference when compared with the sheet-cavity absorber where the pressures on the two sides of the sheet are correlated.
3.5.1 Reflection, Transmission, and Absorption With reference to Figure 3.18, a harmonic plane wave is incident on a plane (infinitely extended) sheet at an angle φ. For a purely resistive sheet and a sufficiently large mass so that the sheet can be regarded as immobile as far as the interaction with sound is concerned, the coefficients of reflection, transmission, and absorption are derived in Section 3.6. Volume absorber, transmission, reflection, and absorption coefficients 2 2+ζ cos φ ζ cos φ R = 1 − τ = 2+ζ
cos φ
φ 2 2 1 − |R| − |τ | = (2+θ cos4θφ)cos 2 +(χ cos φ)2
τ=
α(φ) =
C A
=
(3.15)
ζ = θ + iχ : Normalized equivalent impedance (Eq. 3.33). φ: Angle of incidence, Figure 3.18. (See also Section 3.6, Eq. 3.75.) The computed normal incidence coefficients of reflection, transmission and absorption vs the normalized flow resistance θ of the sheet are shown in Figure 3.19. The absorption coefficient has a maximum value of 0.5 for a normalized sheet resistance θ = 2. The corresponding power reflection and transmission coefficients are both 0.25. Under these conditions, half of the incident power is absorbed and the other half is divided equally between reflected and transmitted power. Actually, these results hold true for a wave with an angle of incidence φ if we replace θ by θ cos φ.
3.5.2 Absorption Spectra, Infinite Sheet On the assumption of a rigid, purely resistive sheet in the previous section, the results in Figure 3.19 are independent of frequency. In reality, a frequency dependence enters in several ways. First, if the sheet is mobile, there is a frequency dependence resulting from the induced motion of the sheet. Second, a sheet is not purely resistive but has a mass reactive component in the interaction impedance, and third, with
84
NOISE REDUCTION ANALYSIS
Figure 3.19: Normal incidence coefficient of acoustic power reflection, transmission, and absorption of an infinite, purely resistive, rigid sheet in free field.
a sheet of finite size, diffraction introduces frequency dependence. This last effect is usually the most important and will be considered separately at the end of this section. The reactive part of the interaction impedance will be neglected to start with, but the induced motion and the corresponding mass reactance will be accounted for. As indicated earlier, the assumption of a purely resistive sheet is often quite good at low and middle range frequencies. In this idealized case of an infinite sheet in a diffuse field, the absorption area is the product of the diffuse field absorption coefficient and the exposed or ‘sonified’ area, which includes both sides of the sheet. For an infinite sheet, this area is of course infinite, and the absorption area per unit area of sheet material will be used. It is this quantity, which is plotted in Figure 3.20. The normalized frequency in the figure is f/fm , where fm = r/(2π m). The parameters 0.4 to 6.4 refer to the normalized flow resistance of the sheet. For example, with a sheet of weight 0.05 lb/ft2 (≈ 0.025 g/cm2 ), the characteristic frequency fm = r/2πm = θρc/2πm for a normalized flow resistance of θ = 1 will be fm ≈ 42/(2π · 0.025) ≈ 268 Hz. To establish frequency scales in Hz in the figure for a sheet of this weight, the actual frequencies that correspond to f/fm = 1 will be 107, 214, 428, 858, and 1716 Hz, respectively, for the flow resistances from 0.4 to 6.4. If the weight of the sheet is doubled, these frequencies are reduced by a factor of 2. If f/fm >> 1, the absorption cross section becomes frequency independent and depends only on the sheet resistance. As was the case for normal incidence, there is an optimum value of the flow resistance for maximum absorption also in a diffuse field. In the high frequency limit, where the sheet is essentially immobile, this value is found to be ≈ 3.2. The corresponding maximum absorption cross section is ≈ 0.95 per unit area of sheet material. The corresponding one-sided diffuse field absorption coefficient of the sheet is half of this value, i.e., 0.475. These values, 3.2 for resistance and 0.475 for the absorption coefficient, should be compared with 2 and 0.5 for normal incidence on one side of the rigid sheet in Figure 3.19.
SHEET ABSORBERS
85
Figure 3.20: Absorption area per unit area of sheet material for a purely resistive limp sheet in a diffuse sound field, accounting for the fact the sound is incident on both sides of the sheet. The frequency is normalized with respect to fm = r/2πm, where r is the flow resistance and m the mass per unit area of the sheet material. (An absorption cross section of unity then corresponds to an absorption coefficient of 0.5 for the sound that is incident on one side.) Normalized flow resistance: 0.4, 0.8, 1.6, 3.2, and 6.4 (see Eq. 3.75).
3.5.3 Finite Sheet, Effect of Diffraction For a sheet of finite dimensions, the effect of diffraction can reduce the absorption significantly at wavelengths large compared to the dimensions of the sheet. Qualitatively, this can be understood by considering steady, inviscid flow about an object such as a circular, impervious, immobile sheet. At normal incidence, the velocity and pressure distributions over the disk will be the same on both sides so that there will be no pressure difference across the sheet that tends to force the flow through the sheet. For oscillatory flow, the pressure difference will not be zero, although it does go to zero as the frequency goes to zero. The reason why there will be a pressure difference and a net force amplitude on the sheet in oscillatory flow has to do with the distortion of the flow by the sheet and is dynamically equivalent to an increase of the inertial mass of the fluid in the region of the sheet. This ‘induced’ mass is similar to that experienced when moving an object (your hand, for example) back and forth in water. With the induced mass denoted by mi , the force amplitude on the disk will be fi = ωmi u, where u is the amplitude of the relative velocity of the disk and the fluid. Our approximate analysis along these lines2 involves a circular porous disk at wavelengths, λ, large compared to the disk diameter d and can be regarded as a low frequency approximation in the calculation of the absorption cross section to supplement the previous analysis, where the sheet dimension was assumed large compared
2 P. M. Morse and K. U. Ingard, Linear Acoustic Theory, Volume XI/1 in Handbuch der Physik, Springer-
Verlag, 1961.
86
NOISE REDUCTION ANALYSIS
to the wavelength. The absorption cross section given here accounts for absorption of sound impinging on both sides of the disc. Absorption cross section, Low frequency approximation σa ≈ A|ur /u|2 (64/π 3 ) (ka)2 θ/|ζ + iχi |2
(3.16)
a: Radius of porous disc (sheet). k = ω/c. A: Area of disc. m = mass per unit area of disc. r = θρc = flow resistance. ur /u: = 1/(1 + ir/ωm) (see Eq. 3.85). ζ : Equivalent normalized disc impedance (see Eq. 3.33). χi = −i(8/(3π ))ka: (k = ω/c) (see Section 3.6, Eqs. 3.86 and 3.87). Whether or not the flow will go through the disc rather than around it depends on the ratio of the normalized average mass reactance ωmi /π a 2 per unit area and the equivalent impedance of the sheet. For a purely resistive and immobile sheet, the condition for a substantial flow-through to occur is ωmi /ρc > θ, where θ is the normalized flow resistance of the sheet. The induced mass of the finite sheet plays about the same role as the mass per unit area of the infinitely extended sheet as far as the reduction of absorption cross section at low frequencies is concerned. Actually, for a sheet with a diameter of 1 ft, the induced mass per square foot corresponds to a weight of 0.08·4/3π ≈ 0.035 lb/ft2 (≈ 5 oz/yd2 ), where we have used 0.08 lb/ft3 for the density of air. This is about the same as the weight of a typical woven sheet, and the induced motion of the sheet then plays the same role as diffraction in reducing the low frequency absorption. Thus, in this case, no significant improvement in the low frequency absorption would result by making the sheet heavier than 0.035 lb/ft2 since the absorption would be dictated by effect of diffraction. For more details, we refer to Section 3.6. The optimum values of the flow resistance discussed in connection with the results for infinite sheets are no longer valid for a finite sheet. The optimum normalized resistance in the low frequency regime for a sheet of radius a is approximately equal to ka = 2πa/λ, i.e., directly proportional to the diameter of the sheet. Figure 3.21 shows an example of the calculated frequency dependence of absorption cross section per unit area of sheet material for a finite sheet, and in the same figure is also shown the corresponding result for an infinite sheet. At low frequencies, the lower curve applies and at high frequencies, the upper. These curves have to be joined smoothly to yield the complete absorption curve. The weight is 0.05 lb/ft2 , typical for a woven cloth, and the normalized flow resistance of the material is 1.0. The diffraction limits the absorption at frequencies below ≈ 800 Hz. In order for the effect of diffraction to be relatively small, the diameter of the sheet in this case should not be less than 3 ft. It is interesting to compare the performance of a thin sheet with that of a solid porous layer with an impervious barrier in the center so as to create two porous half layers backed by the rigid barrier. With a total layer thickness of 1 inch, the two sided absorption area will not be much different from that of the sheet.
87
SHEET ABSORBERS
Figure 3.21: Two-sided absorption cross section per unit area of a circular porous disc with a diameter of 2 ft, a weight of 0.05 lb/ft2 , and a normalized flow resistance of 1.0. The lower curve is the low frequency approximation (see Eq. 3.16), in which diffraction limits the absorption at low frequencies. The upper curve refers to the infinite sheet considered in Figure 3.20 (see Eq. 3.75).
A rigorous analysis of the absorption and scattering from a finite porous limp sheet is not available, as far as we know, and it represents a challenge in mathematical acoustics; a challenge which is not only interesting for its own sake but because it also happens to be of some practical importance.
3.6 MATHEMATICAL SUPPLEMENT 3.6.1 Rigid Single Sheet Cavity Absorber Impedances The (normalized) interaction impedance ζ of the sheet is considered to be known either from measurements (Appendix A) or from analysis (Chapter 3). For a plane wave at normal incidence, wavelength λ = c/f , the pressure field in the cavity behind the sheet will be a standing wave with the pressure maximum at the rigid wall. The complex amplitude distribution will be of the form p(x, ω) = A cos(kx),
(3.17)
where x is the distance from the wall. The sheet is located at x = −L. The corresponding velocity field is obtained from ρ∂ux /∂t = −∂p/∂x, which for harmonic time dependence yields3 ux (ω) = iAk sin(kx)/ωρ. 3 The time factor used in this book in the definition of a complex amplitude is exp(−iωt).
(3.18)
88
NOISE REDUCTION ANALYSIS
The ratio of the amplitudes of pressure and velocity at x = −L is then iρc cot(kL) with the normalized value i cot(kL). Adding the impedance ζ of the sheet yields the total input impedance ζi ≡ θi + iχi = ζ + i cot(kL),
(3.19)
where k = ω/c = 2π/λ, ζ = θ + iχ : For a resistive sheet, χ = 0. L: Layer thickness. The input reactance χi is zero (resonance) at the frequencies determined by χ + cot(kL) = 0. For a purely resistive sheet, χ = 0, and hence cot(kL) = 0, which means that kL = (2n − 1)π/2 or L = (2n − 1)λ/4 (n = 1, 2 . . .), the layer thickness being an odd number of quarter wavelengths at the nth resonance. The input resistance θi is then the same as the sheet resistance θ if we neglect any losses within the cavity. The assumption of a purely resistive screen is normally quite good over a wide frequency range. (As mentioned before, one finds, for example, that for cloth-like sheets, the magnitude of the normalized reactance typically is |χ | ≈ (f/5000)θ, where f is the frequency in Hz.) For a wire mesh screen the frequency dependence of θ and χ , in terms of the geometrical parameters involved, is discussed in Chapter 3. For a locally reacting boundary, the impedance is independent of the angle of incidence (Figure 3.19) but for a nonlocally reacting boundary, it is not, ζi ≡ θi + iχi = ζ + i(1/ cos φ) cot(kx L),
(3.20)
where (see Eq. 3.19), φ: Angle of incidence (see Figure 3.1), kx = k cos φ, ζ : Sheet impedance. For nonlocal reaction, the directions of the waves incident on the wall and reflected from the wall will be the same as those of the waves incident and reflected from the absorber. The distance between adjacent maxima in the standing wave within the air layer in the normal direction will be λx = λ/ cos φ and the corresponding propagation constant is kx = 2π/λx = k cos φ. The factor 1/ cos φ = k/kx in front of cot(kx L) in Eq. 3.20 is the normalized wave impedance ‘in the x-direction’ of the incident plane wave, (1/ρc)p/ux = 1/ cos φ. Absorption Coefficient Because of the change in impedance encountered by the sound wave as it reaches the absorber, reflection will occur. If the x-dependence of the complex amplitudes of the incident and reflected pressure waves are expressed as pi = exp(ikx x) and pr = R exp(−ikx x), the corresponding velocity fields are uix = (1/ cos φ) exp(ikx x) and uxr = −(1/ cos φ) exp(−ikx x). With the screen at x = 0, the total pressure amplitude at the screen will be p = pi + Rpr and the total velocity ux = uix + urx = (1/ρc cos φ)(pi − pr ). From these relations it follows that p/ux = (ρc/ cos φ)(1 + R)/(1 − R). Equating this with the known input impedance, the reflection coefficient can be expressed in terms of ζi , R = (ζi cos φ − 1)/(ζi cos φ + 1).
(3.21)
89
SHEET ABSORBERS The corresponding absorption coefficient is α(φ) = 1 − |R|2 =
4θi cos φ , (1 + θi cos φ)2 + (χi cos φ)2
(3.22)
where ζi = θi + iχi : Eqs. 3.19 and 3.20, φ: Angle of incidence. In a diffuse sound field, the intensity of the sound is the same in all directions. For an angle of incidence φ (with respect to the normal, the polar angle), the acoustic power, which strikes a surface element of unit area will be proportional to cos φ, the projected area normal to the incident intensity. The probability of having intensity striking the boundary in an angular interval dφ about the angle φ is proportional to the solid angle 2φ sin φdφ on a unit sphere with the center at the surface element. π/2 Thus, the total power striking the element will be proportional to 0 cos φ sin φ dφ. The absorbed power is obtained by inserting the absorption coefficient α(φ) as a factor in the integrand. Thus, the average absorption coefficient in a diffuse sound field becomes π/2 π/2 α(φ) cos φ sin φ dφ 0 =2 αst = π/2 α(φ) cos φ sin φ dφ, (3.23) 0 cos φ sin φ dφ 0 where α(φ): Eq. 3.22. It is often called the statistical average or, as in this book, the diffuse field absorption coefficient. The integration over angle of incidence, which is involved in computing the diffuse field absorption coefficient, is based on the assumption that the material is isotropic so that the impedance is independent of the azimuth angle of the incident wave. For a nonisotropic material, as obtained if the air backing is partitioned only in one direction, the integration involves also the azimuth angle, as for the slot absorber in Chapter 3 and the general anisotropic material discussed in Chapter 5. For the locally reacting absorber, the integral in Eq. 3.23 can be expressed in closed form in terms of the normalized input impedance ζi = θi + iχi , 2 − χ2 θ θi 1 8θi χ i 1− ln((1 + θi )2 + χi2 ) + i 2 i arctan( ) , (3.24) αst = |ζi |2 |ζi |2 |ζi | χi 1 + θi where ζi = θi + iχi : Eq. 3.19. As already indicated, the sheet interaction impedance ζ often can be considered to be purely resistive. Then, for the locally reacting absorber, the resonance frequencies are given by cot(kL) = 0, independent of the angle of incidence, and for the nonlocally reacting, by cot(kL cos φ) = 0. If the mass reactance of the sheet is not negligible, which is often the case when the sheet includes a perforated thin plate or ‘facing,’ the resonance frequencies will be reduced accordingly. It follows from Eq. 3.22 that at resonance, with χi = 0, the absorption coefficient is (α)max =
4θ cos φ . (1 + θ cos φ)2
(3.25)
90
NOISE REDUCTION ANALYSIS
With θ = 1, the resonance absorption coefficient at normal incidence is 1. With θ > 1, the normal incidence absorption coefficient will be less than 1 but for oblique incidence, on the other hand, it will be unity for waves with an angle of incidence given by cos φ = 1/θ . With θ < 1, the absorption coefficient will be less than unity for all angles of incidence. It follows that to optimize the diffuse field average absorption coefficient, the input resistance should be greater than 1 (for local reaction, it is about 1.8). At a given frequency and χi , there is an optimum resistance for maximum absorption. This optimum and the corresponding maximum absorption coefficient for the locally reacting absorber is given by √ θi cos φ = 1 + χi cos φ αmax = √ 2 . 2 1+
(3.26)
1+(χi cos φ)
For normal incidence and at wavelengths much greater than the cavity depth so that kL << 1, maximum absorption is obtained when the resistance equals the reactance, i.e., θopt = 1 + (1/kL)2 ≈ 1/kL. (3.27) 1/3 and 1/1 Octave Band Average Absorption The frequency width of a band, which is 1/Nth of an octave and with a (log scale) center frequency fc , is f = fc (21/2N − 2−1/2N ).
(3.28)
To calculate the average absorption in such a band, the interval f is divided into a number of equal parts (logarithmically), each with a frequency width δf (except at the ends of the interval f , where the width is δf/2). For an octave band average, 13 and for a third octave band, 6 intervals have been used (including the elements at the ends). If the center frequency of such an interval is fi , the average absorption coefficient is 1 αav = α(fi )δf, (3.29) f i
where δf is the width of a sub-interval obtained by analogy with Eq. 3.28.
3.6.2 Flexible Sheet Cavity Absorber The velocity amplitude of the sheet is u and the air velocity just in front of (and behind) the sheet is u. The relative velocity of the air with respect to the sheet is then u − u . From the definition of the interaction impedance z = ζρc, the difference in sound pressure amplitude across the sheet is p = p1 − p2 = z(u − u ).
(3.30)
91
SHEET ABSORBERS
With the structural impedance of the sheet being zs , the equation of motion of the sheet takes the form p = z(u − u ) = zs u . (3.31) For a limp sheet of mass m per unit area, zs = −iωm. The ratio of the velocities u and u follows from these equations, u /u = z/(z + zs )
(3.32)
and insertion into Eq. 3.30 yields z ≡ p/u = zzs /(z + zs ) z = r + ix =
r(ωm)2 r 2 +(ωm−x)2
− iωm rr 2−x(ωm−x) +(ωm−x)2 2
(limp sheet),
(3.33)
where z = ζρc: Eq. 3.19. For a limp sheet, zs = −iωm. m: sheet mass per unit area. In terms of an electrical circuit analogy, the equivalent impedance z can be thought of as the parallel combination of the interaction impedance z and the structural impedance zs of the sheet. Unlike z = r + ix, the equivalent impedance z = r + ix
accounts for the motion of the sheet. The second part in the equation refers to a limp sheet for which zs = −iωm. In the idealized case of a purely resistive screen, x = 0, r
increases monotonically with frequency to reach the rigid sheet value r asymptotically, and the reactance goes from zero to a maximum of r/2 at the frequency ωm = r/m and then goes back to zero. At ω = r/m, r and x have the same magnitude. Effect of Bending Stiffness and Structural Resonances For a limp sheet, the bending stiffness is zero and the structural impedance is simply the mass reactance ζs = −iωm/ρc, as already mentioned. This approximation is justified for many sheets used in sound absorption applications. If the bending stiffness is accounted for, the structural impedance of a thin plate is used and at frequencies above the lowest mode of the plate, the impedance of an infinitely extended plate can be used as an approximation. This is known to be
−iωm ω 2 4 ζs = (3.34) 1 − ( ) sin φ , ρc ωc where the ‘critical’ frequency is √ ωc = 2πfc = c2 / B
(3.35)
and the ‘bending stiffness’ B=
Y h2 . 12ρs (1 − σ 2 )
(3.36)
In this expression, Y is the Young’s modulus, h the thickness of plate, ρs the mass density of the plate, σ the Poisson ratio, and ρc the wave impedance of the surrounding fluid. For most sheets of interest, fc is so high (bending stiffness so low) that the effect of stiffness is negligible in the frequency range normally of interest. For example,
92
NOISE REDUCTION ANALYSIS
with a thickness of the order of 0.1 cm, the critical frequency ωc /2π usually exceeds 10,000 Hz. If the sheet has internal damping, it can be accounted for by using a complex value of the Young’s modulus, with Y replaced by Y (1 − i), where is the loss factor for the material. The phase velocity√of a freely propagating bending wave on the sheet can be expressed as vb = c ω/ωc (c is the sound speed in the surrounding fluid), and the fundamental resonance frequency of the sheet will be of the order of vb /2s , where s is the linear dimension of the sheet. The bending stiffness B, and hence the critical frequency ωc of a sheet, can be determined experimentally from measurements of the fundamental frequency of oscillation of a strip of the sheet clamped at one end and free at the other. This frequency is √ B fr = 0.56 2 , (3.37) s where s is the length of the strip. If the strip is clamped at both ends, the fundamental frequency is 6.36 fr and for a circular sheet with a diameter s clamped along the perimeter, the fundamental frequency is 23.1 f1 . (See, for example, Morse and Ingard, Theoretical Acoustics, p 182, McGraw-Hill, 1968.) To get an idea of the numerical values involved, the fundamental frequency fr of a porous sheet (Typar) with s = 10 cm was measured and found to be 4 Hz. This means that the fundamental frequency of a circular sheet with a diameter of 10 cm will be 92 Hz. The samples used in the apparatus described in Appendix A for measurements of the acoustic flow impedance typically have a diameter of about 5 cm, and the fundamental frequency of such a sheet will be 4 × 92 = 368 Hz, since the frequency is inversely proportional to s 2 . The critical frequency, defined above, can be expressed as fc =
c2 c2 0.56 c2 = 0.089 . √ = 2π s 2 fr s 2 f1 2π B
(3.38)
If sheet resonances are present in the frequency range of interest, the structural impedance of the sheet in the vicinity of a resonance will be of the form
ωm 1 ωr ω2 ζs ≡ θs + iχs = −i 1 − r2 + i . (3.39) ρc ω Q ω As before, to account for the losses in this expression Y is replaced by Y (1 − i) and ωc2 by ωc2 /(1 − i) ≈ ωc2 (1 + i), where is the loss factor. This defines θs and the corresponding Q = ωr m/θs ρc, the ‘Q-value’ of the resonance. A similar expression can be used also for a limp sheet under tension with a resonance frequency fr . As far as the numerical analysis is concerned, a computer program for the limp, resistive screen, can be used by replacing the mass m with the complex mass m ˜ = m[1 − ωr2 /ω2 + i(1/Q)(ωr /ω)] to make the program apply to a sheet with a resonance frequency ωr .
(3.40)
93
SHEET ABSORBERS A Low Frequency Resonance
The normalized input impedance of a limp sheet air layer combination at normal incidence is ζi = θi + iχi = ζ + i cot(kL) ≈ ζ + i/kL, (3.41) where ζ = z /ρc is the (equivalent) sheet impedance in Eq. 3.33. The corresponding normal incidence absorption coefficient is α=
4θi . (θi + 1)2 + χi2
(3.42)
The interaction impedance ζ of the sheet is assumed to be purely resistive, ζ = θ. The normalized equivalent mass reactance χ = x /ρc is then due solely to the motion of the screen, and we get from Eq. 3.33 θ =
θ (km )2 , + (km )2
θ2
χ = −
θ2
θ 2 km , + (km )2
(3.43)
where k = ω/c and m = m/ρ. The total reactance of the input impedance, χi = χ + cot(kL), becomes zero in the long wavelength approximation (cot(kL) ≈ 1/kL) when χ = −1/kL, which signifies resonance. From the ratio of these two equations it follows that the input resistance of the absorber at resonance becomes θi = θ ≈
m θL
(resonance).
(3.44)
The normal incidence absorption coefficient at resonance (see Eq. 3.41) then becomes 4(m /Lθ) α0 ≈ (resonance). (3.45) [1 + (m /Lθ)]2 For a given value of mr ≡ m /L = m/ρL, 100 percent absorption is obtained if the normalized flow resistance of the sheet is chosen to be θ0 = mr = m/ρL
(resonance).
(3.46)
χ
The resonance frequency is obtained from ≈ −1/kL, as indicated in the discussion of Eq. 3.43, and it follows then from this equation that ω0 ≈ √
c/L c =√ L(m − L) mr − 1
(α = 1).
(3.47)
Since the analysis is based on the assumption kL << 1, self-consistency requires that mr = m/ρ >> 1 so the term 1 can be neglected compared to mr in the denominator. Then, recalling that the quarter wavelength resonance frequency f1/4 = c/4L, the new low frequency resonance is expressed in terms of it. f0 ≈
1 c/L 2 f1/4 = √ , √ 2π mr π θ
where f1/4 = c/4L, θ ≈ mr = m/ρL.
(3.48)
94
NOISE REDUCTION ANALYSIS
For a rigid sheet, the lowest resonance frequency is the quarter wavelength resonance at f1/4 and the lowest resonance of the limp sheet resonator is approximately √ (π/2) mr times lower. Absorption Coefficient The absorption coefficient for the flexible sheet absorber follows from the expressions for the absorption coefficient of the rigid sheet absorber by replacing the interaction impedance ζ by the equivalent sheet impedance ζ given in Eq. 3.33.
3.6.3 Uniform (Periodic) Lattice This section can be considered to be an exercise in wave propagation in a periodic structure or lattice. It turns out to yield some interesting results about sound absorption by a multisheet absorber. The absorber consists of N identical, equidistant, limp, resistive sheets placed in front of a rigid wall, as shown in Figure 3.13. Each sheet is specified acoustically by the equivalent sheet impedance ζ , defined in the previous section (see Eq. 3.33). The separation of adjacent sheets is much larger than the acoustic thermal boundary layer so that the compressibility of the air in the lattice can be assumed to be isentropic. This should be kept in mind when the absorption spectra of a lattice absorber are compared with those of uniform porous layers in which heat conduction leads to isothermal compressibility at low frequencies. Unit Cells The analysis starts with a choice of a unit cell of the lattice, and in Figure 3.22 two possibilities are indicated. In the first, the beginning of the cell is at a point just in front of one sheet at x = nd, and the end of the cell is just in front of the adjacent sheet at x = (n + 1)d, where n is an integer. The complex amplitudes of sound pressure and velocity at these locations are denoted by p(xn ), u(xn ) and p(xn+1 ), u(xn+1 ), respectively. This unit cell is asymmetrical in the sense that it is not the same for sound waves traveling in the positive and negative directions. It contains two elements, an air layer and a sheet. Asymmetrical cell ζ′
ζ′
pn
pn+1
Symmetrical cell ζ′
ζ′
2
2
pn
ζ′ 2
ζ′ 2
pn+1
Figure 3.22: The asymmetric cell consists of two elements, one sheet and an air layer. The symmetric cell has three elements, a half-sheet, an air layer, and a second half-sheet.
95
SHEET ABSORBERS
The second unit cell is symmetrical, however. Each sheet is considered to be a combination of two sheets, each with the acoustic impedance equal to half of that of one sheet, and the reference planes are placed between these sheets. Thus, in this case the unit cell consists of three elements, two half-sheets and an air layer. Propagation Constant and Wave Impedance For a single harmonic traveling wave, with the time dependence expressed by the factor exp(−iωt), the space dependence of the field variables at the sheet locations is expressed by the factor exp(iqx xn ). The ratio p(xn+1 )/p(xn ) = exp(iqx d) then defines the propagation constant qx in the x-direction of the lattice. The same ratio applies also to the complex velocity amplitude of the fluid. In general, qx is complex and contains both a phase shift and an amplitude decay per cell. To calculate the propagation constant qx , a relationship between the complex amplitudes of the field variables at the two ends of the unit cell must be derived from the dynamics of the components of the cell, i.e., the sheet and the air column between the sheets. The linear relationship between the field variables at the two ends of the lattice can be expressed as p(xn ) = T11 p(xn+1 ) + T12 ρcu(xn+1 )
(3.49)
ρcu(xn ) = T21 p(xn+1 ) + T22 ρcu(xn+1 ),
(3.50)
where u is the x-component of the complex velocity amplitude and Tij the elements of the transmission matrix of the unit cell of the lattice, which have to be determined in terms of the frequency ω and the lattice parameters. The wave impedance is Zρc = p(xn )/u(xn ) = p(xn+1 )/u(xn+1 ) and with p(xn+1 )/p(xn ) = u(xn+1 )/u(xn ) = exp(iqx d), it follows from Eqs. 3.49 and 3.50 that e−iqx d = T11 + T12 /Z = T22 + T21 Z Z = (T11 − T22 )/2T21 ∓ (i/T21 ) 1 − [(T11 + T22 )/2]2 ,
(3.51)
where, in the last expression, the elements Tij are not all independent but related by T11 T22 − T12 T21 = 1, a general property of a passive ‘four-pole,’ in the language of circuit analysis. From these relations it follows that (3.52) e−iqx d = (T11 + T22 )/2 − (i/T21 ) 1 − [(T11 + T22 )/2]2 . Expressing exp(iqx d) in a similar manner yields cos(qx d) = (eiqx d + e−iqx d )/2 =
T11 + T22 . 2
(3.53)
The propagation constant and the wave impedance can be expressed explicitly in terms of the lattice parameters after the sheet impedance has been specified and the matrix elements Tij for a unit cell of the lattice have been calculated. For the first
96
NOISE REDUCTION ANALYSIS
of the unit cells in Figure 3.22, they are obtained as the product of the matrix of an air layer of thickness d and the matrix for the sheet. For a sound wave with an angle of incidence φ on the lattice (the angle measured from the normal to the surface), kx = (ω/c) cos φ and the matrix elements for the air layer are A12
A11 = A22 = cos(kx d) = −i(1/ cos φ) sin(kx d), A21 = −i cos φ sin(kx d)
(3.54)
and for the sheet S11 = S22 = 1 S12 = ζ , S21 = 0,
(3.55)
where ζ is the normalized equivalent impedance discussed earlier. From the product of these matrices we obtain the matrix elements for the unit cell, Tij = k Sik Akj , i.e., T11 = cos(kx d) − iζ cos φ sin(kx d) T12 = −i(1/ cos φ) sin(kx d) + ζ cos(kx d) T21 = −i cos φ sin(kx d) T22 = cos(kx d).
(3.56)
The corresponding expressions for the propagation constant qx and the normalized wave impedance Z, as given by Eqs. 3.53 and 3.51, are then cos(qx d) =
T11 + T22 ζ cos φ = cos(kx d) − i sin(kx d) 2 2
and Z± =
ζ
sin(qx d) ± , 2 sin(kx d) cos(φ)
(3.57)
(3.58)
where sin(qx d) = 1 − cos2 (qx d) has been used together with Eq. 3.57. The plus and minus signs in the last expression refer to propagation in the positive and negative x-directions, respectively. The reason for the difference in the wave impedances for propagation in the two directions is that the unit cell is not symmetrical. For propagation in the positive direction, the cell starts just in front of a sheet and the impedance refers to the ratio of pressure and velocity amplitudes at this point. For propagation in the negative direction, the cell starts with the air column between two sheets. The ratio of the pressure amplitude and the negative velocity amplitude at this point is [sin(qx d)/ sin(kd) cos(φ)]−ζ /2. The difference between this impedance and the wave impedance in Eq. 3.58 is ζ , as it should be. If we had chosen the symmetrical unit cell in Figure 3.22, the expression for the wave impedance in Eq. 3.58 would have contained only the second term, with the magnitude of the impedance being independent of the direction of wave travel. The propagation constant would have been the same as before. The wave impedance becomes infinite if kx d = nπ (cot(kx d) = ∞), i.e., if d cos φ = nλ/2, which is the well-known Bragg condition for constructive interference
97
SHEET ABSORBERS
of the waves reflected from the different layers in a lattice as mentioned in the main text in the discussion of the absorption spectrum for a periodic lattice. Input Impedance and Absorption Coefficient In terms of the wave impedances Z± , the complex amplitudes of pressure and velocity at the nth sheet become p(xn ) = Aeiqx xn + Be−iqx xn ρcu(xn ) = (A/Z+ )eiqx xn + (B/Z− )e−iqx xn .
(3.59)
(In the second term of the last equation, a negative sign is contained in the expression for the wave impedance Z− .) If the lattice is backed by a rigid wall at x = L = N d, where N is the number of sheets in front of the wall, the boundary condition is u(L) = 0, which establishes the relation B/A = −(Z− /Z+ ) exp(i2qx L). After a little algebra, the amplitudes of pressure and velocity at the nth sheet can be expressed as ζ
x d) + p(xn ) = C i cot[qx (L − xn )] cossin(q φ sin(kx d) 2 sin[qx (L − xn )] ρcu(xn ) = C sin[qx (L − xn )],
(3.60)
where C = −(2iA/Z+ ) exp(iqx L). The first sheet in the lattice is located at x1 = 0, and it follows that the input impedance is ζi ≡ θi + iχi =
p(0) u(0)ρc
=
ζ
2
x d) + i cossin(q φ sin(kx d) cot(N qx d),
cos(qx d) = cos(kx d) − i ζ
cos φ
2
sin(kx d),
(3.61)
where kx = k cos φ, ζ : 3.33, d: Figure 3.13, φ: Angle of incidence, N : Number of sheets, qx : Eq. 3.57. Since qx has an imaginary part, the factor cot(N qx d) → −i as N → ∞, and the input impedance becomes the same as for an infinite lattice as given in Eq. 3.58, i.e., ζi → Z+ . As another test, consider the case of a single sheet, i.e., N = 1. From Eq. 3.57 for cos(qx d), the input impedance becomes the familiar ζi = ζ + i[1/ cos(φ)] cot(kx d). Having obtained ζi , the absorption coefficients follow from Eqs. 3.22 and 3.23. Some computed results are shown in Figure 3.14. Field Distribution Within a Cell To compute the field distribution in the region xn < x < xn+1 between two sheets at the location x + ξ , we merely apply the transmission matrix for the air column between xn + ξ and xn + d of thickness d − ξ to obtain p(xn + ξ ) = p(xn+1 ) cos[kx (d − ξ )] − i(ρc/ cos φ)u(xn+1 ) sin[kx (d − ξ )] ρcu(xn + ξ ) = −ip(xn+1 ) cos φ sin[k(d − ξ )] + ρcu(xn+1 ) cos[kx (d − ξ )]. (3.62)
98
NOISE REDUCTION ANALYSIS
The input impedance in Eq. 3.61 contains sin(qx d) = 1 − cos2 (qx d), which can be computed with the help of Eq. 3.57 and cot(N qx d) can be calculated from cot(qx Nd) = i[exp(iN qx d) + exp(−iN qx d)]/[exp(iN qx d) − exp(−iN qx d)] with the use exp(iqx d) either as cos(qx d) + i sin(qx d) or directly from Eq. 3.52. Alternate Choice of Unit Cell The symmetrical unit cell begins at the midpoint between two sheets and ends at the midpoint between the adjacent sheet pair, as indicated in Figure 3.22. In this case, the wave impedance should be the same (except for sign) and independent of the direction of wave travel. It is left as an exercise to show that the elements of the transmission matrix elements are (for normal incidence)
S11 = cos(kx d) − i ζ2 sin(kx d)
S12 = −i sin(kx d) + ζ cos2 (kx d/2) S21 = −i sin(kx d) + ζ sin2 (kx d/2)
S22 = cos(kx d) − i ζ2 sin(kx d),
(3.63)
where kx = ω/c (normal incidence) and c the sound speed. Similarly, the wave impedance becomes (see first part of Eq. 3.58) Z± = ∓
sin(qx d) sin(qx d) =± . S21 sin(kx d) + i(ζ1 /2) sin2 (kx d/2)
(3.64)
The magnitude of this impedance is independent of the direction of wave travel, as it should be. The expression for the propagation constant is the same as that obtained for the asymmetrical unit cell considered above.
3.6.4 Nonuniform Lattice The sheets need no longer be identical and the distance between them need not be the same so that a nonuniform (nonperiodic) lattice is formed. Transmission Matrix Each cell of the lattice consists of two elements, an air column and a sheet. The transmission matrix Cn of the nth cell (starting the count from the rigid wall) can be written as the matrix product of the transmission matrices A and S of the nth air column and the nth sheet Cn = (SA)n , (3.65) and the transmission matrix T of the entire multisheet absorber of N sheets is then
N T11 T12 = Cn , (3.66) T ≡ T21 T22 1
where n = 1 is the cell at the rigid wall.
99
SHEET ABSORBERS
The transmission matrices of the air layer and the sheet and have already been calculated (Eq. 3.55). Input Impedance and Absorption Coefficient In terms of the matrix elements Tij in Eq. 3.66, the relations between the sound pressure p and normal velocity component u at locations (1) and (2) at the beginning of the absorber and at the wall are p(1) = T11 p(2) + T12 ρcu(2) ρcu(1) = T21 p(2) + T22 ρcu(2),
(3.67)
where ρc is the wave impedance of the ambient fluid (air). Since the backing wall is rigid, i.e., u(2) = 0, the normalized input impedance of the lattice becomes ζi ≡ θi + iχi =
p(1) T11 . = ρcu(1) T21
(3.68)
Having obtained ζi , the absorption coefficients can be computed as for the singlesheet absorber.
3.6.5 Sheet as a Volume Absorber Plane Wave Reflection, Transmision, and Absorption, Infinite Sheet With reference to Figure 3.18, consider a harmonic plane wave incident in the xy-plane of an infinite sheet located at x = 0. The angle of incidence is φ. The spatial dependence of the wave is determined by the factors X(x) = eikx cos φ ,
Y (y) = eiky sin φ ,
(3.69)
where k = ω/c. The complex amplitudes of the incident, reflected, and transmitted pressure waves are then AX(x)Y (y), BX(−x)Y (y), and CX(x)Y (y), respectively, and the x-components of the corresponding velocity amplitudes are A cos φX(x)Y (y)/ ρc, −B cos φX(−x)Y (y)/ρc, and C cos φX(x)Y (y)/ρc. The pressure and velocity fields to the left and the right of the sheet are p1 = pi + pr = [AX(x) + BX(−x)]Y (y) p2 = pt = CX(x)Y (y) u1 = ui + ur =
1 ρc
cos φ [AX(x) − BX(−x)] Y (y)
u2 = ut =
C ρc
cos φX(x)Y (y).
(3.70) (3.71) (3.72) (3.73)
The boundary conditions at the sheet, p1 − p2 = ζ ρcu1 and u1 = u2 , then lead to the relations A + B − C = Cζ cos φ A − B = C, where z = ρcζ is the equivalent sheet impedance, as defined earlier.
(3.74)
100
NOISE REDUCTION ANALYSIS
Addition of the two equations gives C = 2A/(2 + ζ cos φ), the pressure transmission coefficient is τ = C/A, and the corresponding power transmission coefficient |τ |2 . The pressure reflection coefficient is R = B/A = 1 − τ and the corresponding coefficient for power is |R|2 . From conservation of energy it follows that the fraction of the incident power, which is absorbed by the sheet is α = 1 − |R|2 − |τ |2 . Thus, to summarize, τ=
C A
=
2 2+ζ cos φ
R =1−τ = α(φ) = 1 − |R|2 − |τ |2 =
ζ cos φ 2+ζ cos φ
(3.75)
4θ cos φ , (2+θ cos φ)2 +(χ cos φ)2
(3.76)
where ζ = θ + iχ : Eq. 3.33, φ: Angle of incidence. This coefficient can be determined also directly from a calculation of the friction losses in the sheet by first determining the x-component of the velocity amplitude at the sheet C 2A cos φ u= cos φ = , ρc ρc 2 + ζ cos φ and then the power loss per unit area (1/2)|u|2 ρcθ = |A|2 /2ρc α(φ) α(φ) =
4θ cos φ , (2+θ cos φ)2 +(χ cos φ)2
(3.77)
where α is the absorption coefficient. For a sheet with a purely resistive impedance (χ = 0) and a sufficiently large mass so that the sheet can be regarded as immobile as far as the interaction with sound is concerned, the normal incidence coefficients of reflection, transmission, and absorption depend on the sheet resistance as shown in Figure 3.19. Infinite Sheet, Diffuse Field, Effect of Induced Motion The sheet is assumed to be purely resistive. As was mentioned earlier, this is normally a good approximation. With sound incident only on one side of the sheet, the average absorption coefficient is α1 = 2
π/2
α(φ) sin φ cos φ dφ.
(3.78)
0
With the expression for α(φ) given in Eq. 3.77 and with η ≡ cos φ, the average absorption coefficient can be written 1 2 α1 = 2θ 0 (1+θ η/2)η2 +(χ η/2)2 dη
= |ζ8θ |2 1 − |ζ2θ |2 log(1 + θ + 14 |ζ |2 )
χ
+ |ζ8θ |2 (θ 2 − χ 2 ) χ |ζ2 |2 arctan( 2+θ ) ,
(3.79)
101
SHEET ABSORBERS
where ζ = θ + iχ : Eq. 3.33, η = cos φ, φ: Angle of incidence, m: Sheet mass per unit area. The corresponding expressions for the average value of the transmission and reflection coefficients |τ |2 and |R|2 based on acoustic power are π/2 |τ |2 = 2 0 |τ |2 sin φ cos φ dφ
χ
= |ζ4 |2 log(1 + θ + |ζ |2 /4) − 2θ χ arctan( 2+θ ) |R|2 = 1 − α1 − |τ |2 .
(3.80)
Effect of Diffraction An approximate analysis for a circular porous disk4 shows that for wavelengths large compared to the disk diameter, the drop of the sound pressure amplitude across the disk (radius a) is given by √ k a2 − r 2 p = −ip0 (4/π) cos φ , (3.81) 1 − (8/3π )ikaβ where p0 is the pressure amplitude of the incoming wave, β = 1/ζ , the normalized equivalent admittance of the screen, ζ , the corresponding equivalent sheet impedance, and φ the angle of incidence. Integration over an impervious rigid disk (β = 0) results in the amplitude of the acoustically induced force on the disk, a 8 f = p2πr dr = −ip0 cos φ (π a 2 ka). (3.82) 3π 0 If we introduce p0 = ρcu0 , where u0 is the velocity amplitude in the incoming wave and consider normal incidence (φ = 0), the force becomes f = (−iωu0 )(8a/3π ) ρπ a 2 ≡ −iωmi , where mi = (πa 2 )(8a/3π )ρ. (3.83) Thus, the induced mass mi created by the disk can be thought of as the mass of a fluid cylinder with the area of the disk as a base and with a length 8a/3π = 4d/3π , where d = 2a is the diameter of the disk. Whether or not the flow will go through the disk rather than around it depends on the ratio of the normalized average mass reactance ωmi /π a 2 per unit area and the impedance of the sheet. For a purely resistive and immobile screen, this condition is ωmi /ρc > θ , where θ is the normalized flow resistance of the sheet. Introducing the expression for the induced mass in Eq. 3.83, we can write the condition as (4/3π)kd > θ or d > (3λ/8)θ, (3.84) where we have used k = ω/c = 2π/λ. In other words, for values of the flow resistance of the order of unity, the diameter of the sheet should exceed approximately one-third 4 P. M. Morse and K. U. Ingard, Linear Acoustic Theory, Volume XI/1 in Handbuch der Physik, Springer-
Verlag, 1961.
102
NOISE REDUCTION ANALYSIS
of a wavelength in order that the flow will begin to have a preference for going through the disk rather than around it. Thus, a relatively heavy disk, such that the induced motion can be ignored, with a diameter of 1 ft should begin to respond like an infinitely extended sheet for wavelengths less than ≈ 8/(3θ) ft. For a sheet resistance of the order of unity, this corresponds to frequencies above ≈ 420 Hz. It should be noted that the induced mass of the finite sheet plays about the same role as the mass per unit area of the infinitely extended sheet as far as the reduction of absorption cross section at low frequencies is concerned. Actually, for a sheet with a diameter of 1 ft, the induced mass per square foot corresponds to a weight of 0.08 · 4/3π ≈ 0.035 lb/ft2 (≈ 5 oz/yd2 ), where we have used 0.08 lb/ft3 for the density of air. This is about the same as the weight of a typical woven sheet, and the induced motion of the sheet then plays the same role as diffraction in reducing the low frequency absorption. Thus, in this case, no significant improvement in the low frequency absorption would result by making the sheet heavier than 0.035 lb/ft2 since the absorption would be limited by the effect of diffraction. To investigate in more detail the frequency dependence of the absorption cross section of the disk, we use the expression for the pressure drop across the sheet in Eq. 3.81. The pressure drop produces an oscillatory flow relative to the disk, and if this velocity amplitude is denoted by ur , the energy absorption per unit area becomes |ur |2 θ ρc. Integration over the area of the disk and over the angle of incidence in a diffuse sound field yields the average power absorbed per unit area of one side of the disk. The amplitude of the relative velocity through the disk is ur = u − u , where u
is the amplitude of the induced velocity of the disk and u = p/(ρcζ ) the absolute velocity amplitude of the air just outside the disk. Quantity ζ is the equivalent disk impedance. Furthermore, for a purely resistive limp disk so that ζ = θ, we get u /u = θ/(θ − iωm/ρc) (see discussion of the single limp sheet cavity absorber). The amplitude of the relative velocity ur = u − u can then be expressed as ur /u =
−iωm r f/fm , = 1/(1 + ir/ωm) = r − iωm i − f/fm
(3.85)
where f is the frequency and fm = r/(2πm), the characteristic frequency. The power absorbed per unit area of the disk is |ur |2 θρc. From the expressions for ur /u, u = p/(ρcζ ) and p in Eq. 3.81, integration over the disk and the angle of incidence yields the average power absorption of a disk of area A in a diffuse sound field (for one-sided exposure to the sound) |p0 |2 ur 2 32 (ka)2 θ , A| | 2
ρc u π |ζ + iχi |2
(3.86)
where χi = −(8/3π)ka is the average normalized induced mass reactance per unit area of the disk. In order to relate this to an absorption cross section or absorption area in a diffuse sound field, we have to introduce the average intensity in a diffuse field, which is known to be Iav = |pav |2 /4ρc, where |pav |2 = 4π |p0 |2 . Thus, the factor |p0 |2 /ρc in Eq. 3.86 can be expressed as Iav /π . Thus, if we replace π 2 in the denominator of
103
SHEET ABSORBERS
Eq. 3.86 by π 3 , we obtain the average absorption cross section of the disk for onesided exposure in a diffuse field. For two-sided exposure, we replace the factor 32 by 64 in the equation to obtain the total absorption cross section σa σa ≈ A|
ur 2 64 (ka)2 θ | , u π 3 |ζ + iχi |2
(3.87)
where |ur /u|: Eq. 3.85, ζ : Eq. 3.33, θ: Normalized flow res. χi : Eq. 3.86. In the low frequency approximation, as expressed by Eq. 3.87, there will be an optimum flow resistance for a purely resistive immobile disk (ζ = θ) and a fixed value of ka given by θ = χi = (8/3π)ka. The corresponding two-sided maximum absorption cross section is (8/π 2 )(ka)πa 2 . The optimum value of the flow resistance for the infinite sheet, discussed in connection with Figure 3.20, is not valid for the finite sheet; the optimum resistance is now approximately equal to ka = 2π a/λ, i.e., directly proportional to the diameter of the sheet. Figure 3.21 shows an example of the calculated frequency dependence of the absorption cross section per unit area of sheet material, as obtained from Eq. 3.87 (with the modifications explained in the text following the equation).
Chapter 4
Resonators 4.1 INTRODUCTION AND SUMMARY Normally, an acoustic resonator refers to a cavity (bottle) resonator rather than a ‘mechanical’ one, such as a plate or a bell, and this convention applies also to this chapter. Generally, all dimensions of a cavity resonator are assumed large compared to the visco-thermal boundary layer thickness but small compared to the wavelength of the incident sound (‘acoustical compactness’). The walls are acoustically hard so that sound absorption is due solely to visco-thermal losses at the walls. Then, the absorption will be significant only in the vicinity of the resonance frequencies of the cavity unless porous materials, such as screens, are added. Actually, additional damping is obtained without such material if the incident sound pressure is high enough to make nonlinear effects significant. Then, the oscillatory flow in an orifice or at a (sharp) corner will separate so that vorticity is created. The corresponding energy will be drawn from the sound wave and create a damping, which often is larger than the visco-thermal effects. Similarly, if steady flow is present, the coupling between the flow and the sound will produce acoustic losses. The simplest acoustic resonator, at least geometrically, is a straight tube. If the tube is closed at one end and open at the other, the fundamental resonance frequency occurs at a wavelength which is 4 times the acoustic length of the tube. Higher resonances occur whenever the tube length is an odd number of quarter wavelengths. If a resonator is exposed to an incident sound wave, as shown in Figure 4.1, a certain portion of the incident acoustic power will be absorbed and the rest will be scattered. The amounts of absorbed and scattered power are conveniently expressed in terms of the absorption and scattering cross sections of the resonator, Aa and As , respectively. This means that if the intensity of the incident sound is I , the absorbed and scattered powers will be I Aa and I As , respectively, as discussed in some detail in the next section. The maximum possible absorption cross section of a resonator at resonance is λ2 /2π when the resonator is set in a rigid wall (baffle) and λ2 /4π without a wall, where λ is the wavelength of the incident sound. It should be remembered that the resonator is acoustically compact so that the opening is small compared to the wavelength and
105
106
NOISE REDUCTION ANALYSIS
Figure 4.1: Tube resonator in a wall (infinite baffle) and normalized radiation impedance (resistance and reactance) of a piston source in a wall vs ka, where k = 2π/λ and a is the radius of the resonator orifice. the absorption and scattering cross sections (as defined here) are independent of the angle of incidence of the sound. In the tube resonator, the potential and kinetic energies are distributed throughout the tube, but in the Helmholtz or bottle resonator, the two forms of energy can be considered to be spatially separated or ‘lumped,’ the kinetic energy residing in the neck of the resonator and the potential energy in the cavity. The Helmholtz resonator is then analogous to an ordinary mass-spring oscillator with the mass being the mass of the air in the neck and the spring being provided by the stiffness of the air in the cavity. If a pulse in the form of a harmonic wave train strikes a resonator, the resonator mode will be excited and then decays exponentially with time. The frequency in this decay or reverberation is the resonance frequency of the resonator. If the frequency of the incident sound is close to but not equal to the resonance frequency, distinct beats will occur between the incident and decaying fields. These can be clearly audible, as discussed in connection with the experiments referred to in Figure 4.5. A (tube) resonator is often used as side-branch in a duct to attenuate sound in narrow bands around the resonance frequencies of the tube but the discussion of it is deferred to Chapter 8, which deals with sound propagation in ducts. The transmission and insertion loss measures of such a side-branch resonator are then illustrated by specific examples. To cover a broad frequency range, several resonators can be used in parallel. Interaction of a fluid flow with a resonator can lead to either damping or excitation. Both can be of considerable practical importance.
4.2 ABSORPTION AND SCATTERING A single resonator in the form of a tube and mounted in a wall is shown schematically in Figure 4.1. Most of what we have to say about it applies equally well to any other type of cavity resonator. The tube resonator is convenient, since its acoustical properties, such as the input impedance, can readily be obtained from the material in Chapter 3. In our analysis of the sound absorption and scattering by a resonator, it has been
107
RESONATORS
assumed that the air layer in the resonator opening moves with uniform velocity so that it can be regarded as a plane, mass-less piston. It will generate sound both inside and outside the resonator, and the motion of the piston will be impeded by the radiation forces from the two sides. The resulting force is proportional to the sum of the input and radiation impedances of the resonator. At low frequencies, the radiation resistance increases as the square of the frequency but levels off to the constant value of unity (normalized) at high frequencies. The normalized reactance first increases linearly with frequency, reaches a maximum of 0.5 at ka = 1, where k = 2π/λ and a is the radius of the resonator opening, and then decreases to zero with increasing frequency, as shown in Figure 4.1. The frequency dependence of the input reactance is stiffness-controlled below the resonance frequency and mass-controlled above. At resonance the reactance is zero and the input impedance is purely resistive and, as shown in Section 4.5, proportional to the square root of the resonance frequency. This follows from the frequency dependence of the viscous boundary layer thickness. With all the walls of the resonator being acoustically hard, only the visco-thermal boundary losses contribute to sound absorption. The viscous losses along the tube walls dominate. With only the fundamental acoustic mode involved in the tube, there will be no viscous losses on the rigid wall at the end of the tube. However, there will be a thermal boundary layer on this wall, which contributes a small amount to the overall losses. Actually, as shown in Section 2.6, the boundary layer can be accounted for acoustically by replacing it with an equivalent acoustic admittance so that an acoustically hard wall in effect is replaced by one which is somewhat softened. In terms of this equivalent admittance ηt , the input impedance takes the form Tube resonator in a wall, input and radiation impedance ζi ≡ θi + χi =
cot(qL)−iηt −i+ηt cot(qL)
≈ i cot(qL)
ζr = θr + χr ≈
(at resonance ≈ π dvh /4a)
(4.1)
(ka)2 /2 − i(8/3π )ka
where q: see Eq. 4.21, ηt : End wall admittance (<< 1), Eq. 4.23, k = ω/c, L: Length, a: Tube radius, Figure 4.1, dvh : See Eq. 4.22. (See also Section 4.5, Eqs. 4.24 and 4.29.) By using the acoustical length L = L + δ, the sum of the physical length and an end correction δ = (8/3π)a, the reactive part of the total impedance ζi + ζr can be expressed as i cot(kL ) = i cot(kL) − ikδ for small values of δ and the resonance frequencies are then obtained from cot(kL ) = 0. As shown in Section 4.5, the normalized input resistance of the tube resonator at √ resonance is θi = (kL )dvh /2a, where dvh ≈ 0.31/ f is the visco-thermal boundary layer thickness in cm, with the frequency f expressed in Hz. The radiation resistance is θr ≈ (ka)2 /2, where k = 2π/λ and a is the radius of the tube. As already indicated in the introductory section, the acoustic power absorbed by the resonator is conveniently expressed in terms of its absorption area (absorption cross section) Aa so that if I is the incident intensity (power per unit area) the absorbed power is I Aa . Similarly, the power scattered by the resonator back into free field is I As , where As is the scattering cross section.
108
NOISE REDUCTION ANALYSIS Absorption and scattering cross sections 4θi 4θi Aa = A |ζ +ζ |2 at resonance = A (θ +θ 2 r r) i i 4θi r As = A |ζ 4θ at resonance = A 2 2 (θ +θ ) +ζ | r
i
i
(4.2)
r
(Aa )max = A/θr = 2A/(ka)2 = λ2 /2π
for θi = θr )
where ζi , ζr : Eq. 4.1, A = πa 2 , θr = (ka)2 /2, k = 2π/λ. (See also Eqs. 4.29, 4.30.) Normally, the absorption by the resonator is limited to a rather narrow frequency band at the resonance frequency, and the resonance absorption depends on the input resistance of the resonator. If it is made equal to the radiation resistance, the resonance absorption cross section will be a maximum λ2 /2π . For a resonator in free field without a wall, the maximum value is λ2 /4π , where λ is the wavelength of the incident sound. Under these optimum conditions, the scattering cross section equals the absorption cross section at resonance. For the tube resonator, the input resistance at a resonance frequency is (π/2)(dvh /2a), and the normalized radiation resistance is θr = (ka)2 /2, where k = 2π/λ and a is the radius. The maximum resonance absorption cross section is obtained when these resistances are equal, and this occurs for the first resonance when a ≈ 5.2/(f1 /100)5/6 ) cm, with f1 (in Hz) being the first quarter wavelength resonance frequency. As already explained, the resonance occurs when cot(kL ) = 0, i.e., when kL = π/2, i.e., L = λ/4. As an example, consider a tube resonator with a length L = 85 cm. It will have its first resonance close to 100 Hz with a wavelength of λ ≈ 3.4 m. The maximum resonance absorption cross section is then λ2 /2π ≈ 1.84 m2 , obtained when the radius is 5.2 cm with a corresponding tube area of ≈ 0.0085 m2 . In other words, the resonator then ‘soaks’ up an amount of acoustic power from the incident wave, which corresponds to an area which is 216 times the area A of the resonator opening. The frequency dependence of the absorption and scattering cross sections Aa and As of a tube resonator of this length are shown in Figure 4.2 for two different tube radii, 1.5 and 6 cm, i.e., the first is smaller than and the second larger than the optimum value of 5.2 cm for the maximum absorption cross section. In the first
Figure 4.2: Absorption and scattering cross sections Aa /A and As /A of a tube resonator in a wall (see Figure 4.1, where A is the area of the tube and tube length: L = 85 cm).
RESONATORS
109
case, a = 1.5 cm, the maximum absorption cross section is approximately 200 times the tube area, which corresponds to an actual absorption cross section of 0.1 m2 . It should be noted that the scattering cross section is smaller than the absorption cross section at all frequencies. However, with a = 6 cm, which is somewhat larger than the optimum radius of 5.2 cm, the scattering cross section at resonance is larger than the absorption cross section. The latter is approximately 150 times the tube area, which means an actual absorption cross section of about 1.7 m2 , a little smaller, as expected, than the maximum value of λ2 /2π , which is 1.84 m2 (with a = 5.2 cm) at the resonance frequency of 100 Hz.
4.2.1 Q-Value The sharpness of the resonance, as measured by the Q-value of the resonator, discussed in Chapter 3, also has a maximum value, which is obtained when the total resistance, i.e., the sum of the radiation resistance and the viscous resistance, is a minimum. This occurs at a radius, which is somewhat smaller (by a factor 0.51/3 ≈ 0.79) than the optimum radius for maximum absorption cross section, given above. The corresponding minimum normalized resistance is θmin = (3/2)(ka)2 and the maximum Q-value is π π = . (4.3) Qmax = 4θmin 6(ka)2 As an example, consider a tube with the first resonance at 100 Hz, so that dvh = 0.031 cm and the equivalent tube length L = 85 cm. It follows that the optimum radius for maximum Q, as given above, is a ≈ 4.1 cm. With λ ≈ 340 cm (at 100 Hz), the θmin is ≈ 0.0086 and the maximum Q-value ≈ 91. It is important to realize that the Q-value will be reduced when the nonlinear resistance is significant, as discussed later in this chapter. In the low frequency region, ka << 1, the expressions for the absorption and scattering cross sections are independent of the angle of incidence. In this region, the assumption that only the fundamental mode is propagating in the tube is fulfilled. In the presence of higher modes (the first cut-on frequency corresponds to a wavelength ≈ 3.4a), the absorption and scattering problem is beyond the scope of this text. For a single resonator in free field rather than in a wall, the low frequency approximation of the radiation resistance is only half of what it is for the wall mounted resonator, and the maximum resonance absorption cross section is found to be half of that for the wall mounted resonator, as already mentioned.
4.2.2 Helmholtz Resonator In the uniform tube resonator, with its first resonance occurring when its acoustic length is a quarter wavelength, the kinetic and potential energies in the sound field in the tube are distributed throughout the tube. At resonance, their average values are equal. If a constriction is introduced in the opening of the tube, as shown in Figure 4.3, the resonance frequency will be reduced because the kinetic energy, and hence the equivalent inertial mass in the system will be enhanced. (It is a related effect that is
110
NOISE REDUCTION ANALYSIS
Figure 4.3: Examples of resonators. A tube resonator with constriction and a Helmholtz resonator.
responsible for the induced mass and the structure factor in a porous material to be discussed in Chapter 5.) An extreme version of this kind of resonator is the bottle or Helmholtz resonator, also shown in the figure. If, at the lowest resonance frequency of this resonator, the wavelength is much larger than the dimensions of the cavity, it is a good approximation to claim that the kinetic and potential energies are localized or lumped, the kinetic energy residing in and close to the orifice and the potential energy in the cavity. This acoustically compact resonator is analogous to an ordinary mass-spring oscillator with the mass being the mass of the air in and close to the orifice and the spring being represented by the stiffness of the air in the cavity. The mass and stiffness are now ‘lumped’ rather than distributed quantities. As shown in Section 4.5, the resonance frequency of the Helmholtz resonator can be expressed in a simple manner. Resonance frequency, Helmholtz resonator f0 = (c/2π) A0 /V
(4.4)
where V : Volume, A0 = πa02 : Orifice area, ≈ + 2a0 : Acoustic orifice length (see Eq. 4.44). As long as the cavity dimensions are much smaller than a wavelength (acoustical ‘compactness’), this expression is valid for any cavity shape. This should always be kept in mind when the frequency is calculated from this formula; the wavelength should be checked to make sure that it is consistent with the assumption of acoustical compactness.
4.2.3 Resonator Absorber in a Diffuse Sound Field We have mentioned briefly the properties of a diffuse sound field in a room and earlier in this chapter the absorption and scattering of a single incident wave by a resonator, both in free field and when mounted in a rigid wall. We now supplement these studies by considering the sound absorption by a resonator in the wall of a room which contains a diffuse sound field. In a rigorous study, this problem should be formulated in terms of the coupling of two cavities, the room and the resonator. In the present approximation, the resonator is assumed to be much smaller than the room and is treated as a perturbation.
111
RESONATORS
Characteristic of a diffuse sound field is that the sound intensity is the same in all directions and that the elementary wave contributions coming from different directions are uncorrelated so that their mean square pressure contributions add. The aim of the study is to determine the decay of the mean energy density in the room caused by the resonator. We denote the total mean square sound pressure at a point in the room by p 2 . It is the sum of the contributions from the elementary waves in the room. We surround the point by a spherical surface corresponding to a solid angle of 4π and denote the elementary pressure squared contribution per solid angle by p02 , so that 4π p02 = p2 .
(4.5)
Starting with p 2 as the measured quantity, this relation, in essence, defines the amplitude p0 of an elementary plane wave contribution per solid angle and the corresponding intensity I0 = p02 /ρc. In terms of this quantity, the power incident per unit area of a wall in the diffuse field is π/2
I0 2π
0
sin φ cos φ dφ = π I0
(4.6)
in which the elementary solid angle in the integration is 2π sin φ dφ and the projection of unit wall area normal to the incident wave is cos φ. Because of the pressure doubling at the hard wall (see Section 4.5.2), the pressure amplitude at the wall will be 2p0 and with a resonator of area A, the power absorbed by it will be A [2p0 /|ζ |ρc]2 θi ρc = 4A I0 θi /|ζ |2 .
(4.7)
Division by the incident power πI0 (Eq. 4.6) defines a resonator absorption cross section in a diffuse field A(4/π)θi /|ζ |2 . (4.8) (Recall that for normal incidence and a single plane wave, the cross section is A4θi /|ζ |2 .) The total energy density in the sound field is E = p 2 /ρc2 and the contribution to it per solid angle from the diffuse field is I0 /c, which means that 4πI0 /c = E or
I0 = cE/4π.
(4.9)
Decay Rate With the room volume being V , the total acoustic energy in the room is V E. With reference to the discussion above, the rate of energy absorption by the resonator is
Therefore,
4AI0 θi /|ζ |2 = Aθi /|ζ |2 Ec/π.
(4.10)
V dE/dt = −A(θi /|ζ |2 )(c/π )E.
(4.11)
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NOISE REDUCTION ANALYSIS
In other words, dE/dt = −βE, where β = (Ac/π V )(θi /|ζ |2 ). For further details about the decay, we refer to discussions of sound waves in ducts. It should be emphasized that a diffuse sound field implies a large number of modes, which are contributing to the measured field in a selected frequency range. The number of modes depends on the bandwidth, which should be kept in mind particularly in experimental studies of resonators.
4.2.4 Two-Dimensional Arrays of Resonators In a Wall Next, let us consider a uniform distribution of (tube) resonators mounted in a wall as a two-dimensional square lattice with the area of a unit cell being A. As before, the radius of each tube is a and the internal normalized input resistance θi . At normal incidence, we can account for the resonators in terms of an average wall input resistance at resonance equal to (A/A0 )θi , where θi is the input resistance of one resonator. If this is made equal to unity, i.e., if the cell of a A is chosen to equal A0 /θi , there will be no reflection from the wall and all the incident sound will be absorbed, as discussed in Chapter 2. Under these conditions, the absorption area of one resonator in the array will be A, independent of the wavelength, whereas for the single resonator in the wall, the maximum absorption cross section is λ2 /2π . How can this apparent difference be reconciled? Actually, there is really nothing to be reconciled since the optimum internal resistances are different in the two cases. In both cases, maximum absorption is obtained when the internal resistance equals the radiation resistance. For the single resonator, the radiation resistance is wavelength dependent, being (ka)2 /2, where k = 2π/λ and a is the resonator radius, as we have already mentioned. For the array, however, the radiated (reflected) sound travels out in a direction normal to the boundary, and due to symmetry (interaction between the array of resonators) the reflected sound field from each resonator is equivalent to that in a square tube with hard walls. Then, if the wavelength is larger than twice the resonator separation, only a plane wave will be reflected (apart from a local inertial field around each orifice, which decays exponentially with distance). The normalized radiation resistance of a single orifice is then (A0 /A) times the wave impedance of the plane wave, which is 1, and if the internal resistance is chosen to equal the radiation resistance, the average input resistance over the wall will be 1, and 100 percent absorption results. In Free Field With reference to Section 4.2.4 dealing with an array of resonators in a wall, carry out an analogous analysis of the absorption by a two-dimensional array of resonators in free field for a sound wave at normal incidence to the plane of the array. (One might be tempted to suggest (erroneously) that if the area of one cell in the lattice is λ2 /(4π ), i.e., the maximum absorption resonance absorption cross section of a resonator in free field, all the incident sound will be absorbed at resonance.)
113
RESONATORS
Due to symmetry, the problem is equivalent to that of the absorption by a single resonator in a square duct with hard walls for an incident plane wave. We proceed in the same way as for the single resonator in free field. The only difference is that the radiation impedance of the ‘piston’ in the resonator opening is now different from that in free field. As before, the wavelength is assumed to be much larger than the radius of the resonator opening and the scattered or re-radiated sound from the resonator is equally divided between the upstream and downstream directions in the duct. Thus, if the velocity amplitude of the piston is u0 , the velocity amplitude of the plane wave in each of these directions will be u = (A0 /A)u0 /2, where A is the duct area and A0 the orifice area. The wave impedance is ρc and corresponding power in the duct is A(ρc)(A0 /A)2 |u0 |2 /4.1 There is an equal amount in the other direction of the duct. The sum of these must equal the power radiated by the orifice, (ρc)A0 θr |u0 |2 , which leads to θr = (1/2)A0 /A. There is also a mass reactive part of the radiation impedance ζr . Then, if the internal impedance is ζi , the total impedance of the orifice piston is ζ = ζr + ζi . In the long wavelength approximation, the driving pressure on the piston is the incident sound pressure (Borne approximation) so that the velocity amplitude becomes u0 = pi /ζ . The power absorbed by the resonator is then W = A0 θr ρc|u0 |2 = A0 |pi |2 /ρc
θr . |ζr + ζi |2
(4.12)
The value W0 at resonance is obtained by replacing ζr and ζi by θr and θi . The amplitudes in this expression are rms values. The maximum absorption at resonance is obtained with θi = θr = 1/2(A0 /A) and is then (|pi |2 /ρc)A/2, i.e., half of the incident power. The other half is divided equally between the sound reflected upstream and (transmitted) downstream.
4.2.5 Three-Dimensional Lattice of Resonators Consider a rectangular uniform lattice of acoustic resonators in which the unit cell is small compared to the wavelength. The volume of a resonator occupies a fraction β (fill factor) of the unit cell. In the analysis of sound propagation through the lattice the resonators are accounted for in terms of an average compressibility in the lattice. At frequencies much below the resonance frequency ω0 of the resonator, the air in the resonator responds to a compression in the same way as in the rest of the gas, and the resulting average compressibility is the same as in free field. At frequencies above the resonance, the compression of the gas in the resonator is out of phase with that in the rest of the gas, and the average compressibility will be reduced and the wave speed increased (the real part of the propagation constant is decreased). The effect of the losses in the resonator can be accounted for by a complex compressibility. The wave speed and attenuation, of course, are not determined solely by the compressibility. They depend also on the inertia and friction of the air in the lattice. 1 We have then assumed that only the plane wave in the duct propagates, which means that the wave-
length should exceed twice the separation of the resonators in the array.
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NOISE REDUCTION ANALYSIS
The volume occupied by the resonators forces the air to be accelerated as it is squeezed between adjacent resonators. As a result, there will be an induced mass, and the equivalent or average inertial mass density in the lattice is increased. Furthermore, the friction along the surface of a resonator also will affect the motion of the air. As discussed in Chapter 5, in connection with sound propagation in a porous material, these effects can be accounted for in terms of a complex density ρ, ˜ and the same holds true for our present special kind of porous material, the resonator lattice. In terms of the complex compressibility andthe complex density, the propagation constant for a wave can be expressed as q = κ˜ ρ˜ (see Chapter 3) from which the normalized value Q = q/k readily follows. Average compressibility and propagation constant, resonator lattice κ/κ ˜ =1−β +
β 1−2 −iD
Q ≡ Qr + iQi = q/k = (κ/κ)( ˜ ρ/ρ) ˜
(4.13)
where k = ω/c, κ = 1/ρc2 , β: Fill factor, = ω/ω0 , D: Damping factor. (See also Section 4.5, Eq. 4.51.) Figure 4.4 shows an example of the calculated real and imaginary parts of the normalized propagation constant in which the induced mass in the lattice has been ignored, i.e., ρ/ρ ˜ = 1. The real part is the index of refraction, i.e., the ratio of the sound speed in free field and the wave speed in the lattice; in other words, it is the index of refraction of the lattice. The maximum and minimum values of Qr are ≈ 1.42 and ≈ 0.53. The imaginary part yields the attenuation in the lattice expressed by the factor exp(−kQi x) for the x-dependence of the amplitude. The maximum of Qi occurs slightly above the resonance frequency and is ≈ 0.82. As explained above, the wave speed below and above the resonance frequency of the resonator is less than and greater than the free field velocity, respectively, i.e., the index of refraction is larger than and less than 1. The attenuation is confined to a narrow range in the vicinity of the resonance.
Figure 4.4: The real and imaginary parts of the complex normalized propagation constant Q for a lattice of resonators. The fractional volume occupied by the resonators is β = 0.1 and the damping factor of a resonator (inverse of the ‘quality’) is D = 0.05.
115
RESONATORS
4.2.6 Transient Response and Reverberation When a resonator is excited by an incident transient sound, it will reverberate and emit sound even after the incident wave train has passed. The reverberation is generally dominated by the decay of the fundamental acoustic mode of the resonator and the frequency equals the resonance frequency. In the example illustrated schematically in Figure 4.5, the resonator is a Helmholtz type (a flask obtained from chemistry supplies) with a resonance frequency of 88 Hz. The incident sound is a pulse modulated harmonic wave, i.e., a set of finite harmonic wave trains (‘chirps’). During the time of interaction of a chirp with the resonator, the total sound in and around the resonator will be a superposition of the incident sound and a transient response, with a frequency equal to the resonance frequency, which decays exponentially with time. If the frequency of the incident sound differs from the resonance frequency by a small amount, the total field will exhibit beats with a beat frequency equal to the difference between the incident and the resonance frequency. The response of the resonator is measured with a microphone within the resonator, as shown. In Figure 4.5 (a), the frequency of the incident pulse is the same as the
Square wave modulated tone
Microphone
Gen- Louderator speaker
Microphone signal
Acoustic resonator (a) v = v = 88 Hz 0
(b) v = 82 Hz
Figure 4.5: Demonstration of the response of a resonator to an incident pulse modulated harmonic wave train. (a): Frequency of the incident sound equals the resonance frequency of the resonator (88 Hz). (b): Frequency is 82 Hz, in which case beats occur with resonance frequency.
116
NOISE REDUCTION ANALYSIS
resonance frequency of the resonator and the sound pressure in the cavity increases during the time of a pulse (the pulse is not quite long enough for the sound pressure within the resonator to reach steady state) and then decays exponentially after the termination of a pulse. During this decay the resonator re-radiates sound, and this can be heard outside the resonator as a reverberation. Since the frequencies of the incident field and the response are the same, no beats are observed. In Figure 4.5 (b), the frequency of the incident pulse is 82 Hz, i.e., slightly lower than the resonance frequency. The beats are clearly observed. The sound pressure in the cavity is considerably lower than at resonance, which illustrates the sharpness of the resonance. Actually, this sharpness, expressed by the Q-value, can readily be determined from the decay curve. We recall from the simple harmonic oscillator that the Q-value is π times the number of complete cycles required to reduce the amplitude by a factor of 1/e, and the decay curve in the figure indicates that this number is 15 so that the Q-value is ≈ 47. This is not the highest value that can be reached by a resonator, as shown in our discussion above. Since the reverberation or ‘ringing’ of the resonator can readily be heard, it is an important attribute of a resonator. This may explain why resonators were built into the walls under the seats in some early Greek open air amphitheaters.
4.3 ACOUSTIC NONLINEARITY Nonlinear acoustic effects normally do not play a significant role in noise control problems except where cavity resonators are involved. If the velocity amplitude of the oscillatory flow in an orifice is sufficiently large, typically greater than about 1 m/sec, the flow is known to separate to form ring vortices (compare ‘smoke rings’), two per cycle, one on each side of the orifice. During each half-cycle a ring occurs on the side, which corresponds to the instantaneous downstream side of the oscillatory flow; there will be a change in direction every half cycle so that vortices will emerge from both sides of the orifice. When made visible (with smoke, for example), and viewed under steady illumination, the succession of vortices, at ordinary audio frequencies, appear as jets emerging from the two sides of the orifice. Under stroboscopic illumination, the individual vortices can be seen, as illustrated in Figure 4.6. The energy required for the generation of the vortices is drawn from the sound, and there will be a corresponding nonlinear acoustic resistance of the orifice. Due to this nonlinearity, a harmonic driving pressure will produce a velocity which, in addition to the driving frequency, also contains overtones. However, experiments have shown that as far as the fundamental frequency is concerned, the nonlinear orifice resistance is found to be ≈ρ|uo |, where |uo | is the magnitude of the complex velocity amplitude of the fundamental frequency component. The corresponding normalized value of the orifice resistance is then ≈|u0 |/c, i.e., the magnitude of the oscillatory Mach number amplitude. This nonlinear resistance should be added to the linear orifice impedance discussed above and in Chapter 3, Eq. 2.11. The flow separation affects not only the resistance but also the mass end correction of an orifice. This end correction is related to the kinetic energy in the nonseparated part of the flow and at very high amplitudes it will be reduced approximately by a factor of 2, since only the (nonturbulent) inflow portion of the flow contributes and
RESONATORS
117
Figure 4.6: Separation of the acoustically driven oscillatory flow (frequency 234 Hz) in an orifice leads to the generation of vortex rings, as seen in (a), under steady, and in (b), under stroboscopic light. not the jet flow on the discharge side. In a resonator, this effect means an increase in the resonance frequency with increasing amplitude, and it can be important for large orifice diameters when the mass end correction might be larger than the length of the orifice. However, the most significant effect is the nonlinear resistance. For a perforated plate with a regular array of circular holes, it is necessary to make a distinction between the impedance of an orifice in the plate and the impedance averaged over the area of the plate. The average velocity over the plate is su0 , where s is the open area fraction of the plate, and u0 the velocity in an orifice. The impedance based on the average velocity will be p/(su0 ) = z0 /s, i.e., larger than the orifice impedance by a factor 1/s. As indicated in Chapter 3, the end correction as well as the nonlinear resistance needs to be modified when we deal with a perforated plate. Both of these quantities vanish for a plate with 100 percent open area, i.e., s = 1. To account for this, a semi-empirical correction factor 1 − s is used as was done already in Eq. 2.11.
4.3.1 Perforated Plate With (Porous) Cavity Backing Still another comment is in order. The nonlinear resistance depends on the velocity amplitude u0 in an orifice. For a given incident sound pressure, the velocity will depend on what is behind the orifice plate, an air layer, porous material, or screen, for example. In any event, let the input impedance of this layer be ζb , i.e., the ratio of the pressure amplitude at the surface of the layer and the average velocity amplitude over the surface. This average is su0 , where s is the open area fraction of the perforated plate. At the exit plane of the orifice plate, the velocity is nonuniform, being u0 behind an orifice and zero elsewhere. However, a short distance away, the flow has become almost uniform, so that the velocity is the same as the average velocity through the plate and over the backing layer, i.e., su0 , and it follows that ζb = pb /(su0 ), where pb is the pressure amplitude at the backing surface. Then, if the sound pressure amplitude
118
NOISE REDUCTION ANALYSIS
at the beginning of an orifice is p, so that p − pb = z0 u0 , the input impedance of the orifice will be zi = p/u0 = z0 + szb , where z0 is the orifice impedance as given above. The same holds true if the layer is a screen. If the backing layer is a thin screen in contact with the orifice plate rather than a short distance away in the uniform region of the flow, the flow velocity through the screen will be localized and equal to the velocity u0 in the orifice. Then, if the screen impedance is zs , the input impedance of the orifice will be zi = z0 + zs rather than zi + szs . This makes the difference between what we have referred to as ‘loose’ and ‘hard’ contact. For a uniform porous layer in hard contact with the plate there will be a transition region in the layer where the velocity goes from u0 to the uniform value su0 in the layer. The thickness of this transition region is approximately the end correction on one side of the orifice, ≈ (0.85/2)(1 − s)d. The additional impedance contribution is then approximately half the flow impedance of a layer of this thickness. To estimate the significance of the transition region we recall that for a uniform porous layer at wavelengths much larger than the layer thickness, the input resistance is ≈ Lr/3, where L is the layer thickness and r the flow resistance per unit length of the material. The ratio of the end correction resistance and the resistive part of sζb is then ≈ 1.3[(1− s)/s]d/L. The impedance contribution to the orifice impedance from the backing can then be expressed as ≈ sζb [1+1.3(1−s)d/L] ≡ βzb , where β = s[1+1.3(1−s)d]/L. Although the effect of nonlinearity on the orifice resistance can be expressed simply in terms of the velocity amplitude in the orifice, as indicated above, this description is not particularly useful since this velocity is generally not known a priori. It is better to express the velocity in terms of the amplitude of the sound pressure that drives the oscillatory flow or better still, the sound pressure of the incident wave. The relation between these variables depends on the particular configuration in which the perforated plate is used. The first task in a study of nonlinear resonator response is to determine the velocity amplitude in the orifice. Once that is done, the nonlinear resistance is known and the absorption coefficients and absorption cross sections can be computed. We refer to Section 4.5 for details and report here only the special case of a ‘resonance’ at which the reactance of the backing layer is canceled by the orifice reactance.
|u0 | c
Equation for velocity amplitude in an orifice at resonance ≈ δ/(1 − s)A for small δ << 1 2δ =A 1 + (1−s)A √ 2 −1 ≈ 2δ/(1 − s for δ >> 1
(4.14)
where A = [1/2(1 − s)](θ0 + sθb + s/cos φ), δ = pi /(γ P ), pi : Incident sound pressure, P : Static pressure. (See also Eq. 4.60.) The approximation for large values of δ implies that the nonlinear resistance is considerably larger than the sum of the linear orifice resistance θ0 and the resistance θb of the backing. This puts a lower bound on the factor (1 − s) in the equation since the nonlinear resistance is proportional to 1 − s. An example of the sound pressure dependence of the acoustic orifice resistance including the nonlinear part, as obtained from Eq. 4.14, is shown in Figure 4.7 at two resonance frequencies, 30 and 300 Hz. These correspond to different depths of the
119
RESONATORS
Figure 4.7: The normalized acoustic resistance of a perforated plate-cavity resonator at resonance vs the incident sound pressure level. Orifice diameter: 0.1 inch. Equivalent neck length: 0.1 inch. Frequencies: 300 and 30 Hz, upper and lower curve, respectively. Left: Open area, 10 percent. Right: Open area, 1 percent.
air cavity, of course. The plate parameters are indicated in the figure and the range of sound pressure levels is 30 to 160 dB re 0.0002 dyne/cm2 (rms). In the linear √ regime, the resistance at 300 Hz is larger √ than at 30 Hz approximately by a factor of 10, as expected on the basis of the f -dependence of the viscous surface resistance. For the plate with a 1 percent open area, the nonlinearity at 30 Hz begins to be noticeable already at the relatively low level of 70 dB; at 300 Hz, the corresponding level is about 80 dB. With 10 percent open area, the corresponding values are about 20 dB higher. The average normalized specific resistance of a 10 percent open perforated plate will be 10 times larger than for an individual perforation (orifice). The figure indicates that at 155 dB the orifice resistance is ≈ 0.1 so that the average plate resistance at both 30 and 300 Hz will be approximately 1. Thus, if the plate is backed by an air cavity such that resonance occurs at the frequency considered, the normal incidence absorption coefficient will be unity at that level. The absorption in the linear regime, on the other hand, will be negligible (see Figure 4.8). With an open area of 1 percent, the corresponding level will be ≈ 15 dB, at which the orifice resistance is ≈ 0.01. For sufficiently large values of the sound pressure, the nonlinear (resonance) resistance dominates and the 30 and 300 Hz curves merge. In this regime it follows from Eq. 4.14 that the corresponding nonlinear orifice resistance becomes 2p |u0 | ≈ , (4.15) c (1 − s)γ P where, as before, γ is the specific heat ratio (≈ 1.4 for air) and P , the static pressure. For a perforated plate/cavity resonator with an open area fraction of s, the normalized input resistance at resonance at such high amplitudes is ≈ (1 − s)/s)|u0 |/c. In order to obtain 100 percent absorption at resonance at a given incident sound pressure, the open area then should be chosen such that this resistance is unity, which means 2p s2 = . 1−s γP
(4.16)
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NOISE REDUCTION ANALYSIS
Figure 4.8: The absorption spectra of a perforated plate-cavity resonator for different incident sound pressure levels. Cavity depth: 8 inches. Plate thickness: 0.1 inch. Hole diameter: 0.1 inch. Open area: 10 percent. Weight: 4 lb/ft2 . Left: 80 dB. Right: 155 dB. Curve identification, from the top: Normal incidence (N); Diffuse field, local (DL); and Diffuse field, nonlocal (DNL). For example, at an incident pressure level of 174 dB, we have p/P ≈ 0.1 and s ≈ 0.24 to yield 100 percent absorption at resonance; at 154 dB, it is down to s ≈ 0.09 and at 134 dB, to ≈ 0.03. Frequently, the perforated plate is used in conjunction with a resistive screen or a porous layer backing. Then, the role of nonlinearity depends on whether or not the screen or the porous layer is in ‘hard’ or ‘loose’ contact with the perforated plate, as discussed in more detail in Chapter 5; normally it is not very important where a hard contact is involved. Effect of Induced Motion of the Plate Acoustically induced motion of the perforated plate will affect the performance at small open areas and high sound pressure levels, and it can be included in the same manner as for the flexible screen in Chapter 2. It should be realized though, that the induced motion will affect the velocity amplitude relative to the plate and hence the nonlinear resistance, as shown in Section 4.5.
4.3.2 Nonlinear Absorption Characteristics A perforated plate, which is ordinarily used as a facing for a porous layer, generally has an open area of 20 to 30 percent. For a resonator absorber with such a facing, the linear absorption is too small to be of practical interest. However, if the open area is reduced to a value less than about 10 percent, some useful absorption can be obtained. This is particularly true at high sound pressure levels. This is illustrated in Figure 4.8, where the narrow band absorption spectrum is shown for incident sound pressure levels of 80 and 120 dB for the resonator discussed in reference to Figure 4.7 where the nonlinear resistance was shown. On the basis of these data it was noted that a facing with a 10 percent open area would yield approximately 100 percent absorption at resonance at a sound pressure level of 155 dB but negligible absorption
RESONATORS
121
in the linear regime. We now show the complete absorption curves for both 155 and 80 dB incident sound pressure levels and find the results consistent with the previous observations. A comment was also made concerning the role of the induced motion of the plate. Figure 4.9 illustrates this effect for a perforated plate with only 0.1 percent open area. With such a small open area, the linear absorption, at 80 dB is almost 100 percent at resonance and an increase in level will correspondingly reduce this peak value as is the case at 120 dB. However, at this level, there are two peaks, one below and one above the linear resonance frequency, the former at a larger value of the absorption coefficient. The explanation for this peculiar behavior involves the effects of both nonlinearity and induced motion and goes as follows. For a very heavy plate, with the induced motion being negligible, the velocity amplitude in the orifice at the linear resonance frequency is quite large so that the total orifice resistance will be (considerably) greater than 1 and the absorption will be correspondingly smaller than 100 percent. However, at the two peaks, being somewhat off resonance, the velocity amplitude in the orifice is reduced so that the nonlinear contribution to the resistance is decreased, making the total resistance smaller, thus leading to a higher absorption. Induced motion reduces the equivalent resistance of the orifice so that an even better impedance match is obtained and the absorption increases. The reason for the asymmetry of the curve is that the induced motion is greater at low than at high frequencies due to the inertia of the plate so that the equivalent resistance will be different at the two peaks. The examples in Figures 4.8 and 4.9 indicate that an increase of the sound pressure level, and hence the acoustic resistance of the perforated plate, can lead either to an increase or a decrease of the normal incidence absorption coefficient at resonance depending on whether the linear resistance is smaller or greater than 1 ρc. The effect of the induced motion on the absorption spectrum is more subtle.
Figure 4.9: Absorption spectra of a perforated plate-cavity absorber showing the effect of both nonlinearity and induced motion of the perforated plate. Cavity depth: 8 inches. Plate thickness: 0.1 inch. Orifice diameter: 0.1 inch. Open area: 0.1 percent. Weight: 4 lb/ft2 . Left: 80 dB at the surface. Right: 120 dB. Curve identification. From the top: Diffuse field, local (DL); Normal incidence (N); and diffuse field, nonlocal (DNL).
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NOISE REDUCTION ANALYSIS
4.4 EFFECTS OF FLOW Another form of nonlinearity involves the effect of flow on the acoustical characteristics of resonators. Flow through or grazing the orifice of a resonator can lead to an increase in the acoustic resistance of the orifice and a (small) decrease of the mass reactance; both will affect the absorption spectrum. Under certain conditions, flow can cause self-sustained oscillations in which case the resonator produces rather than absorbs sound. It is one form of flow induced instabilities, sometimes referred to as ‘flute’ instabilities, in which a vortex sheet interacts with the acoustic mode(s) of the resonator. Other flow induced instabilities, sometimes called ‘flutter’ and ‘valve’ instabilities, not considered here, involve the excitation of structural elements, which modulate the flow which in turn excite acoustic modes.
4.4.1 Flow Induced Acoustic Resistance A steady flow through an orifice produces a static pressure drop of the form P = CρU02 /2,
(4.17)
where U is the velocity in the orifice and C a constant (of the order of unity), which can be expressed in terms of the orifice geometry. It is assumed that the discharge is in the form of a jet. With a sound wave present, it will modulate the flow so that the velocity in the orifice becomes U + δu, where δu is the acoustic contribution. It is assumed that the process is quasistatic. The modulation gives rise to a change δ(P ) of the pressure drop, and it follows from Eq. 4.17 that δ(P ) = CρU0 . δu
(4.18)
Treating δP and δu as acoustic variables, their ratio is the resistance of the orifice and dividing it by ρc, the normalized value becomes θ0 = CM0 ,
(4.19)
where M0 is the Mach number of the steady flow through the orifice. For most purposes, the approximation C ≈ 1 is adequate but for refinement, experimental data, found in standard texts on fluid flow, on steady flow through an orifice should be consulted. There is another approach to the derivation of a flow induced resistance of the orifice, which is based on energy considerations. The kinetic energy flux per unit orifice area of the flow emerging from the orifice is ρ(U0 + δu)3 /2. With harmonic time dependence so that δu = u0 cos(ωt), the time average over one cycle becomes T (U0 + u0 cos(ωt)3 )/2 dt = ρU03 /2 + 3ρU0 u20 /4. (4.20) W = (1/T )Cρ 0
The second part is due to the presence of the sound field. In terms of an acoustic resistance r, this part can be expressed also as (1/2)ru20 , the factor 1/2 arising from
RESONATORS
123
time averaging (u0 is the true amplitude and not an rms value). It follows then that r = (3/2)CρU0 , which we associate with a flow induced orifice resistance. However, this is larger than the result in Eq. 4.18 by a factor of 1.5. The resolution of this apparent paradox is the following. Due to the nonlinearity of the relation between static pressure drop and the velocity, the harmonic velocity perturbation gives rise to a term Cu20 cos2 (ωt) in the pressure with a time average (1/2)Cu20 . This means that in order to maintain the mean velocity at the value U0 in the presence of a sound wave, the driver of the mean flow, for example a fan, has to supply an extra power (1/2)CU0 u20 . This is not drawn from the sound wave but from the flow source and is included in the power W in Eq. 4.20. Therefore, to obtain the contribution from the sound wave, it has to be subtracted from W . This leaves a power CρU0 u20 /4 ≡ ru20 /4 and a corresponding acoustic resistance r = CρU0 , as before. Grazing Flow Grazing flow over the opening of a resonator also causes an increase of the acoustic resistance of the resonator. The mechanism is not as clear as for the flow through the opening and depends to a great extent on the degree of turbulence in the flow. For example, if the pressure spectrum of the turbulence has a strong low frequency portion, the resulting low frequency ‘pumping’ of flow through the opening will have an effect analogous to the flow induced resistance by steady flow through the orifice. Furthermore, as described below, the possibility of flow excitation of a (resonator) absorber adds another element of complexity. We shall not pursue this topic further here and merely point out that empirically, the flow induced resistance by grazing flow can be estimated from Eq. 4.19 by letting the Mach number in this equation be a fraction of the Mach number of the grazing flow.2
4.4.2 Flow Excitation of Pipes and Orifices One of the practical problems and risks associated with the use of resonators as absorbers in the presence of steady flow is the possibility that the resonators can be excited by the flow and act as (high intensity) whistles. This problem is complex, and the notes presented here contain only a few observations and examples and some means of eliminating the problem. In the typical resonator discussed above, an orifice is connected to a closed pipe section or cavity. The pipe can also be open in which case the resonance frequencies of the system will be altered. Figure 4.10 shows one example of an acoustic resonator in which an orifice is connected with two open pipe sections, one on each side. The flow through the orifice, which is assumed circular, separates at the entrance and forms a vortex sheet with circular symmetry. Like the two parallel shear layers behind a blunt body, this sheet also can form periodic vortices, which now takes the form of rings with a characteristic frequency proportional to the flow velocity, fU = SU0 /D , where D is some
2 For a resonator in the wall of a duct, a fraction of 0.5 has been reported.
124
NOISE REDUCTION ANALYSIS
PIPE MODE ORIFICE MODE
Figure 4.10: Flow excitation of orifice and pipe tones. characteristic length, a combination of the orifice diameter D and the length L0 of the orifice, and S is a constant (a Strouhal number). At least for sufficiently short lengths of the orifice, we assume here that D = D. The value of the constant S, according to our experiments, is approximately 0.5. If L0 is considerably larger than D, the influence of L0 cannot be ignored, however, since our experiments indicate that the orifice whistle does not seem to occur when L0 is greater than ≈ 4D. An explanation might be that the vena contracta of the flow then falls well inside the orifice, and as the flow expands it will strike the wall of the orifice and possibly ruin the coherence of the sheet oscillations. Even for small values of L0 , it has an effect on the instability, albeit indirectly, since with L0 less than ≈ D/4, flow induced instability does not seem to occur. A likely reason is that the acoustic frequency and the flow velocity are then so high that the acoustic radiation resistance and the flow induced resistance prevent the development of the instability. If the frequency of any of the modes of the pipe system in Figure 4.10 is sufficiently close to the vortex frequency, acoustically stimulated self-sustained oscillations can occur. Only axial modes will be considered here. The lowest frequency is approximately that of an open-ended pipe with a wavelength approximately twice the length of the pipe. The high frequency end of the spectrum starts with the first mode of the orifice itself with a wavelength approximately twice the acoustic thickness L ≈ L0 + δ of the orifice plate, where L0 is the physical orifice length, δ ≈ (1 − σ )0.85D, the two-sided end correction, and σ , the ratio of the orifice area and the pipe area. In this high frequency regime, the orifice modes are essentially decoupled from the pipe modes but at lower frequencies, the presence of the orifice will affect the frequencies of the pipe modes. Instead of a pipe, any other resonator will produce ‘pipe tones,’ such as a Helmholtz resonator with low frequency through it, as will be discussed in an example below. The simplest example, familiar to all, is the mouth whistle.3 If the characteristic vortex frequency is denoted by fU = SU0 /D, the flow velocity at which an acoustic mode of frequency fa can be stimulated by the flow into a selfsustained oscillation is given by fU ≈ fa = c/λa , i.e., M0 = U0 /c ≈ D/(Sλa ). The lowest pipe mode has a wavelength λa ≈ 2Lp , where Lp is the pipe length, and with D/Lp << 1, the flow velocity, frequency, and sound intensity, will be correspondingly 3 Speech production is different. Here, the time varying acoustic modes of the vocal tract are excited
by a periodic pulsation of air through the glottis.
RESONATORS
125
small. However, in some applications involving perforated plates containing a large number of orifices, the pipe tones can be quite intense. As the flow speed increases, higher order modes of the pipe system will be excited until the pure orifice mode is involved. With the acoustic length of the orifice being L 0 , as given above, the frequency of the lowest mode is fa = (c/2L )(1 − M02 ), and the overtone frequencies are fn = nf1 , where the factor 1 − M02 is due to the wave speeds in the upstream and downstream directions being c − U0 and c + U0 , respectively. The critical flow Mach number for the lowest orifice tone follows from f = fa , i.e., from the equation M0 ≈ (D/2L 0 S)(1 − M02 ). For example, with S ≈ 0.5 and D = L0 , we get M0 ≈ 0.44. From extensive measurements, we have found the Mach numbers thus obtained and the corresponding frequencies indeed cluster around the predicted values. As a rule of thumb, the excitation of intense orifice tones usually can be expected to occur in the Mach number range between 0.25 and 0.5. The conditions for the excitation of a higher orifice mode are obtained in an analogous manner. The vortex shedding, although periodic, is not harmonic, and overtones of the fU exist and can be involved in the stimulation of acoustic modes. Whistle Efficiency In experiments with an orifice having a diameter D = 2r0 = 0.5 inch and a thickness L0 = 0.5 inch, the sound pressure level of an orifice tone at a free field distance r = 100 cm from the orifice was found to have a maximum value of 115 dB, obtained when the pressure drop across the orifice was ≈ 0.13 atm, corresponding to a Mach number of ≈ 0.44 in the orifice. (An orifice with D ≈ L0 seems to give the highest intensity.) To determine the corresponding acoustic efficiency of the orifice, defined as the ratio of the radiated acoustic power and the flow losses, we express the latter as Wf ≈ AρU 3 /2 = A(γ P )2 M03 /2ρc, where we have treated the flow as incompressible and where P = ρc2 /γ is the static pressure, M0 = U0 /c and A = π r02 . On the assumption of an omni-directional source, the acoustic power radiated into free field half-space can be written as Wa = 2π r 2 p 2 /ρc, where p is the rms value of the sound pressure at the distance r from the source. The acoustic efficiency is then ηa = Wf /Wa = 4(r/r0 )2 (p/P )2 /(γ 2 M03 ). The observed sound pressure level of 115 dB corresponds to p/P ≈ 10−4 . Then, with r = 100 cm, r0 = 0.64 cm, γ = 1.4, and M0 = 0.44, we get ηa ≈ 6 · 10−3 . If the orifice is placed in a duct as in Figure 4.10, simulating a valve, for example, we can estimate the sound pressure level in the pipe using this efficiency. Elimination of Orifice/Pipe Tones The results given above express only necessary conditions for the occurrence of the orifice tones. Other factors, such as the uniformity of the shear layer at the entrance and flow induced sound absorption at the exit end are also important. At Mach numbers larger than 0.5, the latter becomes so large as to prevent resonances from occurring. This can readily be demonstrated by exciting an open ended pipe by random
126
NOISE REDUCTION ANALYSIS
U film
Figure 4.11: Example of flow generated tone in a film dryer facility.
noise from a source outside and measuring the response by a microphone placed at the center of the pipe. The flow through the pipe can be obtained by connecting the pipe to a plenum chamber, which is connected to a pump. With the microphone placed at the center of the duct and with no flow through the duct, the spectra obtained clearly show the odd number duct mode resonances as narrow spikes. As the flow speed is increased, the resonances are broadened and at a Mach number of 0.5, they are essentially gone. (No organ music with a Mach number above 0.5 in the pipes!) For a conical orifice, as obtained by countersinking a circular orifice, no whistling occurs if the apex angle of the countersink is larger than 60 degrees, regardless of the direction of the flow. If the vortex sheet in the separated flow at the inlet of the orifice is broken up by making the edge of the orifice irregular, the chance of whistling is markedly reduced,4 and a simple means of eliminating the whistle is to place a wire mesh screen across the entrance (the screen can be rather coarse; a few strands across the opening is usually sufficient). The effect is similar to that of a helical wire wound around a cylinder used to ruin the coherence of the periodic vortices behind a cylinder. Example: Industrial Dryer In many applications, such as in various forms of industrial dryers, flow through orifice plates is often used, and this can give rise to problems associated with whistling. One example is illustrated schematically in Figure 4.11. It involves a film dryer in which the film is transported below a set of plenum chambers which supply air at different temperatures to the film through perforated plates. This facility turned out to generate an intense tone at the resonance frequency of the plenum-orifice combination through excitation of the shear flow in the orifices of the perforated plate. Not only was the tone an environmental noise problem it also affected the film drying process. The acoustic oscillations turned out to modulate the drying rate so that the film came out with striations at a spacing that corresponded to the frequency of oscillation. Similar dryers are used in many other applications in processing facilities; for example, in the textile industry for drying fibers. One way to eliminate such a tone is to place a wire mesh screen on the upstream side of the orifice plate. 4 It is hard to whistle with chopped lips!
RESONATORS
127
4.4.3 Resonator in Free Field With Grazing Flow Like most problems in acoustics, the flow excitation of a resonator has a long history of attempts to understand the mechanism going back to Helmholtz (1868). Since then, numerous papers have been written on the subject, and there probably will be more to come. Our own studies of the problem go back to 1958 (Ingard and Dean) in which Schlieren photographs were taken with stroboscopic and high intensity flash illumination of the flow around the mouth of a tube resonator in free field, identifying periodic vortex generation. The experiments were limited to only one tube, however, 2 cm in diameter and 30 cm long. The air stream was uniform over an area of 3 cm by 3 cm with speeds up to 3500 cm/sec. The angle of attack of the flow could be varied over the range 0 to 50 degrees, and it was found that for angles less than 15 degrees, no oscillations occurred. A wire mesh screen was used in the resonator to provide damping, which could be changed by varying its location in the resonator. Actually, the screen was a package of three screens, each with an open area of 29 percent and a diameter of the strands of 0.1 mm. Only the fundamental mode was considered. With the screens placed at the rigid end wall of the resonator, essentially no damping was obtained, and with the screen at the open end, maximum damping. In this manner, the Q-value of the resonator could be varied over the range from 10 to 43. The frequency of the flow induced oscillations was close to the acoustic frequency of the fundamental mode of the resonator, fa ≈ c/4L , where L = L + 0.32d is the acoustic length of the pipe. With L = 30 and d = 2 cm, this frequency is ≈ 277 Hz. The flow velocity U at which the mode was excited extended over a range about a mean value of ≈ 1150 cm/sec. The simple kinematic model we use for the excitation mechanism is that a perturbation of the shear layer, which starts at the leading edge of the orifice, is convected on the vortex sheet with a speed U = βU ≈ 0.5U.5 As the disturbance reaches the downstream edge of the orifice, an acoustic signal is fed back to the upstream edge to stimulate the vortex sheet. Thus, the roundtrip time of this fluid oscillator will be ≈ d/U + d/c ≈ d/U . Assuming that the self-sustained oscillation occurs when this time equals the period of the acoustic mode, the corresponding flow velocity will be U ≈ cd/(4βL ) = fa d/β. With β ≈ 0.5, d = 2, L = 30.64 cm, and fa ≈ 277 Hz, as given above, this velocity becomes U ≈ 1109 cm/sec, in good agreement with our experiments. Although the highest emitted sound occurs close to this predicted flow velocity, oscillations are maintained over a range of velocities, the range depending on the damping of the resonator. Figure 4.12 shows a stability diagram of the oscillator, showing the critical flow velocity plotted as a function of the Q-value of the resonator. The region of instability is to the right of the curve and the region where no oscillations occur is to the left. It should be noted that the velocity scale is linear with the value 100 corresponding to a flow velocity of 1150 cm/sec.
5 An intuitive mechanical model of a shear layer is a board moving with the fluid velocity U on roller bearings on a plane boundary. The relative velocity of the contact point of a roller with the boundary is zero, and the center of the roller bearing, corresponding to the average speed of the vortex sheet, moves forward with a velocity U /2.
128
NOISE REDUCTION ANALYSIS 160
WIND SPEED (100 CORRESPONDS TO 1150 CM/SEC)
150
140
130 Stable 120 Unstable 110 screen 100
90
80
70
0
10
20
30
40
50
Q
Figure 4.12: Stability contour in a velocity Q-value space for flow excitation of a tube resonator in free field. The velocity scale is linear and the value 100 in this example corresponds to a flow velocity of 1150 cm/sec. The curve divides the space into a stable (left) and unstable region (right).
It is clear from the diagram, that below a Q-value of 10, no oscillations occur, and that the flow velocity range of instability increases with an increasing Q-value. The diagram refers to an angle of attack of the flow of 35 degrees. Starting at the lower bound of the critical velocity curve and moving toward the upper bound at a constant Q-value, the amplitude of oscillation starts from zero, reaches a maximum, and then goes back to zero. As indicated in the diagram, a sufficiently high damping and a corresponding low Q-value will prevent the resonator from being excited by flow. This is familiar from using a soft drink bottle as a whistle by blowing over its mouth. Assuming that the bottle is half full, say, it is fairly easy to make it whistle. However, after shaking the bottle so that a foam is formed, it is generally not possible to excite the bottle because of the sound absorption provided by the foam. In our experiments, the sound pressure was measured not only outside the resonator but also inside, at the end wall. Even in the stable region, weak sound pressures at the resonance frequency were detected inside the resonator, responding to the turbulence in the incident flow and signifying a linear response of the resonator to an oscillating driving force.
4.4.4 Flow Excitation of a Side-Branch Resonator in a Duct Instead of a cavity resonator in free field, we now consider a side-branch cavity resonator in a duct. The model of flow excitation used in the previous section will be
129
RESONATORS
FREQUENCY f (kHz)
7 6 5 4 3 2 1 0
0.1
0.2
0.3
0.4
0.5
0.6
MACH NUMBER M
Figure 4.13: Data points are the measured frequencies of flow excited acoustic modes in a side-branch cavity in a duct. The dashed lines are the acoustic resonance frequencies and the solid lines the Mach number dependence of the shear flow frequencies for m = 1 and m = 2. The predicted instability frequencies are represented by the intersection of the dashed and solid lines, but as for the flow excitation of the resonator in free field (see Figure 4.12), there is a range of flow velocities in which the tone production is obtained.
used also here. Thus, a shear layer is started at the upstream edge of the resonator opening. If U is the free stream velocity, a flow perturbation of the shear layer is carried by the shear layer at a speed U = βU across the opening and interacts with the downstream edge, which feeds back to the upstream edge (not unlike an edge tone oscillator). This defines a characteristic roundtrip time and frequency for the shear layer. If the feedback is assumed to be carried by the speed of sound, the roundtrip time will be D/U + D/c and the corresponding frequency fU = (βU/D)/(1+βM), where M = U/c. As in the previous section, the coefficient β is approximately 0.5. The frequency of the nth mode of the resonator is fn = (2n − 1)c/4L , where the acoustic length of the resonator is L = L + δ and δ ≈ 0.43D (one-sided end correction). The flow frequency fU has overtones and a condition for instability or coupling between the mth fluid mode and the nth acoustic is mfU = fn , where m and n are 1,2,3… The data points in Figure 4.13 show measured frequencies of flow excited tones of a side-branch resonator in a duct as a function of the Mach number in the duct. Both the duct and the resonator had square cross sections of width w = 0.75 inch and the length of the resonator tube was L0 = 3 inches. With an end correction δ = 0.3w, the corresponding acoustic length of the cavity was L 0 ≈ 3.23 inches. The resonance frequencies thus obtained are 1040, 3120, 5200, 7280 Hz, etc. These frequencies are shown as dashed lines in the figure. Duct length was 84 inches and the area, 3/4 inch by 3/4 inch. The resonator was placed 11 inches from the beginning (flow entrance) of the duct.6 6 Ingard and Singhal, unpublished. Experiments carried out in the M.I.T. Gas Turbine Laboratory in
the 1970s.
130
NOISE REDUCTION ANALYSIS
The Mach number dependence of the frequencies of the shear layer of the first two modes, corresponding to m = 1 and m = 2, as given in the discussion above, are drawn as solid lines. As for the resonator in free field, flow excitation occurs over a range of Mach numbers centered around the values predicted by equating the characteristic frequencies of the flow and of the sound, represented by the intersection of the dashed and solid lines. Although most of the data points are consistent with this view, there are others that fall outside. This deviation will be discussed shortly. Over most of the range of Mach numbers, more than one frequency is usually excited. For example, at a Mach number of M = 0.2, the measured acoustic spectrum contained two pronounced peaks close to the predicted first and second resonances. At this Mach number, the levels of the peaks were about the same. The relative strength of the tones depends on the Mach number, however, and for M = 0.22, the second peak was approximately 20 dB above the first, and a weak third peak at the third mode was present. This shift of amplitude toward higher modes with the Mach number was consistent at all locations of the resonator along the duct. Mode Coupling The situation is more complex than what has been implied above, however, since unexpected frequencies can occur in the spectrum, particularly at higher Mach numbers. Thus, with the resonator placed at the center of the duct, the spectrum shown in Figure 4.14 contains ‘satellite’ frequencies around the major peaks in the spectrum. The difference between these frequencies turned out to be the fundamental frequency of the duct. With a length of 84 inches, and a Mach number of 0.3, this frequency is ≈ 7 Hz. In other words, the frequencies of the satellites are fa ± s73, where s is an integer, showing that the resonator mode couples to the axial modes in the main duct. An even more spectacular example of mode coupling is shown in the graph on the right in Figure 4.14 at a Mach number of 0.5. In this case, interaction between w
50
d
dB
w
60
f1
M
40
L
L
d
f1+1
50 dB 40
30
f3
M
f3-1 f5-1
L1
f3+1 f5
30
20
L2
f3+3
f5+3
20
10 10
0
2.5 5.0 FREQUENCY f (kHz)
7.5
0
1
2
3 4 5 6 7 FREQUENCY f (KHz)
8
9
Figure 4.14: Flow excitation of a side-branch resonator in a duct showing coupling of acoustic modes. Duct length: 84 inches. Area: 3/4 inch by 3/4 inch. Resonator placed 11 inches from (flow) exit end of pipe. The reference level on the dB scale is not specified (0 typically corresponds to 70 dB re 0.0002 dyne/cm2 at 12 inches from duct opening). Left: Coupling between resonator mode and the axial mode of the duct (Mach number ≈ 0.3). Right: Coupling of resonator modes (Mach number ≈ 0.5).
131
RESONATORS
the modes in the resonator is involved. In addition to the first three resonances, corresponding to n = 1, 3, and 5, and denoted by f1 , f3 , and f5 , there are combination frequencies f3 − f1 , f1 + f1 , f5 − f1 , f3 + f1 , etc. This coupling effect is not the same if the resonator is placed at the flow entrance rather than at the flow exit of the duct, in this case 11 inches from the end. (The reference level on the dB scale in the figure is not specified. The value 0 dB typically corresponds to 70 dB re 0.0002 dyne/cm2 at 12 inches from the end of the duct in our experiments.) Sound Pressure of Cavity Tones Measurements of the sound pressure level outside the pipe in the experiments referred to above were found to reach a maximum value of 125 dB at 12 inches in front of the pipe opening at a Mach number of 0.5 in the duct. However, this refers merely to one resonator-duct configuration. As far as we know, no data are available from which the acoustic power generated by the cavity tone can be reliably predicted. In the case considered here, the empirical formula SP L = 169 − 10 log(Ad /A) + 30 log(M) (Ad , duct area, A, resonator area) for the prediction of the sound pressure level in the pipe has been used with a corresponding expression for the acoustic power level. Slanted Resonator in a Duct, Effect of Flow Direction Normally, there is no difference between the upstream and downstream edges of the orifice in a side-branch resonator in a duct, and the direction of the flow then does not influence the excitation of the resonator. However, if the resonator or any other side-branch, such as an air bleed vent, is slanted with respect to the duct axis as shown
SOUND PRESSURE LEVEL 30 cm FROM DUCT INLET dB re 0.0002 dynes/cm2
130
FLOW DIRECTION
100 FLOW DIRECTION DUCT ALONE
8.1 cm 70 12.6 0
M
3.5
12.6 cm
0.5
Figure 4.15: Flow excitation of a slanted side-branch resonator in a duct. Sound pressure level, at a distance of 30 cm in front of the duct opening, vs the Mach number in the duct. Upper curve: Flow from right to left. Lower curve (crosses): Flow from left to right. Slant angle: 45 degrees. Length of resonator: 8.1 cm. Length of duct: 28.7 cm. Cross section: 1.9 cm × 2.5 cm. Duct cross section: 1.9 cm × 1.9 cm.
132
NOISE REDUCTION ANALYSIS
in Figure 4.15, the direction of flow is very significant in regard to flow excitation of acoustic resonances. With the flow going from right to left, i.e., against the pointed downstream edge of the opening, the resonator is excited strongly creating a sound pressure level of about 125 dB at a distance of 30 cm from the duct entrance. In the other direction, i.e., with the flow striking the blunt downstream edge, no tone is produced. The excitation mechanism, as indicated earlier, is reminiscent of the jet edge oscillator.
4.5 MATHEMATICAL SUPPLEMENT 4.5.1 Impedance of a Tube Resonator If the thermo-viscous boundary layer thickness is much smaller than the cross sectional dimensions of a tube, the propagation constant in the tube, as discussed in Chapter 3, can be expressed as dvh q ≈ k(1 + i ), (4.21) D where k = ω/c and D = 4A/S is the ‘hydraulic’ diameter of the tube (A being the area and S the perimeter). For a channel of width d between two parallel walls the hydraulic diameter is d/2 and for a circular tube it is the diameter of the tube. We repeat here the expression for the visco-thermal boundary layer thickness dvh , given in Chapter 3, √ dvh = dv + (γ − 1)dh ≈ 0.31/ f √ √ dv = 2ν/ω ≈ 0.21/ f √ dh = 2K/ρCp ω ≈ 0.25/ f . (4.22) The numerical values refer to air with dv , dh expressed in cm and f in Hz. When applied to a circular tube, Eq. 4.21 is the well-known approximation originally due to Kirchhoff. If the tube is terminated by a rigid wall, the tangential velocity component along the wall will be neglected since we deal here only with the fundamental mode. The thermal effect at the wall will be considered, however. As is shown in an example in Appendix C, this can be accounted for in terms of an equivalent normalized boundary admittance ηt ≡ 1/ζt = (1 − i)(γ − 1)kdh /2,
(4.23)
where, as before, dh is the thermal boundary layer thickness and k = ω/c. Since we assume dvh /D << 1, the wave impedance in the channel can be put equal to ρc, corresponding to a normalized value of unity. Then, in terms of the variables ρcu and p, the transmission matrix elements of the pipe are T11 = T22 = cos(qL) and T12 = T21 = −i sin(qL), and the normalized input impedance of a tube of length L is then given by Normalized input impedance of tube ζi =
cos(qL)−iηt sin(qL)) −i sin(qL)+ηt cos(qL)
where q: Eq. 4.21, ηt : Eq. 4.23.
=
cot(qL)−iηt −i+ηt cot(qL)
(4.24)
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RESONATORS
At a frequency such that L is an odd number of quarter wavelengths, kn L = (2n − 1)π/2, where n is an integer, and with qi = kdvh /D being the imaginary part of q, we get cot(qL) ≈ −iqi L (qi L << 1), i.e., ζi ≈ qi L + ηt = kn L
dvh π dvh + ηt = (2n − 1) + ηt , D 2 D
(4.25)
where ηt is given in Eq. 4.23. Since the wavelength is large compared to the diameter, it is a good approximation to neglect ηt (which is proportional to dh /λ) in the expression for ζi .
4.5.2 Absorption and Scattering Cross Sections It is assumed that the air layer in the resonator opening in the plane of the wall moves with uniform velocity so that it can be regarded as a plane mass-less piston (see Figure 4.1). It will generate sound both inside and outside the resonator, and the motion of the piston will be impeded by the radiation reaction forces from the two sides. The total force is proportional to the sum of the input impedance ζi and the radiation impedance ζr , respectively. The incident wave is produced by a distant point source so that the wave incident on the resonator can be considered to be the plane. Then, if we use the Green’s function G for a half space bounded by an everywhere rigid plane, the sound pressure field can be expressed as p=
qGdv +
GudS,
(4.26)
where the first term is the volume integral over the source distribution and the second term is the surface integral over the plane, in this case limited to the surface integral over the piston since the velocity is zero everywhere else on the plane. The first term gives the pressure contribution 2pi at the surface of the plane, where pi is the incident pressure amplitude. The pressure doubling results from the fact that G is the Green’s function that satisfies the boundary condition of zero normal velocity on the rigid plane, and accounts for the reflected field (equivalent to the field from an image source), and the second contribution is the field scattered (radiated) from the piston. The total sound pressure amplitude on the outside of the piston is then 2pi + ps , where ps is the contribution from the scattered wave. For a resonator in free field (without the wall) there is no pressure doubling and the driving pressure is simply the pressure of the incident wave (Borne approximation), assuming the resonator is small compared to a wavelength. If the velocity amplitude of the piston is denoted by u, counted positive into the tube, the scattered pressure at the piston is ps = −u(ρcζr ), where ζr ≡ θr + iχr is the normalized radiation impedance of the piston given in Eq. 4.32. The pressure is continuous across the (mass-less) piston so that the total pressure on the outside, 2pi + ps with ps = −ζr ρcu, must equal the pressure on the inside, which is (ρcζi )u, where ζi is given in Eq. 4.24. From these relations it follows that u=
2pi /ρc . ζi + ζ r
(4.27)
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NOISE REDUCTION ANALYSIS
The power absorbed by the resonator tube is then Wa = A|u|2 ρcθi = A
|pi |2 4θi , ρc |ζr + ζi |2
(4.28)
where A = πa 2 is the area of the tube. The amplitudes of the acoustical variables p and u should be regarded as rms values to avoid recurring factors of 1/2. Quantity |pi |2 /ρc ≡ Ii is the intensity of the incident sound wave and the absorbed power can be expressed as Ii Aa , where Aa is the absorption cross section of the resonator Absorption cross section (4.29) i Aa = A |ζ 4θ +ζ |2 r
i
where ζi , ζr : Eqs. 4.24 and 4.32. Similarly, the re-radiated or scattered power can be written as Ii As , where As is the scattering cross section 4θr As = A . (4.30) |ζr + ζi |2 If the heat conduction losses at the rigid termination of the tube are equivalent to ηt = 0, the reactive part of ζi is χi ≈ cot(kL) and for ζr , it is χr ≈ −(8/3π )ka, as will be shown shortly (Eq. 4.32). With a << L, the sum of these becomes zero when kL ≈ (2n − 1)π/2 + (8/3π)ka, which means that the ‘acoustic’ length of the tube, L = L + (8/3π)a, is then an odd number of quarter wavelengths, L = (2n − 1)λ/4. The quantity (8/3π)a is the ‘end correction.’ At these quarter wavelength resonances, Eq. 4.29 becomes Aa = A
θi (θr + θi )2
(resonance).
(4.31)
The radiation impedance ζr ≡ θr + iχr of a plane piston in a rigid wall (baffle) is given in many texts in acoustics and is θr = 1 −
χr = −(4/π )
π/2 0
J1 (2ka) ka
≈ (ka)2 /2
sin(2ka cos(φ) sin2 (φ) dφ) ≈ −(8/3π )ka,
(4.32)
where a is the radius of the piston. The approximate expressions are valid at low frequencies for which ka << 1. With our choice of time factor (exp(−iωt)), the negative sign of χr refers to a mass reactance. The resistance and the magnitude of the reactance have been plotted as a function of ka in Figure 4.1. The straight line portions of the curves, which extend almost up to ka = 1, correspond to the approximations in Eq. 4.32. In the absence of a wall (and pressure doubling), the radiation impedance for the acoustically compact resonator, i.e., ka << 1, the impedance is half the value given above. In the low frequency approximation, with θr ≈ (ka)2 /2 and with A = π a 2 , the expression for Aar can be written, with k = 2π/λ, Aa =
θ r θi x 2λ2 2λ2 = π (θr + θi )2 π (1 + x)2
(resonance),
(4.33)
135
RESONATORS
where x = θi /θr . The maximum value of Aa is obtained for x = 1, i.e., when the internal resistance equals the radiation resistance, and is Maximum absorption cross section at resonance (Aa )max =
λ2 2π
(4.34)
where the corresponding radius for tube resonator: Eq. 4.37. Under these conditions, the corresponding scattering cross section has the same value, λ2 /2π. In the absence of the wall, the maximum absorption cross section (and the corresponding scattering cross section) is λ2 /4π . The analysis can be applied also to an unflanged resonator in free field with the only difference that the radiation impedance is modified. At low frequencies, the radiation resistance is half the value used above, and the corresponding maximum absorption coefficient at resonance becomes (Aa )max = λ2 /4π
(Free field, unflanged).
(4.35)
Optimum Radius for Maximum Resonance Absorption Cross Section In the discussion following Eq. 4.30 it was shown that at low frequencies such that ka << 1, the reactive part of the total impedance ζr +ζi in Eq. 4.29 is cot(qL ), where L = L + (8/3π)a is the acoustic length of the resonator tube. The corresponding real part of ζr is (ka)2 /2. The real part of ζi is the real part of i cot(qL) and it accounts for the internal viscous losses in the tube. However, there is also an ‘external’ part of viscous losses caused by the tangential velocity along the wall (baffle) at the entrance to the tube. This can be accounted for by a viscous ‘end correction’ to the length L and, as an approximation, we let this equal the mass end correction (8/3π )a. This means that we can replace cot(qL) by cot(qL ). The maximum absorption cross section (Eq. 4.34) at the first resonance is obtained when the input resistance of the tube equals the radiation resistance, i.e., when (ka)2 /2 = (kL )dvh /2a. With kL = (2n − 1)π/2, the corresponding optimum tube radius for the first resonance, n = 1, is given by a 3 = (2/π)dvh L 2 .
(4.36)
√ With dvh ≈ 0.31/ f , L = c/4f and c = 34000 cm/sec, the numerical expression becomes 5.2 (cm with f1 in Hz). (4.37) a≈ (f1 /100)5/6 Maximum Q-Value The total impedance expressing the frequency response of the tube resonator in the vicinity of the resonance is ζr + ζi . With reference to the discussion following Eq. 4.30, the reactive part of this impedance is cot(kL)−(8/3π )ka ≈ cot(kL ). In the
136
NOISE REDUCTION ANALYSIS
vicinity of the nth quarter wavelength resonance, we have cot(kL ) ≈ −(ω−ωn )L /c, and the corresponding total input impedance can be written ζi + ζr ≈ θ − ikn L
ω − ωn ωn
(near resonance),
(4.38)
where θ = θi + θr . With ω − ωn = ω, the ‘half power point’ in the response curve is obtained for kn L ω/ωn = θ , and the corresponding Q-value is Q = ωn /2ω =
π kn L
= (2n − 1) . 2θ 4θ
(4.39)
The maximum value of Q corresponds to a minimum value of the total resistance. With the tube radius as the variable, and with θr ≈ (ka)2 /2 and θi = (kL )dvh /2a, this minimum is obtained for a tube radius given by 3 = (1/π )dvh L 2 am
(4.40)
and the corresponding minimum resistance is θmin = (3/2)(kam )2
(4.41)
and the maximum Q-value becomes Qmax =
π . 4θmin
(4.42)
4.5.3 Helmholtz Resonator If the radius of the orifice in Figure 4.3 is a0 , the normalized radiation impedance is again given by Eq. 4.32, with a replaced by a0 . The internal impedance, based on the velocity in the orifice, is the sum of the contributions from the air in the tube and in the orifice. The former is i(A0 /A) cot(qL), which is the same as before except for a factor A0 /A, the ratio of the orifice area and the tube area. The latter is the sum of several contributions resulting from wall friction, the air mass in the orifice, and the air in the close vicinity of the orifice. The orifice diameter is normally large compared to the viscous boundary layer thickness, the contribution from wall friction to the orifice impedance is obtained simply by multiplying the surface impedance for shear flow (see Chapter 3) by the interior surface area of the orifice, 2πa0 , where is the thickness of the orifice plate. The expression for the force-to-velocity ratio is ρcπ kdv a0 . The reactive contribution of the friction force is small compared to the reactance −iωρA0 of air in the orifice and is neglected. To obtain the normalized impedance, we divide this ratio by ρc and by A0 to get −ik(1 + idv /a0 ). There is also an interior as well as an exterior mass reactance due to the air in the close vicinity of the orifice. This air mass is generally expressed as ρA0 δ, where δ is the end correction. If the resonator is set in an infinite wall (baffle), the exterior
137
RESONATORS
mass end correction is obtained from Eq. 4.32 (with a replaced by a0 ) and equals δe = (8/3π)a0 at low frequencies, ka0 << 1. If there is no baffle, the end correction is somewhat smaller, close to that of an unflanged pipe, δe ≈ 0.61a0 . The interior end correction must be zero when A0 = A, and this is achieved by letting the interior end correction be δi ≈ (1 − A0 /A)δe . With = + δi + δe , the normalized impedance of the total air mass in the orifice can be written −ik . In addition to the viscous stress on the interior walls of the orifice, there are also contributions in the vicinity of the orifice on the walls of the orifice plates due to the tangential velocity component along these walls. These are accounted for in terms of a viscous end correction, which is assumed to be the same as the end correction δ for the mass. The total specific normalized input impedance of the orifice then becomes ζ0 ≈ −ik (1 + iδv /a0 ) + i(A0 /A) cot(kL).
(4.43)
For wavelengths much greater than L, cot(kL) ≈ 1/kL. The reactive part of the impedance in Eq. 4.43 then becomes zero at the resonance frequency and is given by k02 = A0 /AL = A0 /V . With k0 = 2πf0 /c this relation yields the resonance frequency f0 . Resonance frequency, Helmholtz resonator f0 = (c/2π) A0 /V
(4.44)
where ≈ + δe + δi , A0 : orifice area, V : resonator volume, c: sound speed. At sufficiently large wavelengths compared to the cavity dimensions, this expression is valid for a cavity of arbitrary shape. This is easily seen if by starting from the mass conservation equation ∂ρ/∂t + ρdiv u = 0, which is expressed as κ
∂p + div u = 0, ∂t
(4.45)
where κ = 1/ρc2 and p is the sound pressure in the cavity, assumed uniform. Integration of this equation over the cavity volume V and with harmonic time dependence, the equation becomes −iωκVp(ω) = A0 u0 , where A0 is the area of the orifice and u0 the velocity amplitude in the orifice. The normalized cavity impedance is then A0 p(ω) =i . ρc u0 (ω/c)V
(4.46)
Adding the orifice impedance −ik = −i(ω/c) yields the total impedance and by putting that equal to zero, the resonance frequency is obtained as given in Eq. 4.44.
4.5.4 Three-Dimensional Array of Resonators The sound absorption by two-dimensional arrays of resonators placed in a wall and in free field are treated in an example in Appendix C. A somewhat different problem involves sound propagation in a medium which is filled with resonators in the form
138
NOISE REDUCTION ANALYSIS
of a three-dimensional array. The wavelength is long compared to the separation of adjacent resonators, i.e., the lattice is acoustically compact. Under these conditions, the effect of the resonators is expressed in terms of an average complex compressibility of the medium. The volume of a lattice cell is V and the volume occupied by the resonator is Vr and let V1 = V − Vr . We start from the mass conservation equation ∂ρ/∂t + ρdiv u = 0, which is rewritten in the form κ∂p/∂t + div u = 0, where κ = 1/ρc2 is the compressibility of air. Integrate this equation over the volume V1 . The volume is bounded by its outer surface of area A1 and the surface of the resonator. The volume integral of divu is expressed in terms of the surface integrals over these bounding surfaces and becomes A1 u1 and u0 A0 , where A0 is the aperture area of the resonator and A0 the area. Then, for harmonic time dependence, the integrated equation becomes −iωκpV1 + A1 u1 + A0 u0 = 0.
(4.47)
The average compressibility of the entire volume V is denoted by κ, ˜ and the equation for it is then −iωκp ˜ + A1 u1 = 0. (4.48) Combine these equations, and introduce u0 = p/ζi ρc, where ζi is the input impedance of the aperture in the resonator. With β = Vr /V and V1 /V = 1 − β, the average complex compressibility in the volume then becomes κ/κ ˜ = 1 − β + i(1/ζi )A0 β/kVr ,
(4.49)
where k = ω/c. If the equivalent length of the orifice in the resonator is (including end corrections) the resonance frequency is ω0 = c A0 /(Vr ), and if the acoustic resistance of the resonator orifice is θi , the input impedance of the resonator can be expressed as
o o ω0 ω0
ζi ≡ 1/ηi = θi − ik0 = −ik0 + iD , (4.50) − − ω0 ω ω0 ω where k0 = ω0 /c and D = θi /k0 is the damping factor, the inverse of the ‘Q-value.’ With these quantities introduced into Eq. 4.49, the explicit dependence of the complex compressibility on the frequency ratio = ω/ω0 can then be written κ/κ ˜ =1−β +
β . − iD
1 − 2
(4.51)
To account for the presence of the resonators in the equation of motion of the air in the lattice we proceed in the same manner as in the study of sound propagation in a porous material (see Chapter 5) by introducing a complex density given by ρ/ρ ˜ = + ir/ωρ,
(4.52)
where is the structure factor and r is the viscous resistance per unit length in the lattice due to friction at the surface of the resonators. The structure factor, which in
139
RESONATORS
an ordinary porous material is of the order of 1.5, accounts for the induced mass, which is caused by the deflection of the flow by the resonators. With reference to Chapter 3, the propagation constant for a plane wave in the lattice can then be expressed as ˜ κ/κ). ˜ (4.53) Q = Qr + iQi = (ρ/ρ)( The real part yields the ratio of the free field sound speed and the phase velocity in the lattice, i.e., the index of refraction, and the imaginary part expresses the amplitude decay through the factor exp(−Qi kx). The wave impedance of the lattice is ζw = ρ/Q ˜
(4.54)
and from it, the reflection, absorption, and transmission coefficients of the resonator lattice of a given thickness can be calculated. For an infinitely thick array, the pressure reflection coefficient at normal incidence is simply R=
ζw − 1 . ζw + 1
(4.55)
If the lattice is not acoustically compact, we have to apply the general technique of wave propagation in a periodic structure along the lines of the analysis of the sheet lattice absorber in Chapter 2.
4.5.5 Acoustic Nonlinearity, Perforated Plate The nonlinear normalized resistance |u0 |/c of an orifice is related to flow separation, which occurs in an orifice, as discussed in Section 4.3. In a perforated plate, there is no separation if the open area fraction of the plate s is 1. This is accounted for by an empirical correction factor (1 − s) in the expression for the nonlinear resistance to make it (1 − s)|u0 |/c. This should be added to the linear impedance ζ0 given in Chapter 2 to yield an orifice impedance ζ0 = ζ0 + (1 − s)|u0 |/c. The corresponding average impedance over the area of the plate is obtained by multiplying by 1/s since the average velocity over the plate is s times the velocity in an orifice. In general, the perforated facing is backed by a layer with an impedance ζb in loose contact with the facing. In writing down the equation for the velocity amplitude in an orifice, we shall refer all impedances to this velocity so that the impedance by the backing material has to be set equal to sζb . The total orifice impedance is then7 ζ0 = ζ0 + sζb + (1 − s)|u0 |/c.
(4.56)
With the complex pressure amplitude at the surface of the plate being p, the velocity amplitude in the orifice is given by u0 = p/(ζ0 ρc).
(4.57)
7 With the assumption of loose contact, the backing material is separated from the perforated plate by
a distance of the order of an orifice diameter.
140
NOISE REDUCTION ANALYSIS
However, the pressure p at the boundary is generally not known a priori, and it is more useful to express it in terms of the amplitude pi of the incident sound wave. The relation between these pressures is p = (1 + R)pi , where R is the pressure reflection coefficient of the perforated surface, corresponding to the input impedance ζi = ζ0 /s =
1 |u0 | (ζ0 + βζb + (1 − s) . s c
(4.58)
The reflection coefficient is R = (ζi cos φ − 1)/(ζi cos φ + 1) (Eq. 3.21) in terms of ζi and the angle of incidence φ. Thus, 1 + R = 2ζi cos φ/(ζi cos φ + 1). The average velocity amplitude over the plate, which is su0 , is then given by (1 + R)pi = ρcζi , and it follows that s (u0 /c) = (2pi /ρc2 )/(ζi + 1/ cos φ), (4.59) where ζi is given in Eq. 4.58. It contains the magnitude |u0 | of the complex velocity amplitude and the equation for u0 generally has to be solved numerically. We consider here the special case of a ‘resonance’ at which the reactance of the backing layer is canceled by the orifice reactance. In that case, we can replace u0 /c on the left-hand side by |u0 |/c and the orifice and backing impedances by their resistive parts. Solving for |u0 |/c then yields 2δ |u0 | for δ << 1 √≈ δ/A (4.60) 1+ −1 =A ≈ 2δ/(1 − s) for δ >> 1, c (1 − s)A2 where A = [1/2(1 − s)](θ0 + sζb + s/cos φ) and δ = pi /ρc2 = pi /(γ P ), P being the static pressure (recall c2 = γ P /ρ). Effect of Induced Plate Motion As before, the normalized orifice impedance is ζ0 = ζ0 + (1 − s)|u0 |/c, the sum of the linear and nonlinear contributions. It is this impedance and the related drag on the plate which is responsible for the induced motion of the plate. The response of the plate is expressed in terms of a structural impedance. For a limp plate of mass m per unit area, the normalized value of this impedance is −iωm/ρc. The velocity u0 in the orifice is influenced by the induced motion of the plate and also by the impedance of the layer behind the plate. With an open area fraction s, there will be 1/s orifices per unit area of the plate, and the equation of motion for it is then (−iωm)u = (1/s)ζ0 u0 , where u and u0 are amplitudes of the velocity of the plate and the relative velocity of the air in the orifice with respect to the plate. The right-hand side of the equation is the drag force per unit area of the orifice plate. We express this equation as ζs u = ζ0 u0 , where ζs = (−iωm)s is the normalized structural impedance referred to as the velocity in the orifice rather than the average velocity over the plate. With reference to the discussion in Chapter 2, the effect of the induced motion of the plate can be accounted for here by replacing ζ0 by ζ0 =
ζ 0 ζs ζ0 + ζ s
(4.61)
141
RESONATORS
in Eq. 4.59. Furthermore, the nonlinear resistance involves the velocity amplitude in the orifice relative to the plate. Therefore, again with reference to Chapter 2, this relative velocity u0 is obtained by multiplying the absolute velocity by the factor ζs /(ζs + ζ0 ). Taking these considerations into account, we obtain, with δ = pi /ρc2 (see Eq. 4.59), u0 2δ ζs =
, c ζ0 + sζb + s/cos φ ζs + ζ0
(4.62)
where ζ0 and ζ0 , given in Eqs. 4.56 and 4.61, depend on |u0 |/c. This equation is solved numerically for |u0 |/c. It should be noted that the induced motion of the plate reduces the velocity in the orifice, and hence the nonlinear resistance. This should be kept in mind when trying to understand numerical results in which the induced motion plays a significant part. Having obtained |u0 |/c, the nonlinear orifice impedance can be expressed as the sum of its linear and nonlinear components.
Chapter 5
Rigid Porous Materials 5.1 INTRODUCTION AND SUMMARY As mentioned in Chapter 3, Rayleigh1 extended Kirchhoff’s study of visco-thermal attenuation in a tube to include the case when the boundary layer thickness was not necessarily small compared to the tube diameter, and his model of a porous material involved a solid block of infinite thickness perforated by circular, narrow channels. He calculated the normal incidence reflection coefficient for this perforated layer in the limits of small and large boundary layer thicknesses compared to the tube diameter. Rayleigh also pointed out the problem with the model that it did not permit the limit of 100 percent open area because of the geometrical constraints caused by the use of circular tubes. He suggested that parallel channels or ‘crevasses,’ as he put it, would in principle eliminate this constraint (assuming zero thickness of the partition walls between the crevasses). In Chapter 3, we analyzed the sound propagation in the channel between two parallel walls in the entire range of boundary layer thicknesses, and we start this chapter with a section on an absorber involving a finite layer of such channels backed by a rigid wall, as shown schematically in Figure 5.2. Realizing that the absorber is not isotropic, we have chosen it to demonstrate how one and the same absorber can be both locally and nonlocally reacting depending on the angle of incidence of the sound. For waves in a plane perpendicular to the channels, it is locally reacting, but for any other plane, it is not. In a plane parallel with the channels, the absorber is purely nonlocally reacting. The absorber also serves to illustrate the calculation of the diffuse field absorption coefficient of an anisotropic material. Furthermore, it is possible to use this absorber also as a model for a porous material with closed pores, by turning the absorber around to make the sound strike from a direction perpendicular to the (thin) walls or sheets that separate the channels. In that case, there will be no velocity parallel with the walls and no viscous boundary losses so that the absorption will be due solely to the effect of heat conduction and the induced motion of the walls.
1 §348–351 in his Theory of Sound.
143
144
NOISE REDUCTION ANALYSIS
In order for the absorber to be effective, the slot width has to be quite small, typically a few thousands of an inch. Therefore, the slot absorber probably has little practical interest since there are simpler ways of producing a porous material. The absorption coefficient of the slot absorber and a rigid porous layer in general is determined mainly by the thickness L of the layer and its flow resistance r per unit thickness. For a given flow resistance, the porosity and the structure factor of the material influence the absorption coefficient only weakly over the range of values normally encountered in practice. It should be realized, though, that these quantities are not independent. Also the frequency dependence of the compressibility of the air in the material with a transition from isentropic to isothermal conditions can be considered to be determined also by the flow resistance, since the viscous and thermal relaxation frequencies are shown to be practically the same. In general, the compressibility is expressed as a complex quantity. In presenting the absorption coefficient as a function of frequency, we need one curve for each value of r and each value of L. However, by introducing the total resistance of the layer, = rL/ρc (which is here normalized with respect to the wave impedance ρc) and using as a frequency variable the ratio L/λ of the layer thickness and the wavelength, we can present the absorption spectra in a ‘universal’ form in a single figure, which contains a set of curves for different values of the flow resistance parameter . From a practical standpoint, it is important to note that the ‘best’ overall performance of the rigid absorber for a given thickness is obtained when the total normalized flow resistance is R ≈ 4, and the absorption coefficient then will exceed 80 percent for both normal incidence and diffuse field when the incident wavelength is less than approximately 10 times the layer thickness. Another useful piece of information concerns the maximum possible absorption that can be obtained by a porous layer at a certain frequency by selecting an optimum value of the total flow resistance. This information can be extracted from the set of the computed absorption spectra in this chapter, and these optimum conditions are summarized in Figure 5.1. In this figure, the left axis with its linear scale from 0 to 1 refers to the maximum absorption coefficient and the right axis with its log scale from 1 to 100 refers to the normalized optimum total flow resistance of the layer. The solid and dashed curves apply to normal incidence and diffuse field, respectively. In the range of flow resistance given here, the results can be used for both the locally and nonlocally reacting layer. Multilayer absorbers. In order to get a higher absorption coefficient for a given layer thickness than that of the uniform single layer, a multilayered configuration should be considered. The relative thickness and flow resistances of the different layers can be chosen so as to optimize the absorption over a certain frequency range. Because of the large number of parameters involved, it is not possible to give generally valid simple rules for optimum design. Such a design has to be found iteratively in each individual case by repeated use of computer programs. Effect of a cover screen and perforated facing. A special case of the multilayer absorber is one in which a porous layer is covered with a thin porous screen and/or
RIGID POROUS MATERIALS
145
Figure 5.1: The maximum absorption coefficient that can be achieved with a uniform rigid porous layer of thickness L at an incident wavelength λ. The corresponding optimum total normalized flow resistance opt is also shown. The solid and dashed curves refer to normal incidence and diffuse field, respectively.
a perforated plate. Such covers can lead to a significant increase in the absorption coefficient at low frequencies at the expense of a reduction at the high frequencies. If the facing is placed close to the porous layer but not in contact with it, the mass of the cover plays a role because of acoustically induced motion. Effect of an air layer. Still another example of a multilayer absorber is one in which a porous layer is combined with an unpartitioned air layer. In that case, for fixed values of the total thickness and flow resistance of the absorber, an increase of the thickness of the air layer always leads to a reduction of the diffuse field absorption coefficient at all frequencies. However, at normal incidence, the effect is less pronounced. In fact, at very low frequencies there is a small increase in the absorption coefficient with increasing air layer thickness. Sheet absorbers vs uniform porous layers. We have compared the absorption spectra of sheet absorbers and those of a uniform porous layer with the same thickness and total flow resistance. As the number of sheets increases, the performance approaches that of the uniform layer, as expected. It is interesting to note, however, that at low frequencies, a locally reacting, single sheet absorber is somewhat better than a multisheet or uniform porous layer. At high frequencies, the situation is reversed, however. The absorption characteristics of a rigid porous material can be regarded as basic for the design of sound absorptive wall treatments (see Figure 5.1). Actually, even flexible materials behave approximately as acoustically rigid over a wide range of frequencies. However, flexibility can influence the absorption coefficient significantly, particularly at low frequencies, and this can be exploited in the design for the enhancement of the absorption in a particular frequency region. We refer to Chapter 7 for details.
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5.2 THE SLOT ABSORBER The slot absorber is shown schematically in Figure 5.2. With reference to the introduction in the previous section, this simple model of a rigid porous material makes possible the calculation of its acoustical characteristics from first principles in terms of the geometrical parameters of the material and the physical properties of air (or whatever fluid is involved). This analysis can be regarded as an introduction to the study of the acoustical characteristics of porous materials in general, such as glass wool or foam, for which empirical parameters describing the material have to be used. Expressing the results obtained for the slot absorber in terms of these parameters makes it possible to establish an equivalence between the general porous material and the present simple model. To avoid redundancy, the analysis for the slot absorber will be somewhat sketchy and the emphasis will be placed on the general rigid absorber to be treated in the next section. The thickness of each plate is h and the width of each air channel or slot between the plates is d = 2a. The geometrical parameters that describe the material are then d, h, and L and the relevant fluid parameters are density ρ, sound speed c, shear viscosity μ, and heat conduction coefficient K. The acoustical properties of this absorber can readily be calculated in terms of these quantities. For porous materials, in general, empirical parameters have to be used in an analogous study of the absorption characteristics. Parameters like d and h describing the ‘micro-structure’ of the material are seldom used, and the description of the geometry only involves the layer thickness and the porosity, i.e., the volume fraction of air (or fluid) contained in the material and the structure factor, which expresses the apparent increase in the inertial mass of the fluid in the material due to the torturous path it is forced to follow. The steady specific flow resistance per unit length of a single channel was determined in Chapter 3 and it was denoted by r0c , the subscript c used to designate ‘channel,’ and it is 12μ/d 2 , where μ is the coefficient of shear viscosity. With account of the ‘porosity’ H , the corresponding flow resistance for the slot absorber will be 1/H times larger because of our definition of the velocity as the average value rather than the velocity in a channel. (The velocity in a channel is (1/H ) times the average value and it is responsible for the pressure drop. The flow resistance r0 of an absorber
x
Front view
Side view
Figure 5.2: The ‘slot absorber’ consisting of parallel plates of thickness h, separation d, and depth L mounted on a rigid wall, as shown.
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is expressed as the ratio of this pressure drop and the average velocity, which is the same as the velocity outside the absorber.) These parameters are not independent. The relations between flow resistance, porosity, and mass density generally have to be determined from experiments but for the slot absorber we can express them analytically. Physical parameter of the slot absorber ρ = ρm (1 − H ) H = d/(d + h)
or d = H h/(1 − H ) 12μ H d2
r0 = r0c /H = = = 1/H
(5.1)
12μ(1−H )2 h2 H 3
where Mass density, ρ m of porous material, ρm , of plate, H : Porosity, r0 : Flow resistance per unit length, h: Plate thickness, d = 2a: Channel width, μ: Coeff. of shear viscosity. The wall thickness and porosity are not independent, and this should be kept in mind were we to inquire about the dependence of the absorption coefficient on the porosity. For a finite wall thickness h, the flow resistance is zero for H = 1, as it obviously should be, but if we ever use H = 1 in a computation involving the flow resistance, the result obtained should be interpreted as the limit when H → 1 and h → 0. For most porous materials used for sound absorption, the porosity is close to unity. Consider, for example, a commonly used glass wool with a weight of 2 lbs/ft3 (ρ ≈ 0.032 g/cm3 ) and a mass density of the glass fiber ρm ≈ 2.5 g/cm3 . According to Eq. 5.7 the porosity is H ≈ 0.99. In our analysis of sound propagation in a porous material in general in the next section, we shall define an average velocity u in such a way that ρu is the mass flux in the material. This choice will be made also in the analysis of the slot absorber in order to make the results easily adaptable to the porous material in general. By a ‘cell’ size in the slot absorber is meant the distance d + h between the center planes of adjacent channels. Then, if the average velocity in a channel of width d is uc , the average velocity over a cell will be u = [d/(d + h)]uc = H uc . In the previous chapter, dealing with wave propagation in a single channel, the velocity uc in a channel was used. Therefore, before we make use of the formulas, which contain this velocity, we must account for the difference between the new average u (over a cell) and the old average uc by introducing the factor H at appropriate places. With our use of the average velocity, the velocity normal to a boundary of the porous material will be continuous across the boundary. Continuity of sound pressure across the boundary is a good approximation, although, due to the constriction of the flow as it enters the material, there is a small pressure drop, which corresponds to a small mass reactance. However, this reactance can be accounted for by adding a small (‘end’) correction to the thickness L of the absorber; this correction is negligible in most cases. If this is done, the boundary condition of continuity of pressure is accurate. In terms of the average velocity u, the kinetic energy density per unit volume of the absorber is written as ρe |u|2 /2, where ρe ≡ ρ. By definition, is the structure
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factor. The actual velocity in a channel is [(d + h)/ h]u and the kinetic energy density of the flow in the absorber is ρ[h/(d + h)][(d + h)/ h]2 |u|2 /2. By equating these expressions for the kinetic energy, it follows that the structure factor for the slot absorber is simply = (d + h)/d = 1/H . This relation cannot be expected to be true for a porous material in general, however. It should be noted that the slot absorber is anisotropic and its response will depend on the orientation of the plane of incidence of the wave. With the plates parallel with the xz-plane (see Figure 5.2), the absorber will be locally reacting for a plane wave incident in the xy-plane. The oscillatory velocity in a channel is then forced to be in the direction normal to the surface (x-direction), and the velocity amplitude at a certain location on the surface will depend only on the local sound pressure at the surface and not on the sound pressure distribution over the rest of the surface. For a sound wave incident in a plane other than the xy-plane, the absorber will be nonlocally reacting. The velocity in a channel is no longer forced to be normal to the surface, and both the direction and the amplitude of the velocity will depend on sound pressure distribution over the entire surface. For a sound wave of normal incidence, we need not make the distinction between local and nonlocal reaction, of course, and we shall begin with that case.
5.2.1 Input Impedance, Absorption Spectra To determine the frequency dependence of the absorption coefficient, i.e., the absorption spectrum of the slot absorber in Figure 5.2 for sound at normal incidence, we start by expressing the input impedance of the absorber in terms of the quantities introduced in the single channel analysis in Chapter 3. The propagation constant is the same as for the single channel, and, apart from a factor 1/H , the same applies for the wave impedance. Input impedance of slot absorber, normal incidence ζi ≡ θi + iχi = iζw cot(qL)
(5.2)
where q: Eqs. 2.6, 2.79, ζw = ζwc /H , Eq. 2.85, L: Thickness. Examples of the frequency dependence of the real and imaginary parts of the input impedance are shown in Figure 5.3, where the normalized values are plotted as functions of the ratio of the layer thickness L and the wavelength λ for two different values, 2 and 8, of the total normalized steady flow resistance = r0c L/(Hρc). The corresponding channel width d = 2a in the absorber is obtained from Eq. 2.35, √ d = 2 3Lν/ρc. For example, with ν = μ/ρ ≈ 0.15 CGS and c ≈ 34000 cm/sec for air, the channel width corresponding to L = 10 cm and = 2 is ≈ 0.0094 cm ≈ 3.8 mil (thousands of an inch). Since in practice the total flow resistance of an effective absorber typically is ≈ 2, we see that the corresponding pore size will be of the order of one mil. For the values of in the figure, the reactance is positive (stiffness like) over the entire frequency range. For smaller values of flow resistance, < 1, corresponding to channel widths larger than 5 mil, the resonances of the layer become more pronounced
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Figure 5.3: The real and imaginary parts of ζi in Eq. 5.2 vs L/λ. ζi is the normalized specific impedance of a slot absorber of thickness L and λ is the free field wavelength of the incident sound. The parameter values 2 and 8 refer to the total normalized steady flow resistance of the absorber and correspond to slot widths of 3.8 and 1.9 mil, respectively. The porosity was chosen to be H = 0.95.
and the reactance becomes mass-like (negative) in regions above the resonances. It should also be noted that as L/λ decreases, the (stiffness) reactance becomes independent of the flow resistance and the input resistance approaches one-third of the total flow resistance, consistent with the low frequency approximation of Eq. 2.34. It is significant to note that at low frequencies the reactance corresponds to a stiffness of an air layer with an isothermal compressibility approaching the value 1/H γ kL, i.e., a factor γ smaller than for an unbounded air layer (k = ω/c and γ ≈ 1.4 is the specific heat ratio).
Normal Incidence Absorption Spectra The normal incidence absorption coefficient can be computed from the input impedance with the well-known expressions in Chapter 2. In presenting the results, we start with a set of ‘universal’ absorption spectra, shown in Figure 5.4, in which the absorption coefficient is plotted as a function of the parameter L/λ for values of the total normalized flow resistance = r0 L/ρc of the layer from 0.5 to 32. These values should cover the range of practical interest, and it is thus possible to summarize the numerical results in a single figure. Although this choice of variables makes for a convenient presentation, it becomes √ awkward when it comes to a calculation of the corresponding channel width d = 12Lν/ρc, since it is not a constant parameter for each curve but varies from one point to the next. For the orders of magnitude involved (typically one mil) we refer to the discussion of Figure 5.3. As has already been said, the porosity H is close to unity for many materials used in practice, and, as will be shown shortly, the absorption coefficients for values of H between 0.95 and 1.0 are practically indistinguishable. We have used H = 0.95 in obtaining the numerical results in Figure 5.5, but from a practical standpoint, they
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Figure 5.4: Normal incidence absorption coefficient of the slot absorber vs L/λ, where L is the absorber thickness and λ the free field wavelength. The parameter values = 0.5, 1, 2, 4, 8, 16, 32 refer to the total normalized steady flow specific resistance of the absorber, = r0 L/ρc. Corresponding slot widths: see text. are valid for H -values between 0.9 and 1.0, a range which covers a large fraction of the porosities for commonly used materials. Values of < 1 make α smaller than for > 1, except possibly at some high frequencies, and < 1 is of little or no interest from a practical standpoint. To establish the frequency scale in the figure, consider as an example a 4 inch thick absorber, i.e., L = 4 inch. The upper end of the scale, corresponding to L/λ = 1, is then 3360 Hz, and the other end of the scale is 33.6 Hz. Effect of Flow Resistance A question of practical interest concerns the optimum choice of the flow resistance for maximum absorption at a certain wavelength (frequency) and absorber thickness. The results in Figure 5.4 show that for a given frequency, there is indeed an optimum value of . For example, at a frequency of 336 Hz (L/λ = 0.1), the optimum value is ≈ 3 (i.e., r0 /ρc = /L ≈ 0.75 inch−1 and the corresponding absorption coefficient ≈ 0.8). With r0 = 12μ/d 2 H , we find that the corresponding optimum channel width is d ≈ 5.4 mil (0.0054 inch). A further discussion of the role of flow resistance, involving, for example, the optimum value for maximum absorption, will be essentially the same as that for the general porous absorber considered below. Effect of Porosity When it comes to a discussion of the role of porosity on the absorption coefficient, the situation is different for the slot absorber and for the general absorber since for the latter we have not attempted to determine analytically the relationship between the flow resistance and porosity. For the slot absorber, however, we now have such a relationship, which enables us to determine what influence the porosity, per se, will have.
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The flow resistance of the absorber = (1/H )r0c L/ρc = 12μ/(H d 2 ), and not the flow resistance r0c of a channel, has been kept constant in these computations leading to the results in Figure 5.4. To maintain this, any decrease in the porosity parameter H has to be accompanied by an increase in the channel width d and a related decrease in the channel resistance r0c . In Figure 5.5 we have presented the results of such computations with the porosity ranging from 0.01 to 1.0. In each case the total normalized total flow resistance of the absorber is = 2 so that the flow resistance per unit length is r0 /Lρc = 0.5. At sufficiently small porosities, the channel width will be large compared to the boundary layer thickness, and the channels then act like quarter wavelength resonators. The resonances are clearly noticeable in the figure and their bandwidth (damping factor) increases with increasing porosity (decreasing channel width) until the resonators become overdamped and act like broad-band absorbers. At the same time, there is a slight shift of the resonances toward lower frequencies. This is because of the decrease of the phase velocity in the channel with decreasing channel width (see Q vs a/dv in Figure 2.1). It is important to note that in the porosity range 0.95 to 1.0 the variation in the absorption coefficient is insignificant from a practical standpoint. (The curve corresponding to H = 0.95 lies between the curves labeled 0.75 and 1.0 in the figure.) Effect of Heat Conduction There are two effects of heat conduction on the absorption coefficient of a porous layer; one is direct and the other indirect. The direct effect is simply the conversion of acoustic power into heat, which was shown earlier to be ωκi |p|2 per unit volume (see Eq. 2.51). Without going into details here (for such we refer to Section 5.7) the
Figure 5.5: Influence of porosity H on the absorption coefficient of a slot absorber with a
fixed total normalized total flow resistance = 2. Values of porosity H range from .01 to 1. Notice that for porosities in the range 0.95 to 1, the absorption curves are essentially the same from a practical standpoint. For very low porosities, 1 to 10 percent, the channel resistance is correspondingly small and the channel width relatively large so that the absorption will essentially be zero except in the vicinity of the quarter wavelength resonances of the channels.
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direct effect in most cases of practical interest is small compared to the friction losses. The indirect effect, however, can be significant. It has to do with the fact that the compressibility will be isothermal (rather than isentropic) in the porous material at low frequencies, which reduces the stiffness reactance of the air. This in turn reduces the reactive part (dominant at low frequencies) of the input impedance thus increasing the velocity amplitude and the viscous dissipation. The corresponding effect on the normal absorption coefficient is illustrated in a special case in Figure 5.6. In each of the pair of curves shown, the label (1) refers to inclusion and the label (2) to exclusion of heat conduction. = 2 and = 32 indicate the total normalized flow resistance of the porous layer, where, as before, ≡ r0 L/ρc, where r0 is the resistance per unit length. The increase in the absorption coefficient resulting from the effect of heat conduction is readily seen and is significant below the characteristic frequency fv = r0 /2πρ, discussed earlier (see Eq. 2.99). The corresponding transition value of L/λ is simply (L/λ)v = /2π . For = 2 this transition value becomes ≈ 0.32 and for = 32 it is about 5 so that isothermal conditions prevail over the entire frequency range considered here. With = 2 we find that at L/λ = 0.01 (133.9 Hz for L = 1 inch), the ‘isothermal’ absorption coefficient (0.020) is about twice the ‘isentropic’ (0.0094) and for L/λ = 0.1 (1339 Hz), the isothermal value (0.76) is larger than the isentropic (0.57) by a factor of 1.33. Oblique Incidence, Refraction, and Diffuse Field Absorption As stated earlier, the parallel plate absorber is anisotropic and the absorption coefficient depends not only on the (polar) angle of incidence φ with respect to the normal but also on the azimuth angle ψ. Therefore, the refraction of sound as it interacts with a porous material in this case will depend on both these angles.
Figure 5.6: In each of the two pairs of curves shown, the label (1) refers to a true absorption coefficient in which both viscosity and heat conduction has been included and label (2) corresponds to viscosity alone. The first pair corresponds to a total normalized flow resistance of the porous layer of = 2 and the second to = 32. The abscissa is L/λ, where L is the thickness of the absorber and λ the free space wavelength. Porosity is 0.95.
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If the wave is incident in a plane perpendicular to the slots, so that the absorber becomes locally reacting, the wave motion in the absorber will be perpendicular to the wall so that the angle of refraction will always be 90 degrees. More interesting is the case when the incident wave vector is in a plane parallel with the plates, corresponding to ψ = 0 (nonlocal reaction). For this case (for ψ = 0), Figure 5.7 shows the angle √ of√refraction φr vs the angle of incidence φ for some different values of a/dv = 3/2 ω/ωv = 0.1, 1.0, and 10. It is interesting to note that even when a/dv = 10, i.e., when the channel width is 20 times the boundary layer thickness, there is a significant refraction for large angles of incidence (the refraction angle is 73 degrees when the incidence angle is 90). With a/dv ≤ 0.1, the refracted wave travels almost along the x-axis, i.e., approximately the same as when ψ = 90 degrees. The absorber is then approximately locally reacting for all directions of incidence. The normal impedance as well as the related absorption coefficient now depends on both the polar and azimuthal angles of incidence and in computing the diffuse field average absorption coefficient, integration over both these angles is required. In Figure 5.8 the results of numerical computations of αav are plotted vs L/λ for different values of the total steady flow specific resistance = r0 L/ρc. They should be compared with the absorption curves for normal incidence in Figure 5.4. Recall that the absorber is anisotropic and is locally reacting only for waves incident in a plane perpendicular to the plates. However, the difference between the absorption curves for a locally reacting and a nonlocally reacting absorber is significant only for small values of , say for < 2. It should be emphasized that for each value of the frequency parameter L/λ there is an optimum value of the total flow resistance to obtain maximum absorption. For example, for L/λ = 0.1, the optimum value is ≈ 4, and the corresponding absorption coefficient is ≈ 0.8 and for L/λ = 0.02 it is ≈ 16 with αav ≈ 0.4. With
Figure 5.7: Angle of refraction vs the angle of incidence for a slot absorber in the case when
the incident wave vector is in a plane parallel with the slots (ψ = 0). The parameter values 0.1, 1, and 10 refer to the ratio a/dv of the half-width a of a channel and the viscous boundary layer thickness dv . They correspond to ω/ωv = 0.00667, 0.667, and 66.7, respectively, where ωv = r0c /ρ.
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Figure 5.8: Diffuse field absorption coefficient αav for different values of the total normalized steady flow resistance of the absorber, = r0 L/ρc = 2, 4, 8, 16. L = 10 cm, the corresponding flow resistance values per unit length are 0.4 and 1.6 ρc per cm and the frequencies 340 and 64 Hz, respectively. To boost the absorption at 64 Hz from 0.4 to 0.8, L/λ has to be increased from 0.02 to 0.1, which means that the thickness of the porous layer has to be increased from 10 cm to 50 cm and the normalized flow resistance has to be reduced from 0.4 to 0.08 per cm.
5.3 ISOTROPIC POROUS LAYER, PHYSICAL PARAMETERS The analysis of the slot absorber in the previous section has provided some insights into the physics of sound absorption due to friction and heat conduction, which will be used to a great extent in the rest of this chapter. Due to the geometrical complexity of the pores, cavities, and channels in commonly used porous materials, such as glass wool, porous metals, ceramics, foams, etc., an analysis similar to that of the slot absorber will not be attempted here. The interaction of a sound wave with a porous structure has to be described in terms of experimentally determined parameters, such as flow resistance, porosity, and structure factors, and we shall start this section by supplementing what has already been said about these quantities with definitions and comments which refer specifically to porous materials in bulk.
5.3.1 Porosity Porosity is defined as the volume fraction occupied by voids (filled with air in our case) in the porous material. In materials like foams, some, if not all, of the voids may be closed. We shall assume for the most part that the voids are open. The porosity H can be expressed in terms of the overall mass ρ per unit volume of the porous structure and the mass density ρm of the material from which the structure
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is made. With the porosity denoted by H , it follows that ρ = (1−H )ρm . For example, a glass wool material with a density of ρ = 0.064 g/cm3 (≈ 4 lb/ft3 ) with the glass fibers in the material having a density of ρm ≈ 2 g/cm3 will have a porosity of ≈ 96.8 percent.
5.3.2 Flow Resistance and Impedance The steady flow resistance of a porous layer is defined as the ratio of the pressure drop P across the layer and the average mean velocity U of the flow through the layer. In the context of acoustics, it is generally implied that the pressure drop across a test sample is low enough so that the densities, and hence the velocities on the two sides of the sample, can be considered to be the same. It should be noted that it is the mass flux ρu, which is conserved and hence the same on the two sides, and the velocities will be the same only if the change in ρ can be neglected. From this standpoint, it would have been better to define a flow resistance in terms of the mass flux rather than the velocity. The unit of flow resistance is sometimes called one ‘rayl’ by acousticians, but this is ambiguous unless the system of units is specified. Thus, one MKS unit (pressure in N/m2 and velocity in m/sec) is one-tenth of a CGS rayl (pressure in dyne/cm2 and velocity in cm/sec), and the flow resistance expressed in MKS will be 10 times the flow resistance expressed in CGS. Frequently, the flow resistance per unit thickness of a porous material is specified, and it is not unusual that mixed units are used, such as CGS or MKS per inch, for example. Typically, a porous material will have a flow resistance of about 50 CGS (500 MKS) per inch. The flow resistance in MKS units per meter is then about 20,000, which is a bit cumbersome. As in Chapter 1, a flow resistance is often normalized with respect to ρc, which is ≈ 420 MKS (42 CGS) units. Thus, the flow resistance of a porous material is often expressed in terms of the normalized value per inch, which then typically is of the order of 1, a convenient number and we use this normalization throughout the book. On a microscopic level, the flow resistance is determined by the equivalent channel width between fibers or pores and the number of such channels per unit area, which, for a given fiber diameter, is determined by the porosity of the material. Actually, for a given porosity, it follows from dimensional considerations that the flow resistance is proportional to the coefficient of the shear viscosity and inversely proportional to the square of the channel width (or the square of the fiber diameter) consistent with Chapter 3 and the results for the slot absorber and the data in Appendix A. In oscillatory flow, as in a sound wave, the pressure drop contains, in addition to the flow resistive component, which is proportional to the velocity and hence in phase with it, also a component, which is proportional to the acceleration, which is 90 degrees out of phase with the velocity. The flow resistance is now replaced by an impedance, the complex ratio of the complex amplitudes of the pressure drop and the velocity, with a resistive and a mass reactive component. The flow impedance per unit thickness of the material was referred to simply as the impedance per unit length of the material in the analysis of the slot absorber. The contribution to this impedance, which arose from the interaction with the porous
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material, was called the interaction impedance per unit length (i.e., not including the effect of the inertia of the air alone).
5.3.3 Structure Factor In our study of sound propagation in a narrow channel in Chapter 3, we noted that the viscous interaction force in oscillatory flow is not in phase with the velocity, containing not only a component proportional to the velocity but also one proportional to the acceleration. The ratio of the corresponding pressure drop per unit length in the channel and the average velocity amplitude was defined as the interaction impedance per unit length. The real part is the flow resistance per unit length. The reactive component could be interpreted in terms of an increase of the inertial mass density of the fluid in the channel expressed as Gv ρ. In a straight channel between two parallel walls, the low frequency limit for Gv was found to be 0.2 and for a circular tube, 0.33.2 There is another, usually more significant inertial effect and a corresponding mass reactance, due to the fact that the air suffers accelerations in a more or less random manner as it is forced to follow a tortuous path through the pores of the material. These accelerations lead to momentum transfer to the structure (as in a turbine) and a corresponding reaction force back on the air. In describing the motion of the air in terms of its average (and one-dimensional) rather than local velocity, this interaction with the structure can be accounted for in terms of an increase in the inertial mass density of the air. The increase is expressed as Gs ρ, where Gs is the induced or virtual mass factor, typically between 0.3 and 1. This phenomenon of an ‘induced mass’ is familiar from the apparent increase in mass when we accelerate a body in water (walking or moving an arm back and forth in water, for example). The induced mass of an oscillating sphere in water is known to be half of the mass of the water displaced by the sphere. Together with the viscous contribution to the reactance, the total virtual or induced mass factor is then G = Gs + Gv . The total induced mass density is then Gρ, and the corresponding equivalent mass density is = (1 + G)ρ, where is called the structure factor.
5.3.4 Mass Density of a Porous Material For a rigid material, its mass does not affect the dynamics of the air in the material and its effect is merely indirect in as much as it is related to the porosity, the flow resistance, and the structure factor. Only for a flexible material will the mass density have a direct effect, at least at sufficiently low frequencies, typically below 200 Hz, where the interaction with the sound wave will induce oscillatory motion in the material. The degree of excitation of the material increases with the ratio of the flow resistance and the mass density. The motion of the material will influence both the resistive and reactive parts of the flow impedance and the corresponding structure factor. For a further discussion we refer to Chapter 6 and the analysis of the flexible sheet in Chapter 2 is also relevant. 2 The fact that the velocity profile in a pore or channel will be nonuniform because of viscosity increases the kinetic energy of the flow in the channel for a given average volume flow rate.
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5.3.5 Compressibility For a rigid porous structure, the compressibility or compliance of the porous frame is assumed to be zero and is of no particular interest in the analysis in this chapter. For a flexible material, however, it will play a role as the coupling with the sound field will induce volume changes of the material as discussed in Chapter 6. The compressibility of the air in the porous material, however, plays an important role and to account for heat conduction, it is expressed in terms of a complex number, as will be discussed later.
5.3.6 Discussion The quantities thus described are not all independent, as already mentioned. The most important quantity from an acoustical standpoint is the flow resistance, and it can vary greatly from one material and configuration to the next. The porosity, on the other hand, does not vary much amongst commonly used materials and typically is 90 to 98 percent. In some special cases, for example for stacks of wire mesh screens used as porous material, the porosity may go down to 30 percent in some cases. For a foam-like material, the cells may not all be open (interconnected), and if one wishes to compare the relationship of flow resistance to porosity in such a material, one should use only that part of the porosity that corresponds to the open cells. Frequently, a porous material has to be used in environments with high temperatures and/or high flow velocities, chemically aggressive gases, soot, oil, and water, etc. Under such conditions, one has to consider not only the flow resistance but also several other nonacoustical factors, which often limits the choice of materials.
5.4 WAVE MOTION The amount of fluid per unit volume of the porous material is Hρ, where H is the porosity. The average velocity u in the material can be defined in such a way that Hρu is the average mass flux. Another definition is based on a mass flux being expressed as ρu. We have chosen the second of these since it will make the equations and boundary conditions somewhat simpler. As described above, the interaction between the fluid and the porous structure involves a viscous part and a part due to the momentum transfer to the structure as the velocity of the fluid will change in its tortuous path in the interstices of the structure. The resistive part is expressed by the flow resistance r and the reactive part by ρe ω, where the equivalent inertial mass density is ρ and , the structure factor introduced above.
5.4.1 Propagation Constant The motion of the air in the porous material is such that the sum of the friction force and inertia force must be balanced by the pressure gradient. The inertia force is the equivalent mass density multiplied by the acceleration and has the magnitude ρωu, where u is the (average) velocity. If the resistance can be neglected (inertial regime)
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the wave motion is much like that in free field, √ except for the increase in the mass and a corresponding decrease by the factor in the average speed. Actually, if heat conduction in the material prevents temperature fluctuations to be created in the compressions and rarefactions so that the conditions will be isothermal, the compressibility of the air in the material will be higher than in free field. This √ contributes to a further decrease in the wave speed, approximately by a factor γ , where γ ≈ 1.4 is the specific heat ratio (for air). On the other hand, if the friction force dominates so that the inertial force can be neglected, the resistive force is balanced by the pressure gradient. In that case, the wave motion degenerates into diffusion so that the penetration of a sound wave into a porous material is similar to the penetration of temperature fluctuations into a material, which has a periodic heat source at the surface. A close analogy to such a change in the character of motion is a harmonic oscillator in which the friction is increased to the point that the oscillator becomes overdamped; the free motion no longer is oscillatory but instead exponentially decaying. Whether or not there is inertial or resistance domination depends on the ratio of the resistance and the inertial reactance, r/ωρe = θc/ ω, where r is the resistance per unit length, θ = r/ρc, the corresponding normalized value, ω the angular frequency, c, the sound speed, ρ, the density of air or whatever fluid is involved, and , the structure factor. Thus, the transition between the resistive and inertial regimes will occur at ω ≈ θ c, and we call the corresponding frequency f = (1/2π )(θ c) the transition frequency in the porous material. Thus, the resistive and inertial regimes correspond to frequencies much lower and much higher than f , respectively. As an example, we note that for a porous material with a normalized flow resistance of 0.5 per inch (210 MKS units per inch or 8400 MKS per meter) and a structure factor = 1.5, f ≈ 711 Hz (using a sound speed of 340 m/s or 1115 ft/sec), so that the resistive regime extends over a relatively large portion of the frequency range of interest in noise control. With reference to the harmonic oscillator analogy above, we note that in the presence of a dashpot damper and/or a viscous resistance on the moving mass, the amplitude of the oscillation will decay exponentially. Formally, this can be accounted for in terms of a complex frequency of oscillation in the factor exp(−i ωt), ˜ which expresses the time dependence of the oscillation. This complex frequency is then
˜ M, ˜ where M˜ is a complex mass and K˜ a complex spring constant obtained from K/ (see Section 5.7). In an analogous manner, it is convenient to introduce a complex density in the description of wave motion in a porous material. Thus, the impedance per unit length of the air in the material, −iωρ + r, is written −iωρ, ˜ which defines the density as ρ˜ = + ir/ωρ. Then, by using also a complex compressibility, κ, ˜ the results obtained for wave propagation in free field can generally be applied directly to sound propagation in a porous material. In this context, a comment should be made about the complex compressibility. In connection with Eq. 2.9, the complex compressibility denoted by κ˜ av2 gave the explicit frequency dependence of the compressibility in terms of the thermal relaxation frequency ωh . We note that this relaxation frequency is very nearly equal to the
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viscous relaxation frequency ωv = r/ω, where r is the flow resistance per unit length. As a good approximation in the study of sound propagation in a porous material we shall use for κ˜ the expression for κ˜ av2 , with ωh in Eq. 2.9 replaced by ωv . Then, in terms of ρ˜ and κ, ˜ we can express the quantities involved, such as the propagation constant and wave impedance. x-component of propagation constant, Wave impedance qx ≡ (ω/c)Qx = q 2 − qy2 − qz2 = q 2 − k 2 sin2 φ = (ω/c) Q2 − sin2 φ Q ≡ q/k ≡ Qr + iQi = (ρ/ρ)( ˜ κ/κ) ˜ ζw = (ρ/ρ)/Q ˜ x
(5.3)
where ρ/ρ ˜ = + ir/ωρ, κ/κ: ˜ Eqs 2.9, 5.44, φ: Angle of incidence, k = ω/c, ζw : Wave impedance. (See also Eq. 5.38.) The real and imaginary parts of Q are shown in Figure 5.9. The real part, Qr , is the ratio of the sound speed in free field and the wave speed in the material. The imaginary part, Qi , expresses the x-dependence of the sound pressure amplitude through the factor exp(−kQi x), which, in a distance of one wavelength, is exp(−2π Qi ). As an example, consider a typical material with r0 = 0.25ρc per inch (≈ 0.1 ρc per cm). With c ≈ 34000 cm/sec, we get fv ≈ 3400/2π ≈ 541 Hz. Thus, at a frequency of 54 Hz (f/fv = 0.1), we have Qr ≈ 3 so that the phase velocity in the material is about one-third of the free field sound speed. The corresponding value of the imaginary part is Qi ≈ 2.8, and the attenuation exp(−2π Qi )/λ per free field wavelength λ is ≈ 2.1 × 10−8 or about 153 dB. At 54 Hz, the wavelength is ≈ 6.3 m and the attenuation per meter is then ≈ 24 dB. √ The wave impedance in free field is ρc = ρ/κ, and to get the wave impedance in the porous material we use ρ˜ and κ˜ instead of ρ and κ.
5.4.2 Penetration Depth In the context of this general discussion, the concept of penetration depth dp of sound in a porous material is useful. At this depth, by definition, the pressure amplitude has decreased by a factor e ≈ 2.73 corresponding to ≈ 8.7 dB, and it follows that dp = 1/(kQi ) = λ/(2πQi ), where Qi is given in Eq. 5.3 and in Figure 5.9. The penetration depth, normalized with respect to λ = c/f , is plotted in Figure 5.10 vs the normalized frequency f/f , where f = r/(2π ρ). In the inertial regime, with f >> f , dp /λ would have been independent of frequency if the flow resistance had been independent of frequency. However, due to √ the increase of the flow resistance with frequency in this region, approximately as f , dp decreases accordingly, as shown. For frequencies below f , the resistance is essentially constant and equal to the value for steady flow. If the thickness equals the penetration depth, the absorption of a porous layer will be nearly the same as for an infinitely thick layer so from that standpoint it makes little sense to make a layer thicker than the penetration depth. This is a useful observation in the design of an absorber.
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Figure 5.9: The real and imaginary parts Qr and Qi of the normalized propagation constant
(Eq. 5.37) in a rigid porous material vs the normalized frequency f/fv , where fv = r0 /2πρ. The flow resistance per unit length in the material is r0 and ρ is the density of air. Porosity H = 0.95, structure factor s = 1.3.
Figure 5.10: Normalized penetration depth dp /λ vs normalized frequency f/f , where f = r/2πρ is the frequency at the transition between the resistive and inertial control of the wave motion in the material and λ = c/f is the corresponding wavelength.
As an example, consider a material with a normalized flow resistance of θ = 0.25 per inch. The characteristic frequency is then f = θc/2π ≈ 357 Hz, assuming = 1.5. The corresponding wavelength is λ = 37.6 inches. From Figure 5.10 it follows that at the frequency f we have dp /λ ≈ 0.3 so that penetration depth is then dp ≈ 0.3λ ≈ 11.3 inches. Refraction In regard to the refraction of sound as it enters a porous material, nothing needs to be added here in the general description beyond what has already been said in connection with the slot absorber, and we merely refer to Section 5.7 for details.
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5.5 ABSORPTION SPECTRA 5.5.1 Infinite Layer If the thickness of porous layer exceeds the penetration depth, the absorption coefficient will be approximately the same as for an infinitely thick layer and is the best that can be obtained with a single uniform porous layer. The input impedance of the absorber is now simply the wave impedance ζw in Eq. 5.3, and the absorption coefficient is obtained from the expressions in Chapter 2. The computed normal incidence and diffuse field absorption coefficients are shown in Figure 5.11. The normal incidence and diffuse field absorption coefficients for an infinitely thick layer are shown on the left and on the right, respectively, in Figure 5.11 for flow resistances ranging from 0.125 to 4.0 per inch (normalized with respect to ρc). It should be realized that in this case, the absorption coefficient increases with decreasing flow resistance, as expected. Although the thickness is infinite and with a flow resistance different from zero, there will always be some reflection from the surface of the material because of the discontinuity in impedance. A noteworthy property of the infinite layer is that in the low frequency regime, the reactive part of the input impedance is found to be 1/H γ kdp . This can be interpreted as the stiffness reactance of an isothermal layer with a thickness equal to the penetration depth dp and backed by a rigid wall. It remains to comment on the angular dependence of the reflection and absorption coefficients. It is sufficient to use the infinite layer for illustration since the results for finite layers are qualitatively quite similar. The example in Figure 5.12 refers to a nonlocally reacting infinite layer with a porosity H = 0.95 and a structure factor s = 1.3. Two important features should be noted. First, for sufficiently low frequencies, in the resistance controlled regime, the input impedance is relatively large and, since the wave impedance ‘in the normal direction’ is ρc/cosφ for the incident wave, the best impedance match will occur at an angle of incidence φ different from zero and it increases with increasing input impedance. Second, the absorption
Figure 5.11: Normal incidence (left) and diffuse field (right) average absorption coefficients of a nonlocally reacting rigid porous layer of infinite thickness. Starting from the top, the curves refer to the flow resistances 0.125, 0.25, 0.5, 1.0, 2.0, and 4.0 per inch, normalized with respect to ρc.
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Figure 5.12: The angular dependence of the absorption coefficient of an infinite porous layer for f/fv = 0.01, 0.1, 1, and 100, where fv = r0 /2πρ. At a frequency f = 100 Hz, the curves, from bottom to top, correspond to flow resistances ≈ 4.7, 0.47, 0.047, and 0.0047 ρc per inch. coefficient goes to zero as the angle of incidence approaches 90 degrees (grazing incidence). The corresponding pressure reflection coefficient then goes to −1. This has an important consequence in regard to the sound field distribution from a source above an absorptive boundary. Since the reflection coefficient approaches −1 for sound of grazing incidence, the direct and reflected waves will tend to cancel each other making the sound pressure close to the boundary essentially zero.
5.5.2 Finite Layer For a porous layer of finite thickness, the stiffness of the air in the layer tends to limit the absorption at low frequencies, i.e., at wavelengths larger than the thickness, and at low flow resistances, ‘quarter wavelength resonances’ are apparent, well-known from the earlier discussions of the sheet/cavity absorber. The absorption characteristics are obtained in the same manner as for the infinite layer, except that the input impedance now depends on the layer thickness. The results are not much different from those of the infinite layer, particularly if the layer thickness is greater than the penetration depth. Having obtained the input impedance, the absorption spectra follow from the formulas in Chapter 2. For further computation details, we refer to Section 5.7 and proceed with the presentation and discussion of numerical results. Normalized input impedance ζi = iζw cot(Qx kL)[≈ /3 + i/(H γ kL) for kL << 1]
(5.4)
where Qx , ζw : Eq. 5.3, L: Layer thickness, : Total flow resistance, H : Porosity, γ = 1.4. (See also Eq. 5.45.) Thus, Figure 5.13 shows the normal incidence and diffuse field absorption coefficients of a rigid, locally reacting porous layer for some values of the normalized total flow resistance of the layer. The frequency variable is L/λ, where L is the
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thickness of the layer and λ the wavelength. The structure factor has been chosen to be s = 1.3 and the porosity H = 0.95; the results depend only weakly on these variables in the ranges of practical interest for s and H , say 1.3 to 2 and 0.9 to 0.98, respectively. Results are shown for values of the total layer resistance from 2 to 32 (normalized with respect to ρc). For less than 2, the absorption is lower over the entire frequency range, except possibly at higher frequencies, at quarter wavelength resonances, which become noticeable for low flow resistances. It should be noted for greater than 2, approximately, the diffuse field average absorption coefficient will be larger than the normal incidence value at low frequencies. The reason is the same as that given for the infinite layer. For each value of L/λ there is an optimum value of the total flow resistance to yield maximum absorption, and this is of particular interest at low frequencies. For example, for L/λ = 0.035, the optimum value is ≈ 8 with a corresponding maximum absorption coefficient of ≈ 0.4. By connecting the maximum values of α for different values of L/λ, we obtain an envelope to the set of curves in the figure. This envelope, together with the corresponding optimum flow resistance, was shown in Figure 5.1 in the summary at the beginning of this chapter. For a layer of finite thickness L, the low frequency input impedance is the same as that of a rigid sheet/cavity absorber in which the flow resistance of the sheet is /3, where is the total flow resistance of the layer, and the reactance corresponds to that of an air layer with isothermal compressibility. We refer to Section 5.7 for support of these observations. For a nonlocally reacting layer, the normal incidence absorption spectra are, of course, identical with those for the locally reacting layer in Figure 5.13, and, therefore, only the diffuse field spectra are shown in Figure 5.14. There is little difference between the performance of the two types of layers for large flow resistances, say > 1. The reason is that the refraction makes the direction of propagation in the material essentially normal to the boundary. This is not the case for < 1, and the
Figure 5.13: ‘Universal’ normal incidence and diffuse field absorption spectra of a locally reacting rigid porous layer of thickness L backed by a rigid wall. The frequency parameter is L/λ, where λ is the free field wavelength. The parameter is the total steady flow resistance of the layer, normalized with respect to ρc, where ρ is the air density and c the free field sound speed.
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absorption coefficient for the nonlocally reacting layer then does not decrease monotonically with decreasing , as is the case for local reaction. Rather, for sufficiently small values of L/λ, say less than 0.03, there will be an optimum flow resistance for each value of L/λ as can be seen from the curves in the figure. Again, this is related to refraction. For large values of the resistance, the normal velocity component in the material accounts for the major part of the dissipation but as the resistance decreases, the contribution from the tangential component increases and becomes dominant. Consequently, at low frequencies, there will be two optimum values of the total flow resistance, one large and one small. Note on Measured Diffuse Field Absorption Coefficients It should be noted that in the reverberation room method, the quantity that is measured is the absorption area (absorption cross section) of a test sample. Due to diffraction, this area can be larger than the physical area of the test sample and the corresponding absorption coefficient, obtained by dividing the absorption area by the physical area, and it can be greater than unity. Compare the discussion of the absorption cross section of a single resonator in Chapter 4 and that for the circular porous sheet or disc in Chapter 2. The calculated diffuse field absorption coefficient is based on the assumption of an infinite test sample (no diffraction effects) and is expected to agree with experimental data, at least approximately, only at high frequencies where the wavelength is small compared to the dimensions of the test sample.
5.5.3 Examples and Comments 1. Absorption coefficient vs flow resistance for rigid porous layer The absorption spectra of a rigid porous layer are shown in the text for different values of the total flow resistance of the layer. In some applications it is of interest to
Figure 5.14: ‘Universal’ curves for the diffuse field average absorption coefficients of a nonlocally reacting rigid porous layer of thickness L backed by a rigid wall. The frequency parameter is L/λ, where λ is the free field wavelength. The parameter is the total normalized steady flow resistance of the layer. Left: = 0.0625 to 2 (log-scale for absorption coefficient). Right: = 1 to 32 (linear scale for absorption).
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see the absorption coefficient at a specified frequency plotted against the resistance so that optimum flow resistance can easily be spotted. Make such a plot for a 4 inch thick rigid porous layer at the frequencies 100 and 1000 Hz and discuss the results. SOLUTION Using typical values of the porosity and structure factor, 95 percent and 1.3, respectively, we obtain the curves shown in Figure 5.15. At low frequencies, the stiffness limits penetration of sound into the material and a total resistance of the layer comparable with the reactance is needed to optimize the performance, and this resistance then becomes relatively high. In this case, the optimum resistance at 100 Hz for normal incidence is close to 2.5 per inch and for diffuse field close to 3 for both D1 and D2, the maximum values of the absorption coefficient being 0.34 and 0.48, respectively. For N and D1, the absorption coefficient decreases monotonically with decreasing flow resistance. For high flow resistances, the curves for D1 and D2 are essentially the same since the refraction of sound in the porous layer makes the wave in the material normal to the boundary even for D2. At low flow resistances, however, the curves are different. This is because there is a component of velocity in the layer parallel with the boundary in D2, and the corresponding friction losses contribute to the absorption. The relative importance of this component increases with decreasing flow resistance and at a certain flow resistance, in this case ≈ 0.05 per inch, the combined effects of the normal and tangential components lead to a maximum absorption coefficient. At higher frequencies, the reactance plays a decreasingly less important role and the reflection of sound from the layer is determined largely by the resistive component of the input impedance. In this case, at 1000 Hz, the optimum resistance is close to 0.4 for normal incidence and 0.5 for diffuse field. The small difference between the results for D1 and D2 occurs at low flow resistances. The result obtained can be made ‘universal’ if the flow resistance is expressed in terms of the total flow resistance of the layer and the frequency expressed in terms of d/λ, where d is the layer thickness and λ, the wavelength. Thus, for an 8 inch layer, the results are valid for a material with half the flow resistance on
Figure 5.15: Absorption coefficient vs flow resistance. Layer thickness: 4 inches.
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the abscissa and a frequency which is half the frequency in the figure. Thus, the maximum absorption of 0.48 at 100 Hz for the 4 inch layer at a flow resistance of ≈ 3 is obtained at a frequency of 50 Hz and a flow resistance of ≈ 3/2 for an 8 inch layer.
5.5.4 Effect of a Perforated Facing, Its Nonlinearity and Induced Motion A porous layer of absorption material is often covered with a perforated plate (usually to provide a protective facing and/or confinement), and it is of considerable practical interest to determine to what extent this facing affects the absorption characteristics of the layer. In the following discussion, we shall use the terms perforated facing and perforated plate as synonyms, and the open area fraction of the plate is denoted by s. We assume the porous layer to be rigid. Therefore, in the discussion of the effect of the acoustically induced motion of the facing, it is implied that there is ‘loose’ contact between the facing and the layer to allow for an acoustically induced motion of the facing. As discussed in Chapters 4 and 8, the normalized nonlinear specific resistance of an orifice in the perforated plate is expressed as (1 − s)|u0 |/c, where u0 is the amplitude of the velocity of the air in the orifice relative to the orifice plate. There is a small nonlinear effect on the mass end correction of the orifice, but this will be ignored (see Chapter 4 for details). From this information, the absorption coefficient can be calculated, including the acoustically induced motion of the plate and a numerical example as shown in Figure 5.16. As expected, due to its mass reactance, the facing causes a reduction of the absorption coefficient at high frequencies. In our example, with an open area of the facing of 25 percent, a commonly used value, the absorption is reduced at frequencies above ≈ 2000 Hz. At some lower frequency, the mass reactance will cancel the stiffness reactance of the porous layer and this results in a resonance, similar to that of a damped Helmholtz resonator, with the resonance frequency decreasing with decreasing open area, as shown. Due to this resonance, the low frequency absorption will be greater than for the bare layer but extends only over a relatively small frequency range about the resonance. The effects of acoustic nonlinearity (see Chapters 4 and 8) and induced motion of the facing are generally important only at relatively small open areas of the facing. At a level of 80 dB, the nonlinear effect is insignificant, even for an open area of 0.1 percent but at 120 dB, there is a considerable broadening of the absorption curve and also a decrease in the resonance frequency. The latter is due to the induced motion of the plate. At a level of 140 dB, the width of the absorption curve is not increased further; on the contrary, it is reduced. The reason is that the increased induced plate motion has caused a decrease in the relative motion between the air and the plate and the corresponding effective resistance of the plate is reduced. The resonance is now largely determined by the mass of the plate and the stiffness of the backing layer. The low frequency value of the normalized stiffness reactance of the layer is ≈ 1/H γ kL (see Eq. 5.46) and the normalized mass reactance of the plate is ωm/ρc,
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Figure 5.16: The diffuse field absorption spectra of a uniform 4 inch thick nonlocally reacting porous layer with a perforated facing. In each graph, the curves (starting from the left) refer to open areas of 0.1, 1, 5, 25, and 100 percent. The thickness of the facing is 0.1 inch, the hole diameter, 0.1 inch, the weight, 4 lb/ft2 , the flow resistance of the layer, 0.5 ρc. The sound pressure levels are 80, 120, and 140 dB, as indicated. so that √ the resonance frequency of this mass-spring oscillator becomes f = ω/2π = (c/2π) ρ/mH γ L. In this case, with m = 4 lb/ft2 (≈ 1.94 g/cm2 ) and L = 4 inches (≈ 10.2 cm), this frequency becomes f ≈ 37 Hz, which is consistent with the location of the absorption peak in the lower plot of Figure 5.16. The bandwidth of this massspring resonance is quite narrow, since the losses in the system are reduced because of the reduced air velocity with respect to the (moving) plate.
5.5.5 Effect of a Screen on a Porous Layer Instead of the perforated plate, we now consider a limp sheet or screen as a cover for a uniform porous layer. In practice, the cover may or may not be in good contact with the porous layer. For this reason, we consider here the two extreme cases when a screen is adhered to the layer and when it is at a small distance from it, so that acoustically induced motion of the screen is possible. The effect on the absorption coefficient of a screen adhered to the layer is illustrated in a special case in Figure 5.17. It involves an 8 inch thick porous layer with a flow resistance of 0.25 ρc per inch and a screen with flow resistances 1, 2, or 4 ρc. It should be emphasized that the porous layer is assumed to be rigid, so that there can be no acoustically induced motion of the screen. We note from Figure 5.17 that the cover screen, even in hard contact (no induced motion of the screen), yields
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an improvement of the absorption coefficient at low frequencies at the expense of a reduction in absorption at the highs, although for the lowest screen resistance shown in the figure (namely 1.0), this reduction is insignificant for the diffuse field absorption coefficient. If there is a small air space between the screen and the layer so that induced motion of the screen can occur, this motion and the corresponding inertial reactance in combination with the stiffness reactance of the air in the porous layer leads to a resonance and a corresponding increase in the absorption coefficient over a small range about the resonance frequency. However, at frequencies sufficiently far below the resonance, the induced motion tends to decrease the absorption coefficient in comparison with the result obtained for a screen in hard contact. This behavior is illustrated in Figure 5.17 for two different values of the mass of the screen, 0.1 and 1 lb/ft2 (≈ 0.05 and 0.5 g/cm2 ). For the lighter screen, the resonance frequency referred to above occurs at about 160 Hz. As the mass increases, the resonance frequency decreases and the resonance is broadened. These results refer to the diffuse field absorption coefficient of the right-hand plot. Typically, the mass of a screen of woven fabric is only of the order of 0.03 g/cm2 (10 oz/yd2 ), and the use of such a material would result in an absorption curve
Figure 5.17: The absorption coefficient of a rigid porous layer (nonlocal) covered with a thin resistive screen in hard contact with the layer. Top: Hard contact. Bottom: Loose contact. Left: Normal incidence. Right: Diffuse field. Layer thickness: 8 inches. Flow resistance: 0.25 ρc per inch. Flow resistances of screen: 0 (no screen), 1, 2, and 4 ρc.
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Figure 5.18: The normal incidence (N) and diffuse field (DNL) absorption coefficients of two configurations of a nonuniform, nonlocally reacting absorber consisting of three 4 inch thick porous layers with flow resistances 0.4, 0.2, and 0.1 in the first configuration and 0.1, 0.2, and 0.4 ρc per inch, in the second, as shown.
essentially the same as for 0.05 g/cm2 , i.e., considerably different from the result, which would have been obtained if the screen had been in hard contact as in Figure 5.17.
5.5.6 Nonuniform Porous Absorbers In Chapter 1, we analyzed characteristics of lattice absorbers consisting of resistive thin screens separated by air layers and backed by a rigid wall. For a uniform lattice of such elements, it was possible to obtain a relatively simple closed form expression for the absorption coefficient in terms of the characteristics of a screen element, the separation of adjacent screens, and the number of screens. However, for a nonuniform lattice, we had to resort to direct numerical calculations involving the multiplication of the matrix elements of the screen-air layer combinations of the lattice. This approach, of course, can be used for a more general multilayer absorber involving other elements such as uniform porous layers, membranes, screens, perforated plates, air layers, etc., for which the transmission matrices are known. As a first example, we consider an absorber made up of three 4 inch thick porous layers with different flow resistances, 0.4, 0.2, and 0.1 ρc per inch and placed against a rigid wall. The example serves to illustrate the effect of the ordering of the layers. Thus, Figure 5.18 shows the calculated normal incidence and diffuse field average absorption coefficients for the configurations in which the flow resistances of the layers are 0.4, 0.2, 0.1, and 0.1, 0.2, 0.4, respectively. There are no honeycomb-like partitions in the layers so that the absorber is nonlocally reacting. There is a significant difference in performance of the two configurations. In the first, the outer layer has the highest flow resistance and the inner layer, the lowest. In the second, the order is reversed so that the incident sound first encounters the layer with the lowest flow resistance, and the impedance discontinuity is reduced. In this respect it simulates the behavior of a porous wedge absorber.
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Figure 5.19: The normal incidence (N) and diffuse field (DNL) absorption coefficients of a nonlocally reacting 12 inch thick layer with a normalized flow resistance of 0.2 per inch, i.e., same total thickness and flow resistance as the nonuniform absorbers in Figure 5.18.
Except for low frequencies, in this case below 150 Hz, the second configuration yields the best overall performance. Actually, it meets the typical criterion for an anechoic absorber at normal incidence, αn ≥ 0.99 (corresponding to a pressure reflection coefficient ≤0.1), at all frequencies above 262 Hz. It should be noted that in both cases αst > αn at low frequencies, which is analogous to the results discussed earlier in this chapter for a single uniform porous layer. At low frequencies, the absorber with the high flow resistance layer in front gives the best performance. For example, at 60 Hz we have αst ≈ 0.6, whereas it is only about 0.45 for the other configuration. It is interesting to compare these results with those of a uniform 12 inch thick layer with a flow resistance of 0.2 ρc per inch, as shown in Figure 5.19. The total flow resistance of the layer is the same as in Figure 5.18. However, the critical frequency above which the anechoic criterion is met at all frequencies is now ≈ 1000 Hz, i.e., considerably higher than the value 262 Hz, which was found for the nonuniform layer in Figure 5.18. Porous Layer Backed By An Air Cavity An air layer is sometimes used between a porous layer and the rigid backing wall. In order to determine its effect, we have to decide how a comparison should be reasonably made. One such comparison is shown in Figure 5.20 involving four configurations. The overall thickness of each is the same but the thickness of the porous layer is varied keeping the total flow resistance constant, in this case 2 ρc. It is close to the optimum value at normal incidence for a uniform layer without an air cavity. The normal incidence and diffuse field absorption curves obtained for an 8 inch thick porous layer with a flow resistance of 0.25 ρc per inch with no air layer is compared with those obtained with a porous layer with air layer backing. Three
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thicknesses of the air layer are considered, 2, 4, and 6 inches. The corresponding thickness of the porous layers are then 6, 4, and 2 inches. The results are shown in Figure 5.20. It is noteworthy that for normal incidence, the influence of the air space on the absorption coefficient is small at low frequencies with the absorption coefficient increasing somewhat with thickness of the air layer. At higher frequencies this dependence is reversed. The first minimum in each curve is close to the half wavelength anti-resonance of the cavity. For the diffuse field average absorption coefficient, on the other hand, the influence of the air space is more pronounced as the performance is improved with decreasing air layer thickness at all frequencies. This is due to the nonlocal character of the absorber. With a partitioned air backing, the results would not be much different than for normal incidence. If the flow resistance r per unit length is kept the same in each case rather than the total layer resistance, there will be a considerable difference in performance of the four configurations depending on the value of r. For example, if the resistance is optimized for the 8 inch uniform layer (no air space) with a value of ≈ 0.25ρc per inch (corresponding to a total resistance of 2 ρc), the performance of this configuration would be considerably better than for a layer with a thickness of 2 inches, which would have a total resistance of 0.5.
5.5.7 Sheet Absorbers vs Uniform Porous Layer Alternatives to fibrous material (e.g., fiberglass) as materials for sound absorption include various other types of materials such as porous metals, ceramics, plastic foams, etc., and if the flow resistance of these materials is properly chosen, the absorption can be made close to that of fiber materials. However, concerns have been raised about
a
b
c
d
Figure 5.20: The absorption coefficient of a nonlocally reacting porous layer backed by an air cavity and a rigid wall. The overall thickness of the layer-cavity combination is 8 inches and different layer thicknesses, 2, 4, 6, and 8 inches have been considered. In each case the total flow resistance of the layer is the same, namely 2 ρc.
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8”
8”
Figure 5.21: Total flow resistance of each absorber is 2 ρc and thickness 8 inches. The three curves in each graph refer to normal incidence (N), diffuse field, local reaction (DL), and diffuse field, nonlocal reaction (DNL). possible toxic fumes emitted by some foams when burning and in regard to the cost, also porous ceramics may not be good alternatives. The use of thin porous sheets, woven materials, wire mesh screens, sintered metals, perforated plates, and the like, may be viable possibilities, however, and it is of considerable practical importance to compare the absorption characteristics of single or multisheet absorbers with those of a uniform porous layer. It is clear, that a uniform porous layer can be regarded as a collection of closely spaced porous sheets, and if we built absorbers out of a sufficiently large number of sheets, we would expect to be able to achieve the same absorption as for a uniform porous layer. But the cost and weight and mounting problems are factors that can limit the number of sheets, and we shall consider here only absorbers with a small number thereof.
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Thus, in Figure 5.21, the absorption characteristics of an 8 inch thick uniform fiberglass layer backed by a rigid wall (bottom figure) are compared with those of equally thick sheet absorbers. In the top figure, only a single sheet backed by an 8 inch air cavity is involved. In the next, the absorber has 4 uniformly spaced sheets, 2 inches apart, and the bottom graph refers to the uniform porous layer, as mentioned. In each case, the total flow resistance is 2 ρc. In other words, the single sheet has a flow resistance of 2 ρc, each of the 4 sheets has a resistance of 0.5 ρc, and the porous layer a flow resistance of 0.25 ρc per inch, which is typical for fiberglass with a density of about 1-2 lb/ft3 . In the figure, a partitioned air backing is shown, which applies to a locally reacting absorber. Removal of the partitions make the absorber nonlocally reacting. For a uniform porous layer, there is generally little difference between the two diffuse field absorption coefficients. The porous material automatically provides its own partitions, so to speak, at least when the flow resistance is sufficiently high. The three curves for the porous layer (the bottom graph in the figure), starting from the top (at frequencies above 200 Hz) refer to normal incidence, diffuse field, with partitions, and diffuse field, without partitions, respectively. As can be seen, there is relatively little difference between the two diffuse field absorption coefficients. For the single sheet absorber, the first quarter wavelength resonance occurs at ≈ 420 Hz and the maximum of the octave band absorption occurs in the corresponding band. The anti-resonances occur at every multiple of 840 Hz (air layer thickness is an integer number of half wavelengths). This is in part responsible for making the absorption coefficient at higher frequencies lower than for the uniform layer. The most significant difference in performance occurs for the nonlocally reacting sheet absorber (no partitions) with a significant loss of absorption in a diffuse field. The case of no partitions is an idealization and somewhat unrealistic since in practice there are always some kind of partitions to support the sheets. The separation of them typically might be 1 ft. Under such conditions, the experimental data typically are closer to the diffuse-field-local values (DL) at low frequencies and to the diffusefield-nonlocal values (DNL) at high frequencies. For the single sheet absorber the DL absorption coefficient at low frequencies is somewhat better than for the uniform porous layer (at 100 Hz the values are 0.64 and 0.53, respectively). At high frequencies, however, between 300 and 8000 Hz, the absorption is not as good as for the porous layer. Nevertheless, it is better than 70 percent in the entire range between 125 and 8000 Hz. As the number of sheets is increased, the absorption approaches that of the uniform porous layer, as can be seen from the results for the absorber with four sheets. The improvement in absorption with an increasing number of sheets is mainly in the high frequency region.
5.6 EFFECT OF REFRACTION IN GRAZING FLOW In our previous discussion of sound absorption by a material on a wall, it was tacitly assumed that there was no mean motion of the air along the wall. Although this is the norm, there are cases, as in a wind tunnel, where the flow velocity can be large enough to significantly affect sound absorption by a wall treatment. A typical example involves the study of the (free field) sound emission from a sound source (such as a
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propeller) in a wind tunnel where reflections from the walls make the simulation of free field conditions very difficult, if not impossible.3 An absorptive wall treatment is called for and normally could make the space ‘anechoic.’ However, in the presence of flow it is important to determine to what extent the effectiveness an absorptive treatment might be affected by the flow. This is the problem to be considered in this section. The particular application, which led to this study, concerned the acoustic evaluation of a model of a propfan for a high by-pass jet engine. Measurements of the radiation field from the propeller were to be carried out in a wind tunnel with a test section approximately 8 ft in diameter and a flow Mach number of 0.8. The boundary layer thickness at the walls of the tunnel was known to be approximately 2 inches at this Mach number. The most important effect of the flow on sound absorption at the boundary is the acoustic refraction in the boundary layer. For sound traveling upstream, the sound is bent away from the wall and, on the basis of geometrical acoustics, total reflection can occur so that the sound does not reach the boundary. Actually, even under this geometric condition of total reflection, sound penetrates into the boundary layer in the form of an ‘evanescent’ wave, which decays exponentially with a distance normal to the boundary and will reach the boundary, although reduced in strength. Then, even though the absorption is not completely eliminated, the effectiveness of the absorption material can be reduced considerably, and it is of interest to obtain quantitative data on this reduction.
5.6.1 View Angle vs Emission Angle In determining the angle of incidence of the sound that reaches a point at the boundary, we must account for the convection of sound by the mean flow. In this context, we are mostly concerned with the effect of flow on the angle of incidence of the sound that reaches a given point at the boundary. To understand this effect qualitatively, we refer to Figure 5.22 and consider a sound pulse, which leaves the actual source S at time t = 0 to form a spherical wave. At t = 0 the actual source is in the center of the sphere. However, the center as well as the entire sphere drifts with the flow and by the time the wave front reaches the point of observation O at the boundary, the center has reached the point E. Thus, the angle of incidence of the wave as it reaches O is determined by the line from E to O, where E is the equivalent emission point. To determine the location of E, we note that the time it takes for the flow to go from S to E is the same as it takes for sound to go from E to O (relative to the flow). Another way of looking at this problem is to note that if the sound emitted from the source S is to reach O, it cannot be emitted in the direction SO because the flow induced drift will land the sound on the downstream side of O. Therefore, the sound has to be aimed in an upstream direction to reach O, and this direction has to be parallel with EO. With reference to Figure 5.22, we introduce a view angle φv and an emission angle φ. The view angle is defined by the direction SO and the emission angle by EO. These angles are measured with respect to the direction of the flow, 3 It should be pointed out in this context that an acoustic intensity probe is not applicable in the presence of (high speed) flow.
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RIGID POROUS MATERIALS
Figure 5.22: Definitions of view angle φv and emission angle φ and the relation between these angles for different Mach numbers M.
i.e., they are both zero in the direction of the flow and 180 degrees in the direction against the flow. To determine the relation between the view angle and the angle of incidence is a strictly kinematical problem. Only the flow component in the plane of propagation of the sound is relevant in this context, and we denote this component by U . If the source is located at a height H above ground, the distance S − E is H cot φv − H cot φ and the distance E − O is H / sin φ. The flow drift time from S to E is obtained by dividing the former by the flow velocity U , and the propagation time from E to O is obtained by dividing the latter by the sound speed c. Equating these times leads to the relation between φ and φv . View angle φv vs emission angle φ (Figure 5.22) cot φv = cot φ +
1 sin φ
or
tan φv =
sin φ M+cos φ
(5.5)
This relation is plotted in Figure 5.22 for M = 0, 0.2, 0.4, 0.6, and 0.8. As an example, we note that for a Mach number of 0.8, a view angle of 90 degrees corresponds to an emission angle of ≈ 143 degrees. It should be noted that the angle of emission is the same as the angle of incidence of the sound at the observation point O, which is defined in the same manner in relation to the direction of the flow, i.e., zero for a ray in the direction of the flow and 180 degrees in the direction against the flow.
5.6.2 The Boundary Layer If the direction of sound propagation is not normal to the mean flow, refraction occurs when the flow speed is nonuniform. In a boundary layer, the flow velocity varies from zero at the boundary to the free field value at a distance from the boundary approximately equal to the boundary layer thickness. The speed (phase velocity) of the sound varies accordingly since it is the sum of the local sound speed (relative to the air) and the component of the flow velocity in the direction of sound propagation. As a result, a sound wave entering the boundary layer will be refracted toward or away from the boundary depending on the direction of propagation.
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NOISE REDUCTION ANALYSIS
Figure 5.23: Left: Concerning the derivation of the ‘law of refraction.’ The intersection point P between the incident wave front and the boundary moves in the x-direction with the trace velocity U + (c/ cos φ). Right: Angle of refraction vs angle of incidence. To derive the ‘law of refraction’ for sound in a moving fluid, we consider the case of transmission through a velocity discontinuity (vortex sheet) as indicated in Figure 5.23. The velocities above and below the sheet are U and 0, respectively. The incident sound wave is in the xy-plane and the component of the flow velocity in the x-direction is denoted by U . Only this component influences the phase velocity of the wave and hence the refraction. The propagation vector of the incident wave makes an angle φ with the x-direction, and it is defined here as the angle of incidence. Similarly, the angle of refraction in the quiescent air is defined as φr .4 The trace velocity of the incident wave along the x-axis is the velocity of the intersection point between a wave front and the x-axis (see Figure 5.23). In the absence of flow, this velocity is c/cos φ and due to the convective effect of the flow the total trace velocity will be c/cos φ + U , where c is the sound speed and U the component of the flow velocity along the x-axis, as before. For the refracted wave the trace velocity is simply c/cos φr since the flow velocity is zero. The trace velocities on the two sides of the vortex sheet must be the same (the wave fronts are connected), and this leads to the law of refraction Law of refraction in the shear layer (Figure 5.23) cos φ cos φr = 1+M cos φ
(5.6)
where Critical angle φc : cos φc = −1/(1 + M), M = U/c. If the velocity varies continuously in a finite layer (see Figure 5.24) the trace velocity will be conserved throughout the layer, and it follows that the relation between the angle of incidence on the entrance side of the layer and the angle of refraction at the exit is given by Eq. 5.6, where U is the total change in the velocity across the layer. This is plotted for different values of the flow Mach number U/c in Figure 5.24.
4 The conventional definition of angle of incidence refers to the angle between the direction of propagation and the inward normal to the boundary. To specify the plane of incidence requires an azimuthal angle ψ. In our case this angle is either 0 or π with the corresponding values 1 and –1 for cos ψ. This sign change is automatically taken care of by using the present definition of angle of incidence.
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Ø
Ø
Ø
Real boundary layer
.
Ør
Ør
.
Ø
Ø
Ø
Vortex sheet model .
Ør
Ør
U
U U = 0.
.
Figure 5.24: Simulation of a boundary layer by a single step change in velocity as in a vortex sheet.
In the present problem, we are particularly interested in angles of incidence greater than 90 degrees, in which case sound is refracted away from the boundary. At the critical angle of incidence φc , which marks the beginning of total reflection, the angle of refraction is 180 degrees. The condition of total reflection is illustrated at the far right in the figure. For example, with M = 0.8, we get φc = 123.8 degrees. The component of the propagation vector along the normal to the boundary is k sin φr . We note that in the range between φc and 180 degrees, the magnitude of cos φr is greater than unity and sin φr = 1 − cos2 φr becomes imaginary. The refracted wave fronts in this range of total reflection are normal to the boundary and the surfaces of constant pressure amplitude are parallel with the boundary, the pressure amplitude decreasing exponentially with distance into the boundary layer (‘evanescent wave’). For a finite boundary layer thickness, the evanescent wave will reach the wall and interact with it. As a result, some sound absorption will occur and the pressure reflection coefficient will be less than unity. As we shall see in the next section, the degree of absorption depends on the angle of incidence and the ratio of the boundary layer thickness and the wavelength. If we replace the continuous variation of the velocity in the actual boundary layer with a simplified boundary layer consisting of an abrupt change in velocity (vortex sheet), the total angle of refraction will be the same regardless of the velocity profile for a given total velocity change. For the vortex sheet model of the boundary layer and from the standpoint of ray acoustics, total reflection of a ray occurs at the velocity discontinuity rather than within the boundary layer, but the critical angle of incidence for total reflection is not changed. Thus, use of the simple model of a boundary layer does not alter the essentials of the problem, but it results in a considerable simplification in the wave analytical study of the reflection process, as will be discussed next. Total reflection in geometrical acoustics from the shear layer does not mean that the sound pressure will be zero inside the layer and hence at the underlying absorptive boundary. The pressure will be reduced but not extinguished. An exponential decay of the sound pressure takes place within the boundary layer, as indicated schematically
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NOISE REDUCTION ANALYSIS
in the figure, and the sound pressure that reaches the boundary depends on the ratio of boundary layer thickness and the wavelength. With our choice of the angle of incidence, the critical angle for total reflection will be larger than 90 degrees, as can be seen in Figure 5.24, and the critical angles are found to be 146.4, 135.6, 128.7, and 123.7 degrees for the Mach numbers 0.2, 0.4, 0.6, and 0.8, respectively. The corresponding view angles obtained from Figure 5.22 are 138.8, 114.2, 98.8, and 73.6 degrees.
5.6.3 Effect on Absorption The most important effect of refraction in the grazing flow boundary layer above a sound absorptive boundary occurs when the sound goes against the flow so that the sound is refracted away from the boundary. On the basis of ray acoustics, as illustrated in Figure 5.24, there is a critical angle of incidence below which an incident ray does not reach the absorber. Actually, in such a case the wave field in the boundary layer will be evanescent, i.e., it will decay exponentially, and this decay becomes more pronounced the shorter the wavelength with respect to the boundary layer thickness, as shown in Section 5.7. Examples of the computed angular dependence of the absorption coefficient are presented in Figure 5.25. In this particular case, the wall is locally reacting and has a purely resistive impedance with the normalized resistance θ = 2. We recall our definition of the angle of incidence (Figure 5.25); φ = 0 and φ = 180 correspond to grazing incidence with and against the flow, respectively, and φ = 90 corresponds to normal incidence. In applying the boundary conditions at the shear layer, we have used continuity of normal displacement. However, the results obtained using continuity of normal velocity are not significantly different. In the absence of flow, M = 0, the absorption characteristics are symmetrical with respect to φ = 90 degrees, and since in this example, θ = 2, the maximum absorption does not occur at normal incidence, but at an angle given by cos φ = 1/θ , where φ
is measured from the normal, as we have seen earlier in this chapter. With θ = 2,
Figure 5.25: Influence of boundary layer flow on the absorption coefficient of a locally reacting boundary with normalized boundary impedance ζ = 2 + i0. Angle φ = 90 corresponds to normal incidence. M = flow Mach number.
RIGID POROUS MATERIALS
179
φ = 60 degrees, and the two corresponding angles for φ are 30 and 150 degrees, consistent with the curves for M = 0 in Figure 5.25. In the presence of flow, we expect asymmetry with respect to φ = 90, with a lower absorption in the angular region 90 to 180 degrees corresponding to propagation against the flow and refraction in the boundary layer away from the boundary. This, indeed, is what we find. The two graphs in Figure 5.25 refer to two different frequencies, as expressed by the ratio of the boundary layer thickness and the wavelength. In each of the graphs, curves are shown for flow Mach numbers M from 0 to 0.8. The graph to the left, with the lower frequency, corresponds to an incident wavelength 10 times the boundary layer thickness and the graph to the right, to 2.5 times. It is clear that the effect of the flow increases with decreasing wavelength. In a subsonic wind tunnel the boundary layer thickness might be of the order of 2 inches, and if this value is used, the two frequencies considered in the figure are 672 and 2688 Hz. On the basis of geometrical acoustics, and at a Mach number of 0.8, the critical angle for total reflection is 123.7 degrees, and the corresponding absorption coefficient should be zero in the angular range between 123.7 and 180 degrees. The actual absorption, obtained from wave theory, as given here, is small and not zero in this range but does indeed decrease with increasing frequency, approaching the prediction by geometrical acoustics. As expected, the flow causes a reduction of the absorption coefficient at angles φ above 90 degrees, particularly in the region of total reflection. For example, with M = 0.8, the absorption coefficient is reduced from 1.0 to less than 0.05 by the flow at an angle of incidence of 150 degrees. An acoustic wall treatment, such as a porous layer, generally has a reactance χ different from zero, and it is interesting to see what its effect on the absorption will be. In the case of a resonator type lining, the reactance will be stiffness-like below the resonance and mass-like above corresponding to a positive and negative value of χ , respectively. Figure 5.26 shows the angular dependence of the absorption coefficient when the normalized resistance and the stiffness reactance of the boundary impedance are both equal to 1.0 (normalized input impedance ζi = 1 + i). It is interesting to note that in this case the flow produces an increase in the absorption in the vicinity of the critical angle of incidence. The explanation is that in the evanescent region, the impedance contribution of the evanescent wave field in the boundary layer is mass-like and dependent on the angle of incidence. When this impedance is combined with the stiffness reactance of the boundary, a ‘resonance’ results and a corresponding impedance match and an increase in the absorption will occur at a certain angle of incidence.
5.6.4 Region of Total Reflection The following remarks are intended mainly as a summary. In the boundary layer of a grazing flow over a boundary, the flow velocity decreases from the free stream value to zero in a relatively short distance expressed by the boundary layer thickness. This is usually relatively small. The speed of sound along
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NOISE REDUCTION ANALYSIS
Figure 5.26: Angular dependence of the absorption coefficient of a locally reacting boundary with a normalized impedance ζ = 1 + i as influenced by the boundary layer in grazing flow with a Mach number M. The boundary layer thickness is d and the wavelength λ.
the boundary is the sum of the local sound speed and the fluid velocity. Hence, it varies with location in the boundary layer, and it can be increased or decreased by the flow dependent on the direction in which the sound enters the boundary layer. In a direction perpendicular to the flow, there is no effect on the sound speed. As a result of the variation of the resulting wave speed in the boundary layer, refraction will occur. The sound will be refracted toward or away from the boundary depending on whether the direction of wave travel along the boundary is with or against the flow, respectively. Consequently, due to refraction, the angle of incidence at the boundary wall will be different than the angle in the free stream outside the boundary layer. Since the acoustic absorption coefficient of a boundary generally depends on the angle of incidence, sound transmission through the boundary layer will affect the absorption. When the sound is refracted toward the boundary, this effect generally is rather small. When it is refracted away from the boundary, however, the effect can be significant as in the regime of total reflection discussed in connection with Figure 5.22. On the basis of ray acoustics, total reflection prevents the sound from reaching the boundary and on that basis, no absorption should occur. In reality, due to the wave nature of the sound, there will be a penetration of sound through the boundary layer, but this penetration is accompanied by an exponential decay (evanescence) of the amplitude so that the absorption will be considerably reduced as illustrated in Figures 5.25 and 5.26. This can be of practical importance, particularly in applications such as in the study of sound radiation from a source in a wind tunnel, referred to in Figure 5.27. The sound emitted from the stationary source S will be incident on the boundary at an angle, which is not the angle of the line between the point of observation and the source but a line drawn to the ‘emission point’ E, which is located downstream of the source. The distance between S and E increases with the Mach number because of the wave ‘drift’ or convection by the flow. The corresponding relation between the ‘view’ angle and the ‘emission’ angle is discussed in connection with Figure 5.22. As the Mach number increases, the emitted
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RIGID POROUS MATERIALS
U M = 0.2
E
S 106º .4
.8
Total reflection (Evanescence)
Figure 5.27: Example showing the angular region in a wind tunnel in which the sound from a stationary source S will be almost totally reflected due to refraction in the boundary layer. Mach numbers: 0.2, 0.4, and 0.8. sound will move closer to the boundary so that total reflection occurs over a correspondingly larger area of the wall. This is illustrated schematically in Figure 5.27, which shows that although the view angle can be as large as 106 degrees (for M = 0.8) the emission point is so far downstream of the source that evanescence occurs even at a point on the boundary, which is downstream of the source. The regions of the boundary, which are affected in this manner due to refraction for the different Mach numbers, are indicated by the arrows. In these regions, the effectiveness of the absorption material is reduced.
5.7 MATHEMATICAL SUPPLEMENT 5.7.1 Slot Absorber Parameter Relations ρ = ρm (1 − H ) H =
d d +h
or d =
(5.7) Hh 1−H
(5.8)
The flow resistance per unit length of a single channel is (see Eq. 2.33) r0c = 12μ/d 2 , and the corresponding flow resistance of the slot absorber is r0 = r0c /H , which can be written 12μ(1 − H )2 r0 = . (5.9) h2 H 3 Similarly, to go from the impedances of the single channel to the corresponding impedances for the slot absorber, we have to use the factor 1/H to account for the difference between the velocity in a single channel and the velocity outside the absorber which equals the average velocity in the absorber, involved in all four calculations involving pressure drops. Thus, ζw = ζwc /H ζv = ζvc /H, ζ = ζc /H, = c /H ρ˜ = ρ˜c /H.
(5.10)
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NOISE REDUCTION ANALYSIS
The propagation constant q in a single channel is the same as in the absorber (see Eq. 2.79) and for a discussion of it, or the normalized version Q = q/k, we refer to Chapter 3. Input Impedance The thickness of the slot absorber is L and the distance from the rigid backing wall is x. The pressure and velocity amplitudes in the absorber are then proportional to cos(q(L − x)) and sin(q(L − x)), respectively, and the normalized input impedance at the surface x = 0 of the absorber is ζi ≡ θi + iχi = iζw cot(qL),
(5.11)
where ζw = ζwc /H . As shown in Chapter 3, the frequency dependence of q and √other related quantities, √ is expressed in terms of the parameter ξv = a/dv = a ωρ/ 2μ. It is now convenient to use kL or L/λ instead of ω in this parameter. Then, if we express μ in terms of the steady√flow resistance per unit length in a channel r0c = 3μ/a 2 (Eq. 2.33), we obtain ξv = 3kL/2H , where = r0c L/Hρc is the total DC resistance of the layer. In terms of these quantities we have √ ξv ≡ a/dv = 3kL/2H = r0c L/(Hρc) r0c = 3μ/a 2 .
(5.12)
With ξv expressed in this manner, the normalized propagation constant becomes a function of , kL, and H , and the same holds true for the argument qL = QkL in Eq. 5.11. For sufficiently small values of the argument X = |qL|, cot(X) ≈ (1/X)(1−X2 /3), and if we use the low frequency approximations also for Q (see Eq. 2.79) and ζw , we have ζw = Q/γ (Eq. 2.85) and obtain H ζi ≈ i/γ kL − i(1/3γ )Q2 kL = i/γ kL + H /3 ζi = i/H γ kL + /3.
(5.13)
In other words, the low frequency input impedance is equivalent to that of an air cavity of length L covered with a sheet with a resistance equal to one-third of the total steady flow resistance of the absorber. It is significant that the reactive part of the impedance corresponds to the stiffness of an air layer with isothermal compressibility, i.e., it is γ times smaller than the isentropic value 1/kL. To further comment on this important fact, we note that the low frequency reactance of a cavity of depth L is ordinarily ρc2 /ωL and the normalized value is 1/kL, where k = ω/c and c the isentropic sound speed. If the conditions are isothermal, √ however, the reactance will be ρci2 /ωL = (1/γ )ρc2 /ωL, where ci = c/ γ is the isothermal sound speed. The corresponding value, normalized with respect to ρc, is then 1/γ kL, as given in the equation above.
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RIGID POROUS MATERIALS
The conditions for the validity of Eq. 5.13 are ω/ωv << 1 and |qL| << 1. The latter can √ be expressed in terms of kL and . Thus, with ω/ωv << 1, we have |Q| ≈ γ /kL and |qL| << 1 if γ kL << 1/ or ω << c/γ L. Thus, the upper highest frequency at which Eq. 5.13 is valid is the smaller of the two values ωv and c/γ L. As an example, consider a 6 inch (≈ 15 cm) thick layer with a flow resistance r0c = 0.25ρc per inch, i.e., ≈ 0.1 per cm, and = 1.5. We have ωv /2π ≈ 541 Hz and c/2πγ L ≈ 172 Hz. Thus, the condition for validity of Eq. 5.13 is f << 172 Hz. Examples of the complete frequency dependence of the input impedance has already been shown in Figure 5.3. Reflection and Absorption Coefficients In terms of the normalized input impedance ζi , the normal incidence pressure reflection coefficient follows from the well-known expression (see Chapter 2) R=
ζi − 1 ζi + 1
(5.14)
and the absorption coefficient is α = 1 − |R|2 =
4θi . (1 + θi )2 + χi2
(5.15)
Results have already been shown and discussed in Figure 5.4. Oblique Incidence As stated earlier, the parallel plate absorber is anisotropic and the absorption coefficient depends not only on the (polar) angle of incidence φ with respect to the normal but also on the azimuth angle ψ. The complex amplitude of the incident sound pressure wave is of the form pi (ω) = Aeikx +iky y+ikz z ,
(5.16)
where, with k = ω/c, kx = k cos φ, ky = k sin φ sin ψ, and kz = k sin φ cos ψ (see Figure 5.28). Consider first an infinitely thick absorber so that there is no reflection from the end. The pressure field in one of the channels in the absorber is then of the form p(ω) = B cos(qy y )eiqx x+iqz z ,
(5.17)
where y is the y-coordinate within the channel, with y = 0 at the center of the channel, and qy is the wave number determined in the previous section. For this propagational mode we have, as in the previous section, (5.18) qx = (ω/c)2 − qy2 − qz2 = q 2 − qz2 , where q is given in Eq. 2.79.
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NOISE REDUCTION ANALYSIS
Figure 5.28: Oblique incidence of a sound wave on a boundary in the yz-plane. Polar angle
is φ, azimuthal, ψ (with respect to the z-axis). The projection of the propagation vector k on the yz-plane is k sin φ and the component along the x-axis (normal to the plane) is k cos φ. The projections on the y- and z-axes are ky = k sin sin ψ and kz = k sin φ cos ψ.
Continuity of the wave number in the z-direction requires that qz = k sin φ cos ψ
(5.19)
so that Propagation constant, Anisotropic material, Oblique incidence qx ≡ qxr + iqxi = k Q2 − sin2 φ cos2 ψ
(5.20)
where Q = q/k: Eq. 2.79, k = ω/c, φ, ψ: Figure 5.28. Refraction The surface of constant phase, qxr x + qyr + qzr z = constant, corresponds to an angle of refraction φr (between the direction of propagation and the x-axis) given by tan φr =
2 + q2 qyr zr qxr
=
sin φ cos ψ Q2
− sin2 φ cos2 ψ
.
(5.21)
In the last step, we have used qy ≈ 0 and qz = k sin φ cos ψ. The angle of refraction vs the angle of incidence has already been shown and discussed in Section 5.7 and we proceed to the absorption coefficient. Angle Dependence of Wave and Input Impedance From the expression for qx (Eq. 5.20), and with the average velocity amplitude in the x-direction being obtained in the same manner as in Eq. 2.81 (apart from a factor
185
RIGID POROUS MATERIALS H ), we can express the wave impedance (p/ux )av as Wave impedance, Anisotropic material ωρ/H √ ρc/H 2 q (1−F ) =
ρcζw = zw (φ, ψ) =
x
v
(1−Fv )
(5.22)
Q2 −sin φ cos2 ψ
where Q = q/k: Eq. 2.79, Fv : Eq. 2.77, k = ω/c, φ, ψ: Figure 5.28, Fv = F is defined in Eq. 2.78. Although the wave impedance ζw and the propagation constant q are now angle dependent, the input impedance of the parallel plate absorber will have the same form as for normal incidence (Eq. 5.13), i.e., Input impedance, Anisotropic layer ζi (φ, ψ) ≡ θi + χi = iζw cot(qx L)
(5.23)
where ζw : Eq. 5.22, qx : Eq. 5.20. It is important to note that the wave impedance now depends on both the polar and azimuthal angles. The same holds true for the propagation constant qx . In terms of these quantities the normal impedance, has the same form as for normal incidence (Eq. 5.13). Similarly, the expression for the angle dependence, when expressed in terms of this input impedance, has the same form as was used in Chapter 2, i.e., α(φ, ψ) = 1 − |R|2 =
4θi cos φ . (1 + θi cos φ)2 + (χi cos φ)2
(5.24)
The dependence on the azimuthal angle enters into both the real and imaginary parts of the input impedance, as given in Eq. 5.20. Diffuse Field Absorption Coefficient In a diffuse sound field, the angle averaged value of the absorption coefficient becomes Diffuse field absorption coefficient, Anisotropic layer π/2 2π
α(φ,ψ) cos φ sin φ dφdψ
0 αav = 0 π/2 2π 0 0 cos φ sin φ dφdψ π/2 2π 1 =π 0 α(φ, ψ) cos φ sin φ dφdψ 0
(5.25)
where α(φ, ψ): Eq. 5.24, φ, ψ: Figure 5.28. The numerator expresses the total power absorbed by a surface element of the absorber and the denominator is the total power that enters a hemispheric control surface centered on the surface element. Quantity dφ(sin φdψ) is the solid angle element on the control surface and the factor cos φ extracts the normal component of the intensity, which is incident under the angle φ. Results of numerical computations of αav have already been shown and discussed in Figure 5.8.
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NOISE REDUCTION ANALYSIS
5.7.2 Isotropic Porous Layer The amount of fluid per unit volume of the porous material is Hρ, where H is the porosity. The average velocity u in the material can be defined in such a way that Hρu is the average mass flux. An alternate definition is based on a flux expressed as ρu, i.e., without the porosity factor. We choose the second of these since it will make the equations and boundary conditions somewhat simpler. The linearized equation for the conservation of fluid mass then becomes ∂Hρ = −ρdiv u. ∂t
(5.26)
In addition to a viscous interaction impedance analogous to that in the slot absorber, we have now another interaction resulting from the irregular path of the fluid in the structure. The changes in direction (and magnitude) of the velocity result in a force on the structure and a corresponding reaction force on the fluid. This interaction can be accounted for in terms of an induced mass density Gs ρ, which has to be added to the regular density ρ, making the total equivalent density s ρ, where s = 1 + Gs is the structure factor. Thus, the momentum equation can be written ∂ρu ∂u = −grad p − zv (t)u − Gs ρ . ∂t ∂t
(5.27)
Here, zv (t) is a linear operator, which accounts for the viscous interaction force per unit volume. The last term contains the induced mass, as explained above. It should be noted that we have assumed the material to be isotropic so that the velocity vector is in the same direction as the pressure gradient. Later, we shall consider the case when this is not so. Under isentropic conditions, the relation between the density and pressure perturbations δ and p is δ/ρ = κp = 1/ρc2 , where κ (= (1/ρ)∂ρ/∂P ) is the compressibility of the fluid involved and c the ordinary (isentropic) speed of sound. The first term in Eq. 5.26 can then be written ρκ∂p/∂t and we get Hκ
∂p = −div u, ∂t
(5.28)
and if the time derivative terms in Eq. 5.27 are combined, we get ∂s ρu = −up − zv (t)u, ∂t
(5.29)
where s = 1 + Gs and s ρ is the equivalent mass density, as explained above. In the special case where zv (t) can be assumed to be purely resistive, so that zv (t) = r, we can eliminate u between these equations (taking the time derivative of the first and divergence of the second) and obtain the equation for p ∂p ∂ 2p = c12 ∇ 2 p, + (r/ s ρ) 2 ∂t ∂t
(5.30)
where c12 = 1/(H s κρ) = (1/ s H )c2 and c, as before, is the isentropic sound speed.
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RIGID POROUS MATERIALS
If r/ s ρ >> ω, the friction force dominates over the effect of inertia, and the system behaves in much the same way as an overdamped oscillator. The first term on the left-hand side in the equation can then be neglected so that the wave equation degenerates into a diffusion equation. In terms of the analysis of the slot absorber, this condition corresponds to a boundary layer, which is larger than the channel width.
5.7.3 Interaction Impedance, Impedance Per Unit Length, and Complex Density For harmonic time dependence, the first term in Eq. 5.29 becomes −iωs ρu. We combine it with zv u into one term zu ≡ [zv (ω) − iωs ρ]u,
(5.31)
where z = zv − iωs ρ is the specific (equivalent) impedance of the air per unit length and zv the viscous interaction impedance. On the basis of this interpretation, it is convenient to introduce a complex density ρ˜ such that z = −iωρ, ˜ i.e., ρ˜ = ρ(1 + iz/ρω) = ρ(s + izv /ωρ),
(5.32)
where s = 1 + Gs is the structure factor. As we noted in the study of the slot absorber, the compression of the air in the material is not isentropic but contains a compressional (internal) friction term, which is accounted for by a complex compressibility κ, ˜ which we shall define in such a manner as to incorporate the factor H . It expresses the compressibility of the fluid per unit volume of the structure and we have κ˜ = H κ1 , where κ1 is the compressibility (which can be complex), which refers to unit volume of the fluid (and not the porous material, which is different by a factor H ). In other words, the tilde symbol accounts for both the factor H and the complex nature of the compressibility. The complex amplitude versions of Eqs. 5.26 and 5.27 can then be expressed as Linearized acoustic equations −iωκ˜ p = −div u −iωρu ˜ = −grad p κ = H κ
(5.33)
ρ˜ = ρ( + iz/ωρ)
Incidentally, the complex compressibility is analogous to the inverse of the complex spring constant, which is used to account for compressional losses in a spring (caused, say, by a dashpot in parallel with the spring). Similarly, the complex density corresponds to the complex mass in a mass-spring oscillator, which is used to account for a friction force on the mass proportional to velocity. The complex density contains the flow resistance and the structure factor and on the basis of the results obtained from this analysis, experiments can be devised for the measurement of these quantities. For example, they can be obtained from the measurement of the phase velocity and the spatial decay rate of a sound wave in
188
NOISE REDUCTION ANALYSIS
a porous material, assuming that the porosity has been determined from another experiment. Returning to Eq. 5.32 for the complex density, we can introduce an induced mass density Gv ρ due to the viscous forces (as discussed in Chapters 2 and 3). Thus, with zv = r + iχv and χv = −iωGv ρ, we obtain a total structure factor = 1 + Gs + Gv to obtain ρ/ρ ˜ = s + izv /ωρ = + ir/ωρ = 1 + G s + Gv ,
(5.34)
where r is the flow resistance. It should be kept in mind that both r and are frequency dependent. However, in most of the numerical studies, described previously, we have used s = 1.3 for the purely structural contribution to the structure factor assumed to be frequency independent. At low frequencies, the viscous contribution Gv approached 0.2 in a straight channel as the frequency goes to zero, and if this value is used, the corresponding total structure factor then becomes 1.5.
5.7.4 Propagation Constant and Wave Impedance Eliminating u between the equations in 5.33, we obtain ∇ 2 p + ρ˜ κp ˜ = 0.
(5.35)
For harmonic time dependence and with a space dependence of the complex sound pressure amplitude ∝ exp(iqx x + iqy y + iqz z), we obtain from Eq. 5.35, ˜ κ˜ /κ), q 2 = qx2 + qy2 + qz2 = k 2 (ρ/ρ)(
(5.36)
where we have introduced, for normalization purposes, the isentropic compressibility κ = 1/ρc2 and k = ω/c. The corresponding normalized propagation constant is Normalized propagation constant Q ≡ q/k ≡ Qr + iQi = (ρ/ρ)( ˜ κ˜ /κ)
(5.37)
where ρ, ˜ κ˜ : Eqs. 5.33 and 5.34. In Figure 5.9, the real and imaginary parts of the normalized propagation constant are plotted and discussed. The front surface of the porous material is located in the yz-plane at x = 0 and a plane sound wave is incident on it. The complex pressure amplitude is expressed as p(x, y, z, ω) = A exp(ikx x + iky y + ikz z), where, from the wave equation in free field, we get kx2 + ky2 + kz2 = k 2 ≡ (ω/c)2 . The direction of the wave is specified by the polar angle φ with respect to the x-axis and the azimuthal angle ψ, measured from the z-axis. In other words, the projection of the propagation vector on the yz-plane has the magnitude k sin φ and we have ky = k sin φ sin ψ and kz = k sin φ cos ψ (see Figure 5.28).
189
RIGID POROUS MATERIALS
Figure 5.29: Angle of refraction vs the angle of incidence for sound incident on a porous layer. The parameter is f/fv , where f is the frequency and fv = r0 /2πρ is the characteristic frequency of a material with a flow resistance r0 per unit length. Quantity ρ is the density of air. Porosity H = 0.95, structure factor s = 1.3.
Similarly, the wave function inside the material is exp(iqx x + iqy y + iqz z), where q 2 = qx2 + qy2 + qz2 . The wave vector components in the y- and z-directions are continuous across the surface of the absorber so that qy = ky = k sin φ sin ψ and qz = kz = k sin φ cos ψ. This is equivalent to saying that the intersection of the incident wave front with the boundary and the corresponding intersection of the wave front in the porous material are always the same. It follows then that x-component of propagation constant qx ≡ (ω/c)Qx = q 2 − qy2 − qz2 = q 2 − k 2 sin2 φ = (ω/c) Q2 − sin2 φ
(5.38)
where Q = q/k: Eq. 5.44, φ: Angle of incidence, k = ω/c. The velocity component in the x-direction is obtained from ux = (1/iωρ) ˜
∂p , ∂x
(5.39)
where ρ/ρ ˜ = s + izv /ωρ. The wave admittance in the x-direction is the ratio ux /p for a traveling wave in the x-direction for which ∂p/∂x = iqx p. It follows from the equations above that the normalized value of the wave admittance and the corresponding impedance are given by Wave admittance and impedance ηw = 1/ζw = ρcux /p = where Qx : Eq. 5.38, ρ: ˜ Eq. 5.34.
Qx ρ/ρ ˜
(5.40)
190
NOISE REDUCTION ANALYSIS
5.7.5 Angle of Refraction If the angle of incidence is φ, the component of the propagation vector of the incident wave along the plane boundary of the absorber is ky = k sin(φ), where k = ω/c. The wave field in the porous material must have the same y-dependence (matching of the wave velocities along the boundary) so that qy = ky = k sin φ. The wave speed in the normal direction in the porous material is determined by the {qx } = k{Qx }, and it follows that the angle of refraction φr is given by qy sin(φ) = . {qx } {Qx }
tan(φr ) =
(5.41)
The relation between φr and φ is plotted in Figure 5.29 for values of the normalized frequency f/fv from 0.1 to 100.
5.7.6 Input Impedance and Admittance, Absorption Coefficient The porous layer under consideration has a thickness L and is backed by a rigid wall. The front surface of the absorber is at x = 0. The pressure field in the material is the sum of an incident and a reflected wave, p(x, ω) = Aeiqx x + Be−iqx x ,
(5.42)
where we have left out the factor exp(iqy y + iqz z). The corresponding x-component of velocity is given by ρc ux (x, ω) = ηw (Aeiqx x − Be−iqx x ),
(5.43)
where ηw is the normalized wave admittance (see Eq. 5.40). The velocity must be zero at the rigid wall, x = L, and it follows that A exp(iqx L) = B exp(−iqx L). The input admittance at the front surface (x = 0) of the absorber is ρcηi = ux /p = ηw
A−B = −iηw tan(qx L). A+B
(5.44)
The corresponding normalized impedance is Normalized input impedance admittance ζi = 1/ηi = i(1/ηw ) cot(Qx kL)
(5.45)
where Qx : Eq. 5.38, ηw = 1/ζw : Eq. 5.40. From Eqs. 5.40 and 5.37 ˜ 2 /Qx . At sufficiently √ it follows that 1/ηw = (κ/κ)Q low frequencies Q ∝ 1/ kL and Qx ≈ Q. Furthermore, with X = QkL << 1, we have cot(X) ≈ (1/X)(1 − X2 /3). Referring to the expression for Q obtained ˜ κ/κ), ˜ ρ/ρ ˜ = + ir/ωρ, and κ/κ˜ ≈ 1/γ , the low frequency earlier, Q2 = (ρ/ρ) approximation for the normalized input impedance becomes (see Eq. 5.13) ζi ≈
i + , 3 H γ kL
(5.46)
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RIGID POROUS MATERIALS
where = rL/ρc is the total normalized flow resistance of the porous layer. It should be noted that this is approximately true independent of the angle of incidence. This is related to the fact that at low frequencies, the angle of refraction is close to zero, independent of the angle of incidence, as was discussed in the previous section. Having obtained the input impedance, the absorption coefficients follow from the equations already given in Chapter 2.
5.7.7 Perforated Facing, Its Nonlinearity and Induced Motion The modification of the input impedance of the absorber caused by a perforated facing or screen is simply obtained by adding the impedance of the facing or the screen. If there is ‘loose’ contact between the cover and the porous layer, the acoustically induced motion should be accounted for, and this is done by using the equivalent impedances of the facing and the (flexible) screen defined in Chapter 2. In the case of a perforated plate, the nonlinear acoustic resistance should also be accounted for, although it is more significant when there is an air backing rather than a porous layer. With reference to Chapter 4, the nonlinear specific resistance of an orifice can be expressed as |u0 |/c, where u0 is the velocity amplitude in the orifice. Then, if the normalized linear specific impedance of the orifice is ζl , the nonlinear impedance is ζnl = ζl + |u0 |/c.
(5.47)
If the plate is treated as limp with mass m per unit area, the structural specific impedance (referring to the orifice area and not the area of the perforated plate), is ζs = −isωm, where s is the open area fraction of the plate, and the equivalent impedance, which accounts for the induced motion, is ze =
znl ζm . znl + ζm
(5.48)
The normalized specific input impedance of the porous layer is denoted by ζb , and when it is referred to the orifice area rather than the average area over the plate, it will be sζb . Consequently, the total input impedance of the orifice is ζnl + sζb . The velocity amplitude of the air just outside the plate will be u=
p , ζnl + sζb
(5.49)
where p is the sound pressure amplitude just outside the orifice plate. The velocity amplitude relative to the plate is then u0 = uζnl /(ζnl + sζb ).
(5.50)
Combining these equations, we can express the equivalent impedance in terms of the sound pressure. The total average specific input impedance of the plate is then ζi = θi + iχi = (1/s)[ζe + sζb ] = (1/s)|ζe + ζb |,
(5.51)
and the absorption coefficient is then obtained from it in the usual manner (see Chapter 2).
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NOISE REDUCTION ANALYSIS
5.7.8 Anisotropic Layer The results obtained in this section so far have referred to an isotropic structure. For many porous materials, however, such as fiberglass boards, the flow resistance and the related structure factor are different in different directions. In the case of the slot absorber, discussed earlier, the flow resistance was infinite in the direction perpendicular to the slots. Using the complex density introduced earlier, ρ˜ = ρ( + ir/ωρ), we now express the anisotropy by assigning different complex densities ρ˜x , ρ˜y , and ρ˜z in the x-, y-, and z-directions. The x-component of the momentum equation then becomes −iωρ˜x ux = −
∂p ∂x
(5.52)
with analogous expressions for the other components. Considering a traveling wave in the material, p ∝ exp(iqx x + iqy y + iqz s), and inserting the expressions for the velocity components into the mass conservation equation iωκp + ∂ux /∂x + ∂uy /∂y + ∂yz /∂z = 0, we obtain, ω2 κ˜ = (1/ρ˜x )qx2 + (1/ρ˜y )qy2 + (1/ρ˜z )qz2 .
(5.53)
The corresponding expression for qx is then, with c2 = 1/ρκ and k = ω/c, qx2 = k 2 (κ/κ)( ˜ ρ˜x /ρ) − (ρ˜x /ρ˜y )qy2 − (ρ˜x /ρ˜z )qz2 .
(5.54)
As before, continuity of the wave number along the surface of the absorber requires qy = ky = k sin φ sin ψ and qz = kz = k sin φ cos ψ so that, in terms of the normalized propagation constants Qx = qx /k, Qx = (ρ˜x /ρ)(κ/κ) ˜ − (ρ˜x /ρ˜y ) sin2 φ sin2 ψ − (ρ˜x /ρ˜z ) sin2 φ cos2 ψ. (5.55) For a locally reacting material, ρ˜y = ρ˜z = ∞ so that Qx and related quantities, such as wave impedance and input impedance, become independent of the angle of incidence. For the slot absorber in the previous chapter, we have ρ˜y = ∞.
5.7.9 Effect of Grazing Flow Taken literally, total reflection from a shear layer above an absorber implies no sound absorption, but total reflection is a concept which refers to geometrical (ray) acoustics and does not account for the ‘barrier’ penetration, which occurs in wave acoustics. When such penetration is accounted for, there will be some absorption even under conditions of total reflection. We use the vortex sheet model of the boundary layer with flow Mach number M above the layer and M = 0 between the sheet and the boundary. The wave equation for the sound pressure p(r, t) in the moving fluid is (
∂ ∂ + U )2 p = c2 ∇ 2 p. ∂t ∂x
(5.56)
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RIGID POROUS MATERIALS
With harmonic time dependence (time factor exp(−iωt)) the space dependence of the complex amplitude of the sound pressure is expressed as exp(iqx x + iqy y). With qx = q cos φ and qy sin φ and Q = q/(ω/c), it follows from Eq. 5.56 that Q=
1 . 1 + M cos φ
(5.57)
Furthermore, Qx = qx /(ω/c) = Q cos φ and Qy = Q sin φ. The complex amplitude of the sound pressure is the sum of an incident and a reflected wave p(x, y, ω) = (Aeiqy y + Be−iqy y )eiqx x . (5.58) The y-component of the corresponding velocity amplitude field is obtained from ρDu/Dt = −grad p, where D/Dt = ∂/∂t + U ∂/∂x, ρcuy =
Qy (Aeiqy y − Be−iqy y )eiqx x . 1 − Qx M
(5.59)
The associated displacement amplitude η in the y-direction, obtained from Dη/Dt = uy , is given by uy −iωη(x, y, ω) = . (5.60) 1 − Qx M In the region between the shear layer and the boundary, where the mean flow is zero, the relation between the displacement and the velocity amplitudes in the y-direction is simply −iωη1 = uy . The sound pressure just above the shear layer is denoted by p1 exp(iqx x), and the y-component of the velocity amplitude by u1 exp(iqx x). The shear layer is placed at y = 0 and continuity of sound pressure then yields A + B = p1
(5.61)
Qy (A − B) = u1 ρc. (1 − Qx M)2
(5.62)
and continuity of displacement
If, instead, we had used continuity of the y-component of the velocity rather than the displacement amplitude, the factor (1 − Qx M)2 would have been replaced by (1 − Qx M). (Although continuity of displacement is appropriate for an idealized laminar shear layer, there is some experimental evidence that continuity of velocity is more realistic in modeling a turbulent shear layer.) If we introduce a normal impedance ρcζ1 = p1 /u1 just above the shear layer, we can express the pressure reflection coefficient, from Eqs. 5.61 and 5.62, as R = B/A =
Qy ζ1 − (1 − Qx M)2 , Qy ζ1 + (1 − Qx M)2
(5.63)
which can be written R=
(1 + M cos φ)ζ1 sin φ − 1 ζ sin φ − 1 ≡ . (1 + M cos φ)ζ1 sin φ + 1 ζ sin φ + 1
(5.64)
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NOISE REDUCTION ANALYSIS
The quantity ζ = ζ1 (1+M sin φ) can then be interpreted as the normal impedance of just below the shear layer. The absorption coefficient can then be calculated from α = 1 − |R|2 .
(5.65)
If the region above the shear layer is infinitely extended, corresponding to d = ∞, we have ζ1 = 1/ sin φr , and if we use the law of refraction (Eq. 5.6, Figure 5.23), the reflection coefficient can be written R=
sin 2φ − sin 2φr . sin 2φ + sin 2φr
(5.66)
We note that in the evanescent region, where sin 2φr is zero or imaginary, the magnitude of the reflection coefficient is unity and the absorption coefficient is zero, as expected. For the infinitely thick boundary layer (d = ∞) we have already used the expression ζ1 = p1 /u1 = 1/ sin φr for the normal impedance just above the shear layer. It is the normal impedance of a wave traveling in the direction specified by the angle φr . It remains to calculate the impedance ζ1 when d is finite. To do that we introduce the amplitudes p2 and u2 of pressure and normal velocity at the boundary and express the relation between p1 , u2 and p2 , u2 by the linear relation p1 = T11 p2 + T12 ρc u2 ρcu1 = T21 pp2 + T22 ρc u2 ,
(5.67)
where the matrix elements are T12 = −i(1/ sin φr ) sin X T11 = cos X T21 = −i sin φr sin X T22 = cos X.
(5.68)
Here, X = kd sin φr and k = ω/c. With the normalized normal impedance of the boundary given by ρcζ2 = p2 /u2 , we can express ζ1 as T11 ζ2 + T12 ζ1 = . (5.69) T21 ζ2 + T22 Numerical results are shown in Figure 5.25.
5.7.10 Computational Considerations The essential quantities we need to evaluate in numerical studies of sound absorption are the expressions for the complex density and the complex compressibility, expressed by ρ/ρ ˜ and κ/κ. ˜ From these we can determine the propagation constant and the wave impedance of the porous material from Eqs. 5.37, 5.38, and 5.40. In order to determine the complex density ratio ρ/ρ, ˜ we must decide how to express the viscous interaction impedance zv . It was found to be −iωρFv /(1 − Fv )
195
RIGID POROUS MATERIALS
for a single channel (see Eq. 2.84). The corresponding average value for a slot absorber with partition walls of non-negligible thickness then was obtained by multiplying by the factor 1/H , where H is the ‘porosity,’ and it is this value that is appropriate here. The complex density ratio then becomes ρ/ρ ˜ = s + (1/H )
Fv , 1 − Fv
and the specific impedance per unit length is
1 Fv z = −iωρ˜ = (−iωρ) s + . H 1 − Fv
(5.70)
(5.71)
The function Fv is determined solely by the ratio a/dv of the half-width a of a slot or channel and the viscous (≈ thermal) boundary layer thickness dv (Eq. 2.78). This parameter is related to the √ steady flow resistance r0c per unit length of a channel in the slot absorber, a/dv = 3ωρ/2r0c , and hence to the steady flow resistance per unit length in the slot absorber, r0 = r0c /H . We now carry over this result to the general rigid porous absorber and introduce an equivalent value of a/dv based on the known steady flow resistance r0 per unit length of the material, i.e., (5.72) a/dv = 3ωρ/2r0 H = 3ω/2ωv H , where in the last step we have introduced the viscous relaxation frequency ωv = r0 /ρ. Thus, using this value for a/dv , we can determine the frequency dependence of the impedance per unit length. For example, the acoustic resistance per unit length, normalized with respect to the steady flow value, becomes ω Fv /H r/r0 = . (5.73) i ωv 1 − Fv This leaves us with s = 1 + Gs , The dependence of the absorption coefficient on this quantity is weak, and it can be considered to be independent of frequency.5 The contribution by the viscous interaction impedance to the induced mass density, Gv ρ, however, will be frequency dependent in the same manner as in a channel, i.e., with Gv going from 0.2 at low frequencies to zero as the frequency increases (see Chapter 3). It is contained in the second term in Eq. 5.71. With a typical value of Gs = 0.3 and hence s = 1.3, the corresponding variation of the total structure factor = s + Gv is from 1.5 to 1.3. Actually, the absorption coefficient depends only weakly in the structure factor, and the results obtained are almost indistinguishable from those obtained with = 1. The resistive part of the impedance per unit length, as given in Eq. 5.73, is more important. 5 Compare the induced mass caused by a succession of orifice plates in a pipe. As long as the wavelength is large compared to the pipe diameter and the distance between the orifice plates, the induced mass can be regarded as constant, independent of frequency.
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NOISE REDUCTION ANALYSIS
It remains to calculate the complex compressibility κ˜ = H κ˜ 1 . Again, we use the equivalent narrow channel result κ˜ 1 /κ = 1 + (γ − 1)Fh ,
(5.74)
where Fh is the same function as Fv with a/dv in the argument replaced by a/dh , where dh is the thermal boundary layer thickness. As shown in Chapter 3, it is related to the viscous boundary layer thickness through dh /dv = 1/Pr ≈ 1.14, where Pr is the Prandtl number (≈ 0.77 for standard air). Thus, having determined a/dv from Eq. 5.72, we can determine Fh and hence the complex compressibility. It should be emphasized that an equally valid (and perhaps more logical) approach would have been to use Eq. 2.98 (see also Figure 2.3) for the compressibility thus circumventing the use of the function F . From the quantities thus obtained, the normalized propagation constant is Q = (ρ/ρ)( ˜ κ/κ) ˜ 2 (5.75) Qx = Q − sin2 φ, and the normalized wave impedance for an isotropic material ζw =
ρ/ρ ˜ Qx
(5.76)
can be expressed in terms of the parameter a/dv or f/fv . Examples of the computed absorption characteristics have been presented and discussed earlier in this chapter.
Chapter 6
Flexible Porous Materials As indicated in Section 1.2.1 on chapter organization, we have attempted to gather most of the mathematical details and derivations in a separate section which can be skipped at the first reading or skipped altogether by the reader who is interested mainly in numerical results. In this chapter, most of of this mathematical analysis is summarized in Section 6.8. Some of the most important results, often the basis for the numerical results presented in graphs, are duplicated (and often framed) in the main part of the chapter.
6.1 INTRODUCTION AND SUMMARY Although sound absorption of a rigid porous material, discussed in the previous chapter, covers most of the essentials for the design of absorbers, the flexibility of a material is of sufficient importance to warrant special consideration, particularly since the analysis involved is quite different than for a rigid material. As was the case for a thin sheet, discussed in Chapter 2, flexibility of a porous material in bulk is expected to be important at low frequencies and high flow resistances. In the present context, the flexibility can be expressed in terms of the compliance, as measured in an apparatus described in this chapter. It yields not only the compliance, the inverse of the Young’s modulus, but also the ‘loss factor’ of the material due to internal damping. It is implied that we are dealing with a material with open cells or pores, and in order for the air in the material not to influence the measurement of the elastic constant and particularly the loss factor in the porous frame per se, the measurement should be carried out in an evacuated chamber. Some data, obtained in this manner, are described later in this chapter. Unfortunately, the compliance and the loss factor are presently not available for most flexible materials used in practice and an extensive analysis of the effect of flexibility will be of limited value without such data. Therefore, manufacturers and users of flexible porous materials are encouraged to try to provide this information. Material with open cells. A material with open cells permits a steady flow through it and a flow resistance can be defined and measured in the same way as for the rigid material. For oscillatory flow, acoustically induced motion will result and an equivalent impedance rather than the interaction impedance will be involved. 197
198
NOISE REDUCTION ANALYSIS
Whether the effect of flexibility is going to be important or not depends primarily on the ratio of the flow resistance r per unit length and the mass reactance ωM per unit volume of the porous material. This ratio defines a critical frequency r/2π M at which friction drag force on the material and the inertial force per unit volume are the same. For example, for a material with a flow resistance of 1 ρc per inch and a weight density of 1 lb/ft3 , this frequency is 407 Hz. Above the critical frequency, the inertia of the material dominates, and the acoustically induced motion of the material is reduced accordingly with increasing frequency, and the material in essence responds as if it were acoustically rigid. Below the critical frequency, on the other hand, a significant motion of the material can be induced and the equivalent inertial mass density of the air-structure system is then enhanced so that the sound speed and the wavelength in the material are reduced. This means that the quarter wavelength resonance of a layer of the flexible material will occur at a lower frequency than for a rigid layer, and this can be taken advantage of in acoustical design. In other words, the flexible porous layer when ‘tuned’ properly can be useful in certain low frequency applications. It should be kept in mind, though, that at large acoustic amplitudes, the induced motion can possibly lead to acoustically induced fatigue within the layer, a problem which at present has not been explored, as far as we know. In this context, we refer to the measurements of the deformation of a flexible material caused by an incident high level acoustic pulse, which is discussed in this chapter. With an incident pulse, with a peak pressure level of the order of 190 dB, the acoustically induced compression of an 8 inch thick layer can be 4 inches or more. Material with closed cells. For a porous material with closed cells, the heat conduction in the trapped air in the cells is the dominant mechanism of sound absorption. The internal compressional losses in the material itself are generally negligible. As will be shown, the absorption by a layer backed by a rigid wall is limited to narrow frequency bands centered at the structural resonances of the layer. Effect of a screen cover. A screen typically improves the absorption at low frequencies at the expense of a reduction at high frequencies. The mass of the screen in combination with the stiffness of the porous layer leads to a resonance at which the absorption can be quite high. For a rigid porous material, there is a similar effect of a flexible screen only if induced motion of the screen is present. If the screen is attached to a rigid material, there will be no such resonance, of course, but for a flexible material, the mass of the screen causes a resonance even if the flow resistance of the screen is negligible. Shock wave reflection. This chapter ends with a discussion of an experiment dealing with the reflection of a shock wave from a flexible porous layer and the associated deformation of the layer.
6.2 COUPLED WAVES In the rigid porous material, the only wave motion that occurs is in the air within the material. In a flexible material, on the other hand, the acoustically induced deformation of the porous structure will propagate so that there will be two interacting
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FLEXIBLE POROUS MATERIALS
waves, one in the structure and one in the air. In steady state, the interaction results in a wave field, which has two characteristic modes of motion; their relative strength depending on the boundary conditions. The acoustic energy is divided between the structure and the air in a manner which is determined by the characteristics of the coupling and the elastic properties of the structure and the air. The basic problem is to determine the amplitudes of the two wave modes and their contribution to the input impedance of the porous layer from which the absorption coefficient is obtained. The problem is similar to the forced motion of two coupled oscillators although, in our case, the coupling is distributed, i.e., nonlocal, whereas for the oscillators, it is local. In other words, we deal with coupling between waves (distribution of oscillators). Coupled wave equations ∇ 2 p + ka2 p = ca p
∇ 2 p + ks2 p = cs p = ˜ ca = iωzK,
ka2
ω2 κ˜ ρ, ˜
(6.1)
ω2 M˜ K˜
= cs = iωzκ˜
ks2
˜ = + iz/ωρ, where p and p : Pressures into air and structure, respectively, ρ/ρ ˜ ˜ ˜ M/M = 1 + iz/ωM, cs = iωzK, cs = iωzκ, ˜ K: Compressibility, structure, κ/H ˜ : Compressibility, air, Eq. 5.33, z ≈ r: Flow resistance per unit length, H : Porosity. (See also Eq. 6.31.) There are two special limiting cases of the flexible material for which the mathematical analysis is simplified considerably and in which we need to deal with only a single wave. One obvious case is the rigid material, which we have studied already, and the other is the limp material. In the latter, the stiffness is zero and the coupling with the material in essence can be described in terms of an equivalent impedance per unit length, which involves both a resistance and a mass reactance. The analysis based on the limp material assumption can be applied as an approximation to a real material as long as we are in a frequency region in which the response of the material is inertia dominated. A porous layer, analyzed in this manner, will lead to a quarter wavelength resonance using a speed of sound, which is now much lower than in free field because of the inertial mass contribution from the material. However, a resonance frequency thus obtained may fall outside the inertial regime of the material and thus may not be self-consistent. Thus, the limp assumption has limited usefulness.
6.3 DISPERSION RELATION A dispersion relation expresses the dependence of the wavelength λ (or propagation constant k = 2π/λ) on the frequency of a harmonic wave. In free field it is simply k = ω/c = 2π/λ, where, c, the wave speed, is independent of frequency. For a rigid porous material, the propagation constant q was found to be complex with frequency dependent real and imaginary parts, as shown in Figure 5.9 and Eq. 5.3. In the flexible layer, we have two coupled waves, Eq. 6.1, which give rise to two modes of
200
NOISE REDUCTION ANALYSIS
propagation, each mode with a unique combination of velocity and pressure fields in the air and the structure and each with its own dispersion relation. In this unique combination forming a mode, the air wave and the structure wave travel at the same speed. Instead of the single curve in Figure 5.9 for the dispersion relation, we now have two curves, one for each mode, as shown in Figure 6.1. Dispersion relation, Flexible porous material Q4 − (Ka2 + Ks2 )Q2 + Ka2 Ks2 + C = 0 = 12 (Ka2 + Ks2 ) ± 12 (Ka2 − Ks2 )2 − 4C
(6.2)
Q21,2
˜ ˜ ˜ where Ka2 = (ρ/ρ)( ˜ κ/κ), ˜ Ks2 = (M/ρ)( K/κ), C = −Ca Cs = (z/ωρ)2 (K/κ)( κ/κ). ˜ (See also Eq. 6.1.) If the coupling between the fluid and the structure is weak, as it is at high frequencies where the induced motion in the structure is small, it makes good sense to think of these modes as a fluid and a structure mode. However, as the frequency decreases and the coupling increases, the identity of these modes may not be so clear, as can be seen in the figure, where the frequency dependence of the normalized propagation constants Q = q/k are shown. The real part of the normalized propagation is the ratio of the free field sound speed in the fluid and the phase velocity √ in the porous material. Thus, for the fluid mode, this ratio approaches unity (actually s H ≈ 1.1) with increasing frequency. The complex compliance of the porous material is K˜ = K1 + iK2 , normalized with respect to the compliance of air 1/ρc2 . In the example in the figure, we have K1 = 0.5. √ The phase velocity v of the uncoupled structural mode is then such that c/v ≈ K1 M/ρ, where M is the mass density of the structure. With M and ρ corresponding to the weights ≈ 2 and .08 lb/ft3 , respectively, we get c/v ≈ 3.5 in the
Figure 6.1: Real and imaginary parts of the normalized propagation constants for the two modes in a flexible porous material. Solid lines: First mode (‘fluid’ mode). Dotted lines: Second mode (‘structure’ mode). Curve identification: Starting from the top at 500 Hz: (a) Real part (structure), (b) Real part (fluid), (c) Imaginary part (structure), (d) Imaginary part (fluid). Flow resistance: 0.5 ρc per inch. Weight: 2 lb/ft3 . Normalized compliance: 0.5, Loss factor: 0.5.
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high frequency limit, which is quite close to the computed result in Figure 6.1, top curve (a). As the frequency decreases, the speed of the fluid mode decreases (its propagation constant increases) and at a certain frequency, in this case ≈ 200 Hz, the wave speeds of the two modes become equal. The coupling is then enhanced, so that the two modes are pulled together and their attenuation constants become essentially the same. The loss factor has been assumed frequency independent and equal to 0.5, and the imaginary part of the normalized propagation constant for the structural mode approaches a constant at high frequencies. A rigid material is characterized by K˜ = 0 (infinite stiffness), and in that special case the normalized propagation constant Q1 approaches the result obtained in Chapter 4 and Q2 = 0. The latter result means that the phase velocity in the structure-borne wave is infinite, which is consistent with an infinite stiffness. Similarly, for the limp material, we get Q2 = ∞, which means that the phase velocity and wavelength are zero. It will also have infinite attenuation so that the wave will disappear as soon as it has been excited and only the fluid-borne wave remains. That is the reason why the analysis for the limp material becomes simple, only one wave is involved and the coupling to the structure is incorporated in this wave through an equivalent mass. However, in some cases, for example, when the surface is covered with an impervious membrane, the structure-borne wave has to be included to satisfy the boundary conditions at the surface.
6.4 FIELD DISTRIBUTIONS 6.4.1 Pressure and Velocity Fields On the basis of the mathematical analysis in Section 6.8 it is tempting to compute and plot a variety of functions. However, although of interest by itself, much of such effort might not be very relevant in practice for inclusion in a book like this. However, the actual motion of the flexible structure when it is acted on by an incident sound wave and the intimately related questions of the relative importance of viscous and compressional losses and their distribution are basic enough to deserve consideration. Furthermore, the results can be used to test whether or not the analysis makes sense physically and they may provide some insights, which otherwise may have been lost. Computed velocity amplitude distributions (Eq. 6.38) in a flexible porous layer of thickness L = 8 inches and backed by a rigid wall, at x = 0, are shown in Figure 6.2 for both the air and the structure. The velocities are normalized with respect to the air velocity amplitude at the surface, at x = L. Plots covering a wide range of frequencies can be readily obtained but for the present purpose we include only two, one at a frequency below the resonance of the structure and one close to the resonance. If the coupling between the air-borne and structure-borne motions is ignored, the wave √ speed in the structure would be cs ≈ c/ K1 M/ρ, where c is the free field sound speed in air, M the mass density of the structure, ρ, the density of air, and K1 , the compliance of the structure, normalized with respect to the compliance 1/ρc2 of air. In this case, with K1 = 0.5 and M/ρ ≈ 25, and cs ≈ 317 ft/sec, the corresponding quarter wavelength structural resonance would be ≈ 119 Hz.
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Figure 6.2: Velocity amplitude distribution in an L = 8 inch thick flexible porous layer at 40 Hz (left) and 120 Hz (right). Solid curves: air. Dotted curves: porous structure. Flexible porous layer data: Thickness: 8 inches. Weight: 2 lb/ft3 . Normalized compliance: 0.5. Loss factor: 0.5. Normalized flow resistance/inch: 3.
At 40 Hz, the wavelength is much larger than the layer thickness and in the absence of wave coupling, the amplitudes of both the air-borne and the structure-borne waves would decrease almost linearly from the value at the surface, x/L = 1 to zero at x = 0. The coupling between the waves ‘pulls’ these straight lines toward each other (keeping the end point at x/L = 1 fixed), which results in a velocity distribution, which bends upwards for the air-borne wave and downwards for the structure-borne wave, as shown. The amplitude of the structure is about 40 percent of that of the air at the surface. Close to the structural resonance frequency, the amplitude of the structure is almost the same as for the air as it has risen to about 72 percent of the air amplitude at the surface. Above the resonance, at 200 Hz (not shown), the surface amplitude of the structure is found to have been reduced somewhat and the maximum value has moved inward, as expected because of the shorter wavelength, and the maximum amplitude is increased to almost 0.8; it has forced the air amplitude to increase as well. Actually, the maximum air amplitude is now found to be larger inside the material than at the surface with a maximum of about 1.07 at x/L ≈ 0.75. Qualitatively, the response of the flexible structure to the sound wave is similar to that of a simple harmonic oscillator if we regard the air as driving the structure wave. Thus, above the resonance frequency of the structural mode, the inertia of the structure dominates its response and makes its velocity lag that of the air. In this case the corresponding phase angle turns out to be close to 70 degrees at the surface but decreases toward the rigid backing. At resonance, 120 Hz, the phase difference is essentially zero and below the resonance, in the stiffness controlled region of the structure, its velocity runs ahead of the velocity of the air. As the wave penetrates into the material, the phase relationship between the waves changes, and one reason is the difference between phase velocities of the air-borne and structure-borne modes. The relative pressure amplitude distributions (Eq. 6.39) that correspond to the velocity distributions in Figure 6.2, are shown in Figure 6.3. The amplitudes are normalized with respect to the pressure amplitude of the air at the surface of the layer.
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Figure 6.3: Pressure amplitude distribution of the air-borne and structure-borne waves (solid and dotted lines, respectively) in the flexible porous layer in Figure 6.2. The amplitudes are normalized with respect to the pressure amplitude of the air-borne wave at the front surface of the layer (x/L = 1). Frequencies: 40 Hz (left), 120 Hz (right), structure resonance). Layer thickness: 8 inches. Weight: 2 lb/ft3 . Normalized compliance: 0.5. Loss factor: 0.5. Normalized flow resistance/inch: 3. At 40 Hz and with no wave coupling, the pressure amplitude distribution of the air would be essentially constant and that of the structure essentially linear, going from zero at the surface to the maximum value at the wall at x = 0. The coupling pulls the curves toward each other so that the lines become curved, as shown. The pressure in the structure at the wall is about 80 percent of the air amplitude at the surface. At 120 Hz, close to the resonance of the structural mode, there is a considerable increase in the pressure amplitude in the structure, reaching a value of about 2.3 at the wall. The amplitude in the air is about uniform through the layer. At frequencies above the resonance, for example at 200 Hz (not shown here), the distribution is qualitatively the same as for 40 Hz, except for a slight wave-like variation due to the shorter wavelength.
6.4.2 Dissipation Function Having obtained the pressure and velocity distributions in the layer, we can readily compute the distributions of the velocity and pressure induced dissipations of acoustic energy within the porous layer. The velocity and pressure induced dissipation functions w/w0 and w /w0 are plotted in Figure 6.4 vs x/L (solid and dashed lines) at the frequencies 40 and 120 Hz (resonance). The total dissipation function w + w is also shown (thin solid line). The viscous dissipation has a maximum at the surface and decreases to zero at the wall at all frequencies. It should be recalled that the dissipation is determined by the air velocity relative to the structure, and, therefore, does not follow the shape of the absolute velocity distribution. It is interesting to find that in the vicinity of the structural resonance, the maximum pressure induced dissipation in the structure at the wall is about the same as the viscous losses at the surface. However, the viscous loss decreases rapidly toward the wall, where it is zero, of course. It is also noteworthy that the total dissipation has a minimum, in this case at x/L ≈ 0.7.
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Figure 6.4: The normalized dissipation functions w/w0 (solid line) and w /w0 (dotted line) due to the velocity and pressure induced losses in the porous layer referred to earlier, where w0 is the viscous loss at the surface of the layer. The thin solid line is the total dissipation function, (w + w )/w0 . The two graphs refer to the frequencies 40 and 120 Hz, as before.
As the frequency increases and the structure becomes increasingly immobile, the dissipation is dominated by the velocity induced portion in a relatively thin layer close to the surface. This is due to the fact that the penetration distance of the sound wave decreases with increasing frequency. In principle, it should be possible to measure the temperature distribution resulting from the acoustic heating related to these losses, which at high intensities can be significant. What would be required to set a layer of cotton on fire by sound?
6.4.3 Examples and Comments 1. Absorption coefficient vs flow resistance For a flexible porous layer the role of the effect of the flow resistance on the absorption is more complicated than for a rigid layer (see Problem 1 in Section 5.5.3) since the flow resistance determines the induced motion of the material. At high frequencies, this effect is small but becomes essential in the vicinity of structural resonances. Compared to the rigid layer, how much are the optimum values of the flow resistance changed at 100 Hz for a layer thickness of 1 inch, if the normalized compliance of the material is 0.8 and the loss factor 0.5? The weight of the material is 4 lb/ft3 .
6.5 ABSORPTION SPECTRA 6.5.1 General Comments The mathematical analysis (see Section 6.8) of the absorption spectrum, defined in Chapter 7, of a flexible porous layer is considerably more laborious than for the rigid layer, and the length of the general description in this section will be short by comparison. The mathematical analysis predicts that for a sufficiently high flow resistance of the material, there will be peaks in the absorption curves at low frequencies, which are related to structural resonances in an interesting way, as explained below.
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In presenting the results of numerical studies of the absorption coefficient, it is possible to make use of ‘universal’ curves with the frequency variable expressed in normalized form as L/λ, where L is the layer thickness and λ the free field wavelength. The mass density of the material is then normalized with respect to the mass density of air and if a screen is present, its mass is expressed in an analogous manner. However, such presentations are a bit cumbersome to interpret and will not be used here. Rather, we shall simply plot the absorption coefficient as a function of frequency in a typical example in which the effect of the flexibility on the absorption is demonstrated.
6.5.2 Absorption Peaks But Not at Resonances This seemingly contradictory section heading needs an explanation and for this purpose we shall use the computed absorption spectra shown in Figure 6.5. The flexible porous layer in this example is 8 inches thick, has a normalized compliance of 0.5 (as usual, normalized with respect to 1/ρc2 , the compressibility of air), and a weight of 4 lb/ft3 . We consider first the result shown in the top-left graph in the figure. To understand the results obtained, we start by estimating the quarter wavelength resonance frequency of the porous structure. Ifwe neglect the coupling with the air, the speed of the structural wave will be cs = c 1/K(ρ /ρ), where K is the compliance of the material, ρ the mass density of the structure, and ρ the mass density of air. With K = 0.8 and ρ /ρ = 50 (corresponding to 4 and 0.08 lb/ft3 for the structure and the air, respectively), the structural wave speed will be ≈ 177 ft/sec. With an 8 inches thick layer, the wavelength at the quarter wavelength resonance is 32 inches, and the corresponding quarter wavelength structural resonance frequency is then ≈ 66.4 Hz. At this resonance frequency, the relative velocity of the air and the structure is essentially zero and the equivalent resistance and the corresponding absorption coefficient then will be close to zero. This, indeed, is what we find in the computed absorption spectrum when the loss factor is sufficiently small. As the frequency is increased above
Figure 6.5: Computed absorption spectra of a flexible porous layer. Layer thickness: 8 inches. Normalized flow resistance/inch: 2. Normalized compliance: 0.8. Loss factor: 0 (left) and 0.5 (right). Weight: 4 lb/ft3 . N: Normal incidence. DL: Diffuse field, local. DNL: Diffuse field, nonlocal reaction.
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the resonance, the relative velocity, the equivalent resistance, and the absorption coefficient increase until an optimum impedance match is obtained for maximum absorption. If the total resistance of the layer is large enough, in this case 16, this impedance match will be better than for the rigid material, and the absorption at the peak will be considerably larger than for a rigid material. It should be emphasized, however, that the absorption peak does not occur at the structural resonance, where, instead, it is a minimum, close to zero for the reason stated above. This, then, is the explanation for the seemingly contradictory heading of this section. On the low frequency side of the resonance, there is a corresponding increase in the absorption coefficient, but it is generally not large enough to exceed the value obtained for a rigid layer. In the graph to the right in the figure, the loss factor has the more realistic value of 0.5 (rather than 0), and the resonance frequency is not as clearly seen although the absorption peak is still quite pronounced and significantly exceeds the absorption of the rigid material at this frequency. Below 1000 Hz, the lower curve in each graph refers to normal incidence, the upper to diffuse field. The reason why the diffuse field value is larger is the relatively high flow resistance of the layer yields better impedance matching for waves at oblique incidence. Diffuse field coefficients for both locally and nonlocally reacting layers have been computed, but in this example with a relatively high flow resistance there is practically no difference between the two absorption spectra. The only noticeable difference occurs at high frequencies where the nonlocally reacting layer yields a slightly higher absorption.
6.5.3 Effect of a Bonded Perforated Facing The effect of a perforated facing or screen on the absorption by a rigid porous layer was discussed in Chapter 5, Figures 5.16 and 5.17. In carrying out a similar study for a flexible porous layer, we find that the effect of the facing is qualitatively different when the facing is free and when it is bonded to the surface of the porous layer. By free in this context is meant that the facing is not in contact with the flexible layer and thus is not forced to participate in the induced motion of the layer. We have referred to this configuration earlier as ‘loose’ contact. Then, the effect of the facing is not much different than for the rigid material and involves mainly a reduction of absorption at high frequencies. For a typical facing with an open area between 20 and 30 percent, this means frequencies above approximately 2000 Hz. The induced motion of the layer is not affected so that the quarter wavelength resonances of the layer, discussed above, are normally not affected. When the facing is bonded to the surface of the layer, however, it is forced to move with the layer. It is what we have referred to as ‘hard contact.’ Actual bonding with an adhesive material is hard to realize in practice, but it can in effect be simulated by letting the facing keep the layer in compression. The resonances of the system are now intimately related to the mass of the facing and the lowest resonance is in essence the familiar resonance of a lumped mass-spring oscillator. The mass is then the mass of the facing and the ‘spring’ is provided by the flexible porous layer.
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If we assume that the wavelength of the wave in the structure is much longer than the layer thickness, the spring constant of the layer will be ρc2 /(KL), where K is the normalized compliance of the porous structure, as defined earlier, and L is the thickness of the layer. The corresponding resonance frequency is then c f0 ≈ 2π
ρ , mKL
(6.3)
where m is the mass per unit area of the facing. In the example referred to in Figure 6.6, K = 0.8, L = 8 inches, and the weight of the facing is 4 lb/ft2 (m ≈ 2 g/cm2 ). With these values, we get f0 ≈ 34.5 Hz. The absorption coefficient is expected to be zero at the resonance (no significant relative motion of the air and the structure), but it will reach a maximum at a frequency a little above the resonance, where the relative motion is such that the equivalent resistance is reduced to provide a relatively good impedance match to the incoming wave. In order to be able to demonstrate the resonances clearly, the loss factor of the porous structure in the left graph in Figure 6.6 has been chosen to be zero. The first resonance is seen to be slightly lower than the predicted value, but that is to be expected since we did not account for the mass of the porous layer. As the frequency increases above the first resonance, the facing becomes more and more immobile because of its inertia and begins to act like a rigid wall. Then, in essence, the porous layer will be trapped between two rigid walls. The structural resonances are then expected to occur at frequencies such that the layer thickness will be an integer number of half wavelengths in the structure. In this case, the first half wavelength resonance corresponds to a wavelength of 16 inches, and with the wave speed in the structure of 177 ft/sec, as determined above, the corresponding frequency is 132 Hz, which is consistent with the result in Figure 6.6.
Figure 6.6: Computed absorption spectra of the flexible porous layer in Figure 6.5 when covered with a perforated plate facing, which is bonded to the layer. Layer thickness: 8 inches. Normalized flow resistance/inch: 2. Normalized compliance: 0.8. Loss factor: 0 (left) and 0.5 (right). Weight: 4 lb/ft3 . Plate: Thickness: 0.125. Hole diameter: 0.125 inch. Open area: 23 percent. Curve identification: Below 1000 Hz the lower curve refers to normal incidence, the upper, to diffuse field, local reaction.
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In the graph to the right, a relatively large loss factor of 0.5 has been used. The resonance frequency no longer can be clearly identified, but the absorption peaks following the resonances are still quite pronounced. If the flow resistance of the layer is sufficiently high, it is clear that the bonding of the perforated plate to the layer and the resulting low frequency resonance can be used to enhance the absorption in a frequency band above the resonance.
6.5.4 Examples and Comments 1. Effect of a perforated plate; loose vs hard contact with a flexible porous layer Supplement the results given in Figure 6.6 by comparing the effect of a perforated facing on a flexible material when it is in loose contact and when it is in hard contact with the layer. We recall that in ‘loose’ contact, the perforated facing is not attached to the layer and does not participate in the induced motion of the layer. In ‘hard’ contact, it does. Layer thickness: 4 inches. Flow resistance: 0.5 (normalized). Normalized compliance: 0.5. Loss factor: 0.3. Weight: 2 lb/ft3 . Thickness of facing: 0.1 inch. Hole diameter: 0.125 inch. Open area: 23 percent. Weight: 4 lb/ft2 . SOLUTION The absorption spectra obtained from the computer program are shown in Figure 6.7. As usual, the labels N and DL on the curves refer to normal incidence and diffuse field, local reaction. First consider the loose contact. The only significant effect of the perforated facing is the reduction of the absorption coefficient at high frequencies, above ≈ 2000 Hz. In this regime, the absorption is essentially the same as for a rigid layer since, due to the inertia, the induced motion of the layer is insignificant. The effect of the flexibility is the absorption peak at about 200 Hz. As explained in Chapter 5, this peak is related to the quarter wavelength resonance of the porous structure and occurs a little above the resonance frequency; this
Figure 6.7: Absorption spectra of a flexible porous layer with perforated facing. Left: Loose contact. Right: Hard contact.
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is due to the reduction of the equivalent resistance of the layer to a value lower than the flow resistance because of the induced motion thus producing a better impedance match √ to the incoming wave. The wave speed in the structure-borne wave is v = c (ρ/M)/K, where c is the free field sound speed in air, K the normalized compliance of the structure (normalized with respect to that of air), 1/ρc2 , and M/ρ the ratio of the mass densities of the structure and the air. In this case, with M/ρ ≈ 2/0.08 ≈ 25 and with K = 0.8, we get for the velocity of the uncoupled structure-borne wave, v ≈ .28c ≈ 248 ft/sec. With a layer thickness of 4 inches, the corresponding quarter wavelength resonance of the structure is then ≈ 186 Hz. The observed absorption peak is somewhat above this frequency, as it should be (see Chapter 5). Consider next the hard contact between the facing and the porous layer. There is now an absorption peak at ≈ 400 Hz. This is due to the fact that the facing with its mass of 4 lb/ft2 now impedes the free motion of the surface of the porous layer and acts almost like a rigid wall so that the layer is trapped between two walls. The corresponding structural resonance then will have a wavelength which is twice the layer thickness (rather than four times, as for the free plate). The frequency then should be approximately twice the frequency for the free layer and the absorption peak should be somewhat above this frequency, consistent with our results. Another, and from a practical standpoint more important new feature, is that there is now also a ‘mass-spring’ low frequency resonance resulting from the combination of the mass of the plate and the spring provided by the flexible layer. If the wavelength at this frequency is long compared to the layer thickness, the spring constant of the layer is ρc2 /(KL), where K is the normalized compliance, as defined above, c, the sound speed in air, ρ, the air density, and L, the layer thickness. The resonance frequency is then ρ c , (6.4) f = 2π mKL where m is the mass per unit area of the plate. The frequency thus obtained is 49 Hz. This resonance leads to an enhancement of the absorption at frequencies above the resonance, as explained above, and this is the main reason why the absorber with a bonded plate yields a better performance in the frequency range between 50 and 200 Hz. At frequencies above any of the resonances, the performance is about the same as before since the flexibility of the material plays only a small role in comparison with the inertia. 2. Flexible layer with a perforated plate/screen facing Low frequency noise reduction has become more and more important in recent years and as a measure of the performance of an absorber at low frequencies we use here as a measure of the low frequency performance of an absorber the arithmetic mean of the octave band absorption coefficients at 31.5, 63, and 125 Hz, and denote the values for normal incidence, diffuse field local, and diffuse field nonlocal reaction by NRC0’, NRC1’, and NRC2’. (As before, the
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Figure 6.8: Flexible porous layer: See Problem 1. ordinary values, which refer to the average of the absorption coefficients at 250, 500, 1000, and 2000 Hz, are NRC0, NRC1, and NRC2, respectively.) A perforated plate/resistive screen combination is placed in hard contact with a flexible porous layer. The hard contact is obtained by keeping the layer under compression by the facing. Layer thickness: 12 inches. Normalized flow resistance, 0.25. Normalized compliance, 0.8. Loss factor, 0.5. Weight, 4 lb/ft3 . Perforated plate, open area 23 percent, hole diameter, 0.125 inch, thickness, 0.125 inch, weight, 4 lb/ft2 . The mass of the screen is negligible. What should be the flow resistance of the screen to make the NRC1’ and NRC1 equal? SOLUTION Running the program a few times, we find a normalized screen input resistance of 0.98 to yield NRCL - DL = NRC-DL = 0.68. It should be kept in mind, though, that the input resistance used in the program is the equivalent resistance of the screen as modified by the perforated plate when there is hard contact between the two elements. Thus, an input resistance of 0.98 corresponds to a resistance of the bare screen, which is smaller by a factor of σ , the open area fraction of the screen. Thus, with σ = 0.23, the required resistance of the bare screen becomes 0.23. The corresponding narrow band and octave diffuse field (local reaction) octave band data are shown in Figure 6.8.
6.5.5 Porous Material with Closed Cells So far we have dealt with porous materials with open (interconnected) cells or pores, and we now turn to the case when the cells are closed. Acoustically, the essential difference between these two types of porous materials is that in the closed cell material, the relative motion between the air in the cells and the structure is negligible, and there will be no corresponding friction losses. Therefore, the sound absorption is due solely to the compressional losses in the trapped air (or whatever gas is involved) and in the material itself. The latter mechanism turns out to be negligible under normal conditions.
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We used the slot absorber at the beginning of this chapter as an idealized model in our studies of a rigid porous material with open cells having the slots perpendicular to the rigid backing wall. The slot model can be used also to illustrate the absorption of a material with closed cells. To do that, we arrange the plates to be parallel with the rigid backing wall and let the sound be incident on the assembly normal to the channels. In that case there will be no relative tangential motion between the air and the walls and no friction losses. Sound absorption will be caused only by compressional losses due to heat conduction. These losses can be expressed in terms of the complex compressibility κ˜ of the air, as discussed in Chapter 3, and the losses due to the compression of the solid material in the absorber are in terms of a complex compressibility κ˜ of this material. For the average compressibility of the structure as a whole we then use H κ˜ + (1 − H )κ˜ , where H is the porosity, i.e., the fraction of volume occupied by the air channels. Since the compressibility of the air usually is much larger than for the solid, it will influence the average compressibility and the wave speed in the solid considerably, even for small porosities. To apply the results thus obtained to a material with closed cells of arbitrary shape, we assume that the compressibility of the trapped air is the same as that in our model, with the channel width d replaced by the average linear dimension of the cells or voids. Furthermore, we assume that the solid material behaves much like a liquid, so that the average compressibility given above is valid. An analysis based on such a model shows that the absorption is confined to narrow bands centered at the quarter wavelength resonances of the structure and that the absorption peaks are significant only for rather high porosities. The bulk of the compressional losses then are in the thermal boundary layers of the air and the loss factor of the solid material does not matter much. For a given porosity, there will be an optimum channel width for maximum absorption and it turns out, as no surprise, that this optimum corresponds to a channel width approximately equal to the viscous boundary layer thickness.
6.6 NONLINEAR EFFECTS AND SHOCK WAVE REFLECTION When the acoustic amplitude becomes sufficiently high (approximately one percent of the atmospheric pressure), we can expect nonlinear effects to be significant, and the results we have presented so far, most of them based on linear acoustics, can be expected to be only approximately valid. As an illustration, we shall describe in this chapter experiments dealing with large amplitude acoustic pulses or shock waves and their interaction with boundaries, in particular those involving flexible porous layers. The amplitudes involved are of the order of 1 atm (about 194 dB re 20 microPascal). The primary motivation for carrying out this study was to simulate the waves generated in a closed loop pulsed laser in which the gas was energized by an electron beam, pulsed at a rate of 125 Hz. The wave produced by the pulsed electron beam had a
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detrimental effect on the performance of the laser since the acoustic reverberation in the loop made the density in the lasing cavity sufficiently nonuniform to prevent lasing from occurring at the pulse rate of the electron beam. Thus, the problem of attenuating the wave by a substantial amount was essential for the proper functioning of the laser and as a basis for designing an appropriate attenuator, a study of the interaction of shock waves with various (porous) boundaries was called for.
6.6.1 Apparatus The Shock Tube The shock tube is shown schematically in Figure 6.9. Made of steel with a 3 mm wall thickness, the tube was 2 m long and supplied with appropriate flanges and ports for attaching the driver section, transducers, test section, and tube extensions, one 92 cm and the other 213 cm long. One of the extensions was provided with holes over parts of its length to accommodate transducers. The gas in the tube was air at atmospheric pressure. The driver section was terminated with a properly chosen membrane. We experimented with a variety of membrane materials, particularly Mylar films of different thicknesses, to obtain a peak pressure of the shock wave in the range from 0.2 to 2 atm, corresponding to peak pressure levels of ≈ 180 to ≈ 200 dB. (The dB levels given here are based on a reference pressure 0.0002 dyne/cm2 , although this commonly used reference refers to the rms value of a harmonic wave.) For example, Mylar films with thicknesses of 0.013 and 0.025 mm ruptured at driver pressures of 2.45 and 3.7 atm, and another plastic film of thickness 0.0065 mm ruptured at 1.5 atm. The rupture pressure could be made quite repeatable from one membrane to the next with proper experimental procedure. The pressure was monitored by means
Figure 6.9: Experimental arrangement. As shown, the shock tube is terminated by a porous layer, but other terminations, perforated plates, lined ducts, etc., were used.
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of a Heise pressure gauge. If the pressure was raised rapidly until the membrane ruptured, the value thus obtained would vary somewhat from one membrane to the next. To avoid this variation, the pressure was raised slowly to a value just below the rupture value and held there until the Mylar weakened and broke at the preset pressure value. This procedure assured excellent repeatability in producing shocks at a specified peak pressure. A needle valve attached to the driving section served as a simple check valve to minimize the amount of gas entering the shock tube after membrane rupture. This valve was adjusted to yield a very low flow rate so that the gas that entered the tube during data acquisition was only 1 percent of the mass initially flowing into the driving section. The shock tube could be terminated by various elements, such as a rigid plate, various orifice plates or tube extensions containing porous baffles, or lined duct elements. The rigid termination was a 1.5 cm thick steel plate, and the orifice plates were cut from 2.5 mm aluminum stock. Data Acquisition System The transducer, a PCB 1131A21 piezoelectric, was flush mounted with the interior wall in the shock tube with a resonance frequency of 500 kHz and an excellent transient response (no ringing); it was designed for the study of shock waves. The diameter was 5.5 mm, which determined the ‘resolution.’ With a shock wave speed of approximately 480 m/sec, the travel time over the transducer was 12 microseconds, which sets an upper limit of 86 kHz on the meaningful sampling rate of the signal from the transducer. To determine the pressure from the voltage output, we used the pressure-voltage calibration supplied by the manufacturer. The data acquisition system was a Digital Equipment Corporation MINC 11/23 laboratory computer. We used the Data Translation DT 2785/5714 DI 14 bit differential input A/D board for our analog-to-digital conversion system and found that it was possible to drive the board at a 19 kHz sampling rate without introducing sampling errors. (The manufacturer claimed a 10 kHz rate.) This sampling rate was well below the limit resulting from the resolution of the transducer given above. The clock signal for the A/D board was provided by the Data Translation DT 2769 real time clock. A graphic terminal was provided for immediate display of data. The hardware was controlled through FORTRAN subroutines, which were supplied by the Data Translation software. This made it possible to initiate clock pulses by the firing of the Schmidt trigger located on the clock board, have the A/D board take 2046 pieces of data at the 19 kHz clock rate, and display these data on the screen. The data could be extended on the screen for examination of detailed features and then stored on one of the floppy disks in the system for later retrieval. In order to see the leading edge of the pressure pulse, we used a signal other than the signal from the pressure transducer to fire the trigger on the clock board. This trigger signal was produced by a piezoelectric crystal clamped to the flange of the driver section. When the membrane ruptured, the crystal was strained, producing the required trigger signal.
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6.6.2 Amplitude Dependence of Wave Speed At levels ordinarily encountered in acoustic applications, the sound pressure level is less than 120 dB and under such conditions the sound speed is for all practical purposes independent of the sound pressure. In our experiments, however, peak pressures of shock waves of the order of 1 atm were involved corresponding to peak pressure levels of about 190 dB. At these levels the pressure dependence of the sound speed is indeed noticeable. It is demonstrated explicitly in Figure 6.10 in which the travel times of pressure pulses of different peak values, 0.7 and 0.2 atm, can be compared. As can be clearly seen, the stronger pulse arrives sooner than the weaker, demonstrating the pressure dependence of the wave speed.
6.6.3 Reflection From a Flexible Porous layer The studies included a series of measurements dealing with the interaction of a shock wave with a flexible porous material. The peak pressures of the waves ranged from 0.33 to 1.4 atm, as measured at a distance of 1 m from the source. Of particular interest were studies of the interaction with porous flexible layers (Solimide) of thicknesses ranging from 2 to 8 inches with flow resistances ranging from 0.2 to 0.61 ρc per cm. With a layer thickness of 2 inches and an incident peak pressure of about 1 atm, the reflected pulse was essentially indistinguishable from the reflection from a rigid wall. This result was unexpected and difficult to understand at first but became clear after experiments with thicker layers. Layer thicknesses of up to 8 inches were used and in Figure 6.12 are shown the incident pulses and subsequent reflections from a 4 inch and an 8 inch layer. For comparison, in the left graph, is shown also the pulse (dashed line) reflected from a rigid wall termination. The incident peak amplitude is 0.9 atm at the location of the transducer 1.0 m from the membrane in the shock tube. The pressure pulse labeled B is the reflection from the front surface of the material and B’ is interpreted as the reflection from the rigid backing.
Figure 6.10: Demonstration of the amplitude dependence of the wave speed with two pulses
of different amplitudes. The pulse with the larger amplitude, ≈ 0.7 atm peak pressure above ambient, arrives ≈ 4.4 ms, and the weaker pulse, ≈ 0.25 atm, arrives ≈ 5 ms after the trigger.
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215
The graph on the right in Figure 6.12 refers to an 8 inch thick material and an incident peak pressure of 1.4 atm measured at 1 m from the source. This corresponds to a peak pressure of 1.05 atm at the surface of the material. The general character of the reflected wave has not changed much from that obtained for the 4 inch layer except for the time delay between B and B’, which is now larger, as expected. However, this time delay is much longer than would be expected from the roundtrip time in the porous layer based on the speed of sound in free field. An explanation is that the material is compressed and thus dragged along by the wave so that the effective mass density of the layer is much larger than that for air so that a reduction in the wave speed occurs. By comparing the reflected amplitudes B and B’ with A, we can estimate the corresponding pressure reflection coefficients B/A and B’/A, and the results obtained are plotted vs the incident peak pressure in Figure 6.11. The wave B’ has traveled back and forth through the porous layer, and from the difference in amplitudes between A and B’, we can estimate the attenuation of the wave in the layer. However, to make such comparisons accurately, we have to account for the difference in the nonlinear attenuation of the waves A and B along the path between the termination and the transducer. The reflected waves referred to above continue toward the source where they are reflected from the rigid wall and appear as the next set of pulses in Figure 6.11. The wave B’, with a larger amplitude and consequently a higher wave speed than B, has now almost caught up with B. Similarly, A appears a little earlier than B’ because of the amplitude and wave speed difference. To measure the possible compression of the porous material, the following experiment was done. A thin wooden rod was inserted into the material as shown in Figure 6.13 and the surface of the material was stained with ink. A compression of the material would lead to a staining of the rod,
Figure 6.11: Left: Shock wave reflections from a flexible porous layer. The first reflection, amplitude B, is from the surface of the material and the second, amplitude B’, is interpreted as coming from the rigid backing. The amplitude A refers to reflection from a rigid backing without a porous layer. Right: The pressure reflection coefficients B’/A and B/A vs incident pulse incident peak pressure, measured approximately 1 m from the termination. Material: Solimide, layer thickness = 8 inches, flow resistance = 0.61 ρc per cm, and mass density = 0.030 g/cm3 .
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NOISE REDUCTION ANALYSIS
Figure 6.12: Left: Shock wave reflection from a flexible porous layer. (Solimide, 4 inch thick layer, flow resistance, 0.61ρc per cm, density, 0.030 g/cm3 .) For comparison is shown the reflection from a rigid plate termination (dashed curve). Incident peak pressure, 0.9 atm. (≈ 193 dB). Right: Same as above for an 8 inch thick layer and pulse peak pressure, 1.4 atm (≈ 197 dB).
Figure 6.13: Left: Method of measuring compression. A thin rod is inserted in the material and the compression is measured by the marking made on the rod by a dye in the surface of the material. Right: Measured relative compression of a flexible porous layer as caused by an incident shock wave. The compression is expressed as a fraction of the initial layer thickness and plotted vs the peak pressure of the incident pulse. Material: Solimide, layer thickness = 8 inches, flow resistance = 0.61 ρc per cm, and mass density = 0.03 g/cm3 .
as indicated, and the corresponding compression was determined by removing the rod and measuring the length of the unstained portion of the rod. The results indeed proved that a compression took place, and its magnitude, expressed as a fraction of the initial thickness of the material, is shown as a function of the peak pressure of the incident wave in Figure 6.13. Because of the large compression, the flexible material is not likely to withstand repeated exposure to shock waves for an extended period of time. In the closed loop laser application mentioned in the introduction, the pulses were designed to occur at a high rate and under such conditions it is advisable to use a rigid absorption material, for example, porous metal or ceramic, to avoid acoustically induced fatigue failure within the porous layer.
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6.7 MEASUREMENT OF COMPLEX ELASTIC MODULUS As described in the previous section, a large amplitude sound pulse incident on a flexible porous material caused a considerable compression of the material, of the order of 80 percent of the original thickness at a peak pulse pressure of about 1 atm (Figure 6.13). At normally encountered sound pressures, the compression will be considerably smaller, of course, but the flexibility of the material can have a substantial effect on its sound absorption characteristics, particularly at low frequencies, as we have seen in the examples in this chapter. In our mathematical analysis of the interaction of sound with a flexible porous layer, we used a single elastic modulus, the compliance, i.e., the inverse of Young’s modulus, of the porous frame structure. These are usually orders of magnitude different from the corresponding quantity for the visco-elastic material itself out of which the porous structure is made. A valid analogy is the enormous difference in the compliance of a coil spring made of steel and the compressibility of the steel itself. The small compressibility of the material itself has little significance in the context of sound absorption. It should be stressed that we are here interested in the modulus of the porous frame or matrix alone without the influence of the air. In a periodic deformation of the frame, air will be pumped in and out, and this results in damping. This effect, however, is accounted for in our analysis through the coupling of the structure-borne and air-borne waves in the porous material.
6.7.1 Apparatus Apparatus for Measurement of Complex Compliance of a Porous Structure To eliminate the effect of the acoustically driven pumping of air in and out of the porous material and the associated friction damping, the measurement of the complex elastic modulus of a porous frame should be made in an evacuated test cell. The experimental apparatus we have used is shown schematically in Figure 6.14. A porous sample is excited into axial random vibrations by an electromagnetic driver (‘shaker’), and the ratio of the complex velocity amplitudes at the top and the bottom of the sample are measured with the outputs from the transducers A1 and A2 connected to a two-channel FFT analyzer. A typical test specimen consisted of a circular cylindrical sample of the porous material with one thin aluminum disc of the same diameter bonded to the top and one to the bottom surface of the sample to facilitate mounting of an accelerometers A1 and A2 . Supplementary masses (discs), not shown, were added to the top to adjust the axial natural frequency of the test assembly (the lowest mode frequency of the discs was well above the frequency range of the measurements). The experiment revealed a distinct amplitude dependence of the measured compliance and care was taken to make sure that the measured data were within the linear elastic regime of the sample. To identify this regime, measurements were made at successively lower excitation levels until no changes in the compliance were noted.
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NOISE REDUCTION ANALYSIS
Figure 6.14: Apparatus for the measurement of the complex compliance (inverse of the complex Young’s modulus) of the frame of a porous material in a vacuum. The outputs from the transducers A1 and A2 go to a two-channel FFT analyzer.
With reference to the analysis given below, we can determine the complex Young’s modulus E(1 − i) of the frame ( being the loss factor) by comparing the measured frequency response with computed results. The corresponding normalized complex compliance, used in Chapter 5, is K = ρc2 /E, where ρc2 is the modulus for air ≈ 1.4 × 105 N/m2 . With an appropriate computer program, we obtain E and directly from the measured real and imaginary parts of the transfer function. Examples of experimental results: Material
Modulus E, N/m2
K = ρc2 /E
Loss factor,
Soundfoam, 2 lb/ft3 Scottfelt, 8 lb/ft3 Soundglass, 6 lb/ft3
4.9 · 105 1.5 · 105 2.6 · 105
0.29 0.93 0.53
0.21 0.22 0.039
We note that the values for these materials are of the same order of magnitude as for air. The loss factor for soundglass (fiberglass) is considerably lower than for the two other samples, which have a visco-elastic frame. The loss factors measured when air was present were 0.27, 0.24, and 0.057, respectively, i.e., noticeably higher than in a vacuum. The difference is due to the friction effect of the air being pumped in and out of the material when it is deformed.
6.7.2 Data Analysis The propagation constant for longitudinal waves in the structure is √ q = ω ρκ,
(6.5)
where ρ is the density of the porous structure and κ = 1/E the complex compliance (the inverse of the elastic modulus E). The corresponding wave impedance is Z = ωρ/q.
(6.6)
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The transmission matrix elements for longitudinal waves in the structure (based on the variables p and Zu) are T12 = −i sin(qd) T11 = cos(qd) T21 = −i sin(qd) T22 = cos(qd).
(6.7)
These elements define the linear relation between the two pairs of variables p1 , Zu1 and p2 , Zu2 , where the subscripts 1 and 2 refer to the bottom and the top of the sample, respectively. Thus, we have Zu1 = T21 p2 + T22 Zu2 .
(6.8)
If the average mass load at the top of the sample is M per unit area of the sample, the combined mass of the plate and the transducer, we have p2 = −iωMu2 . Then, by introducing the mass ratio μ = M/ρd between the load and the mass of the sample, we obtain from Eq. 6.8 u2 /u1 =
1 . cos(qd) − μqd sin(qd)
(6.9)
The real and imaginary parts of the velocity ratio are obtained directly from the FFT analyzer as a function of frequency, and from this ratio we can determine the complex propagation constant q numerically, and hence the complex compliance κ and the corresponding elastic modulus E. It is of interest to compare κ with the corresponding quantity for air, κ0 = 1/ρo c02 , where ρ0 is the density of air and c0 the sound speed in air. In terms of these quantities we have √ √ q = ω ρκ = (ω/c0 ) K,
(6.10)
where K ≡ Kr + iKi = κ/κ0 is the compliance normalized with respect to the value 1/ρc2 for air.
6.8 MATHEMATICAL SUPPLEMENT 6.8.1 Limp Material As before, the porosity of the porous material is H , and we define the average air velocity u in the material so that the mass flux is ρu, where ρ is the air density. The mass per unit volume of the porous frame is M = (1 − H )ρ , where ρ is the mass density of the solid material, which makes up the porous structure. The average velocity of the structure is u . In the frame of reference of the structure, the fluid velocity is (u − u ), and, as for the rigid material, the force transmitted to the structure per unit volume can be expressed as zv (t)(u − u ) + Gs ρ∂(u − u )/∂t, where zv (t) accounts for the viscous interaction forces and Gs ρ is the induced mass per unit volume. In this frame of reference (which has the acceleration ∂u /∂t), the equation of motion for the structure becomes one of equilibrium between the inertial force −M∂u /∂t, the interaction force zv (t)(u − u ) + Gs ρ∂(u − u )/∂t, and the
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NOISE REDUCTION ANALYSIS
‘buoyancy’ force (1 − H )ρ∂u /∂t, where (1 − H )ρ is the fluid mass displaced by the structure per unit volume. For harmonic time dependence we have ∂/∂t → −iω and with the interaction impedance defined as z = zv − iωGs ρ, the corresponding complex amplitude equation for harmonic motion of the structure is −iωM u = z(u − u ),
(6.11)
where M = M −(1−H )ρ = (1−H )(ρ −ρ). In the special case when the density of the material in the structure is the same as that of the fluid, i.e., ρ = ρ, the structure responds in the same way as if it were part of the fluid and Eq. 6.11 yields u = u , as it should. (This result would not hold if the buoyancy force were omitted, of course.) If the fluid is a gas, so that normally ρ >> ρ, we may put M ≈ M, but for a liquid, the distinction between M and M must be made. In what follows, we neglect the buoyancy force and put M = M. Solving Eq. 6.11 for u , we get u =
z u. z − iωM
(6.12)
The interaction force per unit volume on the structure is then z(u − u ) =
z(−iωM) u ≡ ze u. z − iωM
(6.13)
The last step defines the equivalent interaction impedance, ze =
z(−iωM) z = , z − iωM 1 + iz/ωM
(6.14)
which can be thought of as the parallel combination of z and the impedance −iωM. To obtain the expression for the complex density ratio we simply replace z by ze in Eq. 5.32 to obtain ρ/ρ ˜ = 1 + ize /ωρ. (6.15) The propagation constant and wave impedance are then obtained by the now familiar expressions Q = (ρ/ρ)( ˜ κ/κ) ˜ ζw = (ρ/ρ)/( ˜ κ/κ), ˜ (6.16) where κ˜ = H κ˜ 1 and
κ˜ 1 /κ = 1 + (γ − 1)Fh .
(6.17)
The function Fh = F [(1 + i)a/dh ] and the analogous Fv = F [(1 + i)a/dv ] contain the parameters a/dh and a/dv , and, as explained in Chapters 3 and 5, we use the equivalent value for these parameters obtained from the relation between a/dv and √ the steady flow resistance and a/dh = Pr a/dv ≈ a/dv , where Pr is the Prandtl number. The steady flow resistance is assumed to be experimentally determined.
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The degree of coupling between the sound field and the porous material depends on the ratio of the flow resistance per unit length r0 and the mass M of the porous material. This ratio corresponds to a characteristic frequency fm =
r0 2π M
(6.18)
at which the resistive drag force equals the inertial force of the structure. If f >> fm the inertial force dominates so that the acoustically induced motion will be small and the material then behaves as if it were rigid as far as sound absorption is concerned. To see how this checks with our results, we have to express the condition f >> fm in terms of our frequency parameter L/λ. With R = r0 L/ρc we have f/fm = 2π(M/ρR)L/λ so that the condition f/fm >> 1 for negligible effect of flexibility of the material becomes L 1 R >> >> 1. λ 2π M/ρ
(6.19)
For example, with R = 32 and M/ρ = 50 we get L/λ >> 0.10. Formally, from Eq. 6.13 and by analogy with what was done in Chapter 5, we can now proceed to calculate the absorption characteristics of a limp porous layer backed by a rigid wall. The results can be used for a flexible material in general as long as we are in the mass controlled region of the material, i.e., if any resonance determined by the mass and the stiffness of the material falls well below the region of interest. However, as will be seen in the next section, the lowest resonance of a typical flexible absorber usually does not fall below the region of interest, contrary to the assumption for the validity of the results of the analysis of the limp material. For this reason, numerical results obtained in this section have not been shown.
6.8.2 Equations for Coupled Waves From the idealized cases of rigid and limp materials treated so far, we now turn to the more realistic structure with finite compliance different from zero. A general description of the elastic properties of the structure can be quite involved, but must be used if we wish to account for the excitation of shear and surface waves by an incident sound wave. However, we shall assume that these waves in the structure are unimportant in the present context, and treat the porous material like an isotropic ‘fluid.’ In any event, the analysis will be applicable without qualifications for sound of normal incidence on a porous layer. Even so, we need two elastic constants (which we shall express as ‘compressibilities’ in our analysis) in much the same way as in the description of a coil spring for which one compressibility (although quite small) refers to the material (steel) from which the spring is made and the other to the compressibility of the spring (structure), which is much larger. We shall denote the compressibility of the material in the structure by κ and the normalized (compliance) of the porous frame by K realizing that generally κ is much smaller than K(1/ρc2 ). Each of these constants are assumed to have been determined for harmonic excitation of the material and expressed as complex numbers to account for internal damping.
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NOISE REDUCTION ANALYSIS
If the cells or interstices of the material are all open, the mass per unit volume of the structure will be M = (1 − H )ρ , where ρ is the mass density of the material from which the structure is made. For a urethane foam, the specific density ρ is of the order of 1.2 and for glass fiber about 2.5. The porosity H is the ratio of the air volume in the open cells and the total volume. If the material contains portions with closed fluid filled cells, the mass density ρ
and the compressibility κ are changed accordingly for those portions. Mass Balance As before, we define the average velocity u of the air in the open cells of the material in such a way that ρu is the average mass flow rate. The linearized mass conservation equations for the air and the structure are then, with H = 1 − H , ∂H H ∂ρ ∂t + ρ ∂t + ρdiv u = 0
H
∂ρ
∂t
+ ρ
∂H
∂t
+ H ρ div u = 0.
(6.20) (6.21)
It has been tacitly assumed that there is no steady (i.e., time independent) flow through the porous material. In the presence of such a flow, additional terms must be included in the linearized equations. In free field, with the sound pressure in the fluid denoted by p, we can express (1/ρ)∂ρ/∂t as κ∂p/∂t, where κ is the (isentropic) compressibility. Similarly, with V = 1/H , the term (1/H )∂H /∂t = −(1/V )∂V /∂t can be written K∂p /∂t, where p is the ‘pressure’ in the structure (the negative of the average stress in the structure) and K the compressibility of the porous structure defined by K = −(1/V )∂V /∂P = (1/H )∂H /∂P . (The definition K = (1/M)∂M/∂P is not quite the same because of the possible variation in ρ in addition to the variation in H , and the volume based definition is more in accord with what is measured.) Quantity (1/ρ )∂ρ /∂t becomes ≈ κ ∂p/∂t, assuming that the pressure in the fluid is the main cause of the compression of the structural material itself. As it turns out, this effect generally can be neglected. We shall deal with harmonic time dependence and internal damping will be accounted for in terms of complex compressibilities of the fluid and of the structure. As in Chapter 5, the complex compressibility of the fluid per unit volume of the structure is written κ˜ = H κ˜ 1 , where κ1 is the complex compressibility per unit volume of the fluid. The complex amplitude equations corresponding to Eqs. 6.20 and 6.21 then become ˜ ) (with ∂/∂t → −iω and dH = −dH = −H Kp ˜ − iωκp ˜ = −div u iH ωKp
˜ −iωKp − iωκ˜ p = −div u .
(6.22) (6.23)
As before, the tilde symbol has been used as a reminder that the compressibilities are complex.
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Using the same notation as in the corresponding equation for a rigid material, we have ∂ρu ∂ρGs (u − u ) = −zv (t)(u − u ) − − grad p (6.24) ∂t ∂t with the corresponding complex amplitude equation −iωρu ˜ = zu − grad p,
(6.25)
where, as in Chapter 3, the interaction impedance is z = zv − iGs ωρ,
(6.26)
ρ/ρ ˜ = 1 + iz/ωρ = s + izv /ωρ,
(6.27)
and the complex density ratio
where z is the interaction impedance, s the structure factor, and zv , the viscous contribution to the interaction impedance. In the numerical analysis to be discussed later, we base zv on the result for an equivalent single channel, as was discussed in Chapter 3. The analogous equation for the structure is ˜ = zu − grad p , −iωMu
(6.28)
M˜ = M(1 + iz/ωM).
(6.29)
where
In most cases of interest, it is a good approximation to neglect the first term on the left side in Eq. 6.22 and the second term in Eq. 6.23. Then, to summarize, the linearized equations for mass and momentum balance take the form iωκ˜ p = div u iωK˜ p = div u
−iωρ˜ u = zu − grad p −iωM˜ u = zu − grad p .
(6.30)
Taking the divergence of the momentum equations and eliminating the terms involving the velocity amplitude, making use of the mass conservation equations, we obtain the coupled wave equations below. ∇ 2 p + ka2 p = ca p
∇ 2 p + ks2 p = cs p with ka2
=
ω2 κ˜ ρ, ˜
˜ ca = iωzK,
ks2
=
ω2 M˜ K˜
cs = iωzκ, ˜
(6.31)
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NOISE REDUCTION ANALYSIS
where ρ: ˜ Eq. 6.27, κ˜ = H κ1 , κ1 : Eq. 6.14. If the terms, which were dropped are retained, there will be corrections. Thus, ˜ ca , a factor ka2 will contain a factor (1 − izκ /ωρκ), ˜ ks2 , a factor (1 + iH z/ωM),
˜ δa = (1 − iH ωρ/z), ˜ and cs , a factor δs = (1 + iωM κ˜ /zκ). ˜ These corrections, however, usually have insignificant effects. Dispersion Relation As in Chapter 3, the spatial dependence of the complex amplitude of the incident and refracted pressure wave are expressed as exp(ikx x + iky y + kz z) and exp(iqx x + iqy y + qz z), where kx = k cos φ, ky = k sin φ sin ψ), kz = k sin φ cos ψ, and k = ω/c. Quantities qx , qy , and qz are the components of the propagation constant q in the material with the magnitude q =
qx2 + qy2 + qz2 .
To obtain the dispersion relation for a wave in the porous material we introduce the spatial dependence given above (∇ 2 p = −(qx2 + qy2 + qz2 )p = −q 2 p) into Eq. 5.2 and obtain [−q 2 + (ω/c)2 ka2 ]p = ca p
[−q 2 + (ω/c)2 ks2 ]p = cs p,
(6.32)
where ka , ks , ca , and cs are given in Eq. 6.31. From these relations follows the equation for q, the dispersion relation, in terms of the normalized propagation constant Q = q/(ω/c), (q 2 − ka2 )(q 2 − ks2 ) = ca cs .
(6.33)
In the numerical analysis it is convenient to normalize the propagation constant, and we introduce Q = q/k, Ka = ka /k, Ks = ks /k, Ca = ca /k 2 , and Cs = cs /k 2 , where k = ω/c and c are the sound speed in the fluid in free field. In terms of the isentropic ˜ and compressibility, κ = 1/ρc2 , we can rewrite Ca and Cs as Ca = [z/(−iωρ)]K/κ ˜ The factor z/(−iωρ) can be considered to be the ratio of the Cs = [z/(−iωρ)]κ/κ. interaction force and the inertia force on the fluid. In terms of these normalized quantities, the dispersion relation 6.33 and its solution can be expressed as (see Eq. 6.31) Q4 − (Ka2 + Ks2 )Q2 + Ka2 Ks2 + C = 0 Q21,2 = 12 (Ka2 + Ks2 ) ± 12 (Ka2 − Ks2 )2 − 4C, where 2 ˜ ˜ Ka = (ρ/ρ)( ˜ κ/κ) ˜ Ks2 = (M/ρ)( K/κ) ˜ C = −Ca Cs = (z/ωρ)2 (K/κ)( κ/κ), ˜
(6.34)
where the subscripts 1 and 2 refer to the two characteristic modes of the system, corresponding to the plus and minus signs, respectively. If no terms had been omitted in the original mass conservation equations, the coupling constant C would have contained a factor δa δs but its effect is generally insignificant.
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225
Special Case: Rigid Material A rigid material is characterized by K˜ = 0. We then obtain ˜ ρ/ρ) ˜ Q1 = Ka = (κ/κ)( Q2 = 0
(6.35)
for the two modes. The first is the fluid-borne wave with a propagation constant Q1 , which is consistent with the value obtained for a wave in a rigid material. The second wave mode is the structure-borne; it has Q2 = 0, which corresponds to an infinite wave speed (as expected) since we are dealing with the idealization of zero compressibility. Special Case: Limp Material In the other limit, with K˜ → ∞, Ks2 and C both go to infinity as K˜ does. Then, after having brought out the factor (Ks2 − Ka2 ) from the radical, we get upon expansion of the radical in terms of quantities ∝ 1/K, Q212 = (1/2)(Ka2 + Ks2 ) ± (1/2)(Ks2 − Ka2 )[1 − 2C/(Ks2 − Ka2 )2 ].
(6.36)
Insertion of the expressions for Ka2 , Ks2 , and C in Eq. 6.34 shows that the terms involving (z/ωρ)2 cancel each other and we get z(−iωM) Q21 ≈ Ka2 + C/Ks2 = (κ/κ) ˜ 1 + i ωρ(z−iωM) = (κ/κ)(1 ˜ + ize /ωρ) Q22 ≈ Ks2 − C/Ks2 = ∞,
(6.37)
where the equivalent interaction impedance ze = z(−iωM)/(ze − iωM). This is consistent with the result obtained for the limp material in the previous section. The structure-borne wave now has an infinite propagation constant, which means zero wave speed and wavelength. It will also have infinite attenuation and will disappear as soon as it has been excited so that only the fluid-borne wave remains. Nevertheless, in some cases, for example when the surface is covered with an impervious membrane, the structure-borne wave has to be included to satisfy boundary conditions at the surface.
6.8.3 Pressure and Velocity Fields For each of the propagation constants obtained from the dispersion relation there is a corresponding wave mode in which the spatial variations of pressure and velocity are the same for the fluid and the structure. The general expression for the wave field in the porous material is then a linear superposition of the two modes. To determine the excitation of these modes by an incident plane sound wave, we proceed as follows. The plane wave incident on the absorber is of the form exp(ikx x + iky y + ikz z), where kx = k cos φ, ky = k sin φ sin ψ, kz = k sin φ cos ψ, and k = ω/c, φ and ψ being the polar and azimuthal angles, respectively (Figure 6.32). The field within the material is the sum of one wave traveling in the positive x-direction and one in
226
NOISE REDUCTION ANALYSIS
the negative direction. The wave in the positive x-direction has the same form as the incident wave with k replaced by q, kx replaced by qx , etc. Trace matching of the wave front at the boundary requires qy = ky and qz = kz and we get for Qx = qx /k, Qx =
Q2 − sin2 φ,
where Q = q/k, q 2 = qx2 + qy2 + qz2 , and k = ω/c, as discussed in Chapter 3. We place x = 0 at the wall and x = −L at the surface. The velocity and pressure fields will be linear combinations of contributions from the two characteristic modes of the system, identified by their propagation constant q1 and q2 . Since the velocity components ux and u x of the air and structure are both zero at x = 0 (rigid wall) the characteristic modes for the velocity field will be sine functions, i.e., ux = [U1 sin(q1x x) + U2 sin(q2x x)]eiqy y+iqz z u x = [V1 U1 sin(q1x x) + V2 U2 sin(q2x x)]eiqy y+iqz z ,
(6.38)
where q1 and q2 are the propagation constants for the two modes and V1 = U1 /U1 and V2 = U2 /U2 are the ratios of the velocity amplitudes in the structure and the air for the two modes. The corresponding expressions for the pressure field are p = [P1 cos(q1x x) + P2 cos(q2x x)]eiqy y+iqz z p = [ 1 P1 cos(q1x x) + 2 P2 cos(q2x x)]eiqy y+iqz z ,
(6.39)
where 1 = P1 /P1 as the ratio of the amplitudes in the structure and in the fluid for mode 1, with an analogous expression for mode 2. The pressure ratio is simplest obtained directly from Eq. 6.32 having obtained the propagation constants for the two modes from Eq. 6.36. Thus, in terms of the normalized propagation constant, we obtain 1,2 ≡
Q21,2 − Ka2 P
(−iz/ωρ)(κ/κ) ˜ . = = 2 − K2 ˜ P Q (−iz/ωρ)(K/κ) s 1,2
(6.40)
/U Similarly, the ratios V1,2 = U1,2 1,2 of the amplitudes of the velocity modes in the structure and in the fluid are obtained by insertion of the velocity and pressure fields in the momentum equations in Eq. 6.30 for the fluid and the structure. For each of the characteristic wave modes the spatial dependence of p and p is the same so that the ratio of their gradients will be the same as the ratio of the pressure amplitudes. Thus, we obtain for the corresponding ratio of the velocity amplitudes
V1,2 ≡
U1,2
U1,2
=
iωρ ˜ 1,2 − z , iωM˜ − z 1,2
(6.41)
where 1,2 = (P /P )1,2 is given in Eq. 6.39. Finally, the relation between the pressure and velocity amplitudes P1,2 and U1,2 is obtained by insertion of the velocity and pressure fields for the fluid in the momentum
227
FLEXIBLE POROUS MATERIALS equation for the fluid in Eq. 6.30, P1 = −iρc Z1 U1 P2 = −iρc Z2 U2 Z1 =
1 ˜ Q1x (ρ/ρ
− izV1 /ωρ)
Z2 =
1 ˜ Q2x (ρ/ρ
− izV2 /ωρ).
(6.42)
The complex amplitudes of sound pressure and the x-component of fluid velocity just outside the boundary are denoted by p0 and u0 . From our definition of average fluid velocity within the material explained earlier, continuity of mass flow through the surface of the material at x = −L and sound pressure in the fluid require that u0 = ux (−L) and p0 = p(−L). Furthermore, the stress in the structure at the surface must be zero, p (−L) = 0, which means P2 cos(q2x L) = −( 1 / 2 )P1 cos(q1x L).
(6.43)
This boundary condition, p (−L) = 0, deserves a comment. In our formulation of the coupling between the air and the structure, the interaction force per unit length on the structure is expressed as being proportional to the relative velocity between the air and the structure; the constant of proportionality is the interaction impedance. Consequently, the interaction force on an element of the structure at the surface of the layer goes to zero as the thickness of the element goes to zero, and there is no sound pressure or sound pressure gradient in the air that contributes to the force. Therefore, the strain and the stress (pressure) in the structure are zero at the surface (otherwise the element would have infinite acceleration) unless it is in mechanical contact with another (porous) structure. The situation is analogous to that of a coil spring that interacts through viscous drag with the surrounding air. The strain and stress of the free end of the spring are zero. With P2 /P1 given by Eq. 6.47, the corresponding velocity ratio is obtained from Eq. 6.42, U2 /U1 = (Z1 /Z2 )(P2 /P1 ),
(6.44)
where Z1 and Z2 are given in Eq. 6.42. With this expression for U2 /U1 , the velocity amplitude distribution in the air and in the structure can now be computed from Eq. 6.38. Dissipation Function Having obtained the pressure and velocity distributions in the layer, we can readily determine the distribution of the dissipation of acoustic energy within the porous layer. With the velocity amplitudes being rms values, the distribution of the time average dissipation per unit volume due to the friction drag can be written θρc|u−u |2 , where
228
NOISE REDUCTION ANALYSIS
θρc is the real part of the interaction impedance. We normalize this expression with respect to the dissipation at the surface of the layer, w/w0 = |u − u |2 /|u0 − u 0 |2 w0 = θρc|u0 − u 0 |2 ,
(6.45)
where the subscript 0 refers to the conditions at the surface of the absorber. As explained in Chapter 2, the power dissipation due to the compression of the porous structure can be written as ωK2 κ|p |2 , where K2 is the imaginary part of the complex compressibility K˜ of the structure, and p = P1 [ 1 cos(Q1 kx) + 2 (P2 /P1 ) cos(Q2 kx)], is given in Eqs. 6.42 and 6.43. The factor κ = 1/ρc2 accounts for the fact that K2 is normalized with respect to air compressibility κ. To normalize this expression with respect to w0 (Eq. 6.45), we obtain the velocities at the surface from Eq. 6.39 with x = −L. Then, combining with the velocity ratio U2 /U1 , given in Eq. 6.44, and the ratio P1 /U1 in Eq. 6.42, we obtain for the normalized compressional dissipation function w /w0 =
kL K2 |p |2 , R|u0 − u 0 |2
(6.46)
where k = ω/c and R = θ L is the total normalized resistance of the layer. Examples of the velocity and pressure induced dissipation functions w/w0 and w /w0 , as well as their sum, are plotted in Figure 6.39 vs x/L.
6.8.4 Absorption Coefficients Open Surface We consider first the case when the surface of the porous layer is open; later, the effect of a thin impervious surface film will be studied. Recall Eq. 6.44 for the velocity ratio U2 /U1 = (Z1 /Z2 )(P2 /P1 ).
(6.47)
The sound pressure and velocity u0 at x = −L can be expressed as (see Eqs. 6.38 and 6.39) p0 = P1 (1 − 1 / 2 ) cos(q1x L) = P2 (1 − 2 / 1 ) cos(22x L) u0 = −U1 sin(q1x L) − U2 sin(q2x L) = −i(P1 /ρcZ1 ) sin(q1x L) − i(P2 /ρcZ2 ) sin(q2x L).
(6.48)
The normalized input admittance of the flexible porous layer is then Normalized input admittance, Flexible porous layer, Open surface u(0) = −iη1 tan(Q1x kL) − iη2 tan(Q2x kL) ηi ≡ βi + iσi = ρc p(0) η1 = 2 /Z1 ( 2 − 1 )
(6.49)
η2 = 1 /Z2 ( 1 − 2 ) where Z1 and Z2 : see Eq. 6.43, corresponding normalized input impedance: ζi = 1/ηi .
229
FLEXIBLE POROUS MATERIALS With reference to Chapter 2, the absorption coefficient is α(φ) =
4βi cos φ . (βi + cos φ)2 + σi 2
(6.50)
Closed Surface Some porous materials, such as urethane foams, are processed in such a way as to leave an impervious skin of negligible mass on the surface. The surface can also be made impervious by application of a membrane or plate in contact with the surface. We consider first the case when the mass of the impervious layer can be neglected in which case the boundary conditions at the surface are p0 = p + p
u0 = u = u .
(6.51)
From the second of these relations it follows that U2 (1 − V2 ) sin(q2x L) = −U1 (1 − V1 ) sin(q1x L),
(6.52)
and the velocity at the surface can be expressed as u0 = −
V1 − V2 V1 − V2 U1 sin(q1x L) = U2 sin(q2x L). 1 − V2 1 − V1
(6.53)
With the pressure at the surface being p0 = P1 (1 + 1 ) cos(q1x L) + P2 (1 + 2 cos(q2x L),
(6.54)
it follows that the normalized input impedance p0 /ρcu0 can be written Normalized input impedance, Flexible porous layer, Closed surface ζi ≡ θi + iχi = iζ1 cot(Q1x kL) + iζ2 cot(Q2x kL) ζ1 = ζ2 =
1−V2 V1 −V2 1−V1 V2 −V1
(1 + 1 )Z1
(6.55)
(1 + 2 )Z2
where Z1 and Z2 are given in Eq. 6.43. With reference to Chapter 2, the absorption coefficients can now be computed in terms of the input impedance (Eqs. 3.22 and 3.23). Surface with a Resistive Screen If a screen is attached to the open surface, the boundary condition at the surface is that the velocity of the structure be the same as the velocity of the screen. Furthermore, the force acting on the screen is the sum of the interaction force due to the relative motion of the air and the screen and the force due to the contact with the porous layer.
230
NOISE REDUCTION ANALYSIS
We refer to Chapter 1 for a discussion of the screen. We use the notation z ≡ ρcζz for the interaction impedance of the screen and the mass per unit area is m. The boundary condition requires that the velocity of the screen must equal the velocity amplitude of the porous structure at x = −L, i.e., u x (−L). The velocity amplitude u0 of the air just outside the screen is the same as the amplitude ux (−L) at the entrance to the porous layer. From the definition of interaction impedance z, the flow induced force on the screen is then z(u − u ), and it follows that the equation of motion of the screen is z(u − u ) = −iωmu + p .
(6.56)
This equation now serves as the boundary condition from which we can determine the amplitude ratio U1 /U2 of the two modes in the system. Thus, with zt = z − iωm and Vs ≡ z/zt , it follows from Eqs. 6.41 and 6.42 that [(Vs − V1 )U1 sin(qx1 L) + (Vs − V2 )U2 sin(q2x L)] = (1/zt )[ 1 P1 cos(q1x L) + 2 P2 cos(q2x L)].
(6.57)
Expressing the pressure amplitudes P1 , P2 in terms of the velocity amplitudes U1 , U2 from Eq. 6.43, we obtain for the velocity ratio U2 (Vs − V1 ) sin(qx1 L) + i( 1 Z1 /ζt ) cos(qx1 L) =− . U1 (Vs − V2 ) sin(qx2 L) + i( 2 Z2 /ζt ) cos(qx2 L)
(6.58)
With p0 being the sound pressure amplitude in front of the screen it follows from the definition of the interaction impedance that p0 − p = zs (u − u ) = −iωmu + p
(6.59)
p0 = (1 + 1 )P1 cos(q1x L) + (1 + 2 )P2 cos(qx2 L) −(−iωm)[V1 U1 sin(q1x L) + V2 U2 sin(q2x L)].
(6.60)
or
The velocity amplitude in front of the screen is, from Eq. 6.41, u0 = −[U1 sin(qx1 L) + U2 sin(qx2 L)].
(6.61)
From these expressions for p0 and u0 and the velocity ratio in Eq. 6.58, we obtain the input impedance ζ0 = z0 /ρc ζ0 =
i(1+ 1 )Z1 U1 cos(q1x L)+i(1+ 2 )Z2 U2 cos(q2x L) U1 sin(q1x L)+U2 sin(q2x L) ωm V1 U1 sin(q1x L)+V2 U2 sin(q2x L) −i( ρc ) U1 sin(q1x L)+U2 sin(q2x L) .
(6.62)
To check this expression, we consider first the limiting case when the interaction impedance is zero. We should then recover the result for the open surface in Eq. 6.49. With zs = 0 and m = 0 (zt = 0) it follows from Eq. 6.58 that 2 Z2 cos(q2x L)U2 = − 1 Z1 cos(q1x L)U1 .
(6.63)
231
FLEXIBLE POROUS MATERIALS
Using this relation in the numerator in Eq. 6.62, we obtain for the admittance 1/ζ0 a result which agrees with that obtained earlier in Eq. 6.49. Actually, we can obtain the result for a closed surface if we put zs = zt = ∞ (to make the screen impervious) and m = 0 to simulate a skin with negligible mass. This results in Vs = 1 and Eq. 6.58 yields (1 − V2 ) sin(q2x L)U2 = (1 − V1 ) sin(q1x L)U1 .
(6.64)
If this relation is used in the denominator of Eq. 6.62, we find the impedance to be in agreement with the result in Eq. 6.55. It is important to note that to account for a screen cover attached to an open porous layer it is not merely a matter of adding the impedance of the screen to the input impedance of the bare porous layer. Such an addition is valid only if the screen is placed close to but not in hard contact with the screen. Use of 4 × 4 Matrices An alternate and more straightforward calculation of the input impedance in the presence of a screen can be made by using the 4 × 4 matrices of the screen and the porous layer. Whenever a structure is comprised of flexible layers in mechanical contact with one another, a matrix formulation of a study of the acoustical characteristics of the structure involves 4 × 4 matrices. This obtains where the dynamical description of a structural element can be made with only two field variables, which, when taken together with the two variables for the sound field leads to a total of four variables. As above, we denote by subscript 0 the field variables just in front of the screen and by subscript 2 the variables at the rigid backing wall. Then, if Tij denote the matrix elements of the product of the matrices for the screen and the porous layer, we have, accounting for u2 = u 2 = 0, p0 = T11 p2 + T13 p2
ρcu0 = T21 p2 + T23 p2
(6.65)
so that the normalized input impedance is ζi =
T11 + T13 (p2 /p2 ) . T21 + T23 (p2 /p2 )
(6.66)
The ratio p2 /p2 is obtained by the requirement that p0 = 0 at the front surface where there is not mechanical contact with the screen (see the discussion in connection with Eq. 6.47). Thus, p0 = T31 p2 + T33 p2 = 0
(6.67)
yields p2 /p2 = −T31 /T33 so that the input impedance becomes ζi =
T11 T33 − T13 T31 . T21 T33 − T23 T31
(6.68)
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NOISE REDUCTION ANALYSIS
From the product of the matrices of the screen and the porous layer, we can express matrix elements Tij of the combination in terms of ζe and Mij to obtain T11 = M11 + ζs (M21 − M41 ) T13 = M13 + ζs (M23 − M43 ) T21 = M21
(6.69)
T23 = M23 T31 = −ζs M21 + M31 + ζt M41 T33 = −ζs M23 + M33 + ζt M43 . The input impedance then follows from Eq. 6.68. The matrix multiplication can, of course, be done directly by a computer without the need for explicit expressions for the matrix elements and in a problem involving several layers and screens, this is the only tractable method. General Comments In order to include the frequency dependence of the interaction impedance (flow resistance and structure factor), we again use the results from the equivalent slot absorber and proceed to determine the equivalent parameters a/dv and a/dh from the measured steady flow resistance as discussed in Chapter 3 and express them in terms of the frequency ratio f/fv , where fv = r0 /(2πρ). These parameters determine Fv = F [(1 + i)a/dv ] and Fh = F [(1 + i)a/dh ] (where F is given in Eq. 2.78), which in turn are used to determine the viscous interaction impedance and the complex compressibility. Thus, for the viscous interaction impedance zv , the interaction impedance z, the ˜ complex density ratios for the fluid and the solid, ρ/ρ ˜ and M/ρ, and the complex compressibility of the fluid we have used the expressions Fv zv = (1/H )(−iωρ) 1−F v
z = zv − iωGs ρ ρ/ρ ˜ = 1 + iz/ρ ˜ M/ρ = M(1 + iz/ωM).
(6.70)
The complex compressibility of the fluid per unit volume of the structure is κ˜ = H κ˜ 1 , where H is the porosity and κ˜ 1 /κ = 1 + (γ − 1)Fh .
(6.71)
The induced mass factor Gs and the corresponding structure factor s = 1 + Gs as well as the complex compressibility of the porous structure are empirical constants, which have to be obtained from experiments. The propagation constants Q1,2 (Eq. 6.36) for the two modes can now be calculated together with the corresponding pressure and velocity ratios, input admittance or impedance, and the absorption coefficient, as outlined in this chapter.
Part II
Duct Attenuators
Chapter 7
Duct Acoustics 7.1 PRELIMINARIES As defined here, duct acoustics deals with the propagation of sound in ducts. It is of practical interest in a variety of different contexts ranging from speech and hearing (vocal tract, ear canal) to music (wind instruments), internal combustion and jet engines, wind tunnels and air handling systems in buildings. In some of these applications, the related problem of noise reduction by means of lined ducts and silencers is a small but an essential part; actually, it is the main topic of these notes. When it comes to noise reduction in the ear canal, most of us merely use a plug of cotton or some soft tissue paper or, to be fancier, a commercial brand of ear plugs or muffs, without worrying much about the physics involved. For industrial duct systems, this ear plug approach no longer is possible in general, since the duct usually has to carry flow with only a small pressure drop allowed. Then, a more analytical method is called for since cost and effectiveness of noise reduction play a significant role in the design. Noise reduction in a duct silencer thus obtained is caused in part by sound absorption and in part by reflections and interference effects. Normally, these mechanisms are interrelated and cannot be strictly separated. As a rough classification, however, a silencer in which sound absorption dominates is usually referred to as dissipative and when reflection dominates, as reactive. The latter typically is used for low frequency sound and pulsations. The dissipative silencer is often simply a duct lined with an appropriate (porous) material. The designations ‘silencer’ and ‘muffler’ apply to both dissipative and reactive duct attenuators. Although conceptually, sound transmission through a duct is not much different from sound transmission through a hole in a wall, when it comes to a detailed quantitative analysis, however, additional difficulties arise. Actually, the mathematical analysis of sound transmission through a duct with a complicated shape generally requires the use of tedious finite element numerical procedures. They are comparatively slow and leave much to be desired in terms of providing general insight. Fortunately, there are duct configurations of great practical interest, which do lend themselves to direct mathematical analysis from first principles, and considerable insight can be gained
235
236
NOISE REDUCTION ANALYSIS
and general design principles established by a thorough study of the corresponding solutions. This is the main emphasis in this book. Before proceeding to the ‘nuts’ and ‘bolts,’ we summarize briefly some of the concepts and effects that will be discussed at some length in the rest of the book. Wave modes. We start by illustrating wave modes in a duct with rigid, nonporous walls by a simple experiment. The bulk of subsequent analysis is devoted to the simplest of these modes, the fundamental mode. Although the basic characteristics of higher modes are discussed, inclusion of them in practical cases is often intractible analytically and semi-empirical corrections are used to yield higher order mode corrections. This approach holds true also for the effects of refraction of sound in a nonuniform mean flow and temperature distribution in a duct. Description of silencer performance. A discussion of the quantities used in the description of the performance of a lined duct is next; attenuation, transmission loss (TL and TL0), insertion loss (IL), and noise reduction (NR), and procedures for the measurement of these quantities. The difference between them is apparent particularly at low frequencies and unlined (reactive) duct elements such as straight pipe sections, contraction and expansion chambers, and side-branch resonators. Duct liners, local and nonlocal reaction. To achieve sound absorption and a related attenuation of sound in a duct, the wall is normally lined with a porous material. If the material is placed in compartments separated by closely spaced transverse solid partitions, the sound field in such compartments is determined only by the sound pressure in the duct at the entrance to the compartment, and not at other locations. Such a liner is called locally reacting. The sound pressure produces a ‘pumping’ of air in and out of the compartment, and this leads to energy absorption. Axial motion in the liner is prevented by the partitions. It should be clear that if the liner (and hence the partitions) occupy the entire duct, the transmission of sound is prevented by the partitions (if assumed rigid). With the partitions removed, there will be sound transmission in the axial direction within the liner and the corresponding sound field at a given axial position depends not only on the sound pressure in the duct at that location but on the entire axial distribution of pressure along the duct. Such a liner is said to be nonlocally reacting. In reality, transverse partitions are normally used but their separation is not always so small that the liner qualifies as locally reacting. The corresponding measured duct performance then falls between the values obtained for locally and nonlocally reacting liners. Actually, at low and high frequencies the predicted values for the two types of liners are essentially the same but in the mid-frequency range, local reaction yields a somewhat higher predicted value than nonlocal. It should be added that the mathematical analysis of the locally reacting duct is considerably simpler than for the nonlocally reacting. At high frequencies, higher order modes will affect the experimental data and yield higher values than predicted for the fundamental mode. Fortunately, the problem of noise reduction at high frequencies is typically not critical since the low, and to some extent, the mid-frequencies typically dictate the design. Acoustically equivalent ducts. Considerable attention is paid to parametric studies of parallel baffle type silencers or attenuators (sometimes referred to loosely as ‘splitters’), see Figure 7.6, and the acoustically equivalent rectangular lined ducts or
DUCT ACOUSTICS
237
channels. Their attenuation spectra are shown in Figures 8.1 and 8.3 for locally and nonlocally reacting liners respectively. The effect of perforated facings and screens on such liners have also been determined. Actually, results for other multilayer liners of porous materials, screens, air layers, and perforated plates are presented in this book and can readily be determined with a computer program. The effects of slots in a porous liner are also discussed. From the family of attenuation curves obtained for different parameter values, optimum design and the corresponding maximum attenuation values have been obtained. The results are summarized in Figure 8.2. Effects of mean flow. Normally, a duct carries a mean flow which has at least five effects on sound transmission. The first is convection; the wave speed is the sum of the local sound speed and the flow velocity and is increased or decreased by the flow depending on the direction of sound propagation. The flow speed is nonuniform across the duct with the maximum at the center of the duct. This leads to refraction of sound, toward the boundary downstream and away from the boundary upstream with a corresponding increase and decrease, respectively, of the attenuation. It is particularly important at high frequencies. A semi-empirical correction for this effect is proposed. Another effect of flow is a modification of the boundary impedance caused by soundflow interaction. This is of particular importance when the liner consists merely of a perforated plate backed by an air layer or when cavity resonators are involved. The resistive component of the boundary impedance is increased by the flow and this leads to broadening of the resonances. In addition, flow generated noise in a duct normally is increased somewhat by insertion of a silencer; this tends to decrease its insertion loss somewhat. Still another flow effect is the sound attenuation in turbulent duct flow. An analysis of it is deferred to Chapter 9 and an interesting result is a comparison with visco-thermal attenuation, which shows that the flow induced attenuation usually is dominant, at least at low frequencies. Effects of temperature. The major effect of temperature is its influence on the sound speed (and hence the wavelength at a given frequency) as the performance of a duct liner normally is intimately related to the ratio of the thickness of the liner and the wavelength. This (typically) results in a shift of the attenuation spectrum toward higher frequencies with increasing temperature. Like the flow velocity distribution, the temperature distribution in a duct is not uniform. A higher temperature at the center of the duct leads to refraction of sound toward the boundary, but unlike the effect of flow, it is independent of the direction of the sound. Thus, the refractions by flow and temperature cooperate for sound in the flow direction but are opposed in the opposite direction. The temperature can also affect the boundary impedance of a duct liner since the flow resistance of a liner increases with temperature. Liquid pipe lines. A question often asked is ‘To what extent can the results obtained for air ducts be applied to liquid pipe lines?’ An answer is given in Chapter 9 by two examples, one involving a pipe line with a slightly resilient wall and the other with a highly resilient one.
238
NOISE REDUCTION ANALYSIS
7.2 WAVE MODES 7.2.1 Simple Illustration The sound wave in a duct with rigid walls can have many different forms. The simplest is the fundamental wave mode in which the sound pressure is uniform across the duct, and it behaves like a plane wave in free field. We can imagine such a wave as being generated by an oscillating plane piston (approximated by a loudspeaker) at the beginning of the duct. This wave will travel unattenuated along the duct at all frequencies if we neglect visco-thermal effects at the boundaries and absorption within the gas itself. If the piston is simulated by the two loudspeakers in Figure 7.1, the speakers have to be driven in phase to produce a plane wave. If the speakers are driven 180 degrees out of phase, so that one pushes when the other pulls, the average axial velocity amplitude in the duct will be zero, and there will be no plane wave generated. Sound will still be produced, though, not in the plane wave mode but in the form of ‘higher order’ modes. Unlike the plane wave, however, the character of a higher mode depends strongly on frequency. The wave contributions from the individual speakers travel out into the duct, one with a positive and the other with a negative sound pressure. If there were no phase difference between them proportional to the path difference of wave travel to the point of observation (receiver), they would cancel each other. This is the case in the mid-plane of the duct where the net sound pressure will be zero. Loudspeakers
(0,0)-mode
Oscillator Amplifier
D
Speakers in phase: Plane wave propagation at all frequencies (0,1)-mode
Speakers 180 deg out of phase
Figure 7.1: If the two loudspeakers operate in phase (push-push), a plane wave will be generated. If they are 180 degrees out of phase (push-pull), the (0, 1) higher acoustic mode will propagate if the frequency exceeds the cut-on frequency c/2D, where c is the sound speed and D is the duct width.
DUCT ACOUSTICS
239
At a point in the duct away from the mid-plane, however, there will be a source-to-receiver path difference so that the elementary waves at the receiver are not completely out of phase. In fact, if the difference is half a wavelength, the waves will arrive in phase so that constructive interference results. We then get a wave that travels through the duct with a node at the mid-plane and, unlike the fundamental mode (i.e., the plane wave), it is characterized by zero average oscillatory axial flow in the duct (the flows above and below the mid-plane are 180 degrees out of phase). The wave is called a higher order mode but could also be termed a ‘push-pull’ wave. With reference to the example in the figure, the pressure amplitude variation in the vertical direction shows a nodal plane in the horizontal plane at the center. The pressure amplitude in the x-direction (horizontal) is uniform, i.e., no nodal plane. At higher frequencies than that in the figure, more than one nodal plane can occur. In general, there will be vertical nodal planes also, and in terms of the number of nodal planes m and n in the two directions, the wave is designated as an (m, n)-mode. It can be generated by several pairs of loudspeakers like those in the figure. The plane wave then is the (0, 0)-mode and the wave in the figure is the (0, 1)-mode. For the (0, 1)-mode to be propagated through a hard walled duct without attenuation due to destructive interference between the waves from the two speakers, requires that the wavelength be short enough so that a path difference between these two waves of half a wavelength can be obtained; at sufficiently low frequencies and correspondingly long wavelengths, this is not possible, and the interference between the push and pull contributions to the sound pressure leads to a decrease of the resulting amplitude with distance. The reason is that the path difference decreases with increasing distance from the source so that destructive interference will be more pronounced with increasing distance. As a result, it turns out that the wave amplitude will decrease exponentially with distance. Such a wave is called evanescent. The largest path difference is at the beginning of the duct in the plane of the source where it equals the width D of the duct (from the top of one speaker to the bottom of the other). Then, if half a wavelength equals the duct width D, there will be constructive interference between the elementary wave from the top of one speaker and the bottom of the other, and λ/2 = D signifies the condition of ‘cut-on’ of the higher mode. The corresponding frequency, f0,1 = c/λ = c/2D, is called the ‘cut-on’ frequency of the (0, 1)-mode (it is also called the ‘cut-off’ frequency, the choice depending on from what direction the frequency is approached, we suppose). At this frequency the mode represents a standing wave perpendicular to the duct axis. The arrangement shown in Figure 7.1 is useful as a simple table top demonstration of higher mode generation. The sound source consists of two identical loudspeakers mounted in the wall at the beginning of the duct. They are driven by an oscillatoramplifier, as indicated. To change the speakers from in-phase to 180 degrees out-ofphase operation (from ‘push-push’ to ‘push-pull’) simply involves switching the leads from the amplifier to the speakers, as shown. In a particular experiment, the duct height was D = 25 cm corresponding to a cut-on frequency of 684 Hz. With the speakers operating out-of-phase, increasing the frequency through the cut-on value brings us out of the exponential decay regime and a marked change in sound pressure emitted from the duct is observed. The duct can be said to act like a high-pass filter for the (0, 1)-mode.
240
NOISE REDUCTION ANALYSIS
Because of the evanescence (exponential decay with distance) of the wave below the cut-on frequency, the sound that radiates from the end of the duct is feeble. It is due to what is left of the evanescent wave when it reaches the end. It is also possible that a weak plane wave component may be present because of an unavoidable difference in the speakers so that the average velocity over the total source surface is not exactly zero. In any event, if one of the speakers is turned off in this push-pull operation, a substantial increase in sound pressure is observed because the wave, which is now generated by the remaining speaker, is not destructively interfered with. As will be clear in our subsequent discussion, the wave field in the (0, 1)-mode can be thought of as a superposition of plane waves, which travel at a certain angle with respect to the duct axis, being repeatedly reflected from the duct walls to build up a traveling (0, 1)-mode wave. At the cut-on frequency, these waves are normal to the axis of the duct, but at a higher frequency, the angle φ with the axis is given by sin φ = (λ/2)/D. The phase velocity of this mode will be the speed of the intersection point of a wave front with the boundary (or the duct axis), and this speed is c/ sin φ, i.e., greater than the free field sound speed and the (0, 0)-mode in the duct. Because of the difference in phase velocities, the superposition of a plane wave and a higher mode results in a wave field that varies with position along the duct. Similar arguments show that if the wavelength is smaller than D/2n, where n is an integer, a mode, the (0, n)-mode with n nodal planes and a cut-on frequency f0,n = nf0,1 , can propagate. Again, the wave field in the duct can be regarded as a superposition of plane waves which are reflected back and forth between the boundaries and traveling in a direction which makes an angle φ with the duct axis, where sin φ = λ/(2nD). The phase velocity of a higher mode is always greater than the sound speed in free field and, like the angle φ, is frequency dependent. If the duct wall has an absorptive liner, the reflections from the walls involve not only a reduction of the amplitude but also a phase change and the superposition of the waves no longer leads to ‘clean’ nodal planes and there will be no sharply defined cut-off frequency. The sound field still can be expressed as a superposition of fundamental and higher order modes corresponding to the (0, 0) and (m, n) modes. In most problems of noise control, the fundamental mode is of main interest and most of the numerical results given in these notes refer to it.
7.3 MEASURES OF SILENCER PERFORMANCE 7.3.1 Attenuation If a microphone is moved along the air channel in a lined duct in which the fundamental mode is dominant, we find that the recorded sound pressure level decreases linearly with distance from the source except for some irregularities at the entrance and at the end of the duct. These irregularities are due to higher order modes and reflection. (If the frequency is below the cut-on values for the higher modes, the contribution from these modes to the total sound field, the dominant contribution to the field, is normally from the fundamental mode.) The slope of the linear decrease of the sound pressure level of this mode is, by definition, the attenuation per unit length in the duct, and it is this and related quantities described below that are computed in
DUCT ACOUSTICS
241
the mathematical analysis in Chapter 10. The (total) attenuation of the fundamental mode is the product of the slope and the length of the duct.
7.3.2 Transmission Loss, TL and TL0 Two-Room Method The concept of transmission loss, TL, generally refers to a partition wall between two (reverberation) rooms, a source room and a receiving room. It is normally defined as the ratio, expressed in dB, of the incident acoustic power on the partition wall from the source room and the power transmitted into the receiving room. On the assumption of diffuse sound fields in the rooms, the incident and transmitted powers, and hence the TL, can be expressed in terms of the measured average sound pressure levels in the two rooms, the area of the partition, and the absorption area (determined from measured reverberation time) in the receiver room. In principle, the measurement of the transmission loss of a duct silencer could be carried out in the same way with the duct inserted into an opening of a heavy wall between the two rooms. The transmission loss of the duct then could be determined from the average sound pressures in the two rooms in the same way as for the partition wall. This method does not seem to have been used for ducts, however. One reason is, no doubt, that a silencer normally is not used in this manner, but the method should be seriously considered as an option. ‘Standard’ Method The silencer to be studied is placed in a test duct, as shown schematically in Figure 7.2. Typically, the sound field is produced by one of the two options shown in the figure, which also allows for the generation of a mean flow through the duct. The sound field in the test duct at the entrance of the silencer is a superposition of an incident wave and a wave reflected from the silencer. Similarly, the sound field at the exit contains a transmitted wave and a wave reflected from the end of the test duct. The extraction of the primary incident and transmitted powers (in the test duct) from these fields for the determination of TL as the ratio (in dB) of these powers is by no means obvious and requires further measurements and analysis. This is one of the hidden difficulties with TL, which is seldom thought of. In this context, one might think that if the incident wave is in the form of a short pulse (which has a broad frequency spectrum), the separation of the incident and reflected waves should be straight forward. However, since the pulse generally contains higher modes in addition to the plane wave, the overall pulse shape will change with position because the different modes in the pulse have different phase velocities, and the interpretation of the data can be complicated. Thus, the measurement of TL is not simple in comparison with that of other quantities, such as the insertion loss (see below). However, the calculation of TL is relatively simple, however, and whenever data of TL are presented, they often are calculated rather than directly measured (see Section A.2.3) or inferred from insertion loss measurements.
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Figure 7.2: Top: Sound is injected from a plenum chamber. Reflections from both ends of the main duct will be significant at low frequencies. Bottom: Sound is injected from the sides of the duct, which contains a parallel baffle or wedge absorber to make reflection from the source end negligible. In both cases, air passages are made available for mean flow to enter the test duct.
The great advantage of TL is that it depends only on the characteristics of the silencer and not on system parameters of the test facility, such as the location of the silencer in the test duct, the reflections from the ends of the test duct, and the impedance of the sound source. For this reason, TL is frequently used, as in this book, in comparing (calculated) performance of different silencers and should, in principle, be used in specifying performance in silencer manufacturers’ literature. As if the various measures of silencer performance were not enough, we introduce an additional one, a transmission loss TL0 which, unlike TL, readily can be measured. The incident power is now replaced by the net value, i.e., the difference between the direct incident power from the source and the power reflected from the silencer. The net transmitted power is defined in an analogous manner as the difference between the power transmitted by the silencer and the power reflected from the end of the test duct. The corresponding transmission loss is then defined as the ratio, expressed in dB, of these net powers. We denote this transmission loss by TL0 to distinguish it from TL. This new transmission loss, unlike TL, can be measured by standard intensity probes through integration of the intensity over the area of the test duct on the inlet and exit of the silencer. The difference between the incident and transmitted net powers is absorbed by the silencer. The quantitative analysis of TL and TL0 is given in Section A.2.3. TL0 depends on the termination impedance of the test duct, whereas TL is independent of all system parameters. We refer to Figure 7.3 where various measures of silencer performance for a particular silencer are illustrated. If the ends of the test duct are reflection free, the differences in this example are rather small. The transmission loss is always positive if no sound, such as flow noise, is generated within the silencer. For a source which delivers the same net acoustic power regardless of the acoustical load, TL0 would be the ideal quantity to use in describing silencer
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Figure 7.3: 1/1 and 1/3 OB spectra of TL, TL0, IL, and NR of a channel lined on one side with a porous layer. Liner thickness: 5 inches. Flow resistance (normalized): 0.5 per inch. Channel length: 4 ft. Air channel width: 5 inches. L1 = L2 = 10 ft (see Figure 7.2). The first four graphs refer to a reflection-free source and termination (of main duct). For the last two, the main duct is open ended, as indicated. The dashed line in the TL spectrum includes semi-empirical correction for higher modes.
performance. Actually, for many sources, including jet engines, the only source data that is available is the power level PWL. This level is generally assumed to be independent of the acoustic load. Then, based on the PWL of the source, and the TL0 of a silencer, the output power level of the silencer will be PWL-TL0. For a purely reactive silencer, such as an unlined duct section or expansion chamber, there is no power absorbed and TL0 will be zero. The calculated and frequently
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reported TL for such a silencer is generally not zero but it has limited value since it is rarely measured and difficult to apply in practice. It is then better to use the insertion loss, to be described next.
7.3.3 Insertion Loss, IL In our discussion of silencer performance, we have in mind an experimental arrangement shown in Figure 7.2, where a silencer, in the form of an acoustically lined duct section, is located in a larger duct, which we have called the ‘test duct’ or ‘main duct.’ Sound is injected into it at one end and a fan provides a mean flow, which can be reversed in direction. The noise from the fan is reduced to an insignificant level by means of an appropriate silencer. The insertion loss, IL, is defined as the change of the sound pressure level at a fixed location or as the change of the level of the total radiated acoustic power caused by the insertion of the silencer. The power can be determined by integrating the intensity over the exit end of the test duct by means of an intensity probe or by measuring the average sound pressure level in a reverberation room, which terminates the test duct. Thus, unlike TL, it is simple to measure and is of more direct practical value than TL. The calculation of IL, however, is more involved than for TL since it requires knowledge of system parameters such as the source impedance, the location of the silencer in the test duct, and the reflection coefficients at both ends of the test duct. In mathematical modeling, the source can be characterized by an ‘internal pressure’ and an equivalent source impedance (by analogy with electrical circuits). For example, a fan has a low source impedance, whereas a positive displacement pump or compressor has a high source impedance. Maximum power from the source is obtained when the input resistance of the silencer equals the source resistance and when the corresponding reactive parts cancel each other. Unlike the transmission loss, the insertion loss can be positive or negative depending on the degree of impedance mismatch caused by the silencer (see examples in Figure 7.3). Thus, it is possible (typically at low frequencies) that the insertion of the silencer will improve the impedance match and hence cause increased power output from the source. The input impedance of the test duct is often referred to loosely as the ‘acoustic load’ on the source. Our mathematical analysis of silencer performance, starting from first principles, considers only the fundamental mode, and higher modes are accounted for by semiempirical corrections. The open area of a lined duct silencer normally is smaller than the area of the test duct so that the sound encounters a (sudden) contraction in the area at the entrance of the silencer and a sudden expansion at the exit. These area changes affect TL and IL but have nothing to do with attenuation within the silencer, which, as we have seen, deals only with what happens after the sound has entered the silencer. Thus, it should not be surprising to find that the TL and the attenuation per unit length multiplied by the length of the silencer are not equal. For example, if the absorptive liner is removed in such a silencer, the attenuation (and TL0) will be zero but TL (and IL) will not be because of the effects of reflections. A typical test duct in a laboratory may have a cross section of 4 ft by 4 ft and a length of about 60 ft. It is connected to a fan via a large silencer and a plenum chamber,
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which contains the sound source(s), as illustrated schematically in Figure 7.2. The frequency dependence of the insertion loss typically is expressed in terms of 1/1, 1/3, and sometimes 1/12 octave band averages over a frequency range from 31.5 to 8000 Hz. In one of the source configurations in Figure 7.2, the sound is injected into the duct from one or more loudspeakers placed in an acoustically treated plenum chamber. Both ends of the test duct then will be highly reflective at low frequencies, and the equivalent source impedance is approximately equal to the impedance of an open ended duct. In the second arrangement in the figure, the source end of the duct contains a wedge or parallel baffle absorber designed to give high absorption over the range of frequencies involved and arranged in such a way as to permit air flow through the test duct. The normalized source impedance of such a ‘reflection free’ source will be approximately 1 under ideal conditions. The sound can be injected into the duct from loudspeakers mounted in the wall of the duct, as shown. The arrangement in the figure applies specifically to the kind of silencer used in air handling applications in buildings. However, what has been said about the relation between transmission loss and insertion loss applies in general. A silencer stack, as in a test cell for jet engines, can contain many elements, lined ducts, area transition elements, expansion chambers, side branches, elbows, etc. The test cell typically is U-shaped with an air inlet and a discharge stack. The insertion loss, IL, often refers to the insertion of the entire test cell with all its acoustical interior treatments, the reference power then being the power emitted by the bare jet engine. Another insertion loss may refer to the insertion of an interior acoustical treatment in an existing cell; the reference power is then the power emitted from the untreated cell. If the insertion loss of the untreated cell is IL0 (the reference power being that of the bare jet engine) and the insertion loss of the treated cell is IL, the insertion loss of the interior acoustical treatment is IL-IL0. In the particular case of a reflection-free source (normalized internal impedance = 1), the insertion loss of a straight empty hard duct is zero (see Section 9.1). Thus, in the bottom sketch in Figure 7.2, assuming the source to be reflection-free, the insertion loss of the entire duct is the same as the insertion loss of the interior lined duct section. Multi-Source Environment Effect of ‘Background Noise’ Another, but sometimes overlooked factor, is that with the normally used noise measurement equipment, the measured noise level includes the contribution from all sources including the ‘background’ noise, not only the primary noise source under consideration. Therefore, if the background noise is high enough the measured (total) level will be practically the same after insertion of a silencer, i.e., the insertion loss of the silencer will be practically zero. A typical example concerns the insertion loss of an exhaust muffler for an automobile. Assuming that the car is not moving but with the engine running, the total noise is contributed by several sources, the air inlet, the exhaust, valves, and other engine components. The reference level in this case is the level obtained for the exhaust pipe without the muffler installed. To determine the effect of a muffler on the exhaust noise, the microphone has to be placed sufficiently close to the exhaust so that the
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exhaust noise dominates. As the distance from the exhaust is increased, the noise from other sources (background noise) will be relatively more significant and the insertion loss will decrease with the distance from the exhaust. If a silencer itself produces flow noise, it is possible that the insertion loss will be negative. In one case, the noise from a jet engine test facility was too high to be acceptable and a muffler was added to the top of the exhaust stack. However, no reduction of the total noise output was achieved. The muffler itself produced enough additional flow noise to make its overall insertion loss zero and even negative.
7.3.4 Noise Reduction, NR Noise reduction is the difference between the sound pressure level at the beginning and at the end of an acoustical element or silencer. Unlike TL, this quantity depends not only on the element itself but on system parameters. For this reason, the quantity can be misleading, but it is sometimes used to check the performance of a silencer in situ in an exhaust stack, for example. By measuring the sound pressure levels at the bottom and top of the silencer, L1 and L2 , the noise reduction, NR = L1 − L2 , is obtained. At high and middle frequencies this measure is frequently not too different from TL or IL, but considerable deviations can occur. For example, if the top of the silencer happens to fall in a pressure node in the standing wave between the silencer and the top of the stack, the NR thus obtained can be much larger than the TL or IL. Another example in which NR will be somewhat absurd is in a standing wave field in a bare duct. Depending on the locations of the measured values of L1 and L2 , the NR of the bare duct section can be anything, positive, negative, very large or very small, including zero. Therefore, the concept should be used with care.
7.3.5 Numerical Examples For comparison of the different performance measures of a silencer, we have shown in Figure 7.3 the computed spectra of TL, TL0, IL, and NR of a silencer consisting of a rectangular duct lined on one side with a porous layer. In each graph the 1/1 OB (octave band) and 1/3 OB spectra are shown and the 1/1 OB spectrum for TL also includes the spectrum with a semi-empirical correction for higher order modes.1 As can be seen, there is not much difference in this example between TL, TL0, IL, and NR when the source and the termination of the main duct are reflection-free.2 With the main duct open ended and the reflection from the source assumed to be the same as from the open end of the main duct, the IL and the NR, can differ markedly from the TL at low frequencies and can be negative at some frequencies, as shown in the last two graphs in the figure. 1 It is recommended that the reader use a computer program to produce similar graphs for other values of the silencer parameters. 2 Recall that TL refers to incident and transmitted power and TL0 to the net values of these quantities. Also recall that NR is the difference in sound pressure level at the entrance and exit of the silencer and that TL = IL when the source and termination are reflection-free.
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Figure 7.4: The attenuation per unit length (the air channel width) for the lined duct referred to in Figure 7.3.
Figure 7.5: Transmission loss (TL) and insertion loss (IL) of a circular lined duct. L =
5 ft. L1 = 10 ft, L2 = 10 ft (see Figure 7.2). Inner diameter: 24 inches. Liner thickness: 8 inches. Flow resistance: 0.25 ρc/inch. Perforated plate: Thickness: 0.1 inch. Hole diameter: 0.1 inch. Mat: Normalized flow resistance: 0.1. Weight: 0.1 lb/ft2 . Left: Transmission loss. Right: Insertion loss. Reflection-free source. Open ended main duct.
The attenuation per unit length (the channel width) is shown in Figure 7.4. The peak value of the attenuation is approximately 4.2 dB per unit, with the unit length being the channel width 5 inches and the duct length 4 ft, i.e., 9.6 unit lengths, and the total attenuation will be 40.3 dB. This is consistent with the peak value of the 1/3 OB TL of about 40 dB, and the frequency of the maximum, approximately 500 Hz, is the same for both. Actually, in this case, there is not much difference between the total attenuation and the 1/3 OB TL spectrum. Another example of typical computed IL and TL spectra refer to a circular lined duct as shown in Figure 7.5 in which the source is reflection-free but the main duct is open ended. The porous liner is covered with a resistive mat and a perforated plate. The length is L = 5 ft, the diameter of the air channel 24 inches, and the thickness of the liner 8 inches. There is no mean flow. The distance from the sound source to
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the beginning of the lined duct is L1 = 10 ft and the distance from the end of the duct to the end of the test duct is L2 = 10 ft. The source is assumed to be reflection free, i.e., the normalized source impedance is 1. The mathematical analysis of duct performance is conveniently done by means of transmission matrices (see Appendix A) upon which computer programs are based. In regard to IL, one has to make clear how the reference power is defined. In the examples given above, it refers to the bare test duct of length L + L1 + L2 . The fluctuations in the insertion loss depend on the frequency dependence of the acoustic input impedance at the source and the corresponding impedance mismatch and power output for the test duct both with and without the silencer inserted. Note that the insertion loss can be negative which means, of course, that the power radiated from the end of the test duct is larger with the silencer than without. The transmission loss, on the other hand, is always positive. If both source and termination of the test duct are reflection-free, TL and IL are equal but TL0 will be slightly different. In measurements of duct performance, particular care should be taken in making clear which quantity is measured3 so that a measurement of NR is not taken to mean IL, TL, or TL0.
7.3.6 Pressure Drop and Flow Noise (Self-Noise, SN) The pressure drop in flow ducts and silencers typically is dominated by the contribution at discontinuities in the flow area. For a silencer, the major contribution comes from the exit flow, where flow separation and associated turbulence occurs. Similarly, most of the flow generated noise in a duct typically results from flow separation and related turbulence at area discontinuities. As for the pressure drop, it is the exit flow from a silencer which normally is the dominant contributor to the ‘self-noise’ (SN). However, manufacturers’ recommended limits of the flow speed to be used in silencers usually are sufficiently low so that SN is not much of a problem. However, if SN does become a problem, the insertion loss of the silencer can be compromised. It follows also that a silencer with high SN will yield higher insertion loss when used in conjunction with sources of high acoustic power or when placed close to a given source where the noise level is high. For example, if the sound pressure level is 100 dB and the insertion loss of the silencer is 30 dB, the level after the silencer will be reduced to approximately 70 dB if the self-noise power level is 65 dB, say. If, on the other hand, the noise level to be reduced is only 90 dB (rather than 100 dB), the insertion loss will be limited to 25 dB, although the silencer is capable of 30 dB.4 This problem of self-noise has frequently led to disappointing results in efforts to improve the insertion loss of an existing facility, such as an exhaust silencer in a gas turbine power plant. If an original installation does not meet the specified requirements, an additional length of silencer has frequently been added as a ‘fix,’ but often with little or no effect. In several such cases, the failure most likely was due to flow generated noise, although transmission through the walls is a possibility. 3 Confusion in this regard is frequently encountered in practice.
4 Some manufacturers of silencers include the self-noise power in the description of their products.
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Actually, a self-noise limited insertion loss can be encountered even if there is no mean flow in the duct. Such was the case in some experiments we carried out with high amplitude acoustic pulses (weak shock waves). The study involved simply the insertion of a 1 ft long porous baffle in the extension of a shock tube; half of the area of the tube was occupied by the baffle. The spectra of the pulses emitted from the tube with and without the baffle were compared and the corresponding insertion loss was determined. The insertion loss obtained for an incident wave with a peak pressure of 0.2 atm and a flow resistance of the baffle of 0.3 ρc per cm was consistent with linear theory. The insertion loss for a 0.7 atm wave, however, was considerably lower at high frequencies and a similar result was obtained with a porous material with a flow resistance of 0.6 ρc per inch. This behavior of the insertion loss could be caused either by a wave induced compression of the porous material in the baffle at high incident pulse pressures or by the noise from the (turbulent) exit jet pulse. The flow velocity in this jet is expected to be higher (at least by a factor of 2) when the baffle is present (recall that total acoustic power from a jet is proportional to the 8th power of the velocity). This adds to the transmitted sound in the pulse and the insertion loss will be reduced accordingly. Thus, in such a case, self-noise will be produced by the sound (shock) itself. As an aside, we note that the shock wave used in these experiments has an energy spectrum, which is dominated by low frequencies, controllable mainly by the length of the driver section in the shock tube (remember that this pulse has a ‘DC’ component). Thus, a shock tube should be kept in mind by experimenters as a possible sound source in absorption and transmission measurements when ordinary loudspeakers may not yield sufficient power at low frequencies.
7.4 LINED DUCTS Our analysis of lined ducts in Chapter 8 deals mainly with sound propagation in a rectangular duct having one wall lined with an acoustical material, as indicated on the top left in Figure 7.6. Actually, the results are valid also for the acoustically equivalent configurations shown in the figure. This includes the case with two opposite walls in the duct lined with identical liners (top right in the figure). In the configuration at the bottom left, the two liners are different, and the duct is then acoustically equivalent to the parallel baffle arrangement at bottom right in the figure. We refer to Chapter 10, Figure 10.1, for the analysis of configurations with different liners on the two sides of the duct. Both locally and nonlocally reacting liners are considered. The locally reacting liner contains partitions that force the oscillatory flow in the sound wave to be perpendicular to the liner surface; the velocity amplitude in the liner at any position along the duct is then determined solely by the sound pressure at that location, and the input impedance and the corresponding admittance are then independent of the distribution of sound pressure along the duct; these quantities
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Figure 7.6: Equivalent duct configurations (silencers), which yield the same attenuation for the fundamental acoustic mode. can then be regarded as known a priori either from calculation or from measurements.5 As in the hard-walled duct, there are many wave modes possible, but there can be no mode with a strictly constant pressure amplitude across the duct corresponding to the (0, 0)-mode in the hard duct. The sound pressure at a wall lined with a porous material will not be the same as that at a rigid, impervious wall. Furthermore, as the frequency increases, there are many wave patterns with several minima and maxima that will ‘fit’ across the duct; there will be no pure nodal planes, however, as in the case of the hard-walled duct above. Out of the many possible modes, we shall be particularly interested in analyzing the lowest mode and its spatial decay rate. The method of solution is well established, but we have added some observations and numerical studies, which can be useful. The corresponding analysis for ducts with nonlocally reacting liners yields some new aspects to the attenuation characteristics. The solution to the wave equation for the sound pressure amplitude is composed of wave functions representing ‘standing’ waves in the transverse directions y and z, and a traveling wave in the x-direction along the duct. In what follows, we consider only a wave traveling in the positive x-direction. This is sufficient for the calculation of the attenuation. When it comes to an analysis of the transmission loss and the insertion loss, reflections have to be accounted for and a wave also in the negative direction must be included. When a sound absorptive lining is present, the fundamental mode will decay. The frequency dependence of this decay, which is the main topic of interest, depends on several parameters. For higher order modes, the mode identification is no longer as clear as the hard duct and there is no longer a well defined cut-on frequency signaling a transition from evanescence to propagation. As for the fundamental mode, there will be decay at all frequencies, although the decay of a higher mode will be enhanced below the ordinary hard-wall cut-on frequency. The decay of a higher mode is greater
5 Since for a locally reacting liner the impedance is independent of the angle of incidence, normal
incidence data can be used, obtained, for example, with the two-microphone method in a tube.
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than for the fundamental, and it is the latter that usually dominates the sound that is emitted from the end of a lined duct. However, particularly for short ducts, higher modes can contribute markedly to the overall attenuation. At high frequencies, in the geometric or ray acoustics regime, a higher mode can be visualized as a ray which travels at a certain angle with respect to the duct axis. Every time this wave is reflected from a boundary it loses some of its energy and the amplitude will decay with distance in the duct. An expression for the decay can be derived in terms of the absorption coefficient of the boundary. At least in this regime, there is a unique relation between absorption and sound transmission, which is not the case in general. The same geometrical approach cannot be used for the fundamental mode, however.
7.5 ‘REACTIVE’ SILENCERS The sound reducing mechanisms involved in a silencer are absorption, reflection, and interference. In the previous section dealing with ‘dissipative’ silencers, absorption was dominant; in a ‘reactive’ silencer, reflection and interference dominate. With reference to Figure 7.3 and the related discussion, there is usually relatively little difference between total attenuation, transmission loss, insertion loss, and noise reduction, except at low frequencies. The difference depends on the bandwidth, as indicated, decreasing with increasing bandwidth, and on the source and termination impedance. For a reactive silencer, which, in its purest form contains no absorptive material, the difference between the various measures can be considerable.
7.6 ACOUSTICALLY EQUIVALENT SILENCERS The bulk of the data presented in this book refers to a rectangular duct and to the related and commonly used configurations of lined channels such as parallel baffle or ‘splitter’ silencers as shown in Figure 7.6. Because of the acoustic equivalence of these ducts (equivalent as far as the fundamental acoustic mode is concerned), the numerical work has been concentrated on a rectangular duct with one side lined, as shown at the top left in the figure. The liner is a uniform porous rigid layer of thickness d in a duct with an air channel width, D1 = D/2. The width D is the channel width in the equivalent duct (top right) with two opposite sides lined. Similarly, the two bottom configurations in the figure are acoustically equivalent and refer to the slightly more general case when the thicknesses d1 and d2 of the liners are not necessarily the same. The expression for the attenuation is derived in Chapter 10 for the duct on the left. The figure on the right is an example of a parallel baffle attenuator, which is generally used to accommodate a large volume flow through the attenuator while controlling pressure drop and/or physical size. It should be emphasized that these equivalencies are valid for the fundamental mode because of the symmetry of the pressure amplitude with respect to the dashed planes in the figure. The transverse velocity amplitude is zero in these planes and therefore can be replaced by rigid, impervious walls without altering the sound field. In practice, the focus on the fundamental mode usually does not present much of a limitation since this mode has the lowest attenuation and therefore generally dictates
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the performance of the duct. Semi-empirical corrections for higher modes and flow will be discussed later. The above process of building equivalent duct configurations can be repeated to more than two air channels with the thicknesses of the interior baffles being alternately 2d1 and 2d2 .
7.7 ADDITIONAL COMMENTS ON SILENCER TESTING As already mentioned, a typical laboratory test of a silencer is carried out in a facility illustrated schematically in Figure 7.2 with the open end of the test duct terminating in a calibrated reverberation room.6 The input power to the room is then determined from the average sound pressure level in the room. The transmitted power can be measured also by means of an intensity probe in the duct. The change in the acoustic power in dB resulting from the insertion of the silencer in the test duct is the insertion loss, IL, of the silencer in this particular test facility. Although, normally, insertion loss is considered to be the most relevant from a practical standpoint, it suffers from its dependence on the source impedance and the effects of reflections from both ends of the test duct. This means that the acoustic power output from the source may be different with and without the silencer in the test duct, and the reflections will make the IL depend on the axial position of the silencer; this dependence can be significant, particularly at low frequencies. The crucial difficulty with IL is that the conditions in the laboratory and in the field installation rarely are the same so that the effects of source characteristics and reflections are apt to be different in the laboratory and in the field. A model of a field installation can be made, of course, and tested in the laboratory. This is frequently done in special applications such as jet engine test cells, for example. It is then important that proper scaling parameters are used. The ratios of wavelength and dimensions have to be maintained and the effects of temperature on sound speed, wave impedance, and flow resistance have √ to be accounted for. The sound speed increases with √ the absolute temperature as T , and the wave impedance is proportional to P / T , where P is the static pressure. The flow resistance is proportional to the √ coefficient of shear viscosity, which increases with temperature approximately as√ T and the frequency dependence of the flow resistance is given approximately as f . If the main objective of a silencer test is to make comparisons between different silencers and not to model a field installation, one might consider using performance measures other than IL, such as TL, TL0, or even NR. The effects of reflections from the ends of the test duct can be made small by using properly designed parallel baffle or wedge absorbers at both ends of the test duct. The lower sketch in Figure 7.2 has an absorber only at the source end.
6 The calibration relates the average sound pressure level in the room, and the power input to the room. It depends on total sound absorption in the room, which can be determined by measurement of the reverberation time.
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Figure 7.7: Filled circles: Computed TL (IL). Filled squares: TL0. Open rectangles: NR. Dashed line: TL with semi-empirical correction for higher modes. Refers to a silencer test as shown in Figure 7.2 with both walls of the silencer lined. The ends of the test duct are reflection-free. Length of silencer: 3 ft. Channel width: 4 inches. Thickness of liner: 8 inches. Flow resistance: 0.3 ρc per inch. With both ends of the test duct made reflection-free by means of absorbers, the IL and the TL are identical, independent of the location of the silencer in the test duct. In practice, the reflections can be significant at low frequencies and the useful frequency range of the duct absorbers should be established. IL can be measured as before. The only problem that might be encountered is that if a reverberation room is used for the measurement of power, the sound source has to be powerful enough to compensate for the reduction in power caused by the absorber at the end of the duct. If an intensity probe is used for power measurement, this problem does not occur. TL0 can be measured as described earlier by means of intensity probes for determining the net input power and the net transmitted power. The simplest of all quantities to measure is NR, the difference of the (average) sound pressure levels at the two ends of the silencer, but it is the quantity which usually is most affected by reflections from the end. It is assumed that whenever probes are used within the duct that the signal caused by flow-probe interaction is adequately reduced. The calculated octave band spectra of IL (TL), TL0, and NR in a special case are shown in Figure 7.7. There is only a small difference between TL (IL) and TL0. NR is consistently somewhat higher than TL but has the same general frequency dependence.7 Since the ends of the test duct are reflection-free in this case, each of these quantities is uniquely determined by the silencer characteristics and is independent of system parameters. As mentioned earlier, it should be kept in mind that TL0 will be zero for a reactive silencer (no power absorbed).
7 Recall that for normal incidence on an impervious partition wall the NR is close to 6 dB higher than
the TL because of the pressure doubling at the wall due to reflection.
Chapter 8
Lined Ducts A duct with the interior walls lined with an appropriate porous material is presently the most common form of a general purpose silencer. As far as acoustical characteristics of the liner are concerned, the most important physical property of the liner material is the flow resistance. This depends on the porosity and the ‘structure factor’ of the material. For a compressible material, compliance and other elastic properties are obvious additional quantities of interest, although flow resistance still remains the most important, particularly since the elastic properties are rarely known a priori and seldom measured. It should be realized, though, that in our efforts to improve the low frequency performance of duct liners and silencers, the compliance of the liner material will play a very important role in much the same way as for sound absorption. The effect of compliance on sound absorption is summarized briefly in our chapter on flexible porous materials to explain the sometimes observed high absorption coefficient at low frequencies. It is possible to enhance the low frequency performance of lined duct silencers in a similar manner by proper use of compliant materials.
8.1 ATTENUATION MECHANISMS 8.1.1 Dissipation in Duct Liners As a qualitative description of the mechanisms of sound attenuation in a lined duct, we note that the sound pressure in a wave within a lined air channel produces ‘pumping’ of air in and out of the porous duct liner, and this gives rise to conversion of acoustic energy into heat and hence attenuation of the sound wave. For a locally reacting liner, divided into cells by partitions, the velocity of the air in this pumping action is forced to be perpendicular to the wall. On the other hand, in a nonlocally reacting liner there is an oscillatory velocity component also in the axial direction of the duct and a corresponding wave in the liner. This wave feeds energy back into the air channel of the duct so that, in effect, the liner provides a flanking path. This is what makes the two liners behave differently, particularly for small flow resistances, as can be seen from the results in Figures 8.1 and 8.3. 255
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The amplitude of the air pumping by the sound wave into the liner is inhibited at low frequencies because of the stiffness reactance of the air in the liner, and this is the main reason for the decrease toward zero of the attenuation at low frequencies. In the high frequency region, the sound field in the duct can be thought of as composed of a free running plane wave grazing the boundary, as discussed in Chapter 7. The interaction with the boundary then will be the same as for a plane wave in free field at grazing incidence yielding a pressure reflection coefficient of −1. The incident and reflected waves then cancel each other at the boundary to produce zero pressure amplitude at the boundary and hence no pumping effect and attenuation. This qualitative explanation is strictly speaking applicable to a higher order mode in the duct, which can be regarded as a plane wave bouncing back and forth between the boundaries, as discussed in Chapter 7. For the fundamental mode, we can use the following (qualitative) explanation for the decrease of the attenuation toward zero at high frequencies. A compression in the wave in the center of the duct travels as a signal toward the boundary where it is reflected. The time required for the signal to return to the middle is D/c, where D is the duct width. If this delay time is longer than the acoustic period T , i.e., D/c > T or D/λ > 1, the reflected signal in effect will not be ‘felt’ (on the average) by the wave during the compression period, exactly as if the boundary were not present and no attenuation results. For further discussion, we refer to Section 10.1. At mid-frequencies, with wavelengths of the order of the width of the air channel, we are close to a resonance of a standing wave between the walls, and this tends to increase the sound pressure at the walls. The attenuation typically goes through a maximum in this frequency region before it turns toward zero with increasing frequency, as explained above; the decrease turns out to be approximately as the inverse square of the frequency, only weakly dependent on the liner impedance. Thus, with the attenuation going to zero both in the low and high frequency limits, it is clear that the attenuation spectrum becomes bell shaped, on the average. In this context it should be remembered that the absorption coefficient of a porous material increases with frequency over the entire frequency range of interest, and this is an explicit demonstration that there is no true one-to-one correspondence between absorption coefficient and attenuation, an assumption made (and often used) in early accounts of the subject.
8.1.2 Interference Interference is often thought of as a result of the superposition of two or more sound fields produced by different sources. Such an interference can give rise to cancellation of the sound from one source by the sound from another. When this technique is utilized in practice, it is usually referred to as ‘active noise control.’ An incident and reflected field can also produce interference, the typical example being the standing wave in a tube. Another example is encountered in Section 8.7.2 dealing with sound propagation in a water line with an air layer as a liner.
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In general, the attenuation is a combination of the effects of dissipation and interference. Interference is of particular importance in reactive silencers, as discussed in Chapter 9. After these mainly qualitative considerations, we turn to quantitive aspects of the subject dealing with specific duct geometries and start with lined duct silencers with a rectangular cross section.
8.2 RECTANGULAR DUCTS The main purpose of this section is to present and discuss numerical data on lined rectangular ducts. It was considered appropriate to include Eqs. 8.1 and 8.2, which produced the calculated data even though the equations are not essential for this purpose. The mathematical analysis is given in Section 10.2.
8.2.1 Locally Reacting Liner For the fundamental mode in an unlined duct, the axial dependence of the complex sound pressure amplitude is exp(ikx x), where kx = ω/c = 2π/λ. The magnitude of the sound pressure is then constant and there is no attenuation, neglecting visco-thermal effects. With reference to the analytical supplement in Chapter 10, the propagation constant is modified by the finite boundary impedance from kx = ω/c to kx =
(ω/c)2 − ky2 , where ky , a complex quantity, is determined from the admit-
tances of the boundaries by solving (numerically) a transcendental equation. Thus, kx becomes complex, and the imaginary part ki determines the decay rate and attenuation of the sound pressure, i.e., |p(x)| = |p(0)| exp(−ki x). Propagation constant, Locally reacting liner kx ≡ kr + iki = (ω/c)2 − ky2
(8.1)
ky D1 tan(ky D1 ) = −ikD1 η1 where η1 : Liner admittance (normalized), D1 : Channel width, Figure 10.1, ky : see Eq. 10.9. For a porous liner, the impedance or admittance is determined by the thickness of the liner, its flow resistance per unit length, and to some extent, by the porosity and structure factor, which are not independent of the flow resistance. Computed attenuation spectra for a duct lined on one side with a locally reacting porous liner are shown in Figure 8.1. The attenuation is expressed in dB per unit length, where the unit length is the width D1 of the air channel (top left in Figure 7.6). The frequency parameter is the ratio D1 /λ of the channel width D1 and the free field wavelength λ, and the flow resistance of the porous material is expressed in terms of the normalized value of the total flow resistance of the liner. In this manner, the number of parameters needed to describe the attenuation spectrum is reduced. It is in this sense that the curves might be called ‘universal.’ It should be stressed that the attenuation refers to the fundamental mode in the duct.
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NOISE REDUCTION ANALYSIS
Figure 8.1: Attenuation of the fundamental mode (in dB per length of duct equal to the channel width D1 ) in a rectangular duct with one side lined with a locally reacting rigid porous layer with a total normalized flow resistance (2, 4, ... 32). Liner thickness: d. Fraction open area D1 /(D1 + d) = 20 to 70 percent. The results can be used for a duct with two opposite walls lined with identical liners if the channel width (distance between liners) is 2D1 . In practice, a channel with a width larger than a wavelength normally carries higher wave modes with attenuations higher than for the fundamental mode. Until these higher modes have been attenuated down to the level of the fundamental mode, the average attenuation in the duct for the entire sound field will be larger than for the fundamental mode. Thus, the attenuation of the overall sound field will not be constant along the duct but once the higher modes have been ‘filtered’ out, the
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attenuation for the fundamental mode will be valid. Thus, in regard to the total attenuation, the effect of higher modes will be more important for short ducts than for long. It should be noted that the results can be used, again expressed in terms of the attenuation per unit length D1 , for a duct with two opposite walls having identical liners if the distance between the liners is D = 2D1 . For example, if the channel width in such a ‘two-sided’ duct is D = 8 inches, the width that should be used in the figures is 4 inches in both the abscissa and ordinate to obtain the attenuation in a distance of D1 = 4 inches. Attenuation spectra are shown for values of the open or ‘free’ area fraction of the duct, D1 /(D1 + d), from 20 to 70 percent in steps of 10, where D1 is the width of the air channel and d the thickness of the liner, as shown in Figure 7.6. For each of these values, curves corresponding to normalized total flow resistances of the liner, = 2, 4, 8, 16, and 32, are shown. The spectra are bell shaped, consistent with the qualitative observations made earlier, and it should be noted that when the attenuation is expressed in this manner, the maximum value will be essentially the same, ≈ 4 dB per channel width, independent of the fraction open area of the duct for values less than 60 to 70 percent. However, the width of the curve at the maximum and the attenuation at frequencies below the maximum do depend on the open area fraction; the larger the open area, the narrower the spectrum. The lowest normalized total liner resistance considered in the figures is = 2. Lower values generally are not of practical interest because the low frequency attenuation is poor. As the resistance decreases, the duct liner begins more and more to behave like a quarter wavelength resonator with pronounced attenuation peaks when the liner thickness is close to an odd number of quarter wavelengths. The attenuation at the peaks will exceed the 4 dB values mentioned above but the peaks are correspondingly narrow. Optimum Design Notice that for each value of the frequency parameter D1 /λ, there is an optimum total normalized resistance of the liner. For example, it follows from Figure 8.1 that for an open duct area of 30 percent and D1 /λ = 0.02, the optimum total normalized resistance of the liner is ≈ 6 and the corresponding maximum attenuation per unit length (channel width D1 ) is ≈ 0.5 dB. It is shown in Chapter 10 (Eq. 10.18) that if the porous liner is replaced with a resistive screen backed by an air layer, the attenuation curves will be quite similar in the low frequency regime in which the wavelength is much larger than the channel width. The maximum possible attenuation that can be achieved with the screen liner at low frequencies (d << λ) is obtained when the normalized flow resistance of the screen equals the stiffness reactance of the air layer, i.e., is R1 = 1/kd = λ/2π d, where d is the thickness of the air layer, λ the free field wavelength, and k = 2π/λ. For the uniform porous layer, in the same low frequency regime, the stiffness reactance of the air in the layer is 1/(H γ kd), where H is the porosity and γ , the specific heat ratio, and ≈ 1.4 for air. It is altered by the porosity H (since there is less
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NOISE REDUCTION ANALYSIS
Figure 8.2: Rectangular duct, one side lined with porous layer. Maximum attenuation, dB per channel width D1 vs D1 /λ, for optimum design of liner. Open duct area, 20 to 70 percent in steps of 10. Example, D1 /λ = 0.2, maximum attenuation ≈ 0.5 dB in a length equal to D1 , obtained with optimum total normalized flow resistance of the liner ≈ 6.5.
air in the layer) and by the factor γ (since the conditions in the layer are isothermal rather than isentropic). The optimum total normalized flow resistance of the layer is now 3/(H γ kd) and the corresponding value for the maximum attenuation in length D1 is H /(γ kd), i.e., larger by a factor H γ than for the screen liner. Since typically H ≈ 0.95, this factor is ≈ 1.3. Under more general conditions, the optimum resistances and corresponding maximum attenuations can be obtained from the set of attenuation spectra in Figure 8.1 for different total flow resistance values of the liner. By connecting the points thus obtained we get the envelope to this set of attenuation curves. This envelope then represents the maximum achievable attenuation vs the frequency parameter D1 /λ in a duct with a given open area. A simplified summary of the results thus obtained is presented in Figure 8.2. The abscissa in this figure is the frequency parameter D1 /λ. The left ordinate axis refers to the highest attainable attenuation of the fundamental acoustic mode in a length of duct equal to the channel width D1 . The right axis, labeled opt , is the corresponding optimum total normalized flow resistance of the liner. The open area fraction σ = D1 /(D1 + d), where d is the thickness of the liner and D1 the channel width, covers the range 20 to 70 percent in steps of 10. The width of the 4 dB attenuation plateau in the mid-frequency region decreases with increasing open area until it becomes zero at ≈ 60 percent. For higher values of the open area, the maximum value of the attenuation decreases to values below 4 dB until it disappears, of course, for an open area of 100 percent. In the example illustrated in the figure, we have D1 /λ = 0.2 and a 30 percent open duct. As can be seen, the maximum possible attenuation is about 0.5 dB in a length of duct equal to the channel width D1 and the corresponding optimum total normalized flow resistance of the liner is about 6.5. The optimum resistance decreases as D1 /λ increases and becomes less well defined in the region of the 4 dB plateau, i.e., for
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0.1 < D1 /λ < 0.4, where a value ≈ 2 is a good approximation, independent of the open area.
8.2.2 Nonlocally Reacting Liner Analogous results are shown in Figure 8.3 for a nonlocally reacting porous liner. As indicated in the introduction, the unique feature of the nonlocally reacting liner as compared to the locally reacting is the absence of cross partitions within the liner, and hence the presence of an axial velocity component in the liner. Because of this component, and the related wave propagation within the liner, the computation of the propagation constant and attenuation (Eq. 8.2) is somewhat more involved than for the locally reacting liner (Eq. 8.1). The reason is that the liner admittance is not known a priori, as it is for the locally reacting liner, but has to be expressed in terms of the propagation constant for the wave within the porous material. Propagation constant, Nonlocally reacting liner kx = (ω/c)2 − ky2 ky D1 tan(ky D1 ) = −ikD1 ηi
(8.2)
ηi = −i(qy /k)(ρ/ρ) ˜ tan(qy d) qy2 = q 2 − k 2 + ky2 where D1 : Channel width, k = ω/c, d: Liner thickness, ρ/ρ ˜ = + ir/ωρ, q: see also Eq. 10.29. The axial propagation within the liner becomes particularly pronounced for low flow resistances, and it is then found that the attenuation at low frequencies can be as good as or even better than with a liner of high flow resistance. To illustrate this, we have extended the normalized resistance range of the liner down to 0.25. To keep the number of figures down, the number of open area fractions has been reduced to include only 0.2, 0.3, and 0.4. Another feature of the attenuation spectra is that the attenuation peak is now closer to 3 dB rather than 4 dB for the locally reacting liner but in exchange, the attenuation spectrum is somewhat broader. For relatively large values of the total flow resistance, the maximum attenuation occurs when the channel width-to-wavelength ratio D1 /λ is ≈ 0.5 but as the resistance decreases, the maximum attenuation moves toward lower values. The Porous Plug A nonlocally reacting liner becomes a uniform porous plug when the width of the air channel is reduced to zero. Relatively little attention has been paid to it as an acoustic attenuator because the pressure drop in the plug becomes unacceptably high at the mean flow velocities normally encountered. As an example, consider a plug with a total flow resistance of ρc. If the Mach number in the flow is M, the static pressure drop in the plug will be ρc2 M = Mγ P , where P is the static pressure and γ = 1.4 the specific heat ratio. The transmission loss will be
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NOISE REDUCTION ANALYSIS
Figure 8.3: Attenuation of the fundamental mode (in dB per length of duct equal to the channel width D1 ) in a rectangular duct with one side lined with a nonlocally reacting porous layer with a total normalized flow resistance (0.25 to 32). Thickness of liner: d. Fraction open duct area = D1 /(d + D1 ) = 0.2, 0.3, and 0.4. The results can be used, again, in terms of dB per length D1 , for a duct with two opposite walls lined if the separation between the liners is 2D1 . ≈ 1 dB. With a Mach number of 0.01, the pressure drop will be 0.014P, i.e., 5 inches of water, even for such a small transmission loss. Therefore, the porous plug does not seem to be very promising unless flow channels are cut in the porous material to reduce the pressure drop. But this brings us back to the nonlocally reacting parallel baffle attenuator. In special applications, however, the plug can be a viable option,
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Figure 8.4: Left: The velocity dependence of the normalized acoustic resistance per inch of a porous material with 100, 45, and 10 pores per inch. Porosity 97 percent. Right: Transmission loss of a 5 inch long 100 PPI porous plug with a mean flow velocity of 10 ft/s through the plug at static pressures 1, 10, and 100 atm.
particularly when very high frequencies are involved, as discussed in the example below. The static pressure drop in the plug is in part due to the viscous drag and in part to flow separation and turbulence within the material. As a result, the relation between P and the flow velocity U can be expressed as a sum of terms of the form P ∝ U n . The exponent n is 1 at low velocities at which the viscous drage dominates and approaches 2 at higher speeds, where flow separation and turbulence within the material takes over. For a porous material and modest flow speeds, n often is found to fall between 1.5 and 1.7. The (differential) acoustic resistance r for a given mean flow velocity (and n) is given by the slope of the P vs U curve, i.e., r=
δ(P ) ∝ n U n−1 , δU
(8.3)
where δU is the perturbation in flow velocity caused by the sound wave. An example of the flow resistance of a porous (ceramic) material obtained in this manner is shown in Figure 9.15 for samples with 100, 45, and 10 pores per inch and a porosity of 97 percent. The resistance that enters into the calculation of the transmission loss is the normalized value r/ρc. In the linear (viscous) regime, r is a constant and r/ρc decreases with increasing static pressure at a fixed temperature. The nonlinear component of r, however, is approximately proportional to the density so that r/ρc becomes less dependent of the static pressure. The other important difference between the linear (intrinsic) and flow induced resistance is the frequency dependence. The linear resistance (which is due to viscosity) increases with frequency, whereas the flow induced part (due to turbulence) is almost independent thereof. The combined effect of the two resistance contributions leads to the calculated TL spectra in Figure 9.15 in which the static pressure dependence of the nonlinear resistance has been ignored.
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NOISE REDUCTION ANALYSIS
Figure 8.5: Porous plug attenuator.
The dependence of the transmission loss on the static pressure drop is due to the pressure dependence of the normalized viscous acoustic resistance. With the flow induced resistance assumed to be independent of static pressure, the effect of the flow induced resistance on the TL will be pressure independent. The static pressure drop is dominated by the nonlinear component and on the assumption that this component is proportional to density, the pressure drop will be some constant fraction of the static pressure regardless of the flow velocity. In the TL example in Figure 9.15 the pressure drop is 21 inches of water at 10 ft/s. It is important to realize that the acoustic flow resistance, and hence the TL, increases with the mean flow. The results in Figure 9.15 refer to a flow speed of 10 ft/s and to constant values of the mean pressure. Actually, in a more detailed analysis one should account for the nonuniformity of these quantities in the material. As the static pressure decreases with distance of travel in the material so does the density; the flow speed then increases, as required by the conservation of mass flow. In Figure 9.15 the frequency range extends to 100 kHz, which normally is of little or no interest. There are special applications, however, where it is important. Flow noise (from small jets, for example) with substantial energy in this high frequency range can interfere with the performance of instrumentation in which ultrasound is used, as in flow velocity instruments, for example, and a porous plug attenuator such as shown in Figure 8.5 might then be appropriate. The area expansion is used to reduct the flow speed and the pressure drop in the plug. Normally a porous plug is not a viable option as an attenuator in a duct with flow because of high pressure drop. In special applications involving very high frequencies it could be useful, however.
8.2.3 Locally vs Nonlocally Reacting Liner, An Example As explained earlier, a locally reacting liner contains closely spaced transverse partitions which force the (oscillatory) acoustic fluid motion in the liner to be normal to the duct wall; in the nonlocally reacting liner there is no such constraint. Figures 8.1 and 8.3 show the attenuation spectra of a rectangular duct channel lined on one side with a porous liner; the first figure refers to a locally reacting liner and the second to a nonlocally reacting liner. The ordinate is the attenuation in dB per unit length (the channel width D) and the abscissa, the frequency parameter is D/λ, λ being the free field wavelength.
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Figure 8.6: A TL comparison. Rectangular duct with one side lined with a porous material. Thickness, 4 inches, normalized flow resistance per inch, 0.5, air channel width, 2 inches, duct length, 36 inches. Left, 1/1 OB spectra: Open circles: Locally reacting liner. Open squares: Nonlocally reacting. Filled circles and squares: include semi-empirical corrections for higher modes. Right: 1/3 OB spectra: Circles: Locally reacting liner. Squares: Nonlocally reacting liner. The ‘bandwidth’ in these narrow band spectra is about 1/38 OB.1 At first glance, there does not seem to be much difference between the spectra for the two different liners. It should be borne in mind, however, that a difference of 1 dB in the attenuation per unit length (D) becomes a difference of L/D dB in the total attenuation in a duct of length L. In practice, we are generally interested in 1/1 or 1/3 OB spectra; an example is shown in Figure 8.6 with transmission loss spectra of a rectangular duct channel with a 4 inch thick porous liner on one side. The main purpose of the example is to compare the performance of locally and nonlocally reacting liners. The 1/1 OB transmission loss spectra are shown on the left and the 1/3 OB spectra on the right. The flow resistance of the liner is 0.5 ρc per inch; the channel width is 2 inches and the duct length 2 ft. In each of the graphs the spectra for both locally and a nonlocally reacting liners are shown, and the 1/1 OB spectra include the semi-empirically corrections due to higher modes in accordance with the discussion in Section 8.6. As is typical, local reaction yields higher peak values than nonlocal, which is particularly evident in the 1/3 OB spectrum. The peak values are related to the resonances that occur at frequencies for which the thickness of the liner is close to an odd number of quarter wavelengths (in the porous liner). We note that in this example this resonance is clearly visible in the 1/3 OB spectrum for the locally but not for the nonlocally reacting liner. In reality, a liner is seldom purely locally or purely nonlocally reacting2 and experimental data on ducts in practice generally yield values of the transmission loss between those predicted for locally and nonlocally reacting liners. 1 The width of the graph is 512 pixels on the computer screen and it extends over 13.3 octaves. Thus, 1 pixel (the resolution) covers approximately 0.026 octaves. 2 Recall that the former contains closely spaced transverse impervious partitions (spacing smaller than a quarter of a wavelength) and the latter has no partitions. An example of the effect of partition spacing on the attenuation is given later in this chapter, Section 8.5.5.
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NOISE REDUCTION ANALYSIS
8.2.4 Attenuation vs Flow Resistance of Liner In the design of a lined duct one is often confronted with the problem of optimizing for a particular frequency region, typically at a low frequency. Often the parameter which is available for variation is the flow resistance, since the geometrical parameters are often set by other factors, such as static pressure drop. The brute force approach to handle this problem is to obtain attenuation spectra for a (large) number of values of the flow resistance, as obtained by a computer program, and then from these results determine the attenuation at a given frequency as a function of the flow resistance. The result is obtained quicker by using a modified version of the program, which plots the attenuation vs the flow resistance directly, as shown in Figure 8.7. The input parameters are the thickness of the liner, the width of the air channel, and the frequency. In this case, the liner thickness is 8 inches, the channel width 4 inches, and the frequencies 50 and 200 Hz. Consider first the result for 50 Hz. For a normalized flow resistance above 0.7 per inch, there is practically no difference in the attenuation for the two liners, and there is a maximum attenuation of about 0.3 dB in a length equal to the channel width for a normalized resistance of about 1.5 per inch. However, for lower flow resistances, the nonlocally reacting liner yields a maximum attenuation of about the same value at a resistance of only ≈ 0.06. The corresponding attenuation for the locally reacting layer is substantially smaller. As discussed earlier, this behavior is related to the presence of an axial velocity component (and wave) in the nonlocally reacting liner, the significance of which increases with decreasing frequency; this component is absent in the locally reacting liner. Actually, even the locally reacting layer shows a maximum, albeit considerably lower, at a resistance of about 0.03. This maximum is related to heat conduction losses and occurs in the vicinity of the thermal relaxation frequency. This frequency is approximately equal to the viscous relaxation frequency, which is r/(2πρ) = θc/(2π ), where r = θ ρc. With θ ≈ 0.03, this frequency becomes ≈ 64 Hz.
Figure 8.7: Comparison of the attenuation in dB per unit length (the width of the air channel) vs the normalized flow resistance per inch of liner for locally and nonlocally reacting liners.
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At 200 Hz, we are rather close to the peak of the attenuation spectrum in this case, and the difference between locally and nonlocally reacting liners becomes less pronounced. Now, it is the locally reacting layer which yields higher attenuation for small flow resistances. This is due to the ‘quarter wavelength resonance’ of the liner.
8.2.5 Example: A Proposed Air Intake Silencer for an Automobile Later, in connection with Figure 9.9, a labyrinth resonator assembly for the attenuation of the air induction tone in an automobile is illustrated and discussed. The muffler proposed here, Figure 8.8, is acoustically more ‘rugged’ and much less dependent on source conditions. It is in the form of a flat box to fit under or be an integral part of the hood of the automobile or it could constitute the hood structure itself. It is 1 inch thick and contains the air intake channel in the center, which is 2 inches wide and 3 ft long. The rest of the box is filled with porous material, 20 inches on each side of the channel, and it is covered with a resistive screen and perforated facing on the walls of the air channel, as indicated schematically in the figure. The liners are made approximately locally reacting by means of partitions, as shown. We are interested in finding a combination of parameters of these components that will yield an octave band transmission loss of the attenuator above 20 dB in all bands above and including 63 Hz. The effect of the mean flow in the channel, assumed to be −100 ft/sec (i.e., directed against the sound, of course) should be included. From computer experiments, we find that one possible configuration is as follows. Porous material: Flow resistance of porous material, 0.25 ρc per inch. Screen: Flow resistance, 3 ρc, weight 0.2 lb/sqft. Perforated facing: Open area 23 percent, hole diameter 0.1 inch, thickness, 0.1 inch. The complete octave band transmission loss spectrum is shown in Figure 8.8.
Figure 8.8: A proposed air induction silencer (flat box) for an automobile engine to fit under or be an integral part of the hood. The thickness of the box is 1 inch, the width of the air channel 2 inches, and the length 36 inches. The width of each of the porous liners is 20 inches.
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NOISE REDUCTION ANALYSIS
8.3 ADDITIONAL DUCT SHAPES A mathematical supplement to this section is given in Section 10.3.
8.3.1 Rectangular Duct With All Sides Lined The discussion will now be extended to a rectangular duct in which each side has a different liner, as shown in Figure 8.9. The liners are locally reacting and the normalized admittances are denoted by ηy1 , ηy2 , ηz1 , and ηz2 as shown. Figure 8.14 is an example of the computed attenuation in a 10 ft long rectangular duct with an air channel 2 ft by 4 ft and an 8 inch thick porous liner on all sides. In other words, it is a special case of what is shown in Figure 8.9. The normalized flow resistance is 0.25 per inch. For comparison is shown the attenuation obtained when the two liners on the short walls are absent. The configuration is then acoustically equivalent to a duct with one side lined and with a channel width of 1 ft.
Figure 8.9: Rectangular duct lined on all sides with different locally reacting liners with normalized admittances, as shown.
Figure 8.10: The attenuation in a 10 ft long rectangular duct with a 24 inch by 48 inch air channel and an 8 inch thick porous liner on all sides. The flow resistance is 0.5 ρc per inch. For comparison is shown the attenuation when the two liners on the short sides are removed. The configuration is then acoustically equivalent to a duct with one side lined and with a channel width of 12 inches.
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Figure 8.11: Fundamental mode attenuation in dB per unit length D, where D is the width of the air channel of a square duct lined on all sides with a locally reacting porous layer. Frequency variable is D/λ, where λ is the wavelength. Total normalized flow resistance of the liner: 2, 4, 8, 16, and 32, as indicated. Open area fractions of duct: 10, 20, 30, 40, 50, and 60 percent, as shown, corresponding to liner thicknesses d such that d/D is 1.08, 0,62, 0.41, 0,29, 0.21, and 0.15, respectively.
For a square duct lined on all sides with a locally reacting porous layer the ‘universal’ attenuation spectra are shown in Figure 8.11. Denoting the width of the air channel by D and the thickness of the liner by d, the open area fraction is s = D 2 /(D + 2d)2 . The six graphs in the figure refer to open areas from 10 to 60 percent, and the five curves in each graph correspond to the values 2, 4, 8, 16, and 32 of the total normalized
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NOISE REDUCTION ANALYSIS
flow resistance of the liner (values less than 2 normally are of little practical interest). The frequency parameter is D/λ, where λ is the free field wavelength.
8.3.2 Circular Duct If the thickness of the liner is small compared to the radius of the circular duct in Figure 8.12, it should be a good approximation to treat the liner as locally plane and use the same expression for the input admittance as for the rectangular duct. This would simplify the mathematical analysis. However, to be able to apply our results without this limitation, we use the actual input admittance with cylindrical wave functions both in the air channel and in the annular liner. ‘Universal’ attenuation spectra of a circular duct lined with a rigid porous liner thus obtained are shown in Figure 8.13. The attenuation per unit length D, the diameter of the air channel, is plotted as a function of D/λ, where D is the diameter of the air channel and λ the wavelength. The six graphs refer to open area ratios of the duct of 10, 20, 30, 40, 50, and 60 percent, and each graph contains five curves corresponding to values of the total normalized flow resistance of the liner of 2, 4, 8, 16, and 32, as indicated. Values less than 2 have not been included (because of little practical interest). Although narrow resonance peaks are obtained for low flow resistances, the overall performance for most practical purposes is inferior to that obtained with θ = 2. The open area fraction of the duct is defined as the ratio of the area of the air channel and the total area of the duct, including the duct√liner. Thus, if the open area fraction is s, the thickness of the liner becomes d = (1/ s − 1)D/2, the same as for the square duct. For example, with s = 0.3, we have d = 0.41D. For the range of parameter values considered here, the maximum attenuation is 8 to 10 dB per distance D. For a given open area fraction, the peak attenuation occurs at values of D/λ, which increases with increasing flow resistance of the liner. For example, for an open area of 10 percent it goes from D/λ ≈ 0.15 for a flow resistance = 2 to D/λ ≈ 0.8 for = 32. The peak value is about the same, but the width of the attenuation curve decreases with increasing flow resistance in this range. It should be noted in this context that the first cross mode in a hard duct cuts on when D/λ ≈ 0.59.
Figure 8.12: Circular and annular lined ducts with locally reacting porous liners. The front and back of the center body in the annular duct is closed at both ends. Otherwise, the center portion would provide a by-pass channel without attenuation.
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Figure 8.13: Fundamental mode attenuation in dB per unit length D, where D is the diameter of the air channel of a circular duct lined with a locally reacting porous layer. Frequency variable is D/λ, where λ is the wavelength. Total normalized flow resistance of the liner: 2, 4, 8, 16, and 32, as indicated. Open area fractions of duct: 10, 20, 30, 40, 50, and 60 percent, as shown, corresponding to liner thicknesses d such that d/D is 1.08, 0,62, 0.41, 0,29, 0.21, and 0.15, respectively.
As expected, the results are similar to those in Figure 8.11 for a square duct lined on all sides but the attenuation for the circular duct is somewhat better. It is of interest to compare the results with those of a rectangular channel lined on one side with locally and nonlocally reacting layers shown in Figure 8.1 and 8.3, respectively.
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8.3.3 A Comparison, Circular vs Square Lined Duct The frequency dependence of the attenuation of the fundamental acoustic mode in a circular duct of length 5 ft and a diameter (air channel) of 24 inches is compared with the attenuation in a square duct of width 24 inches with all sides lined. The liner parameters: Thickness: 8 inches, flow resistance: 0.25 ρc per inch, porosity: 0.95, structure factor: 1.3. Programs yield the results shown in Figure 8.14. The square duct is lined on all sides with a liner of the same thickness as in the circular duct. The ‘hydraulic,’ or, in this case, we should perhaps say the ‘acoustic diameter,’ is the same in the two cases; it is defined as 4 times the ratio of the channel area and the acoustically treated perimeter. The attenuation for the circular duct is somewhat better than for the square duct. This is related to the difference in admittance of a circular and a plane liner. Actually, if the admittance for a plane layer is used in the circular case, the attenuation at frequencies below the peak will be essentially the same as for the square duct. But at frequencies above the peak, the circular duct is still somewhat better than the square duct. This may seem a bit paradoxical since the volume of porous material in the square duct is larger than in the circular duct (by a factor 4/π ≈ 1.27). It should be kept in mind, though, that for a locally reacting liner, the portions of the liner in the corners of the square duct are ‘inactive.’ If these volumes are subtracted, the ‘active volume’ in the square duct is somewhat smaller than in the circular duct, the ratio being ≈ 0.96. As an assignment for the reader we recommend that the results be compared with the ‘universal’ curves for circular and square ducts given in the text.
8.3.4 Annular Duct If we add a concentric core to the circular lined duct (see Figure 8.12) the air channel becomes annular, which is another configuration of practical interest. In Figure 8.12
Figure 8.14: Left: The attenuation in a circular duct lined with a rigid, locally reacting porous liner. Right: The attenuation in a square duct lined on all sides. In both cases the duct length is 5 ft and diameter (width), 24 inches. The liner parameters are: Thickness = 8 inches, flow resistance = 0.25 ρc per inch, porosity = 0.95, structure factor = 1.3 (the total normalized resistance of the liner is 2.0).
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the center body consists of a solid pipe, closed at both ends, and covered with a porous liner, but it can also be a uniform porous cylinder. If the width of the annulus is small compared to the diameter, we should be able to approximate the performance by that of a rectangular duct lined on two opposite sides. Under more general conditions, however, it is necessary to carry out the analysis using the appropriate mathematical functions to calculate the attenuation, and this has been done in the numerical analysis using the results from Chapter 10. As usual, we have considered only the fundamental mode; it has no azimuthal angle dependence. Figure 8.15 shows an example of the computed attenuation in an annular duct channel of width 4 inches and with a uniform porous core with a radius of 4 inches. The outer radius of the annulus is 8 inches and the thickness of the liner on the outer wall is 4 inches. For comparison is shown the attenuation in a rectangular channel of the same width with two opposite sides lined with 4 inch thick porous liners. The results are similar. The rectangular duct having a somewhat better low frequency attenuation. One reason is that in the annular duct the core is equivalent to a plane liner with a thickness equal to half the core radius, i.e., half the thickness of the outer liner. If, in this particular case, the channel area of the rectangular duct is chosen to be the same as that of the annular duct, the volumes of porous material in the ducts are the same. The result for the rectangular duct is consistent with the results given in Figure 8.1 for the attenuation per duct width vs the width-to-wavelength ratio in a duct with one side lined. The parameters in this figure are the open areas of the duct and the total normalized liner resistance. The open area of the duct in this example is 33 percent and the total liner resistance is 2.0. The duct width is 2 inches; thus, the length of the duct (24 inches) is 12 duct widths. The first attenuation peak is about 40 dB, which corresponds to an attenuation per duct width of approximately 3.3 dB. The peak occurs at 700 Hz (wavelength ≈ 19.2 inches) and the quarter wavelength is somewhat longer
Figure 8.15: Left: The attenuation spectrum for an annular lined duct with an outer rigid porous liner and with a uniform porous core. The liner thickness, the width of the annulus, and the radius of the core are all 4 inches. Flow resistance: 0.5 ρc per inch. Duct length: 2 ft. Right: Rectangular duct with a channel width of 2 inches and lined on one side with a 4 inch thick layer of the same porous material. It is equivalent to a duct lined on two opposite sides and a channel width of 4 inches.
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than the liner thickness (the sound speed in the porous material being somewhat less than in free field). The first peak in the attenuation occurs at a width-to-wavelength ratio of ≈ 0.10. The second peak corresponds to ≈ 3.75 dB per channel width, and the corresponding channel width-to-wavelength ratio is ≈ 0.26, consistent with the results in Figure 8.1, which were obtained with a somewhat different (and earlier) computer program.
8.4 DUCTS IN SERIES AND IN PARALLEL 8.4.1 Ducts in Series A duct system may consist of unlined and lined ducts, area transitions, and lined elbows. The total transmission matrix can be obtained by multiplying the matrices of the individual elements. The area changes at the entrance and exit of a lined duct can be incorporated in the lined duct matrix with an option to exclude them either at the entrance or the exit. Example: Three Ducts in Series As an example, we consider here a 3 ft long rectangular duct with two opposite sides lined with 8 inch thick porous layers covered with perforated facings with an open area of 23 percent, thickness 0.1 inch, and hole diameter 0.1 inch. There is no area change of the air channel going from the main duct to the lined duct. First, consider a uniform liner with a normalized flow resistance of 0.5 per inch over the entire length of the duct. We wish to compare the transmission loss of this duct with that obtained for a nonuniform duct consisting of three sections in series, each 1 ft long. In the first and third section, the flow resistance of the liner is 0.5, as before, but in the middle section, it is only 0.1. The transmission loss is obtained from the overall transmission matrix of the duct, which in this case is the product of the three individual transmission matrices of the duct sections. The program produces the result shown in Figure 8.16. The nonuniform duct is not quite as good as the uniform at frequencies below 150 Hz but makes up for the difference at higher frequencies. The additional attenuation peaks that are present in the nonuniform duct is the result of the combined effect of the quarter wavelength liner resonance of the weakly damped middle section and the internal reflections that occur at the junctions between the sections. It should be pointed out that we are dealing here with a duct of uniform channel width without any area changes between duct sections and at the entrance and exit of the duct. Actually, when such area changes are present, the transmission loss is somewhat higher at low frequencies due to reflections and the interaction between separated flow and sound at the area changes.
8.4.2 Parallel Ducts, Interference Filter Ducts with different transmission characteristics can sometimes be used as filters analogous to the old Quincke tube in which waves traveling through parallel ducts of different lengths are made to interfere destructively at the exit. Instead of having
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Figure 8.16: The transmission loss of the uniform and nonuniform ducts described in the text. Thin curve: Fundamental mode only. Curve with markers: With semi-empirical correction for higher modes.
different lengths, the ducts can be made to have different sound speeds so that the ‘acoustic’ lengths are different even if the physical lengths are not. The phase velocity in a duct with a porous liner is lower than in an unlined duct, and by putting them together in parallel, an interference filter can be produced if the waves in the two ducts are 180 degrees out of phase at the exit. This approach is particularly appealing in liquid pipe lines since a large reduction in phase velocity can be achieved by means of an air layer liner.
8.5 DUCT LINER CONFIGURATIONS The numerical results presented so far have referred to a duct liner of a uniform porous material. This has been, and still is, considered to be the typical duct liner but interest is developing in other types of liners such as single and multiple layers of resistive sheet material. A uniform porous layer is often covered with a perforated plate (facing) as a protective layer. The open area is then chosen so large, typically 20 to 30 percent, that the attenuation is thought not to be affected by the facing. Nevertheless, it is of interest to determine what the actual effect is as will be done next.
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8.5.1 Effects of a Perforated Facing The modification in the analysis required to incorporate the effect of a perforated plate on a uniform porous liner merely involves adding the facing impedance to the input impedance of the porous liner. Figure 8.17 shows the attenuation thus obtained, for both a locally and nonlocally reacting liner. One side of the duct is lined with an 8 inch thick porous liner having a normalized flow resistance of 0.25 per inch. The width of the air channel is also 8 inches. The liner is covered with a perforated facing. Attenuation curves for open areas of the facing from 0.1 to 100 percent are shown.There is not much difference between the results for the locally and nonlocally reacting liner. The peak attenuation is somewhat higher for the locally reacting liner and the width of the curve is somewhat smaller. A liner without a facing corresponds to the curve marked 100 percent. With an open area of 25 percent, the effect of the perforated facing is relatively small but not quite negligible at high frequencies. As the open area decreases, the frequency of the attenuation peak decreases, and it is interesting to note that with an open area of 0.1 percent, the attenuation peak, about 2 dB per foot, occurs in the frequency range 30 to 35 Hz, i.e., about 10 times lower than the peak frequency obtained without a facing. The width of the attenuation curve, though, is considerably reduced. Thus, facings can be more useful in design than just constraining and protecting the porous material. Acoustic Nonlinearity Next, we investigate the effects of the acoustic nonlinearity and the induced motion of a perforated plate on the attenuation in a duct with a locally reacting liner. Actually, these two factors are not independent since the nonlinearity of the perforated plate depends on the relative velocity of the air and the plate. Further-
Figure 8.17: The attenuation in dB/ft of the fundamental mode in a rectangular duct with one side lined with an 8 inch thick porous layer with a normalized flow resistance of 0.25 per inch and covered with a perforated facing with open areas ranging from 100 to 0.1 percent, as shown. The width of the air channel is 8 inches. Left: Locally reacting liner. Right: Nonlocally reacting liner.
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Figure 8.18: Effect of nonlinearity and induced motion of a perforated plate. Attenuation in dB/ft. Rectangular duct, one side lined with an 8 inch thick porous layer. Normalized flow resistance: 0.25 per inch. Perforated facing, open area: 0.1 percent. Width of air channel: 4 inches. Thickness of facing, 0.25 inches (left) and 0.1 inch (right), corresponding weights, 10 and 4 lb/ft2 .
more, the velocity depends also on the input impedance of the porous layer behind the perforated plate and an account of the nonlinearity makes the computational problem considerably more difficult. For local reaction, the input impedance of the porous layer is known a priori and need not be involved in the numerical search for a solution to the dispersion relation. This is not the case for nonlocal reaction, however, and the computations become quite lengthy and time consuming. Actually, in most cases of practical interest, the results are essentially the same for both cases. At very small open areas of the perforated plate, typically less than 3 percent, and high sound pressure levels, the induced motion can be significant. The plate (which we have treated as limp) acts like a mass-spring oscillator with the air in the backing layer providing the stiffness. Figure 8.18 shows examples of the calculated attenuation in a duct with an 8 inch thick liner covered with a perforated plate with an open area of 0.1 percent. The width of the air channel is 4 inches. In the left figure, the thickness and the hole diameter of the perforated plate are both 0.25 inches and the weight ≈ 10 lb/ft2 ; and on the right, the corresponding values are 0.1 inch and 4 lb/ft2 . The flow resistance of the porous material is 0.25 ρc per inch. The sound pressure levels, 80 and 120 dB, referred to in the figure should be interpreted as average values in the duct. As expected on the basis of linear theory, the frequency at the attenuation peak obtained with the thicker perforated plate with √ its larger orifice mass is lower than with the thinner, approximately by a factor of 2.5, where 2.5 is the ratio of the equivalent orifice lengths. Furthermore, also as expected, the attenuation curve is broader at the higher level due to the nonlinear resistance of the perforated plate. However, there is a less obvious additional shift toward lower frequencies at 120 dB for the thinner plate. This is due to the induced motion of the plate, which adds inertia to the system and hence lowers the resonance frequency (with a slight increase in peak attenuation).
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8.5.2 Effect of Duct Liner Flexibility The flexibility of a porous material can affect the absorption coefficient significantly at low frequencies. This is particularly true when the ratio of the flow resistance and the mass is large so that the acoustically induced motion of the porous structure is pronounced. For example, with a liner thickness of 8 inches, and a weight of 4 lb/ft3 , the effect for a limp layer is essentially negligible unless the flow resistance exceeds 0.5 ρc per inch. Actually, the effect depends on the ratio of the flow resistance and the inertial reactance of the material as discussed already in connection with the sound absorption coefficient, and we refer to it for details. If there is a small space between a perforated facing and the porous layer (loose contact), so that the plate can move independently of the layer, the effect of the plate is minor at frequencies below ≈ 2000 Hz if the open area exceeds ≈ 25 percent. The acoustically induced motion of the porous layer will have a resonance when the structural wavelength of the layer is approximately 4 times the layer thickness (quarter wavelength resonance). The corresponding frequency is fs = cs /λ, where the structural wave speed cs depends on the compliance and mass of the porous material. Normally this frequency is relatively low, often less than 100 to 150 Hz. The attenuation will be affected by this resonance in much the same way as the absorption coefficient. If the plate is adhered to the liner and if it is relatively heavy, say ≈ 4 lb/ft2 , it will keep the surface of the layer from vibrating, and it will act approximately like a rigid wall as far as the structural oscillations are concerned. Then, a structural resonance of the layer will occur at a frequency for which the layer thickness is half a wavelength, i.e., twice that of the quarter wavelength resonance. Using the input impedance of a flexible liner, we can determine the attenuation in a lined channel in much the same way as for the rigid layer, but the computations are considerably more complex.
8.5.3 Multilayer Liners For a locally reacting layer, we can use the input admittance obtained from the analysis of a multilayer absorber to calculate the attenuation and transmission loss of the fundamental acoustic mode in a rectangular duct with such a multilayer liner. This is what has been done in a computer program. It is of practical interest to compare the results obtained for a multilayer sheet liner with that of a conventional uniform porous liner. To make such a comparison meaningful, the parameter of the sheet absorber must be chosen in a reasonable manner in comparison with those of the porous layer. The obvious choice is to make the total thickness and the total flow resistance the same for both. An example involving multiple screens is shown in Figure 8.19, where the transmission loss is expressed in 1/1 octave bands. The total thickness of each liner is 8 inches, the width of the air channel 4 inches, and the length of the lined section 36 inches. The liners are made up of (rigid) identical, uniformly spaced, resistive screens. The total flow resistance is 2 ρc for each liner.
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Figure 8.19: Octave band average transmission loss (fundamental mode only) of a 3 ft long silencer with multiple sheet liners compared with that of a silencer with an 8 inch thick uniform porous liner (top curve). The total flow resistance in each case is the same, 2 ρc, and the total thickness of the each liner is 8 inches. The width of the air channel is 4 inches.
For the single sheet liner, the maxima of the transmission loss occur at frequencies where the liner thickness is an odd number of quarter wavelengths, and there are minima which correspond to the anti-resonances at which the liner thickness is a multiple of half wavelengths. For the double screen liner, the first anti-resonance occurs when the screen separation is half a wavelength. On an octave band basis, as in Figure 8.19, these irregularities in the transmission loss are not obvious and a narrow band analysis is required to display them clearly. As the number of screens is increased, the performance approaches that of a uniform porous layer with the same total flow resistance. The liner with 4 screens yields almost the same performance as the uniform porous layer. It should be noted that for very low frequencies the single sheet liner is slightly better than the uniform layer.
8.5.4 Slotted Liner Figure 8.20 shows examples of slotted absorbers or liners. The first simply has slots cut into a uniform porous layer. As shown, it has an impervious cover on one side of each slot, but this is not necessary. It was present in an installation for nonacoustical reasons. The slot width is D and the slot separation d + D , which is assumed small compared to a wavelength so that the layer can be considered locally reacting. To analyze the absorption or attenuating characteristics, we need the input impedance, and this can be calculated in the following way. Each slot is regarded as an air channel lined on one side with a nonlocally reacting porous liner of thickness d . The depth of this ‘mini’ duct is d. Thus, by applying the analysis for sound propagation in such a duct with a nonlocally reacting liner, we can determine the average input impedance of the mini duct over one spatial period d + D . If this slotted material is mounted on a wall, we can readily determine its absorption spectrum accounting for the fact that the material is anisotropic in the same way as for the slot absorber. The second type of slotted liner shown in the lower part of Figure 8.20 is completely equivalent acoustically under the assumptions made.
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Figure 8.20: The attenuation of the fundamental acoustic mode in a rectangular duct lined
on one side with a slotted porous layer. The open area of the duct D/(D + d) = 0.3 and the total normalized resistance of the liner = rd/ρc = 8. The parameter values 0 to 20 percent refer to the slotted area fraction D /(d + D ). The liner thickness d equals the liner period D + d .
The same is true when the slotted layers are used as liners in a duct, as shown. The wave runs along the axis of the duct perpendicular to the slots and the liner. We can then apply the analysis for sound propagation in a duct with a locally reacting liner in which the liner thickness is d and the width of the air channel D, as indicated in the figure. The input admittance of the liner is obtained from the analysis of the mini duct with the liner thickness d , channel width D , and length d. The slotted layer is described by the total normalized flow resistance = rd/ρc, the slotted area fraction D /(d + D ), and the layer thickness d. The effects of the slots on the absorption and attenuation are quite similar. An example is shown in Figure 8.20. For comparison, we show the attenuation for the uniform porous layer (0 percent), having a total normalized flow resistance of = 8, and we find that a slot area as small as 1 percent produces a noticeable effect on the attenuation. It is clear that the mid-frequency attenuation can be improved by means of slots, but the penalty is a sharp reduction in attenuation at low frequencies. The effect of the slots in many respects is the same as a lowering of the average flow resistance of the liner. In calculating the attenuation, the liner has been treated as locally reacting, and this limits the validity of the result to wavelengths large compared to spatial period of the liner D + d .
8.5.5 Effect of Partition Spacing If the spatial period of the slots in the previous section or if the partition spacing in the air backing of a liner is not small compared to the wavelength, the calculation of the attenuation becomes considerably more complicated than for a locally reacting liner.
281
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O1
O2
LOUDSPEAKER
MICROPHONE
RESISTIVE CLOTH
L1
PARTITIONS
pc
PERFORATED FACING
TERMINATION
(c) 25
l = 24 CM 20
dB/m 15
10
5
0
200
3
4
5
6
7
8 9
1000
1500
Figure 8.21: Top: Simple arrangement for measuring attenuation. Bottom: Measured attenuation spectrum (dB/m) with a partition spacing in the liner approximately half a wavelength at the resonance frequency of the liner.
To be strictly locally reacting, a duct liner consisting of a resistive sheet backed by an air layer must have transverse partitions with a spacing much smaller than a wavelength. In a simple experiment, illustrated in Figure 8.21, the effect of the spacing on the attenuation was studied. The duct was lined on one side with a resistive sheetperforated facing combination backed by a 6.5 cm thick air layer. The width of the air channel was 14.5 cm. The open area of the facing was 8 percent and the resulting effective resistance of the liner was 1.5 ρc. The diameter of the holes in the facing was 0.4 cm and the thickness of the facing, 0.4 cm. The attenuation in the duct was measured with a simple arrangement of a moving microphone, as shown in Figure 8.21, and data were obtained with partitions spacings 10, 20, 24, 30, 55, 110 cm and also without partitions. For a spacing of 10 cm the liner was, to a good approximation, locally reacting, and the peak attenuation occurred at about 660 Hz, close to the resonance frequency of the liner. For a spacing of 24 cm, i.e., about half a wavelength at the resonance frequency 660 Hz, a pronounced double peak in the attenuation spectrum was obtained, as shown, reminiscent of the response of coupled oscillators. In this case, the oscillators or modes would be the transverse quarter wavelength mode of the liner and the half
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wavelength mode in the axial standing wave between adjacent partitions in the liner. The overall effect is a ‘line broadening’ of the attenuation spectrum, which suggests a rule for optimum partitions spacing; it should be chosen to be twice the liner thickness.
8.6 EFFECTS OF HIGHER MODES AND FLOW An analytical supplement to this section is given in Section 10.4.
8.6.1 Higher Modes The discussions so far have involved the fundamental mode in a lined duct, and we shall now make some comments about the role of higher modes and a high frequency estimate of the transmission loss in a lined duct based on geometrical acoustics. If the modal composition were known, it would be possible in principle to compute the total sound field in the duct and its axial dependence, including the transmission loss. In practice, however, this composition is usually not known and we have to resort to empirical or semi-empirical estimates of the role of higher modes. We shall present such an estimate based on ray acoustics. The ‘Z-Modes’ With reference to the duct in Figure 8.22, let the z-direction be into the paper, the ydirection normal to the lined boundary, and the x-axis along the duct. The duct width in the z-direction is Dz and the walls at z = 0 and z = Dz are unlined, rigid, and impervious. So far, the sound pressure in the z-direction has been assumed uniform (fundamental mode). If we allow for a modal pressure variation it must be of the form cos(kz z), where zz = mπ/Dz and m is an integer; m = 0 representing the fundamental mode. To study the effect of the z-modes, i.e., modes with variation in the z-direction, we calculated the attenuation including modes with m up to 20, all assumed to have the same amplitude, but there was no significant change in the computed average attenuation of the sound field. The ‘Y-Modes’ The variation of the sound pressure amplitude in the y-direction is described by cos(ky y) but, because of the lossy boundaries, the quantity ky is a complex quantity and its determination requires the numerical solution of the transcendental equation discussed in the analytical supplement Chapter 10. Once ky is found, however, the propagation in the x-direction is expressed by the propagation constant kx =
(ω/c)2 − ky2 − kz2 from which both the wave speed and the attenuation in the
duct can be found. So far we have dealt with only the fundamental of the many possible y-modes. The attenuation of this lowest mode is lower than for higher modes and, in a long duct, it will be the mode that will ‘survive’ and be largely responsible for the overall
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Figure 8.22: With reference to the geometrical or ray acoustics approximation in estimating transmission loss in a duct at high frequencies. Left: Angular range for rays leaving the duct without reflection from the liner. Right: Angular range for one reflection.
attenuation. The role of the higher modes is difficult to determine precisely since the relative strength of the modes depends on the generally unknown source distribution at the beginning of the duct. It is clear, however, that because of the higher decay rate of the higher modes, the attenuation rate of the total sound field at the beginning of the duct will be greater than the average attenuation over the entire length of the duct. To get an idea of the role of such higher modes, we resort to geometrical acoustics (high frequency limit of wave acoustics), and consider the simple case of a line source placed in the entrance plane of the duct, to the right as shown in Figure 8.22. The duct is lined on both sides with identical liners. Air channel width is D and the duct length L. In the absence of the liners, the power We leaving the duct at the end will be the same as the power W emitted into the left hemisphere of the source. If the liners are totally absorptive so that all rays that strike them will be absorbed and the only sound emitted from the duct is carried by the rays, which are not reflected, i.e., those that are emitted in the angular range between −φ0 and φ0 = arctan(D/2L), as shown. Then, if the source is omnidirectional, the power escaping from the duct will be We = 2[(φ0 /π]W and the corresponding transmission loss T L = 10 log(W/W0 ) = 10 log(π/(2φ0 ). Thus, for small angles we have φ0 ≈ D/2L and the transmission loss becomes T L ≈ 10 log(π/2 arctan(D/2L)) ≈ 10 log(π L/D)
(Geometrical acoustics). (8.4) This expression should be regarded as an upper limit at high frequencies, since we have assumed total absorption by the liners. It will be reduced, of course, if the liner is not perfectly absorbing, as commented on below. Thus, in this geometrical acoustics regime, the transmission loss is not proportional to the length/width ratio but rather to the logarithm of this ratio in the D/L << 1 approximation. As an example, with L = 36 inches and D = 4 inches, we get T L ≈ 14.5 dB. In reality, the liner is not totally absorbing, and we can improve the expression for We by accounting for the number of reflections suffered by a ray along its path to the exit of the channel. This number depends on the angle of emission of the ray. One reflection occurs in the angular range |φ1 | − |φ0 | as indicated on the right in the figure. After each reflection, the intensity is reduced by the factor |R|2 , where R is the
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pressure reflection coefficient. The power carried out through the duct by the sound φ which has suffered one reflection is then We1 = 2 φ01 |R(φ)|2 dφ with analogous expressions for multiple reflections. With L/D >> 1, the angle of incidence of the reflected waves is close to 90 degrees and a reflection coefficient close to unity. Disregarding the reflections, we have found the simple relation in Eq. 8.4 to agree quite well with experimental data3 obtained for lined ducts ranging in length from 3 to 10 ft and in channel widths from 3.5 inches to 12 inches. This geometrical acoustics result tells us nothing about the frequency dependence of the transmission loss except that the analysis is meaningful only at wavelengths considerably shorter than the duct width D. We have used an empirical frequency dependence of the higher mode correction (to be added to the fundamental mode TL) such that 0 for f < f1 (8.5) δ≈ δmax (1 − exp(−0.35 f/f1 )) for f ≥ f1 , where δmax = 10 · log(π/2 arctan(D/2L)) ≈ 10 log(π L/D) and f1 = c/2D. The correction approaches the upper limit discussed above as the frequency increases, starting from zero at a frequency f1 = c/2D and reaching a value close to the limiting value at a frequency ≈ 10 f1 . This behavior has been found to be in fair agreement with experimental TL data for silencers ranging in length from 3 ft to 10 ft with channel widths from 3 inches to 12 inches. The line source model used here as a simple illustration of the idea can be replaced by one in which the sound enters the duct from a diffuse field in a plenum chamber, but this model is not pursued here.
8.6.2 Convection A mean flow in the duct can influence the sound transmission in several ways. The convection of sound by the flow is the most obvious effect. It leads to a reduction of the attenuation in the downstream direction and an enhancement upstream. The refraction of sound, resulting from the transverse flow gradient in the duct, is important only at high frequencies and leads to an increase in the attenuation in the downstream direction and a decrease upstream. Typically, the two effects cancel each other at a frequency where the attenuation is a maximum. More subtle is the effect of flow on the input impedance of a duct liner. The effect is usually small but can be significant if the liner consists of a perforated plate backed by an air layer or a porous layer with small flow resistance. The effect of normal and grazing flow on the flow resistance of a perforated plate is discussed in the next chapter. In a mean flow with velocity U , a plane sound wave will be convected by the flow so that the resulting wave speed with respect to a stationary frame of reference will be c + U , where c is the sound speed (relative to the fluid). U is positive (negative) for downstream (upstream) propagation. 3 Courtesy of Industrial Acoustics Company, Inc., New York.
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The attenuation in dB is proportional to the distance of wave travel expressed in terms of the wavelength (or the number of ‘pumping’ periods into the boundary during the passage of the sound). The wavelength downstream will be longer than upstream by a factor (1 + M)/(1 − M), where M = U/c is the flow Mach number. Thus, the distance of wave travel, as measured in wavelengths, will be greater upstream than downstream, and the corresponding attenuation will be greater by the same factor (1 + M)/(1 − M). As an example, with a flow speed of 5000 fpm, corresponding to M ≈ 0.07 at room temperature, this factor is ≈ 1.16 and an attenuation of 30 dB in the downstream direction corresponds 34.8 dB upstream. However, this result is valid only at relatively low frequencies when the fundamental mode dominates in the duct and convection is the important flow effect. When the wavelength is smaller than the channel width, it turns out that the effect of flow is reversed, leading to an increase of the attenuation in the downstream direction and a decrease in the upstream direction. This is a result of refraction, as will be discussed next.
8.6.3 Refraction The semi-empirical higher order mode correction given above was based on ray acoustics and applied to a uniform gas at rest. Under such conditions, the acoustic rays were straight and the angle φ0 below which rays emerged from the duct without striking the duct wall was simply φ0 = arctan(D1 /(2L)), as shown in Figure 8.22. In a moving hot gas there will be lateral gradients of velocity and temperature and hence of the wave speed. With the maximum temperature and velocity assumed to be at the center of the duct channel, the flow refracts the sound toward the boundary for a wave in the downstream direction and away from the boundary in the upstream direction. The thermal gradient refracts toward the boundary, independent of direction. Thus, in the downstream direction, the flow and temperature gradients cooperate in refracting the sound toward the boundary but oppose each other upstream. Under steady state operating conditions of a silencer, the wall temperature is expected to be essentially the same as the gas temperature and in our computer program only the effect of a flow gradient is accounted for. In turbulent duct flow, the velocity goes to zero at the walls in a non-uniform manner, most of the decrease occurring in a relatively thin boundary layer with a thickness depending on the flow velocity. Outside this layer the decrease of the velocity from the center of the duct can be approximated as linear, for the present purpose. In such a flow field, a sound ray will follow a circular path with a radius of curvature given by4 1/R = −(1/c)d(U + c)/dy, where y is the vertical coordinate. A positive value of the curvature corresponds to a ray that turns upwards. The effect of a temperature gradient depends of the sound speed, (1/c)dc/dy = (1/2T )dT /dy, where T is the absolute temperature. The geometrical considerations in Figure 8.22 in estimating the transmission loss now have to be reexamined to account for curvature of the sound rays.
4 See, for example, Uno Ingard, Waves and Oscillations, Cambridge University Press, 1988, Section 12.6.
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Figure 8.23: Effect of refraction in a duct due to nonuniform flow, which shows how the limiting ray in Figure 8.19 is now curved, reducing the critical angle below which sound escapes from the duct.
Figure 8.24: An illustration of the effect of flow direction on the transmission loss including semi-empirical corrections for flow and higher modes. Thin lines: No flow. Solid line: Sound in downstream direction. Dotted line: Sound in upstream direction. Flow speed in the silencer channel: 100 ft/sec.
Thus, Figure 8.23 shows the ray (solid line) in the downstream direction emitted under the critical angle φ0 below which emitted rays will escape through the duct without being reflected from the boundary. The dashed, curved line represents the corresponding ray in the upstream direction. This figure shows half of a duct, which is lined on one side and which contains a centered line source at the beginning of the duct. For flow in the downstream direction, the curvature will reduce the critical angle and increase the attenuation. The opposite holds true for sound in the upstream direction. Applying a little geometry, we can express the critical angle in terms of the radius of curvature, and hence in terms of the flow gradient (see Eq. 10.79). Having obtained
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φ0 , a higher order mode correction can thus be obtained as described in connection with Figure 8.22. Figure 8.24 shows an example of the effect of flow on the transmission loss. The equivalent average velocity gradient was estimated to be U/(12D). It should be noted that at frequencies below ≈ 2000 Hz, the insertion loss is decreased by the flow for sound propagation in the downstream direction and increased at higher frequencies due to the effect of convection. At higher frequencies, the insertion loss is increased in the downstream direction due to refraction.
8.6.4 Scaling Laws A nonuniform distribution of temperature in a duct gives rise to refraction, as discussed earlier. There are more direct effects of temperature, however. The obvious√and most important is the increase of the sound speed with temperature, c ∝ T , where T is the absolute temperature. At a given frequency, the wavelength then will increase with temperature and the ‘acoustic’ thickness (measured in terms of wavelengths) of a duct liner will be reduced. This generally results in a decrease in the attenuation. Another√effect is the increase in the shear viscosity with temperature (approximately as T ) and the corresponding increase in the flow resistance of a porous material. This effect√is compounded by the decrease of the wave impedance ρc with temperature (∝ 1/ T ) so that a flow resistance normalized with respect to ρc will increase approximately in direct proportion to the absolute temperature T . As a commonly encountered example of duct liner design, we note that a typical temperature in the exhaust stack of a gas turbine power plant can be about 1000◦ F (T ≈ 811 K), and the effect of temperature on the attenuation then can be quite substantial. Thus, it is important that model experiments’ data of duct performance obtained at room temperature be appropriately corrected when used in applications at other temperatures. The static pressure does not influence the sound speed or the coefficient of shear viscosity, but it does affect the wave impedance, and hence the normalized flow resistance and the kinematic viscosity. In a duct with flow, the temperature dependence of the kinematic viscosity, and hence the Reynolds number will influence the static pressure drop. So far it was tacitly assumed that the gas was air at static pressure of 1 atm. In some applications, typically involving compressors, both the gas and the static pressure can be different, and this can have a substantial effect on duct liner performance through the influence on wave impedance, shear viscosity, and hence the normalized resistance. The input values in our computer programs refer to a measured normalized flow resistances in air at room temperature (293 K) and a pressure of 1 atm. Appropriate corrections are made within the programs. The scaling laws of interest for duct design can be summarized as follows using the following notations. Molecular weight: M, density: ρ, absolute temperature: T , static pressure: P , shear viscosity: μ, kinematic viscosity: ν. The reference values of these quantities are given the subscript ‘0’ and refer to air at P0 = 1 atm and T0 = 293 K.
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NOISE REDUCTION ANALYSIS
Density ρ/ρ0 = (P /P0 )(T0 /T )(M/M0 )
(8.6)
Sound Speed c/c0 =
(T /T0 )(M0 /M)
(8.7)
Wave Impedance ρc/ρ0 c0 = (P /P0 ) (M/M0 )(T0 /T )
(8.8)
For example, with P /P0 ≈ 5, M/M0 ≈ 4, and T /T0 ≈ 1.18 (150◦ F), we get ρc/ρ0 c0 ≈ 5 · 2/1.09 ≈ 9.2. Shear Viscosity The coefficient of shear viscosity μ is approximately independent of density and approximately proportional to the square root of temperature. Thus, for a gas with a reference value μ0 we have μ/μ0 = T /T0 . (8.9)
Kinematic Viscosity The kinematic viscosity, ν = μ/ρ, expresses the ratio of viscous and inertial forces in a sound field and determines, for example, resonance in √ the sharpness of an acoustic √ a cavity. With μ being proportional to (T ) and ρ/ρ0 = (P /P0 ) (T /T0 ), it follows that ν/ν0 = (T /T0 )3/2 (P0 /P ).
(8.10)
Reynolds Number The Reynolds number of the flow with speed U in a duct of width D is R = U D/ν. For a given mass flow rate Q, the velocity is proportional to Q(T /P ) so that R/R0 = T0 /T .
(8.11)
An increase in temperature, and hence a decrease in R, tends to decrease the intensity of instabilities in a flow and it will affect the wall friction in turbulent duct flow.
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Normalized Flow Resistance θ = r/(ρc), where r is the specific flow resistance. Thus, for a gas with a molar mass M, θ/θ0 = μ(T )/μ0 (T0 )(ρ0 c0 )/(ρc) ≈ μ(T0 )/μ0 (T0 ) T /T0 (ρ0 c0 )/(ρc) = μ(T0 )/μ0 (T0 ) M0 /M(P0 /P )(T /T0 ). (8.12) For example, with μ(T0 )/μ0 (T0 ) ≈ 0.75 and with the values of other parameters, M/M0 ≈ 4, etc., given above, we get θ/θ0 ≈ 0.09. In other words, in order for a liner in the muffler to be effective, a flow resistance approximately 11 times the value in normal air would be required. In the computer program, all the input resistances refer to the values obtained for normal air. To be able to apply the program to other gases than air, the quantities, M/M0 , μ/μ0 , in addition to pressure and temperature, are also input parameters. The scaling factors are incorporated in the program. Example As an illustration we consider a rectangular duct lined on one side with an 8 inch thick porous layer, covered with a perforated facing. The flow resistance of the porous layer is 0.5 ρc per inch, measured at room temperature. The perforated plate has an open area of 23 percent, thickness, 0.1 inch, and hole diameter, 0.1 inch. The width of the air channel is 4 inches. Determine the spectrum of the attenuation per unit length (unit length being the channel width) at room temperature, 70◦ F, and at a temperature of 1000◦ F. The result is shown in Figure 8.25. (It is applicable also to a duct with two sides lined (identical liners) and a channel width between the liners of 8 inches, i.e., twice the original channel width.) The result in the figure is expressed in dB per channel width in a duct with one side lined, and if we apply the program to a duct with two walls lined and use the new channel width as unit length, the attenuation per channel width will be twice that in the figure.
Figure 8.25: Effect of temperature on attenuation in a lined duct. For parameter values: see text.
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NOISE REDUCTION ANALYSIS
The attenuation spectra at the two temperatures are shown. The markers signify octave band average values assuming the source spectrum to be uniform. The spectrum at 1000 degrees is displaced toward higher frequencies in comparison with the spectrum at 70 degrees and the attenuation at mid-frequencies is lower. This is an effect which can be of considerable practical importance. For example, with a duct length of 10 channel widths, an increase in temperature from 70 to 1000◦ F results in a decrease in attenuation at 500 Hz from 32 to 14, i.e., approximately 18 dB, a substantial amount.
8.6.5 Static Pressure Drop in Ducts The Reynolds number5 is R = U D/ν, where ν = μ/ρ is the kinematic viscosity, ≈0.15 CGS for normal air (μ is the coefficient of shear viscosity and ρ the density). For a rectangular channel of width W and height H , we get D = 4W H /2(W + H ), which, if W << H , becomes ≈ 2W. At a Reynolds number R ≈ 2000 there is a transition from laminar to turbulent flow, and sufficiently far from the entrance to the pipe (typically 10 pipe diameters) and with increasing Reynolds numbers, the flow becomes turbulent and ‘fully developed,’ which means that the flow profile and the statistical properties of the flow are independent of the axial position. For laminar flow, the axial pressure gradient in the duct is proportional to the mean velocity U , and we have for a circular duct with a diameter D, dP /dx = −ψL U = −(32μ/D 2 )U,
(8.13)
where μ is the coefficient of shear viscosity.6 In turbulent duct flow, the pressure gradient is closer to a square law dependence on the velocity, and it is customary to express the average shear stress on the wall around the perimeter as (8.14) τ = f (ρU 2 /2), where U is the mean velocity in the pipe and f is a ‘friction coefficient.’ If the cross sectional area of the pipe is A and the perimeter S, the pressure drop per unit length can be expressed in terms of the shear stress from the relation A(dP /dx) = −τ S as follows, Axial pressure gradient in turbulent pipe flow dP /dx = −(S/A) τ = −(ψ/D) (ρU 2 /2 = −(ψ/D)(γ /2)P M 2 )
(8.15)
D = 4A/S: hydraulic diameter of pipe (A = area, S = perimeter), ψ = 4f : wall friction factor (see Figure 8.26), M = U/c: Mach number, P = ρc2 /γ : static pressure, c: sound speed, γ : specific heat ratio (≈ 1.4 for air).
5 Based on the hydraulic diameter D = 4A/S (A: area, S: perimeter), which equals the physical diameter for a circular cross section. 6 See, for example, Uno Ingard, Acoustics, Infinity Science Press, 2008.
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Figure 8.26: Friction factor ψ in pipe flow. The solid curve refers to pipes with ‘smooth’ walls. Wall roughness increases ψ as indicated schematically by the dashed curves for different values of D/, where is the wall roughness and D = 4A/S, the hydraulic diameter.
If P is expressed in atm, we have to multiply by ≈ 105 to get it in N/m2 ; with ρ = 1000 kg/m3 and g = 9.81 m/s2 , one atm corresponds to the weight of a water column of height 105 /ρg ≈ 10.2 m (401.6 inches). If we bring the expression for the pressure drop in laminar flow into the same form as for turbulent flow, it follows from Eqs. 8.13 and 8.15 that the friction factor ψL for laminar flow can be written, ψL = 4f = 64/R, where R = DU/ν is the Reynolds number (based on the hydraulic diameter 4A/S). This friction factor is inversely proportional to the Reynolds number. In the turbulent regime, the Reynolds number dependence is not as strong; in fact, ψ approaches a constant value at high Reynolds numbers. The study of pressure drop in a pipe has a long history and the most extensive measurements of ψ were made about 60 to 70 years ago.7 In Figure 8.26, the solid curve refers to a pipe with ‘smooth’ walls. A good empirical expression for this function is8 ψ = 0.0054 + 0.396/R 0.3
(smooth walls).
(8.16)
For a rough pipe wall, the measured Reynolds number dependence of ψ, the dashed curves in Figure 8.26, is qualitatively different than for a smooth wall. Although ψ initially decreases with R as for a smooth duct, it reaches a minimum and then increases with R asymptotically to a constant value, as shown schematically in Figure 8.26. The minimum (both value and location) as well as the asymptotic value ψr depends on the roughness of the wall, being the average size of the protrusions from the wall. Thus, in the high Reynolds number regime, the friction factor depends only on the roughness parameter /D and the values obtained from Figure 8.26 7 For further details see, for example, Ludwig Prandtl, Essentials of Fluid Dynamics, 1952, or Sidney Goldstein, Modern Developments in Fluid Dynamics, Vol. II, 1938. 8 Due to R. Hermann, presented in a dissertation, Leipzig, 1930. An even older and frequently quoted formula, ψ ≈ 0.3164/R 1/4 , is due to Blasius, 1913.
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NOISE REDUCTION ANALYSIS
range from 0.02 to 0.04 for values of /D from 0.001 (i.e., D/ = 1000) to 0.03 (i.e., D/ = 330). The roughness depends on the material and values for different materials and can be found in engineering texts on fluid dynamics. The following data are listed to provide some idea of the range of values to be expected for (in cm) for various surfaces: • Riveted steel: 0.09-0.9 • Concrete: 0.03-0.3 • Wood: 0.018-0.09 • Cast iron, uncoated: 0.018-0.027 • Cast iron, asphalt dipped: 0.012 • Commercial steel or wrought iron: 0.0045 • Centrifugally spun cement: 0.00015-0.0045 In principle, ψ can be determined from measurements of the total pressure drop over a number of ducts of different lengths. Assuming that the pressure drop due to the exit and entrance flows are independent of the duct length, the pressure drop contribution due to the length alone can be obtained and from it, the friction constant ψ. However, there are complications; one is that fully developed turbulent flow in a duct occurs only beyond a certain distance from the entrance; another is that as the pressure and density decrease with distance in the duct, the flow speed increases. For flow in a duct with a porous liner covered with a perforated facing, the roughness is less well defined and more work is required to determine it and the corresponding friction factor in terms of the parameters of the liner and the duct. For a typical facing with a 23 percent open area and hole diameter of about 1/8 inches on a porous layer, a friction factor ψ ≈ 0.025 − 0.030 has been found to be a reasonable approximation. An interesting fact is that the interaction of the flow with a boundary with a perforated facing frequently leads to an instability, which is revealed as a tone with a frequency of the order of U/D, where U is the velocity and D the axial separation of the holes. Even if a pure tone is not generated, the spectrum of the flow noise in a duct generally has a peak in the vicinity of this frequency. In addition to the pressure loss due to wall friction, there are also losses due to the area changes and the related flow separation at the inlet and the exit of the duct. The exit loss is generally the most important and for it we use the expression9 Pressure loss, exit Pressure loss ≈ (ρU 2 /2)(1 − σ )2 = (γ /2)M 2 P (1 − σ )2
(8.17)
(See Eq. 8.15 for notation.) where σ is the ratio of the air channel area and the area of the main duct. This result refers to a silencer with an abrupt change in cross section at the exit. With a gradual change the loss is somewhat smaller, although a long transition is required to achieve 9 See, for example, Uno Ingard, Acoustics, Infinity Science Press, 2008.
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a substantial reduction. Even then it is difficult to avoid flow separation from the wall. For an exit with rounded edges, a reduction of 5 to 10 percent might be achieved. At the inlet, as the flow enters the channel from the main duct, it separates and is known to form a stream with an area, which first contracts and then expands. The minimum area occurs at the ‘vena contracta.’ The corresponding pressure loss is Pressure loss, inlet Pressure loss, inlet ≈ (γ /2)M 2 P (A1 /Av − 1)2
(8.18)
where A1 is the channel area and Av the area of the stream at vena contracta. It is known from experiments10 that as the open area fraction σ of the silencer goes from 1 to 0.2, Av /A1 goes from 1 to 0.63 and the pressure loss from 0 to 0.34 (γ /2)M 2 P ).11
8.7 LIQUID PIPE LINES, ELEMENTARY ASPECTS An analytical supplement to this section is given in Section 10.5. A detailed analysis of sound transmission in a liquid pipe line is, in several respects, more difficult than for an air duct. The main reason is that the wave impedance of water is much higher than for air, by a factor of about 3400. Thus, the acoustic coupling between the liquid and the walls of the pipe involved is much stronger than in air and a detailed analysis should include the coupling of the waves in the liquid and in the pipe walls. Instead of a general analysis along these lines, we consider two special cases. The first refers to the propagation in a hose assumed to have locally reacting and slightly compliant walls. The meaning of this will be clarified shortly. In the second example, the liner is highly compliant since it consists of an air layer which is separated from the liquid by a limp membrane.
8.7.1 Liquid Pipe Line with Slightly Compliant Walls In a project dealing with noise generation from a centrifugal pump, the liquid pipe line loop contained long sections of rubber hose, which were used instead of orifice plates or valves for the purpose of providing ‘noise free’ throttling. The sound attenuation was also of interest since it was desirable to eliminate disturbing wave reflections in the test loop; the comments on the attenuation made here came out of this project. A general treatment of sound transmission would involve a study of the coupling of the wave motion in the liquid and in the pipe wall similar to the study of the sound transmission in a flexible porous material. Such a study is beyond the scope of this book. However, if the pipe wall is assumed to be locally reacting, a calculation of the attenuation can be carried out in the same manner as for the circular lined air duct, discussed earlier. It is then merely a matter of expressing the acoustic admittance of 10 See standard engineering texts and handbooks on fluid flow.
11 Noise generation by the flow is largely due to the (jet) exit flow; the noise power can be estimated from the discussion of jets. See, for example, Uno Ingard, Acoustics, Infinity Science Press, 2008.
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NOISE REDUCTION ANALYSIS
the wall in terms of dimensions and elastic properties of the pipe wall, as discussed in Chapter 10, and using these data in the computer program developed for the circular duct. The response of the wall to the sound pressure in the liquid is a displacement of the wall in the radial direction, which depends only on the local sound pressure.12 To make a rough estimate of the attenuation and to obtain an approximate explicit expression for the attenuation in terms of the pipe line parameters, the simplifying assumption is made that the wall compliance is small so that the sound pressure amplitude is approximately constant across the pipe in its fundamental acoustic mode. In a circular pipe with rigid walls, the first higher mode can propagate only if λ < 1.7D, where λ is the acoustic wavelength in water and D the diameter of the pipe. For a 3 inch pipe this means that λ < 0.13 m. With a sound speed in water of approximately 1440 m/sec, the corresponding frequency range for propagation of higher modes is f > 11076 Hz. Generally, we shall be interested in frequencies below f1 = 11076 Hz, and we shall deal only with the fundamental mode here. For an air layer liner, to be treated in the second example, the various modes of propagation will be discussed in some detail. With only the fundamental mode present in the pipe, it will be driven in a ‘breathing’ mode and its impedance will be that of a resonator with a resonance frequency equal to the ‘ring’ frequency v/(2πa) of the boundary wall, where v is the longitudinal wave speed in the material and a the pipe radius. It is assumed then that the wall thickness is small compared to a. The corresponding period is the roundtrip time of the longitudinal wave in the boundary wall (for details, see Chapter 10). The internal damping in the wall is accounted for by assigning to it a complex elastic modulus and a corresponding loss factor of the material so that the elastic modulus is expressed as E(1 − i). As shown in Chapter 10, the phase velocity and the damping in the water can then be expressed in terms of and the parameter β = (D/d)(ρ0 c02 )/E, where D is the pipe diameter, d, the wall thickness, E ≡ ρ1 c12 , (the real part of) the elastic modulus of the wall material, ρ0 c02 , the inverse of the compressibility of water (ρ0 the density and c0 the wave speed). As an example, consider a steel pipe with D = 3 inches and a wall thickness d = 1/4 inches. Then, the ring frequency is f0 = v/(2π a) ≈ 2.1 × 104 Hz, and with ρ0 /ρ1 ≈ 1/7.8 and c0 /c1 ≈ 1440/5000, we find that the sound speed is decreased by approximately 6.5 percent as a result of the compliance of the wall independent of frequency if ω << ω0 . For a material such as Plexiglas, with ρ0 /ρ1 ≈ 0.87 and c0 /c1 ≈ 1440/2600 ≈ 0.55, the sound speed will be reduced by approximately one half. The pressure reflection coefficient at the interface between two pipes of different materials will be R = (c − c0 )/(c + c0 ). Thus, on the basis of our assumption of a locally reacting wall, this estimate gives, for the junction between a steel tube and a Plexiglas tube, R ≈ 0.33.
12 The wall would be locally reacting if the pipe were segmented in the axial direction or prevented from
carrying waves in the axial direction in some other manner.
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Figure 8.27: Left: The sound speed ratio c/c0 , where c is the phase velocity of the wave in a pipe with a slightly compliant wall and c0 is the free field sound speed in water (the phase velocity in the pipe if the wall had been rigid). The parameter β = (D/d)(ρ0 c02 /ρ1 c12 ), D = pipe diameter, d = wall thickness, ρ0 , density of water, and ρ1 , c1 , the density of the wall and the longitudinal wave speed in the wall. Right: The attenuation per free field wavelength in water. The three curves correspond to the loss factors = 0.1, 0.05, and 0.025 in the wall. There is no noticeable difference in the wave speed (phase velocity) for these three values.
8.7.2 Liquid Pipe Line With Air Layer Wall Treatment As the second example, we consider a wall which is almost a pressure release boundary. It involves an attenuator, which, in essence, is a pipe section with a gas layer as a wall lining, the gas being contained by an impervious membrane or by a tube or bladder with flexible walls, as shown schematically in Figure 8.28. The static pressure in the gas is kept the same as the pressure in the water so that there will be no tension in the membrane or tube walls, which we shall assume to be limp. The results given here refer to the configuration shown in Figure 8.28, which is a rectangular duct with two opposite walls lined with air layers contained by limp membranes. The two other walls are assumed to be rigid. The results obtained are qualitatively valid also for other pipe geometries. The basic idea of using air layers is to simulate a pressure release boundary since for such a boundary a wave with a frequency less than the cut-on frequency of the first higher mode cannot propagate but will decay exponentially. The cut-on frequency is f01 = c0 /2D, where D is the channel width and c0 the free field sound speed in water, ≈ 1440 m/s. Thus, with D = 10 cm, the cut-on frequency will be f01 = 7200 Hz. The pressure distribution in this mode is described by a cosine function with the pressure amplitude going from a maximum at the center of the channel to zero at the boundaries. Below the cut-on frequency, the wave decays and the transmission loss would be very high. Since the cut-on frequency often is quite high, as in the example above, the idea of a pressure release boundary is very appealing as far as noise control is concerned. The important question is: “To what extent does our liner simulate a pressure release boundary?” The membrane and the air behind it form a resonator and the impedance of the liner will be stiffness controlled below the resonance frequency. In this regime there is an acoustic mode that propagates without attenuation. Unlike the idealized case of a pressure release boundary, the pressure profile no longer has
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NOISE REDUCTION ANALYSIS
Figure 8.28: A rectangular water line with two opposite walls covered with air layers.
Figure 8.29: Left: The ratio c0 /c of the free field sound speed c0 and the phase velocity c in the water channel. Right: The transmission loss. The three curves in each graph correspond to static air pressures of 1, 10, and 100 atm. The membrane mass is 0.2 g/cm2 . Thickness of air layer: 1.27 cm. Width of pipe: 10 cm. Length of attenuator: 20 cm. The dotted curve in the transmission loss graph corresponds to a pressure release boundary.
a maximum at the center but rather a minimum with maxima at the walls. In fact, the profile is described by a hyperbolic cosine function rather than a cosine function, and we call this mode the hyperbolic mode. When the frequency exceeds the resonance frequency of the liner, i.e., in the mass controlled region of the boundary, the mode decays in much the same way as for a pressure release boundary. Thus, the practical problem is to keep the resonance frequency as low as possible to simulate a pressure release boundary over a wide frequency range. However, this can present problems if the pressure in the air has to be high, which makes the corresponding resonance frequency high for a fixed mass of the membrane. We have considered this influence of static pressure in Figure 8.29. The left graph in this figure shows the frequency dependence of the phase velocity of the hyperbolic mode (actually the ratio of the free field sound speed in water and the phase velocity). In this particular case, the resonance frequencies of the liner for the pressures 1, 10, and 100 atm in the air are approximately 380, 1020, and 3300 Hz, respectively. Above these frequencies the hyperbolic mode will decay. It should be noted that the air layer causes the phase velocity in the channel to be considerably lower than the free field sound speed. For example, it is more than 60 times lower at a pressure of 1 atm.
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The corresponding transmission loss for a 20 cm long attenuator is shown on the right in the figure. Below the resonance frequency for each static air pressure, the hyperbolic mode propagates without attenuation. The periodic variations in transmission loss are due to the reflections from the ends of the attenuator. Then, at every frequency for which the length of the attenuator is an integer number of half wavelengths, the transmission loss will be zero.
Chapter 9
Reactive Duct Elements 9.1 UNIFORM DUCT SECTION As indicated in Chapter 7, the mechanisms involved in an attenuator1 are absorption, reflection, and interference. Whereas the previous chapter dealt mainly with the effects of energy absorption in porous duct liners. The present chapter focuses on ‘reactive’ silencers in which reflection and interference dominate but with little or no sound absorption involved. Before reading this section it is recommended to review the discussion of measures of silencer performance in Section 7.3. For a dissipative silencer, there is usually relatively little difference between total attenuation, transmission loss, insertion loss, and noise reduction, except at low frequencies. For a reactive silencer, which, in its purest form contains no absorptive material, this is no longer the case. The transmission loss, TL, which is determined by the ratio of the primary incident acoustic power and the transmitted power is then due to the effects of reflection and interference. The net incident power is the difference between the primary incident power and the power reflected from the silencer. Recall from the previous chapter that we have denoted the transmission loss based on the net incident power by TL0. It will be zero only if there is no energy loss (dissipation) within the silencer. Perhaps there is no more direct a way to illustrate the difference between attenuation, transmission loss (TL), and insertion loss (IL) than to consider an unlined pipe element with acoustically hard walls. For such an element, the attenuation is zero (neglecting the visco-thermal, flow induced, and nonlinear attenuation). If the pipe section has a smaller cross sectional area than the main duct, thus forming a constriction of the duct, a wave incident on the pipe will be reflected so that only a portion of its acoustic power will be transmitted; the reflection then determines the transmission loss since there are no losses within the pipe. The transmission loss is independent of the location of the pipe element and is always positive. The insertion loss, as mentioned, depends on the (multiple) reflections from both the source end 1 The designations ‘attenuator’ and ‘silencer’ are synonymous in the present context. An analytical
supplement to this section is given in Section 10.6.
299
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NOISE REDUCTION ANALYSIS
and the termination of the main duct and depends on the location of the pipe element; the insertion loss can be positive at some frequencies and negative at others. As already discussed in Section 7.3, the most relevant quantity in noise control applications is the insertion loss. However, the quantity normally presented in textbooks, handbooks, and manufacturers’ product descriptions is the transmission loss, and it should not be mistaken for the insertion loss. For the dissipative attenuator, the difference is usually small, except at low frequencies, but for a mainly reactive element, such as a constriction, expansion, or side-branch resonator in a duct, the insertion loss can vary greatly from one installation to another. Therefore, as a general rule, the use of reactive duct elements as attenuators is risky; unlike the transmission loss, the insertion loss can be negative and thus results in more noise output from the system than without the ‘silencer.’ However, combined with careful analysis of the overall system involved and a backlog of practical experience (or both, of course), the reactive elements can be quite useful.
9.1.1 Role of Source Impedance From elementary electrical circuits, we know that the power output from a battery, for example, is at a maximum when the resistance of the external circuit equals the internal (source) resistance of the battery; this is often referred to as ‘impedance matching.’ There are related experiences from other areas. For example, to produce maximum power transfer in an elastic one-dimensional collision of two bodies, the target being initially at rest, maximum energy (all of it) is transferred if the two masses are the same. If the target mass is much larger than the projectile mass, almost all of the energy is ‘reflected,’ and if it is much smaller, the energy transfer is slight, since the motion of the projectile is essentially unaffected by the collision and will carry most of the energy after the collision. Actually, if the target is initially at rest and with the masses of the projectile and the target being m1 and m2 , the energy of the target after the collision will be the fraction 4m1 m2 /(m1 + m2 )2 of the incident projectile energy. The situation is much the same in the interaction of a sound source with its ‘load.’ The load on a loudspeaker, for example, will be quite different when it radiates into a partially evacuated chamber and when it is attempted to drive a wall with it. In the latter case, the membrane of the loudspeaker will not move and no acoustic power will be delivered even though an oscillatory force will be applied to the wall. In the former case, the reverse is true, the velocity amplitude will be high but the force low. In either case, practically no acoustic power will be delivered. It corresponds to a very low and a very high mass in our collision analogy. Less obvious is the effect of applying a straight pipe to a loudspeaker. If the pipe is much shorter than the wavelength involved, the effect is minor; the sound pressure and the velocity in the pipe will not vary significantly along its length and the load on the loudspeaker will be approximately the same as without the pipe. However, when the pipe length becomes of the order of a quarter of a wavelength or larger, the situation can be quite different. If the pipe diameter is much smaller than the wavelength, the pressure reflection coefficient will be close to -1 and a pressure node occurs at the end (or close to it) as the incident at reflected pressures interfere
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destructively. In the resulting standing wave in the pipe the pressure maxima (velocity minima) occur at distances of an odd number of quarter wavelengths from the end. This means that the acoustic impedance, the ratio of the pressure and the velocity, will be high at these locations. Thus, if the pipe length is a quarter wavelength, the load impedance on the speaker will be (much) higher than without the pipe. On the other hand, if the pipe length is an integer number of half wavelengths, the sound pressure at the beginning of the pipe will be the same as at the end, and the pipe will not alter the load on the speaker. In other words, the pipe acts like an impedance transformer with an impedance transformation, which depends on frequency (or rather on the ratio of the pipe length and the wavelength). The corresponding variation of the power output from the source then depends on its ‘internal impedance.’ A high impedance source would generate maximum power for the quarter wavelength pipe, whereas a low impedance source would do the same for the half wavelength pipe. If the radiating area of a loudspeaker is covered with a highly resistive porous plug, the equivalent source impedance will be high. The velocity amplitude in the plug then would be comparatively independent of the external load and in the limiting case of infinite resistance of the plug, independent of the load. The source is then often referred to as a ‘constant velocity’ source. In the other limit, when the source impedance is zero, the pressure drop in the corresponding plug is zero, and the pressure applied to the load will be independent of the load. The source is then called a ‘constant pressure’ source. A fan is an example of a low impedance sound source, and the piston in an internal combustion engine or a positive displacement compressor is an example of a high impedance source. Then, a quarter wavelength pipe section applied to the latter source would normally increase the load impedance, as explained above, and more acoustic power would be generated. In other words, the insertion loss of the pipe would then be negative; it ‘amplifies’ the sound by providing better impedance match (recall that the insertion loss of an acoustical element is the decrease in radiated power resulting from the ‘insertion’ of the element). Figure 9.1 shows the computed insertion loss of a straight pipe section vs the pipe length-to-wavelength ratio L/λ for a constant pressure and a constant velocity source (low and high source impedance, respectively). The insertion loss of the pipe for the constant velocity source is negative for all pipe lengths, i.e., the source delivers more power with the pipe than without it. In the idealized case of a constant pressure source, the source impedance is zero so that the power output will go to a maximum when the load impedance goes to zero.2 We note that the insertion loss in Figure 9.1 goes toward a maximum at low frequencies (wavelengths long compared to the pipe length). In this regime, the input impedance of the pipe is due to the air mass in the pipe (moving approximately like a uniform plug) and the mass reactance at the end of the pipe, which is proportional to the pipe radius (≈ 0.61a).3 This creates an impedance mismatch and a reduction of acoustic output power corresponding to an insertion loss of the pipe ≈ 20 log[1 + L/(0.61a)].
2 The velocity amplitude goes to infinity in our idealized example. 3 See acoustics texts.
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(With L/a = 20, as in Figure 9.1, this value is ≈ 30.3, i.e., close to the maximum in this figure.) As the wavelength decreases, i.e., L/λ increases, the velocity in the pipe will be affected by the compressibility of the air so that its mass reactance will decrease and when L/λ = 0.5, the insertion loss will be zero, according to the figure. For L/λ = 0.5, the input impedance of the pipe is the same as the radiation impedance at the end of the pipe, and this means that addition of the pipe will not alter the radiation load on the source and the insertion loss will be zero. This corresponds to the zero point labeled ‘1’ in the left plot. Actually, for the constant pressure source, there is an additional, slightly lower value of L/λ, labeled ‘2,’ for which the insertion loss is again zero. The impedance is here the same as at L/λ = 0.5 except for a change in sign of the reactance; this does not alter the power output and the insertion loss again becomes zero. Between these values, the input impedance will be zero, so that a perfect impedance match will be obtained with the source since it has zero internal impedance. The pipe then causes an increase of the power output, making the insertion loss negative. This explains the peculiar behavior of the insertion loss curve in the vicinity of the frequency where the pipe length is an integer number of half wavelengths. A reflection-free source has an internal normalized source impedance ζi = 1, and the power reflected from the end of the pipe will not be reflected again by the source regardless of the length of the pipe. The radiated power from the pipe will then be independent of the pipe length and the insertion loss will be zero, an important characteristic of the pipe. The addition of the pipe means an increase in the load impedance (except in the vicinity of L = nλ/2, as explained above) and for the constant velocity source, with ζi = ∞, an improved impedance match results causing an increase in radiated power so that the insertion loss will be negative. If the length is an odd number of quarter wavelengths, the matching to the high impedance source will reach its optimum with a correspondingly large negative insertion loss.
Figure 9.1: Left: The insertion loss of a pipe for a constant pressure (upper curve) and for a constant velocity source (lower curve), zero and infinite internal impedance, respectively. Right: The corresponding input impedance of the pipe. L = pipe length, D = pipe diameter, λ = wavelength, D/L = 0.1.
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Such is the case, for example, in the air induction system of an internal combustion engine in which the periodic air flow through the valves is responsible for the air induction tone. In the case of an automobile engine, the air inlet pipe containing the air filter, typically is about 2 ft long. Measurements have shown that the equivalent sound source placed at the throttle body has a high internal impedance over most of the frequency range of interest, say 60 to 150 Hz in a 4 cylinder engine. The insertion loss of the inlet pipe will then be negative, and the corresponding amplification will be a maximum when the pipe length is a quarter wavelength (see Figure 9.1), in this example at a frequency of about 140 Hz. Thus, the insertion of a straight pipe section as a noise control measure is not to be recommended for a high impedance source; it is viable only when the source impedance is close to zero, which is the case for many aero-acoustic sources, such as fans. In many applications of noise control, notably in the design of test cell silencers for jet engines, the source is specified by its acoustic power level for radiation into free field. Without further readily available information, a reasonable assumption is that the power level of the source is the same even in the presence of a silencer and having made this assumption, the rational procedure is to calculate the transmission loss, TL0, of the silencer, which then yields the reduction in acoustic power radiated from the source-silencer combination. It is then assumed, of course, that the flow generated noise within the silencer is negligible.
9.2 EXPANSION CHAMBER With reference to Figure 9.2, the expansion chamber considered here is simply an expanded section of a circular duct with a larger diameter than the rest of the duct. Actually, since we are dealing with wavelengths larger than the cross sectional dimensions of the duct, it is not essential that the duct is circular. The important parameter is the ratio of the areas of the main duct and the expanded section.
Figure 9.2: Expansion chamber and an example of the calculated transmission loss, TL. The diameter of the chamber is D and the length L. In this particular case D = L. The area ratio, referred to in the figure, is the ratio of the chamber area and the duct area.
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9.2.1 Transmission Loss For a discussion of the transmission loss and the difference between TL and TL0, we refer to Section 7.3.2. Since the expanded section is unlined, the power loss in it and hence TL0 will be zero. Because of the area changes, however, an incident wave will be reflected from the chamber and the transmission loss, TL, will not be zero. The same holds true for the insertion loss, which accounts also for the reflections from the source and from the end of the main duct. An example of the computed frequency dependence of the TL is shown in Figure 9.2 in which the frequency parameter is L/λ, where L is the length of the chamber and λ the wavelength. Since typically D ≈ L, there is no reason to extend the frequency range above L/λ ≈ 1 since the analysis here assumes that only a plane wave is propagating in the chamber. It should be recalled that the first higher mode in a circular tube is cut-on when D/λ ≈ 0.59. If there were no reactive impedances associated with the expansion and contraction at the ends of the chamber, the input impedance of a chamber of length λ/2 would be independent of the area, and the transmission loss would be zero. In reality, however, there are mass reactive components due to the area discontinuities and the values of L/λ that correspond to T L = 0 are altered somewhat. For more on this point, we refer to the discussion of Figure 9.4.
9.2.2 Insertion Loss Although the transmission loss characteristics, such as in Figure 9.2, are often presented in texts and handbooks, the insertion loss, IL, is of more direct practical interest. It depends on the characteristics of the duct with respect to which the insertion loss is evaluated and accounts for the reflections from both ends of this duct, the source impedance, and the axial location of the expansion chamber. In this case the main duct is a uniform hard duct of length L + L1 + L2 with an area equal to the smaller duct area in the figure.
Figure 9.3: The insertion loss of an expansion chamber for a high impedance source. The length of the chamber is L and the overall length of the main duct is 3L. For definition of quantities in the figure, see Figure 9.2.
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The numerical results in Figure 9.3 of the insertion loss of an expansion chamber of length L refers to a duct with the overall length 3L of which the length L will be occupied by the chamber. The reference system, therefore, is the bare uniform duct of length 3L. We have considered the insertion loss for two different locations of the chamber in which the distance L1 from the source to the beginning of the chamber is 0 and 2L. As in the example of the transmission loss, the diameter of the chamber is D = L, and two values of the area expansion ratio, 2 and 4, have been considered. We have chosen a high impedance source with an internal resistance of 10ρc and zero reactance and a termination impedance of an open ended duct. The frequency dependence of the insertion loss is much more complicated than for the transmission loss. To a great extent, the reason is that the frequency response of the reference system is also complicated, and we refer to the previous subsection for details. For example, when the pipe length is an odd number of quarter wavelengths, the input impedance of the pipe is high and is then a good match to the source impedance in Figure 9.3 resulting in a corresponding large power output. Then, anything that is done to the duct is likely to destroy this match and reduce the power output so that a positive insertion loss is obtained. Since in this example the length of the duct is 3L, this is expected to occur when L/λ = 1/12, 3/12, 5/12, 7/12 . . ., and we note that for these values the insertion loss indeed is at a peak or close to it. The best overall performance is obtained when the expansion chamber is placed close to the source, corresponding to the left curve in the figure. This is to be expected since the input impedance of the expansion chamber is relatively low, thus providing a large impedance mismatch.
9.3 ‘CONTRACTION’ CHAMBER The ‘opposite’ of an expansion chamber might be called a ‘contraction chamber,’ for lack of a better name (or perhaps an expansion chamber with a chamber-duct area ratio less than 1). This element is illustrated in Figure 9.4. It can be thought of as a conventional duct silencer without a liner. The diameter of the contracted section is D and the length is L. The diameter of the main duct is D0 .
9.3.1 Transmission Loss Again, we refer to Section 7.3.2 for a discussion of the two transmission loss measures TL and TL0. The frequency is assumed low enough so that only the fundamental mode is propagating; for a circular chamber of diameter D, this means a wavelength larger than 1.7 D. All higher modes created around the area discontinuities are evanescent and can be accounted for in terms of mass reactances assigned to the entrance and exit of the chamber. The general behavior of the transmission loss curve is similar to that for the expansion chamber, but for the values of the area ratios considered here, the transmission loss is relatively small. The maxima of the transmission loss occur when L/λ is somewhat smaller than an odd number of quarter wavelengths. The mass reactances mentioned above are equivalent to ‘end corrections,’ which in effect make the ‘acoustic length’ of the duct channel greater than the physical. Maxima of the TL-curve will
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Figure 9.4: Transmission loss, TL, of a ‘contraction chamber.’ The diameter of the chamber is D and the length L. In this particular case, D = L/4. The area ratio is the ratio of the contracted area and the duct area (0.25 and 0.5). occur when the acoustic length is an odd number of quarter wavelengths. Similarly, the TL becomes zero when the acoustic length is equal to an integer number of half wavelengths. For the expansion chamber, the mass end corrections produce a correction in the other direction, so that the L/λ values at which the zeros occur are somewhat larger than an integer number of half wavelengths (see Figure 9.2). This can be thought of as a result of end corrections of the actual duct, which penetrate into the chamber to make its acoustical length shorter than the true length.
9.3.2 Insertion Loss To complete the comparison with the results for the expansion chamber, we have calculated also the insertion loss, and examples of the results are shown in Figure 9.3. As for the expansion chamber, two locations of the duct element have been considered to correspond L1 = 0 and L1 = 2L, where L1 is the distance from the source to the beginning of the element and L is the length of the contracted section. Also, as before, we have chosen a high impedance source with a normalized resistance of 10 and a reactance of 0. The duct following the contraction is assigned a termination impedance equal to that of an open ended pipe radiating into free field. The reference power in the calculation of the insertion loss is that obtained for a uniform pipe of length L1 + L + L2 with an area equal to the larger pipe area in the figure. When the attenuator is placed close to the source, it is not surprising to find that the insertion loss is negative over a substantial portion of the frequency range. Because of the constriction, the input impedance of the element is relatively large so that a better impedance match with the high impedance source is to be expected along with
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Figure 9.5: Insertion loss of a ‘contraction chamber.’ Normalized source resistance = 10, reactance = 0. Length-diameter ratio L/D = 4 of constriction chamber. The distance from source to beginning of the chamber is L1 and from the end of the chamber to the duct termination, L2 . a corresponding increased power output. When the constriction is placed at the end of the duct, corresponding to L2 = 0, the radiating area is smaller than the duct area, and this has been taken into account in calculating the insertion loss. As in the case of an expansion chamber, the TL and IL characteristics are quite different and depending on placement of the contraction and other system parameters, the insertion loss can be negative and cause amplification of the sound rather than reduction.
9.4 SIDE-BRANCH RESONATOR IN A DUCT In many noise control problems, a duct is often an integral part of the system, and the control of noise requires the insertion of noise control elements in the duct. One such element is a side-branch resonator, as illustrated in Figure 9.6. It consists simply of a straight pipe element of length L with a cross sectional area As , and it is closed at the end. Damping of the resonator can be achieved by filling it with porous material with a low flow resistance or by means of a screen across the open end, as indicated. Damping is provided also by the sound-flow interaction at the opening of the side resonators. The separation of the flow and the related turbulence is modulated by the oscillatory flow in the sound, which leads to enhanced turbulence with the corresponding energy drawn from the sound field, which leads to damping. A perforated plate backed by a partitioned air space can be regarded as side-branch resonators in parallel thus acting as a locally reacting duct liner often used in automotive muffler applications. Experimental studies by Rao and Munial have shown4 that the flow induced normalized acoustic resistance of a perforated liner with air backing is ≈ 0.5 M/σ , where M is the Mach number of the mean flow in the duct and σ the fraction open 4 K. Narayana Rao and M.L. Munial, Journal of Sound and Vibration (1986), 108(2), 283-295.
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Figure 9.6: Side-branch resonator in a duct with the opening covered with an optional resistive screen. The source and termination impedances are denoted by ζs and ζt . area of the perforate. The measurements covered Mach numbers between 0.05 and 0.2 and open area fractions from 0.03 to 0.1. The thickness of the perforate ranged from 1 to 3 mm and the hole diameter from 1.75 to 7 mm. At high sound pressure levels, nonlinear resistance due to oscillatory flow separation at the entrance to the resonator also have to be accounted for. The corresponding equivalent normalized resistance is ≈|u|/c, where |u| is the oscillator velocity amplitude at the entrance.
9.4.1 Transmission Loss As before, we refer to Section 7.3.2 for a discussion of TL and TL0. Figure 9.7 shows the computed transmission loss, TL, of a side-branch tube vs Ls /λ, where Ls , the acoustical length, is the sum of the physical length and an end correction approximately equal to the radius of the tube. Results for different area ratios As /A from 0.25 to 2 are shown. We note that unless the area of the side-branch pipe is relatively large, a substantial transmission loss is obtained only in a narrow frequency range and for values As /A < 1, the band width generally is too small to be of interest in practice. On the other hand, with the pipe area equal to twice the duct area, the transmission loss will exceed 10 dB over a frequency band, which is about 40 percent of the first resonance frequency. With the assumption of no losses in the pipe, the transmission loss theoretically goes to infinity when the pipe length Ls is an odd number of quarter wavelengths but in reality, the visco-thermal losses in the pipe or in a screen or the nonlinear resistance will limit the insertion loss. In the presence of flow in the duct, there is also a flow induced contribution to the loss, as already mentioned (see also Chapter 4). The addition of even a relatively course wire mesh screen across the opening of the resonator provides a larger input resistance than that due to the visco-thermal losses on the walls of the pipe. For example, the normalized resistance of a screen with a wire diameter of 0.0045 inches and 100 mesh is θ ≈ 0.02. This should be√compared with the input resistance of a pipe, 2π(L/λ)(dvh /d), where dvh ≈ 0.31/ f cm, f , the frequency in Hz, and d, the pipe diameter. Thus, with d = 10 cm, and L/λ = 1/4 with the first quarter wavelength resonance chosen to be 100 Hz, the normalized input resistance corresponding to visco-thermal losses is only ≈ 0.005.
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Figure 9.7: Left: The computed transmission loss TL of a side-branch pipe vs Ls /λ, where Ls is the acoustical length of the resonator (see Figure 9.6). Right: The effect of a damping screen over the opening of the resonator. Normalized resistances: 0, 0.2, 0.4, 0.8. The effect of the resistance of a screen for values of the normalized resistance from 0.2 to 0.8 is also shown in the figure. Apparently, there is not much advantage in this case of adding a resistive screen across the opening. In the absence of a screen and losses in the pipe, no power is absorbed by the resonator and TL0 = 0.
9.4.2 Insertion Loss As for the duct elements discussed earlier in this chapter, the insertion loss (IL) generally is of more direct practical interest than the transmission loss. It depends not only on the characteristics of the resonator but also on its location and on the impedances of the source and termination of the main duct. Unlike the transmission loss, the insertion loss can be positive or negative. It is imperative, therefore, that a careful study of the insertion loss be undertaken in each particular case in order to determine whether the insertion of a side-branch resonator can be expected to be an effective noise control measure. Before a full scale installation is implemented, model experiments should be considered. The termination impedance ζt in Figure 9.6 is assumed to be that of an open-ended duct in free field. If both the source and termination are reflection-free, the insertion loss becomes equal to the transmission loss, TL, and we refer to the results already presented. For the source impedance ζs , we consider the two special cases of high and low values. High Impedance Source As a next step, let us consider the insertion loss when only the duct termination is reflection-free, i.e., ζ2 = 1 and the source impedance is high, in this case ζi = θi = 10. We choose a resonator tube length of 3 ft, which corresponds to a quarter wavelength resonance frequency of ≈ 93 Hz. In this case, with a high internal impedance of the source, the strategy in choosing a location for the resonator is to make the load impedance at the source as small as possible to provide as large an impedance mis-
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Figure 9.8: Insertion loss of side-branch tube in a duct (see Figure 9.6). Tube: Length, L = 3
ft (res. freq. ≈ 93 Hz). Diameter, 0.3 ft. Main duct: Diameter 0.3 ft. Length: 10 ft. Location of side branch: L1 = 0, L2 = 10 ft. Source impedance ζi = 10. Left: Main duct, reflection-free. Right: Main duct, open ended.
match as possible. An obvious location is as close to the source as possible. With this location and a total length of the main duct of 10 ft and a duct diameter of 0.3 ft (the same as the diameter of the resonator tube), we obtain the insertion loss shown in Figure 9.8, on the left, for a reflection-free termination (impedance = 1), and on the right, for an open ended main duct. If the resonator is placed half a wavelength from the source, the impedance at the source will be approximately the same as at the resonator, i.e., very low at the resonance frequency. Thus, we expect a relatively good performance of the resonator and this is confirmed by computations. The result is not much different from that in Figure 9.8 except for small regions with negative insertion loss just below and above the resonance frequencies. On the other hand, if the resonator is placed a quarter wavelength from the source at resonance, the impedance at the source will be high, and the insertion of the resonator is then not expected to yield significant insertion loss in comparison with those above. This, again, is confirmed by the computations. The situation is quite different when reflections from an open ended main duct are accounted for. The reflection coefficient at low frequencies is then ≈−1. The corresponding insertion loss curve is shown on the right in Figure 9.8. As expected, there is a substantial insertion loss in the vicinity of the quarter wavelength resonance at 93 Hz. However, the overall frequency dependence is quite irregular since we now have reflections from both the resonator and the duct termination. It should also be borne in mind that the insertion loss is based on the empty duct as a reference so that the frequency response characteristics of it enter into the picture. For example, the positive insertion loss peak at ≈ 28 Hz in the figure to the right, with the resonator placed close to the source, is largely due to the frequency response of the bare duct. The 10 ft bare duct is a quarter wavelength long at a frequency of 28 Hz. At this frequency, the input impedance of the bare duct will be high and well matched to the high impedance source used here. Consequently, the acoustic power generated in the bare tube will be high at this frequency, and anything that will interfere with this impedance match will produce a reduction in the radiated
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power, and hence a positive insertion loss. The insertion of the resonator provides such an interference, even though 28 Hz is not a characteristic frequency of the resonator. At a still lower frequency, the stiffness reactance of the resonator combines with the mass reactance of the main pipe in parallel, which tends to produce an anti-resonance, a corresponding high impedance, and a negative insertion loss. With the side-branch resonator placed at the end of the duct, it has practically no effect since this location represents a pressure node. Example: A Labyrinth Resonator Assembly Although the insertion loss at the resonance frequency of the resonator can be quite high, it extends only over a narrow frequency range. However, with several ‘overlapping’ resonators with different resonance frequencies used in parallel, the range can be improved. The high impedance source used here mimics the situation in an air induction system in an automobile engine. The length of the air inlet tube is typically about 2 ft, and it becomes a quarter wavelength at a frequency of about 140 Hz, which corresponds to the firing frequency at ≈ 4200 rpm in a 4 cylinder, 4 cycle engine. The equivalent source impedance at the throttle body of the engine is relatively high so that the pipe (unfortunately) provides maximum amplification of the sound at that frequency. The insertion of a side-branch resonator at the throttle body with the same length as the air duct then results in a substantial insertion loss at the quarter wavelength frequency of 140 Hz. To cover a broad range of frequencies, corresponding to 2000 to 5000 rpm in the engine mentioned (≈ 67 to 167 Hz in a 4 cylinder, 4 cycle engine), several resonators of different lengths are called for. The practical problem is that the resonator tubes required to cover this frequency range are too long to be placed under the hood of a car in a conventional manner. However, it can be achieved by means of a ‘labyrinth’ resonator assembly containing four or five folded resonator tubes with their openings placed close to the throttle body of the engine. The resonators are packaged in a flat or curved panel, which can be placed in contact with or made an integral part of the hood. An example of such an arrangement is shown in Figure 9.9. An alternate (and better) approach is discussed in connection with Figure 8.8. Low Impedance Source As before, the strategy in optimizing the performance of a side-branch resonator is to place it at a location to provide as much impedance mismatch at the source as possible. With a low source impedance, the input impedance of the duct at the source should be as high as possible, which is achieved if the resonator is placed an odd number of quarter wavelengths (at resonance) from the source. In this example, with a resonator length of 3 ft, the optimum distance from the source is a quarter wavelength at the resonance frequency, i.e., 3 ft. The resulting duct impedance at the source is then high, providing optimum mismatch with the source impedance (in this case 0). The insertion loss obtained is shown on the left in Figure 9.9.
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NOISE REDUCTION ANALYSIS LABYRINTH SILENCER PANEL TOP REMOVED TO REVEAL CHANNELS AND AIR INLET
CHANNELS
CHANNELS
AIR INLET
AIR FILTER
FILTER BOX
Figure 9.9: An air inlet assembly for an automobile engine with five ‘labyrinth’ resonators packaged in such a manner that it will fit under or be made an integral part of the hood of the car.
The left curve in this figure refers to a reflection-free duct termination and the right to the standard conditions at the end of an open pipe for which the reflection coefficient at low frequencies is ≈−1. Both curves are not much different from those obtained under optimum conditions for the high impedance source in Figure 9.8 but it is important to note that the optimum locations of the side branch are quite different in the two cases. A Warning The main duct is often far from reflection-free at the low frequencies where resonators might be used, and the insertion loss curves will have considerable fluctuations. Therefore, resonators and other reactive elements should not be used indiscriminately without a careful study of the effect of source impedance and the location of the resonators, factors which are often overlooked. Disappointing performance is often the result. As we have seen in the examples above, the resonators may work very well in a chosen frequency range under some conditions, but not at all, under others.
9.5 PERFORATED PLATE 9.5.1 Effect of Mean Flow On the Acoustic Resistance The steady flow resistance of porous materials generally is a nonlinear function of the flow velocity. This is due to flow separation and turbulence, and the effect is particularly pronounced in a perforated plate. Due to the nonlinear relation between pressure drop and flow velocity, P ∝ U 2 ∝ (U0 + u)2 , a superimposed oscillatory flow u, as in a sound wave, will make the corresponding oscillatory pressure drop
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Figure 9.10: Insertion loss of a side-branch resonator. Length: 3 ft (res. freq. ≈ 93 Hz). Diameter: 0.3 ft. Main duct: Diameter: 0.3 ft. Distance to source: 3 ft (a quarter wavelength at resonance). Distance to the end of main duct: 7 ft. Left: Reflection-free termination of duct. Right: Open ended duct. Low impedance source (constant pressure, zero impedance).
contain a contribution ∝ U0 u proportional to the mean flow velocity. In other words, the mean flow, in effect, provides the perforated plate with an acoustic resistance, which in most cases far exceeds the viscous contribution. The steady flow through an orifice plate or in a duct with sudden changes in cross section is discussed in most texts on fluid flow. Because of flow separation and turbulence, the problem is difficult to analyze in all its details from first principles, and empirical coefficients usually have to be introduced to express the relation between pressure loss and flow velocity. For isentropic flow through an orifice plate in a uniform duct, the pressure falls to a minimum at the location of maximum velocity in the orifice but then completely recovers to its upstream value sufficiently far from the orifice on the downstream side. However, due to flow separation and turbulence, this recovery is not complete and the pressure loss will be proportional to U02 /2, where U0 is the velocity in the orifice. The constant of proportionality depends on the open area fraction of the orifice plate, i.e., the ratio of the total orifice area and the duct area. When this fraction is sufficiently small, the constant is approximately 1. For the purpose of the present discussion, the pressure loss is expressed simply as P ≈
ρU02 (1 − s)2 , 2
(9.1)
where s is the open area fraction, the ratio of the orifice area, and the duct area. We now treat a superimposed sound wave as a quasi-static modulation δU0 in the velocity and a corresponding variation p in the pressure loss, p ≈ ρU0 δU0 (1 − s)2 .
(9.2)
The corresponding acoustic resistance of the orifice is then r0 ≈ p/δU0 = ρU0 (1 − s)2 .
(9.3)
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If we define the acoustic resistance based on the velocity variation δU = sδU0 in the duct (i.e., averaged over the orifice plate), the corresponding resistance becomes 1/s times the value in Eq. 9.3, with the corresponding normalized value given by (1 − s)2 (9.4) M0 , s where M0 = U0 /c is the Mach number in the orifice. It is illuminating to approach this problem from another point of view, considering the increase in the average energy loss caused by a superimposed harmonic acoustic perturbation in the velocity U0 . Thus, we start from the kinetic energy flux in the (jet) flow from the orifice, W0 = ρU03 /2. With a superimposed acoustic velocity perturbation u(t), the corresponding flux is W = ρ[U0 + u(t)]3 /2. Expanding the brackets, we get U03 + 3U02 u(t) + 3U0 u(t)2 + u(t)3 . With the time average of the acoustic perturbation being zero, u(t) = 0, the increase in the time average flux caused by the perturbation is W − W0 = (3/2)U0 u(t)2 . In terms of the acoustically induced increase in the flux, the corresponding acoustic resistance would be obtained from ru(t)2 = W − W0 . This yields an acoustic resistance (3/2)ρU0 , i.e., 50 percent higher than before. This apparent paradox is resolved when we realize that the acoustic perturbation increases the static pressure drop required to maintain the average flow.5 In the case of harmonic time dependence, the product of this static pressure increase and U0 will account for the 50 percent increase in power. In other words, the source of the mean flow has to supply this additional power to maintain the flow velocity U0 when a sound wave is present. The remaining contribution is the power drawn from the sound wave, and this leads to the same value for the acoustic resistance as before. As an example, we consider here the interaction of a sound field with a perforated plate in a tube with mean flow; in particular, we are interested in the reflection, transmission, and absorption coefficients, R, τ , and α. This analysis was inspired by an inquiry about the means of quenching an instability in a combustion chamber in which the axial acoustic modes in a pipe were involved. The use of a perforated plate as a damper turned out to be a feasible approach in eliminating the instability. We shall make use of our study of the interaction of a sound wave with a resistive sheet. Thus, with reference to it, we have θ ≈ r/ρc =
R = ζ /(2 + ζ ),
τ = 2/(2 + ζ )
α = 1 − |R|2 − |τ |2 .
(9.5)
In this case, the impedance ζ should be that of a perforated plate. A semi-empirical expression for this impedance is ζ = θp + iχp
1 θp ≈ s 16kx 2 [1 + 8x(1+x 3) ] 10x χp ≈ −(k/s)[4/3 − (1/3) 1+10x ]
= (1 − s)(8/3π)d ≈ + 0.85(1 − s)d,
(9.6)
5 A harmonic perturbation in velocity leads to a nonharmonic perturbation of the pressure drop with a
mean value different from zero.
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where k = ω/c, , √the plate thickness, d, the hole diameter, s the open area fraction, x = dv /d, dv = 2ν/ω, the viscous boundary layer thickness, and ν, the kinematic viscosity. In the presence of a mean flow through the plate, we have to add to this impedance the normalized resistance (1−s)2 M0 /s due to the sound-flow interaction, as described above. For grazing flow, as in a duct lined with a perforated plate with partitioned air backing, forming a resonator type liner, we refer to the discussion in Section 9.4. It is assumed that the orifice diameter d and the plate thickness are much smaller than a wavelength. In this example, = d = 0.2 inches and s = 0.2. The computed coefficients of power reflection |R|2 , transmission |τ |2 , and absorption, α = 1−|R|2 − |τ |2 are shown in Figure 9.11 as a function of frequency for flow Mach numbers 0 and 0.4 in the orifice. The mass reactance of the orifice will dominate at high frequencies and the reflection coefficient then approaches 1. With no flow, the absorption is due solely to viscosity, and, as can be seen, is less than 2 to 3 percent over the entire frequency range. The absorption coefficient increases with increasing Mach number to 0.5 at M0 = 0.4. This turns out to be close to the maximum in this particular case; for values of M0 > 0.6, the absorption decreases with increasing M0 . Nonlinear Reflection, Transmission, Absorption We consider again the perforated plate in the previous section, this time focusing on the effect of the acoustic nonlinear resistance of the plate on the reflection, transmission, and absorption of an incident sound wave. A more general liner is one in which the perforated plate is backed by a porous layer. The induced motion influences the velocity amplitude in the orifices of the plate; the velocity amplitude relative to the plate is given by 2δ ζs u0 , (9.7) =
c ζ0 + 2s ζ0 + ζs
Figure 9.11: The power absorption (A), reflection (R), and transmission coefficient (T) of a perforated plate for a plane wave at normal incidence with superimposed mean flow. Plate thickness, 0.2 inches, hole diameter, 0.2 inches, open area fraction, 20 percent. Left: No flow. Right: Mach number in orifice, M = 0.4. In each case, the reflection coefficient is close to 1 at 10,000 Hz.
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where δ = |pi |/γ P , ζ0 = ζ0 + (1 − s)2 |u0 |/c, ζs = s(−iωm/ρc), γ = 1.4 (specific heat ratio), and ζ0 = ζ0 ζs /(ζs + ζ0 ). The quantity ζ0 is the linear orifice impedance, which is sζ , where ζ is given in Eq. 9.6. The pressure magnitude of the incident wave is |pi | and P is the static pressure. Having solved this equation (numerically) for |u0 |/c, we obtain the reflection and transmission coefficients R and T for pressure and the coefficient α of sound absorption within the plate from the relations R = 2(ζ0 + s)/(ζ0 + 2s) T =1−R α = 1 − |R|2 − |T |2 .
(9.8)
The coefficients for acoustic power reflection and transmission are |R|2 and |τ |2 . For the perforated plate in Figure 9.12, the nonlinearity does not significantly affect the coefficients for incident pressure levels less than ≈ 120 dB. Results obtained for levels of 100 dB (essentially linear regime) and 160 dB are shown in the figure. The absorption coefficient in Figure 9.12 at 160 dB remains constant, essentially independent of frequency at low frequencies. This is due to the fact that the perforated plate was assumed to be rigid. In reality, it is flexible and will be induced to move by the sound, particularly for small open areas and a sufficiently light plate.
9.5.2 Shock Wave Interaction With an Orifice Plate In addition to studies of the interaction of shock waves with porous materials, we considered also the reflection from and the insertion loss of a porous layer inserted as a centered partition in a circular tube and also of an orifice plate used as a termination of the tube. In both instances, the insertion loss was found to decrease with increasing amplitude of the incident wave. Most likely this reduction has to do with the noise generated by the jet flow caused by the incident pulse, which prevents the sound to reach its normally low level thus reducing the insertion loss. With the orifice plate at the end of the tube, the amplitude of the reflected wave is expected to be between what is obtained for a rigid wall and an open end termination,
Figure 9.12: Power reflection (R), transmission (T), and absorption coefficients (A). Orifice plate: Thickness = hole diameter = 0.2 inches. Open area = 20 percent. Right. 160 dB.
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Incident pulse
Pressure, atm Experiment 1 Theory
Time 25 ms
Figure 9.13: Shock wave reflection from an orifice plate with an open area fraction of 0.6. The pressure is measured in the shock tube 1 m from the membrane in the driver section, peak pressure 0.94 atm (≈ 193 dB). Compare the reflections from closed and open ends.
reflection coefficients being +1 and ≈−1, respectively. Measurements were made with orifice plates with open area fractions s = 0.22, 0.42, 0.60 and 0.79. Indeed, the positive portion of the reflected wave dominated the negative portion for small open areas but for S = 0.79, the roles were reversed. At s = 0.6, the positive and negative portions were about the same and the reflected pulse amplitude was at a minimum with a reflection coefficient of ≈ 0.1 (Figure 9.13). This referred to an incident pulse amplitude at the plate of 0.73 atm (0.94 atm at the microphone position 1 m from the plate) and a pressure in the driver section of the shock tube of 3.7 atm.
9.6 ATTENUATION IN TURBULENT FLOW IN DUCTS A mathematical supplement to this section is given Section 10.7. We have already discussed the flow effects of convection and refraction on sound attenuation in a lined duct; refraction has a significant effect at high frequencies and convection is dominant at low frequencies. This section addresses a different question related to flow, namely the attenuation that might result from the direct interaction of sound with the turbulent duct flow. In turbulent flow, the static pressure drop is approximately proportional to the square of the velocity at sufficiently high Reynolds numbers, and when the flow is modulated by the acoustically driven oscillatory flow, there will be a time dependent perturbation in the pressure drop proportional to the product of the acoustic velocity amplitude and the mean velocity. The product of these perturbations corresponds to an acoustic energy loss. The mechanism is simply that during the half cycle when the fluid velocity in the sound wave adds to the steady velocity there will be a larger increase in friction losses than the decrease during the other half of the cycle, when the velocities are in opposition. This leaves a time average increase in the loss, which results in attenuation. Another attenuation in a hard-walled duct is due to the visco-thermal losses at the walls of the duct, but we shall find that turbulent flow usually causes a higher attenuation, at least at low frequencies. There are other effects, which should be considered in a more detailed analysis than that given here. For example, the spatial variation of mean pressure, density, and
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NOISE REDUCTION ANALYSIS
velocity along the duct leads to a decrease in sound pressure in the flow direction even if the acoustic energy were conserved along the duct. This is due to the pressure loss in the mean flow due to friction. As a result, conservation of mass flux requires the mean flow velocity in the duct to increase with distance as the mean pressure and density decrease. If the duct is long enough, the flow will be choked at the end of the duct. The decrease of sound pressure with altitude in sound propagation in the atmosphere is a similar effect, now resulting from the altitude dependence of density and temperature, and hence of the wave impedance ρc. Since the intensity in a sound wave is p 2 /ρc or u2 ρc, a decrease of the density (or sound speed, or both) requires p2 to decrease and u2 to increase with altitude in order for acoustic energy to be conserved.
9.6.1 Static Pressure Drop See discussion in Chapter 8.
9.6.2 Sound Attenuation As shown in Section 10.7, the attenuation over a travel distance x due to the interaction of a fundamental acoustic mode in a duct with turbulent flow is Attenuation in turbulent duct flow Attenuation ≈ 8.7 ψ
|M| x 1+M D
dB
(9.9)
(M: Mach number, x: Distance, D: Hydraulic diameter) This is valid for sufficiently large Reynolds numbers. The decay in a circular hard duct due to visco-thermal boundary losses at the boundary is known to be exp(−kdvh x/D), √ where dvh = dv + (γ − 1)dh isthe visco-thermal boundary layer thickness, dv = 2μ/ρω, the viscous, and dh = 2K/(ρωCp ) the thermal boundary layer thickness. The quantities μ and K are the coefficients of shear viscosity and heat conduction, Cp , the specific heat per unit mass, and ρ, the density. In air at room √ temperature, the frequency dependence of dvh can be expressed as dvh ≈ 0.31/ f cm, where f is the frequency in Hz. Thus, the decay of the √ amplitude in a distance equal to the diameter D then becomes kdvh ≈ 5.7 × 10−5 f , and the corresponding attenuation in dB is obtained by multiplying by 20 log(e) ≈ 8.7. Visco-thermal attenuation in a duct p(x)/p(0) = exp(−kdvh x/D) √ kdvh ≈ 5.7 × 10−5 f √ Attenuation in dB, air ≈ 49.6 × 10−5 f (x/D) (x: Length, D: Diameter, f : Frequency in Hz)
(9.10)
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To compare this with the flow induced attenuation, we use a typical value for the friction constant ψ ≈ 0.02. We then find that the flow induced attenuation becomes equal to the visco-thermal counterpart at a frequency
f ≈ 1.2 × 10
5
M 1+M
2 .
(9.11)
Below this frequency, the flow induced attenuation dominates. As an example, with M = 0.1, the frequency is ≈ 1200 Hz and the attenuation from each of the contributors is then 0.017 dB for x = D, i.e., negligible in comparison with the attenuation obtained for a lined duct.
9.6.3 A Proposed AeroAcoustic Instability As discussed in the previous section, the pressure gradient in the flow is expressed as dP ρU 2 = −(ψ/D) . dx 2
(9.12)
There is a pressure gradient also in a harmonic traveling sound field, which for a plane wave has the magnitude |
dp | = (2π/λ)|p|, dx
(9.13)
where |p| is the sound pressure amplitude and λ the wavelength. Over half of the temporal period of the wave this gradient will be in opposition to the gradient of the mean flow. This adverse gradient might influence the turbulence in the flow, and hence the sound attenuation. In order for such an effect to occur, the pressure gradient in the sound field must be greater than the gradient of the mean pressure, i.e., |dp/dx| > |dP /dx|. Then if we introduce the expression P = ρc2 /γ for the static pressure and the Mach number M = U/c, it follows from Eqs. 10.128 and 10.129 that |dp/dx| > |dP /dx| corresponds to |p|/P < (γ /4π) ψ(λ/D)M 2 ≈ 0.11(λ/D) ψ M 2 .
(9.14)
As an example, we choose ψ ≈ 0.02 in which case Eq. 10.123 becomes |p|/P <≈ 0.0024(λ/D)M 2 . Then, with M = 0.5 and λ/D = 0.1, we get |p|/P < 6 × 10−5 . This means that if the sound pressure level in this example exceeds ≈ 110 dB it is possible that the nonlinear effect referred to above will occur. This question does not seem to have been investigated, but could be settled with some simple experiments.
9.7 NONLINEAR ATTENUATION Figure 9.14 shows an example of the recorded pressure pattern resulting from multiple reflections of pulse waves in a shock tube when it was terminated by a rigid wall. The distance from the membrane in the driver section of the shock tube to
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NOISE REDUCTION ANALYSIS
2
Pressure, atm.
1 50 Time, ms
100
Figure 9.14: Left: The initial and successively reflected pressure pulses in the shock. The amplitude of the initial pulse (at 1 m from the membrane) is 0.94 atm corresponding to ≈ 193 dB. Right: The corresponding attenuation in dB vs travel distance.
the rigid termination was 2 m and the length of the driver section 0.1 m. The pressure transducer was located a distance of 1 m from the membrane and the absolute peak pressure at that point in this example was 1.94 atm (0.94 atm above ambient) corresponding to ≈ 193 dB. Sixteen reflections are shown in the figure corresponding to a total travel distance of 32 m, and several more could readily be observed. The average pressure in the tube, indicated by the dashed curve, increases slowly toward the asymptotic value P + (Ld /L)P , where Ld is the length of the driver section, L the length of the shock tube (including the driver section), and P the initial over-pressure in the driver section. In our case P was 3.67 atm so that the asymptotic pressure should be 0.17 atm above ambient, which is in good agreement with the observed value. Figure 9.14 shows also the attenuation 20 log[p(1)/p(x)] as a function of the travel distance x (in m). The slope of the attenuation curve in the figure yields the attenuation per unit length, and it is clear that this attenuation decreases with travel distance and hence with decreasing amplitude. Initially, when the amplitude is high, about 0.93 atm above ambient, the attenuation is almost 2 dB/m, which is much higher than the visco-thermal attenuation.
9.8 ON AIR INDUCTION ACOUSTICS Earlier in this chapter we discussed a resonator assembly designed to reduce the air inlet noise of an internal combustion engine (see Figure 9.9). The testing of such an assembly on a 4 cylinder engine revealed not only the anticipated reduction of the air induction tone over the typical speed range of the engine but also an unexpected increase of engine performance at low speeds. It was this latter observation that motivated a more detailed study of the acoustics of air induction. In fact in regard to the resonator assembly, the interest of the engine manufacturer was shifted from noise reduction to a potential increase in engine efficiency resulting from the resonators. Actually, the effect of acoustic resonances in the inlet pipe on engine performance has a relatively long history but is generally not well known and in regard to the understanding of the mechanism and quantitative aspects of the problem, there was
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room for further work. We present here a modification and extension of an analysis by P.M. Morse and others6 and include some numerical results for a specific engine. Our study shows how to calculate the radiated acoustic power and the effect of the acoustical properties of the inlet system on engine efficiency in terms of the engine parameters. Although the numerical results presented here refer to the simple case of a straight inlet pipe, the result can be applied to a more general inlet including, for example, one with side-branch resonators. Furthermore, the exhaust system of an internal combustion engine, including a muffler, can be analyzed in a similar manner, and it is found that the sound field also in the exhaust pipe system can react on the engine to affect its performance. Much of the work on the exhaust has been focused on the role of the muffler. However, since the pressure drop of the catalytic converter in the exhaust normally is many times that of the muffler, the role of the converter should be included in such studies. The exhaust problem will not be pursued here, however.
9.8.1 Sound Pressure and Radiated Power We consider a 4 cycle single cylinder engine with a straight inlet air pipe (‘runner’); it runs from the valve-port and opens at the other to free field, as shown schematically in Figure 9.15. In an engine with several cylinders, the runners from the cylinders terminate in a plenum of a manifold to which is connected an external inlet pipe with its throttle body and air filter. The model used here is approximately valid for each of the cylinder pipes if we assume the reflection from the end of a runner to be the same as if it ends in free field rather than a plenum. The rest of the air inlet system, the throttle body and the external air inlet pipe, is to some extent decoupled acoustically from the runners by the plenum, but is of importance nevertheless from the standpoint of noise emission. The air velocity in the inlet pipe at the valve is denoted by U and is represented by the solid portion of the curve in Figure 9.15. Since we are dealing with a 4-cycle engine, the valve is open only during a quarter of the period during which air is injected. During the rest of the cycle (dashed), i.e., during compression, combustion, and exhaust, the valve is closed. With the origin of the time axis chosen as shown in the figure, the valve is open in the interval −T /8 < t < T /8. During the fundamental period T of the valve motion and the velocity function U (t), the shaft and the piston have gone through two complete periods. Thus, if the fundamental period of the valve motion is T , corresponding to an angular velocity ω = 2π/T , the period of the shaft rotation is T /2 and the angular velocity is 2ω = 2π N/60, where N is the number of revolutions per minute of the shaft. The corresponding frequency is f = N/120. The motion of the piston in the cylinder is periodic and assumed to be harmonic; it has the same frequency 2ω as the shaft. Then, with the displacement amplitude denoted by D, the piston displacement from the central position will be D sin(2ωt)
6 P. M. Morse, R. H. Boden, and Harry Schecter, “Acoustic Vibrations and Internal Combustion Engine
Performance,” Journal of Applied Physics, Vol. 9, 1938.
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NOISE REDUCTION ANALYSIS
X
U
T
t
Figure 9.15: The time dependence of the periodic air velocity U into a cylinder through its valve in a 4-cycle internal combustion engine. The period of this velocity (and the valve) is T . The valve is open only during one quarter of its period; as illustrated, this occurs between −T /8 and T /8.
and the velocity U1 (t) = (2ωD) cos(2ωt), counted positive in the downward direction, i.e., in the negative x-direction in Figure 9.15. The volume displacement amplitude is denoted by σ V , where V is the volume of the cylinder between the central position of the piston and the valve. With the air velocity in the pipe just outside the valve port in the inlet pipe being U (t), as already indicated, we have U ≈ (Ac /Ap )U1 , where Ac and Ap are the areas of the cylinder and the pipe and U1 , the velocity within the cylinder (piston velocity). We have then neglected the effect of compressibility, which will be accounted for shortly. The acoustical role of the inlet pipe is expressed in terms of its impedance as seen from the valve port. It is denoted by ρcζ ≡ ρc(θ + iχ ). At a quarter wavelength resonance of the pipe, the impedance is ρc/θt , where θt is the equivalent termination resistance of the pipe including the effect of flow induced acoustical losses. Typically, θt ≈ 0.05, as will be discussed further below. At an engine speed of 3000 rpm (fundamental valve frequency of 25 Hz) and with U1 = 2ωD cos(2ωt), as given above, the estimate of the pressure amplitude ρcU/θt in the pipe at the valve, is readily found to be of the order of 1 atmosphere for a typical displacement amplitude of the order of 1 inch. If compressibility of the air in the cylinder is accounted for, this amplitude will be reduced. However, in order to stay within the bounds of linear acoustics, we have to keep in mind that the sound pressure amplitude obtained will be well below the static pressure. To obtain the relation between the velocity in the pipe, U , and the velocity of the piston, U1 , accounting for compressibility, we start from the conservation of mass equation ∂ρ/∂t + div (ρu) = 0 and integrate it over the control volume V bounded to surface of the piston and a horizontal plane just above the valve. Then, with (1/ρ)∂ρ/∂t = κ∂p/∂t, where κ = 1/ρc2 is the compressibility and c the sound
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speed, and with the volume integral of div (u) expressed as a surface integral, we obtain ∂p + Ac U1 − U Ap = 0, (9.15) κV ∂t where we have put the control volume equal to the cylinder volume V , as introduced above (thus neglecting the volume of the valve opening and the time dependence of the volume between the piston and the valve, using the average value of that volume). Furthermore, the pressure has been assumed uniform throughout the volume, which is valid as long as the acoustic wavelength is large compared to the linear dimensions of the cylinder. In the continued analysis, we introduce a normalized sound pressure P = p/ρc2 = κP . Then, from Eq. 9.15 it follows that the velocity outside the valve port becomes ∂P ∂P = (Ac /Ap )(2ωD) cos(2ωt) + (V /Ap ) . (9.16) ∂t ∂t We have assumed that the pressure drop across the valve port can be neglected so that the pressure in the pipe just outside the valve port is the same as in the cylinder. This approximation will be discussed later. We expand the pressure in a Fourier series U = (Ac /Ap )U1 + (V /Ap )
P (t) =
∞
Pn cos(nωt − φn ).
(9.17)
0
During the time interval −T /8 < t < T /8, when the valve is open, the expression for U (t) then becomes nωPn sin(nωt − φn ). (9.18) U (t) = (Ac /Ap )(2ωD) cos(2ωt) − (V /Ap ) n
To be able to make use of the known impedance relation between frequency components of U (t) and P (t), we express U (t) in its Fourier components. The Fourier expansion of the function cos(2ωt) for −T /8 < t < T /8 F (t) = (9.19) F (t) = 0 for |t| > T /8 in the first term in Eq. 9.18 is F (t) = Cn =
∞
n=0 Cn cos(nωt) 4 cos(nπ/4) − π(n2 −4) (n = 2, 0)
C0 = 1/2π,
(9.20)
C2 = 1/4
and the expansion of S(t) = sin(nωt − φn ), defined in the same time interval, is +
sin(φn ) S(t) = − sin(nπ/4) nπ q=1 (Sq sin(qωt − φn ) − Tq sin(qωt + φn )
∞
Sq =
sin(q−n)π/4 π(q−n)
Tq =
Sq = 1/4
sin(q+n)π/4 π(q+n)
(q = n).
(q = 0) (n = 0)
(9.21) (9.22)
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NOISE REDUCTION ANALYSIS
The first term is the time average of S(t) (corresponding to q = 0) over the period T . For q = 0, the dominant term in the expansion of S(t) corresponds to q = n and is Sn = 1/4. All other terms will be neglected. Rewriting the term Sn sin(nωt − φn ) in Eq. 9.21 as Sn cos(nωt − φn − π/2), it follows from Eqs. 9.18 to 9.20 that the expression for the total contribution to the nth harmonic component of U (t) is Un (t) = (Ac /Ap )(2ωd)Cn cos(nωt) − (V /Ap )nωPn Sn cos(nωt − φn − π/2) = {Un (nω)e−inωt } Un (nω) = (Ac /Ap )(2ωD)Cn − i(V nω/4Ap )Pn (nω),
(9.23)
where we have introduced the complex amplitude Un (nω) and corresponding complex pressure amplitude Pn (nω) ≡ Pn exp(iφn ). (It should be noted that the complex amplitude of the nth harmonic component of velocity has been denoted simply by U (nω) and the magnitude by Un with an analogous notation for the pressure.) The pressure in the pipe just after the valve is the pressure in the cylinder minus the pressure drop p over the valve. Then, if the input impedance of the pipe is denoted by ζ (ω)ρc, we have ζρc ≡ [p(ω) − p(ω)]/Ux (ω) = p(ω)/Ux − ζv ρc, where ζv ρc is the valve impedance and x refers to the direction indicated in Figure 9.15. Recalling that the velocity U (t) is counted positive in the negative x-direction and that pn (ω) = Pn (ω)ρc2 , we obtain −ζ (ω)ρcUn (nω) = Pn (nω)ρc2 for the calculation of the complex amplitude Pn (nω) of the normalized sound pressure, where ζ = ζ + ζv . With Un (nω) given by Eq. 9.23, it follows that −[(Ac /Ap )(2ωD)Cn − i(κV nω/4Ap )Pn (nω)]ρcζ (nω) = Pn (nω)ρc2 ,
(9.24)
i.e., Pn (nω) ≡ Pn eiφn =
−(Ac /Ap )(2ωD/c)Cn ζ (nω) . 1 − i(κV nωζ (nω)/4cAp )
(9.25)
We rewrite this equation in a somewhat more compact form by introducing the V = Ac , where is the distance from the center position of the piston and the valve. Furthermore, we express the displacement amplitude as a certain fraction σ of , D = σ . Then, with = (Ac /Ap ) and k = ω/c, Eq. 9.25 can be written Pn (nω) ≡ Pn eiφn = −
2σ (k )Cn ζ (nω) . 1 − i(nk )ζ (nω)/4
(9.26)
(Cn : Eq. 9.20, ζ = ζ + ζv , ζ : Eq. 9.27.) The impedance ζv can be treated as an orifice plate impedance. In the vicinity of a quarter wavelength resonance of the pipe, the input impedance of the pipe will dominate and ζv can be neglected. Of this resonance, particularly in the vicinity of a frequency for which the pipe length is an integer number of half wavelengths, and it will be important. For numerical results, we refer to Figures 9.16 and 9.17.
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Figure 9.16: (a): Pressure amplitude (rms, μPascal) in the pipe at the valve, (b) radiated power (μW), and (c) supercharge efficiency contribution of each of the first 10 harmonic components of the valve frequency at an engine speed of 2000 rpm. Pipe length: 4 ft. Cylinder diameter: 4 inches. Pipe diameter: 3 inches. Distance between center position of the cylinder and the valve: 3 inches. Piston amplitude: 2 inches.
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NOISE REDUCTION ANALYSIS
Figure 9.17: The same set of curves as in Figure 9.16 except that the engine speed is now 2800 rpm.
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9.8.2 Pipe Impedance In the presence of mean flow, the major acoustic losses in the pipe generally occur as a result of the interaction of sound with the separated flow at discontinuities. For example, the discharge of a flow from the end of a pipe at a mean flow Mach number M is equivalent to an acoustic resistance at the discharge end with the normalized value approximately equal to M. Often the Mach number is in the range 0.05 to 0.2 and the visco-thermal losses at the boundary of a straight pipe are then often negligible. Even in the absence of a mean flow, losses can occur at the ends of a pipe or at discontinuities if the amplitude of the oscillatory flow in the sound field is large enough to cause flow separation. The corresponding equivalent resistance at the end of the pipe then has the normalized value |u|/c, where |u| is the local velocity amplitude in the sound field. In the present case, flow separation is likely to occur at the valve port and at the air inlet end of the pipe. In the absence of losses, the termination impedance ζt of the pipe is simply the radiation impedance ζr of the open end of the pipe in free field. When the losses in the pipe are sufficiently small, they can be accounted for as an additional resistive component of the termination impedance. Thus, with a normalized equivalent impedance at the end of the pipe denoted by ζt , the normalized input impedance of a pipe of length L, as seen from the valve port is ζ ≡ θ + iχ = (ζt − i tan(kL))/(1 − iζt tan(kL)) i.e., θ = (θt (1 + tan2 (kL)))/(1 + θt2 tan2 (kL)) χ = −((1 − θt 2 tan(kL))/(1 + θt 2 tan2 (kL)),
(9.27)
where k = ω/c = 2π/λ. This result follows directly from the transmission matrix of a pipe and the expression for the impedance. As the frequency increases, the resistance goes from θ = θt to a maximum value θmax = 1/θt when kL = (2n − 1)π/2, i.e., when the pipe length is an odd number of quarter wavelengths. This corresponds to the resonances of a pipe with one end open and the other closed. For kL << 1, the reactance starts out being mass-like (χ < 0), corresponding to the mass of the air in the pipe (oscillating at approximately the same amplitude throughout the pipe) and remains mass-like until kL = π/2 when it changes sign and becomes stiffness-like. The maximum magnitude |χ |max = (1 − θt2 )/2θt ≈ θmax /2 occurs when tan(kL) = 1/θt . Similar behavior applies in the vicinity of kL = (2n − 1)π/2, where n is a positive integer. The velocity amplitude in the pipe is then nonuniform with the maximum value at the end of the pipe being larger than the amplitude at the valve port by a factor 1/θt at the resonances, kL = (2n − 1)π/2. This factor generally is larger than 10. As an approximate expression for the radiation into free space we use 2 /4
ζr ≡ θr + iχr ≈ 1 − e(ka) where a is the pipe radius.
−i
0.61ka , 1 + (ka)2
(9.28)
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NOISE REDUCTION ANALYSIS
To account for the distributed losses along the pipe due to visco-thermal effects or the interaction of sound with turbulent flow in the pipe we use k = kr + iki , where kr ≈ ω/c and ki accounts for the losses per unit length. In the cases of interest we have ki L << 1, and the visco-thermal losses can then be accounted for by an additional termination resistance ki L. To study the role of a side-branch resonator or any other element in the inlet pipe on radiated power or engine efficiency, we merely have to determine the corresponding change in the input impedance of the inlet pipe. If a side-branch resonator is located at the beginning of the pipe, the total input impedance of the pipe is the parallel combination of the pipe impedance in Eq. 9.27 and the side-branch impedance.
9.8.3 Radiated Power If there are no losses in the pipe, the radiated power by a harmonic component can be calculated from Wn = (1/2)Un2 ρcθ (πa 2 )where Un = Pn /(|ζ |ρc),
(9.29)
where θ is the normalized input resistance of the pipe and Un , the velocity amplitude in the pipe at the valve. In the presence of losses, however, we first determine the velocity amplitude at the end of the pipe and then use Eq. 9.29 with Un replaced by the velocity amplitude U2n at the end and θ by the radiation resistance θr . It follows from the transmission matrix of a pipe that U2n = Un /| cos(kL) − iζr sin(kL)|.
(9.30)
For numerical results, we refer to Figures 9.16 and 9.17.
9.8.4 Acoustic ‘Supercharge’ The basic reason why the sound field in the pipe might affect the average gas flow into the cylinder is that the valve is open only during a portion of the period so that the average effect over one period may not necessarily be zero. If a reflected sound wave at the valve represents a compression when the valve is open, it will increase the mass flow into the cylinder; if it is negative, the opposite will be true. The primary outgoing sound wave from the air pulse induced into the cylinder by the piston will be a rarefaction wave. When this wave is reflected from the open end of the pipe, its sign will change since the reflection coefficient (at long wavelengths) is close to −1. Thus, the wave will return to the valve as a compression after the roundtrip time 2L/c, where L is the length of the pipe. If this occurs when the valve is open, a supercharging effect will take place. If the valve is closed at that time, the wave is reflected from the valve as a compression, but the next time it returns to the valve it will be a rarefaction after the sign reversal at the end of the pipe. As an example, consider the nth harmonic of the fundamental frequency of the valve motion. The wavelength is then λn = λ/n. If L = λn /4 = λ/4n the roundtrip time in the pipe is 2L/c = λ/2nc = T /2n. For the fundamental, n = 1, with a
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roundtrip time of T /2, a positive portion of the wave will return when the valve is closed. Thus, although the pressure amplitude at the valve will be large at the quarter wavelength resonance, the supercharging effect from the reflected wave will not be positive. In the same manner, one can proceed to determine qualitatively the effect of multiple reflections. In the analytical treatment of the problem, we determine the volume of air that flows through the inlet valve per cycle, by integrating the velocity U (t) in Eq. 9.16 over the time interval −T /8 < t < T /8 when the valve is open to get
1 U dt = 2σ V 1 + [P (T /8) − P (−T /8)] , 2σ −T /8
Q = Ap
T /8
(9.31)
where we have used Ac D ≡ σ V . The change in Q due to the effect of the sound pressure is expressed by the factor E = Q/Q0 = 1 +
1 [P (T /8) − P (−T /8)], 2σ
(9.32)
where Q0 is the value without the effect of sound. We shall term this the acoustic supercharging efficiency or loosely, the engine efficiency. As we have seen, the pressure in the pipe is caused by the periodic flow through the valve, and it will be periodic also with a frequency spectrum that depends on the Fourier spectrum of U (t) and the frequency response of the pipe. If we express P in terms of the Fourier series in Eq. 9.17 and note that ωT /8 = π/4, cos(nωt − φn ) = cos(nωt) cos φn + sin(nωt) sin φn , and that cos(nωt) − cos[nω(−t)] = 0, we can rewrite E as ∞ 1 Pn sin(nπ/4) sin(φn ). (9.33) E =1+ σ 1
where Pn , φn . It should be noted that since Cn ∝ cos(nπ/4), the contribution of the nth harmonic of the pressure to the efficiency factor E will be proportional to sin(nπ/4) cos(nπ/4) ∝ sin(nπ/2). Thus, the contribution from n = 4 will be zero. This is to be expected, since the fourth harmonic has a complete period in the time interval −T /8 < t < T /8 when the valve is open, yielding a zero time average value of the velocity over this interval. Numerical results will be discussed in the following example.
9.8.5 Numerical Example In Figures 9.16 and 9.17 are shown the calculated pressure amplitudes, radiated power, and engine efficiency of the first ten harmonic components in the respective series expansions; two engine speeds, 2000 and 2800 rpm have been considered, as indicated. The corresponding fundamental frequencies of the valve motion are 16.7 and 23.3 Hz. In this particular case, the diameters of the cylinder and the pipe are 4 inches and 3 inches, respectively. The distance between the center position of the cylinder and the valve is = 3 inches and the amplitude of oscillation is D = 2 inches,
330
NOISE REDUCTION ANALYSIS
corresponding to σ = 2/3. As discussed earlier, an equivalent termination resistance at the end of the tube was added to account for flow induced losses. It was chosen to be θt = 0.05, corresponding to a Q-value of the pipe of about 30. This resistance is dominant at low frequencies, nka << 1. The length of the pipe is 4 ft, and, at 2000 rpm, it is close to a quarter wavelength at the fourth harmonic component, which is 4.19 ft. Actually, due to the mass reactive part of the radiation impedance at the end of the pipe, its equivalent acoustic length is somewhat larger than 4 ft so that the fourth harmonic is quite close to the resonance. As can be seen in Figure 9.16, the pressure amplitude of this harmonic component is indeed the largest, even though the amplitude of the velocity is 2.9 times smaller than that of the fundamental. The peak sound pressure level is 175.6 dB. It should be noted that the maximum radiated power does not occur at the fourth harmonic, close to the quarter wavelength resonance, but at the third harmonic. The peak power level (re 10−12 watt) is ≈ 99.5 dB. As indicated in our qualitative discussion of the acoustic supercharge effect, we noted that it will be positive when the pipe length is less than λ/8. At a pipe length equal to a quarter wavelength, the effect is generally negative, except in this special case when it occurs at a harmonic, which is a multiple of four; the effect is then zero, as explained earlier. The total acoustic supercharge in this case is ≈ 3.8 percent. The corresponding results at an engine speed of 2800 rpm are shown in Figure 9.17. Now, the third harmonic is the one closest to quarter wavelength resonance of the 4 ft pipe, and the corresponding pressure peak is quite pronounced. Actually, due to the end correction mentioned above, a speed 2700 would produce a better match with the resonance. For this reason, the peak sound pressure is about the same as at 2000 Hz, ≈ 176.3 dB. As was the case at 2000 rpm, the maximum power does not occur at the quarter wavelength resonance but at a lower harmonic. The supercharge effect at the quarter wavelength resonance shows the normal behavior of being negative. In fact, in this case this negative contribution dominates so that the total effect is negative, ≈−5.6 percent. By introducing side-branch resonators in the pipe, like the labyrinth assembly referred to above, the frequency dependence of the input impedance of the pipe system and the corresponding supercharge effect can be altered considerably, as was noted in the experiment with the labyrinth resonator assembly, mentioned above. It should be pointed out that at a level of about 175 dB, nonlinear effects, particularly in the valve opening, no doubt will play a role and will act to reduce the sound pressure. This has not been accounted for.
Chapter 10
Mathematical Supplements and Comments As already mentioned in the introduction, this chapter should be of interest to those readers who wish to pursue the computational aspects of the subject of Duct Acoustics. It assumes that the reader is familiar with the basics of the theory of sound including the use of complex amplitudes as presented, for example, in some books on Acoustics.
10.1 SUPPLEMENT TO SECTION 8.1 10.1.1 High Frequency Attenuation of Fundamental Mode in Lined Duct, Average Compressibility We start from the mass conservation equation ∂ρ/∂t + ρ0 div u = 0. By introducing the compressibility of air, κ = (1/p)ρ/ρ = 1/ρc2 , this equation can be expressed as κ∂p/∂t + div u = 0.
(10.1)
The air channel in the duct has a width D1 , and it is lined on one side where the normalized admittance is η. (The admittance or the corresponding impedance is calculated and discussed for several porous layers.) Integrate Eq. 10.1 over a volume element D1 dx of length dx and unit height. We consider harmonic time dependence with angular frequency ω so that ∂/∂t → −iω. Then, if the average pressure in this volume is denoted by pa , the first term in Eq. 10.1 becomes (−iωκpa )D1 dx. The volume integral of the second term is converted into a surface integral of the normal velocity u over all the surfaces of the volume element. The contribution from the surfaces normal to the x-axis is D1 [u(x + dx) − u(x)] and the contribution from the lined surface is (η/ρc)pa dx, where we have expressed the normal velocity into the liner as (η/ρc)pa and approximated the pressure at the surface by the average pressure. Eq. 10.1 then reduces to (−iωκD1 + η/ρc)pa + ∂u/∂x = 0. 331
(10.2)
332
NOISE REDUCTION ANALYSIS
This has the form a the one-dimensional equation for mass conservation with a complex compressibility κ˜ (−iωκ)p ˜ a + ∂u/∂x = 0,
(10.3)
κ˜ = κ(1 + iη/ρcκD1 ) = κ(1 + iη/kD1 ),
(10.4)
where where k = ω/c. The imaginary part of κ˜ and hence the attenuation goes to zero with increasing frequency, which was to be shown.
10.2 SUPPLEMENT TO SECTION 8.2 10.2.1 Locally Reacting Liners The bulk of the numerical results in Section 8.2 are the basic attenuation spectra referring to a rectangular duct with one side lined (and equivalent duct configurations). The analysis presented here is a little more general as it deals with a duct with two opposite walls with different liners. The width of the air channel between the liners is D, and the surfaces of the liners are at y = 0 and y = D, as shown in Figure 10.1. The two remaining walls, at z = 0 and z = D1 (the z-direction is perpendicular to the paper plane), are assumed to be rigid and impervious. The liners are locally reacting, as indicated by the partitions represented by the dashed lines, and the corresponding normalized input admittances η1 and η2 are then independent of the distribution of sound pressure along the duct and can be regarded as known a priori. The problem at hand is to determine the wave field in the duct, subject to the boundary conditions at the walls. We shall be mainly interested in the fundamental acoustic mode and its attenuation. Harmonic time dependence is assumed, and, as before, the time factor exp(−iωt) is used in the definition of complex amplitudes. The solution to the wave equation for the sound pressure amplitude is composed of wave functions representing ‘standing’ waves in the transverse directions y and z, and a traveling wave in the positive x-direction. This is sufficient for the calculation of the attenuation. When it comes to an analysis of the transmission loss, insertion loss, and noise reduction, reflections from the end of the duct and from the sound source
Figure 10.1: Rectangular duct with two opposite walls lined with a locally reacting material with the normalized admittance η1 and η2 , respectively.
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MATHEMATICAL SUPPLEMENTS AND COMMENTS
have to be accounted for and a traveling wave also in the negative x-direction must be included. We let the two boundaries perpendicular to the z-axis be rigid. Later, these boundaries will be lined also. Then, considering only a traveling wave in the positive x-direction, the complex amplitude of the sound pressure in the channel between the two liners has the form p(ω) = [A cos(ky y) + B sin(ky y)] cos(kz z)eikx = A[cos(ky y) + R sin(ky y)] cos(kz z)eikx x .
(10.5)
With ∂/∂t → −iω, the corresponding velocity components are obtained from −iωρuy = −dp/dy and −iωρuz = −∂p/∂z, k
y uy (ω) = A iωρ [− sin(ky y) + R cos(ky y)] cos(kz z)eikx x
kz uz (ω) = −A iωρ [cos(ky y) + R sin(ky y)] sin(kz z)eikx x .
(10.6)
The boundary condition uz = 0 at the rigid z-boundaries at z = 0 and z = Dz of uz = 0 leads to the requirement kz Dz = nπ,
(10.7)
where n is an integer. To determine ky , we apply the boundary conditions at the liners. In Eq. 10.6, uy is the amplitude of the velocity in the positive y-direction. The admittance is defined as the ratio of the velocity amplitude in the direction into the boundary and the amplitude of the pressure. Therefore, at y = 0, this velocity is in the negative y-direction, which has to be accounted for, and the boundary conditions become ρcuy /p = (ky / ik)
− sin(ky D)+R cos(ky D) cos(ky D)+R sin(ky D)
ρcuy /p = (ky /ik)R = −η2
= η1
(y = D)
(y = 0).
(10.8)
Elimination of R between these equations yields an equation for ky . Introducing the normalized quantity Ky = ky /k, we find that, after a little algebra, Equation for Ky , Local reaction η1 +η2 Ky tan(Ky kD) = −i 1+(1/K )2 η η y
(10.9)
1 2
where Ky = ky /k. k = ω/c, η1 , η2 : Normalized liner admittances, D: Channel width. See Figure 10.1. In general, this equation has to be solved numerically for the complex quantity Ky in terms of the frequency parameter kD = 2π D/λ. The special case when only one side of the duct is lined is obtained by putting η1 = 0, in which case Ky tan(Ky kD1 ) = −iη2 , (10.10)
334
NOISE REDUCTION ANALYSIS
where we have changed the notation from D to D1 , to be consistent with the notation used in Figure 7.6, top left. Having obtained ky and kz , we find kx from Axial propagation constant kx ≡ kr + iki = k 2 − ky2 − kz2
(10.11)
where ky = kKy : Eq. 10.9, k = ω/c. This is the propagation constant of ultimate interest since it contains the attenuation constant as the imaginary part of kx . The x-dependence of the complex sound pressure amplitude is given by |p(x)|/|p(0)| = exp(−ki x) and the corresponding attenuation in dB at a distance x is 20 log10 |p(0)/p(x)| = 20 log10 (e)ki x ≈ 8.72kxi .
(10.12)
Most of this chapter will be devoted to the fundamental acoustic mode in which the sound pressure is uniform in the z-direction (n = 0) and the lowest order of the solution of Eq. 10.9 for ky . It is shown later that a presence of higher z-modes will not make the overall attenuation much different than for the fundamental mode. This question will be discussed in more detail later. Wave Impedance of the Duct For future reference, we point out that the wave impedance of the fundamental mode in the duct is p/ux , where p and ux are the complex amplitudes of pressure and axial velocity in a traveling wave. With ux =
1 ∂p = pkx /ωρ, iωρ ∂x
(10.13)
the normalized value for the wave impedance and admittance becomes ζw =
1 1 p = = k/kx , ηw ρc ux
(10.14)
where k = ω/c. Since the imaginary part of kx is positive, it follows that the reactive part of the wave impedance is negative, i.e., mass-like. This is due to the wave field having a transverse component of the fluid velocity in addition to the axial so that, for a given axial velocity amplitude, the kinetic energy per unit length, and hence the equivalent mass density, will be increased by the transverse motion. Low Frequency Approximation Although Eq. 10.9 generally has to be solved numerically, we can obtain approximate formulas for Ky at low and high frequencies. The algebra involved is rather tedious, but there are several reasons for carrying out this work. First, the results obtained show explicitly the dependence of the attenuation on the physical parameters involved,
MATHEMATICAL SUPPLEMENTS AND COMMENTS
335
which help to provide some physical insight. Second, and equally important, is the use of these approximations as starting points for the iterative root-finding routines, which are involved in the numerical analysis. At low frequencies, kD << 1 and ky D << 1 and, generally also |η1 |, |η2 | << 1, the left-hand side of the equation can be replaced by (Ky )2 so that Ky2 ≈ −i(η1 + η2 )/kD.
(10.15)
Then, from Eq. 10.11, it follows that Low frequency approximation √ kx ≡ kr + iki ≈ k 1 + i(η1 + η2 )/kD
(10.16)
(For additional information, see Eqs. 10.9 and 10.11.) As a first approximation for the attenuation constant ki we then get ki ≈
ηr , 2D
(10.17)
where ηr = η1r + η2r . To proceed further, we have to know the characteristics of the duct liner, in particular the input admittance or the corresponding impedance. First, consider a liner consisting of a resistive screen backed by an air layer and then a uniform porous liner. The total flow resistance is the same in both cases, normalized value , and the thickness of the layers are the same, d. The normalized input impedance of the sheet liner is ζi ≈ + i
1 , kd
(10.18)
and the corresponding admittance is ηi = 1/ζi . The result for the porous layer is obtained by replacing by /3 and kd by H γ kd, H being the porosity and γ , the specific heat ratio (1.4 for air). For the resistive screen liner, the real part of the admittance is η1r = η2r = (kd)2 /(1 + (kd)2 ). If this is used in Eq. 10.17, the first approximation for ki becomes ki ≈ k
d kd . D 1 + (kd)2
(10.19)
For a given frequency, this has a maximum for = 1/kd, and the corresponding maximum value of ki D is kd (ki D)max ≈ . (10.20) 2 In other words, the corresponding attenuation in a distance D will be 20 log(e) (ki D)max ≈ 8.7kd/2 dB, i.e., directly proportional to kd and hence to d/λ. The same result is obtained for the porous layer with d replaced by H γ d.
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NOISE REDUCTION ANALYSIS
We can improve on this result if we keep the complete expression in Eq. 10.16, which can be written in the form 1/2 √ √ kr /k = (1/ 2) X 2 + Y 2 + X 1/2 √ √ ki /k = (1/ 2) X 2 + Y 2 − X , (10.21) where η1 + η2 = ηr + iηi , X = 1 − ηi /kD, and Y = ηr /kD. The maximum of ki /k is obtained in a straight-forward manner in terms of and kd but the algebra is a bit cumbersome. The results obtained reduce to those given above for small values of d/D. High Frequency Approximation The frequency is now considered to be high enough, kD >> 1, to reduce Eq. 10.9 to tan(ky D) η1 + η 2 1 ≈ −i . ky D η1 η2 kD
(10.22)
The right-hand side goes to zero as kD increases. This means that for the fundamental mode, ky D will lie in the vicinity of π . Thus, if we put ky D ≈ π + (with << 1), we find from Eq. 10.22 that ≈ −iπ/ηkD, where η = η1 η2 /(η1 + η2 ). Thus, inserting ky D ≈ π − iπ/ηkD into Eq. 10.11 (with kz = 0 for the fundamental mode), we find, for kD >> π, (10.23) kx D ≈ kD 1 + 2iπ 2 /η(kD)3 ≈ kD + iπ 2 /η(kD)2 . At sufficiently high frequencies, ω >> r0 /ρ, where r0 is the flow resistance per unit length in the porous layer, the input admittance of a layer is ≈1 so that η = η1 η2 /(η1 + η2 ) ≈ 1/2. Thus, with kx = Kr + iki we get High frequency approximation for attenuation ki D ≈ 2π 2 /(kD)2
(10.24)
(Two walls lined, admittances η1 , η2 D: Channel width. See Eq. 10.22.) In other words, the attenuation ki D is inversely proportional to the square of the frequency and, in this approximation, dependent only on D/λ, and not on the liner resistance. This is related to the frequency dependence of the average compressibility in the air channel, discussed in Section 10.1.1, which shows that the boundary in effect becomes decoupled from the air channel at high frequencies. The attenuation in dB in a distance equal to the width D of the air channel is 20 log(exp(ki D) ≈ 8.72 ki D. The complete frequency dependence of the attenuation requires a numerical solution to Eq. 10.9. The results shown in Figure 8.1 refer to a duct with only one side lined. From these results we note that there is a region at high frequencies where the attenuation decreases faster with frequency than the asymptotic inverse square dependence. This can be understood from the frequency dependence of the admittance.
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MATHEMATICAL SUPPLEMENTS AND COMMENTS
√ ρ/ρ ˜ ≈ s + ir/ωρ and in the high frequency regime, the admittance is shown to be η ≈ 1/ ρ/ρ, ˜ where r is the flow resistance per unit length of the material. Thus, there is a√frequency regime in which ω < r0 /ρ such that η is approximately proportional to ω. This means that ki in Eq. 10.23 will be inversely proportional to the 5/2 power of the frequency before it turns into the inverse second power when kD >> 1. This is consistent with the attenuation spectra in Figure 8.1.
10.2.2 Nonlocally Reacting Liner Duct Lined On One Side The bottom liner in Figure 10.1 is now removed (d1 = 0), and the width of the air channel is denoted by D1 . The partitions within the liner are also removed so that it becomes nonlocally reacting. As we have seen in Section 8.2 there is not much difference in the attenuation spectra for locally and nonlocally reacting liners, and in hindsight this section may seem unnecessary, particularly since it is algebraically much more cumbersome than for the locally reacting liner. It is included nevertheless for the record and for those readers who may want to study the approach used and check the analysis. In constructing the sound field, we note that the sound pressure must have a maximum at the rigid walls of the duct (at y = 0 and y = D1 + d), and the complex amplitudes of the sound pressure in the air space and within the liner then must be given by p(ω) = A cos(ky y)eikx x p (ω)
= B cos(qy
y )eiqx x
(air)
(10.25)
(liner),
(10.26)
where y = D1 + d − y and A and B are constants. For the wave in the air, the components of the propagation constant are kx and ky , kx2 + ky2 = (ω/c)2 . In the liner the corresponding quantities are qx and qy with qx2 + qy2 = q 2 . The propagation constant q in a porous material is ˜ (10.27) q = κ˜ ρ, where κ˜ and ρ˜ are the complex values of the compressibility and the density of the air within the porous material. The corresponding normalized propagation constant is Q = q/k = (κ/κ)( ˜ ρ/ρ). ˜ Since the spatial variation of the fields in the air and in the liner must be the same to satisfy the boundary conditions for all values of x, we must have qx = kx =
(ω/c)2 − ky2 .
The expressions for the y-components of the velocity amplitudes (counted positive in the positive y-direction) in the air and in the porous layer are ikx x uy = (1/ωρ) ∂p ∂y = −A(ky /ωρ) sin(ky y)e
u y = −(1/ωρ) ˜ ∂p ˜ sin(qy y )eiqx x , (10.28) ∂y = B(qy /ωρ) √ where we have made use of ∂/∂y = −∂/∂y. The quantity ρ˜ = ρ s + ir/ωρ is the complex density of air within the porous material, where s is the structure factor and r, the flow resistance per unit thickness of the liner, as mentioned above.
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NOISE REDUCTION ANALYSIS
The boundary conditions at the surface of the liner at y = D1 and y = d are p = p and uy = u y and from these, the following relation obtains (from Eqs. 10.25 to 10.28) Equation for ky , nonlocal reaction ky D1 tan(ky D1 ) = −(ρ/ρ)q ˜ y D1 tan(qy d) ≡ −ikD1 ηi
(10.29)
˜ tan(qy d). Qy = qy /k. where ηi = −iQy (ρ/ρ) This equation for the determination of ky D1 can be brought into the same form as the corresponding equation for the locally reacting liner in Eq. 10.9 if we introduce the normalized input admittance of the liner in the y-direction. It follows from the equation of motion for the porous material that the normalized wave admittance for a wave traveling in the y-direction in the porous material is Qy (ρ/ρ), ˜ where Qy = qy /k. The corresponding input admittance of the liner of thickness d is then ηi = −iQy (ρ/ρ) ˜ tan(qy d). If this expression is used in Eq. 10.9, we obtain Eq. 10.29, which illustrates that the analyses in the two cases are consistent. Since the wave velocity in the x-direction must be the same in the air and in the material, the wave numbers kx and qx must be the same also. Accounting for this equality and making use of the wave equations in the two regions, we have, with k = ω/c, Axial propagation constant, kx = kxr + ikxi kx2 = qx2 = k 2 − ky2 = q 2 − qy2
(10.30)
within the porous material, q = The quantity q is the (total) propagation constant κ˜ ρ˜ with the normalized value Q = q/k = (κ/κ)( ˜ ρ/ρ). ˜ Combining Eqs. 10.29 and 10.30 we can solve (numerically) for ky and then obtain kx from Eq. 10.30. The imaginary part of kx yields the attenuation with the spatial variation of the pressure amplitude expressed by exp(−ki x). Wave Impedance If one wishes to determine the insertion loss of a finite lined duct element, we need an expression for the wave impedance of the fundamental mode in the duct and we proceed to derive an expression for it. We are interested in a plane wave incident on the lined duct element, and use for the wave impedance the ratio of the average sound pressure amplitude across the duct and the average axial velocity amplitude. With the y-dependence of the sound pressure amplitudes in the air and in the liner given by p = A cos(ky y) and p = B cos(qy y), and the open area fraction by σ = D1 /(D1 + d), the average pressure over the total duct area becomes pav = Aσ
sin(ky D1 ) ky D1
+ B(1 − σ )
sin(qy d) qy d
= A cos(ky D1 )[σ T (ky D1 ) + (1 − σ )T (qy d)] T (x) ≡ tan(x)/x,
(10.31)
where we have made use of A cos(ky D1 ) = B cos(qy d), which follows from the boundary condition p = p for y = D1 (y = d).
MATHEMATICAL SUPPLEMENTS AND COMMENTS
339
Returning to Eqs. 10.25 and 10.26 for the pressure amplitudes in the air channel and in the porous material, we obtain the corresponding axial velocity amplitudes
/∂x. The average values of these ˜ from ux = −(1/ωρ)∂p/∂x and ux = −(1/ωρ)∂p quantities are obtained by integrating cos(ky y) and cos(qy d) over the width D1 of the channel and the thickness d of the porous layer, respectively, and dividing by d + D1 . Then, making use of B/A = cos(ky D1 )/ cos(qy d), obtained from the boundary condition at the porous surface, the average axial velocity amplitude is found to be ˜ − σ )T (qy d)], uav = A(kx /ωρ) cos(ky D1 )[σ T (ky D1 ) + (ρ/ρ)(1
(10.32)
where ρ/ρ ˜ is the complex density ratio discussed earlier. For a wave traveling in the positive x-direction, the ratio ζd = (1/ρc)pav /uav
(10.33)
is then the normalized wave impedance. Low Frequency Approximation At sufficiently low frequencies such that the wavelength is large compared to d and D1 , we expect that ky D1 << 1 and qy d << 1 so that tan(ky D1 ) ≈ ky D1 and tan(qy d) ≈ qy d. With these approximations used in Eqs. 10.29 and 10.30, we find, with Kx = kx /k, Ky = ky /k, and Q = q/k, Kx2 ≈
2 ρ/ρ+(d/D ˜ 1 )Q ρ/ρ+d/D ˜ 1
Ky2 = 1 − Kx2 =
(d/D1 )(1−Q2 ) , ρ/ρ+d/D ˜ 1
(10.34)
where Q = q/k is defined in Eq. 10.27. As a check, we note that in the special case d = 0, i.e., when there is no liner in the duct, we get Kx2 = 1, as it should be. In the other limit, D1 = 0, the duct is filled completely with porous material, and in that case we get Kx2 = Q2 , also as it should be. For the purpose of determining the optimum flow resistance for maximum attenuation, to be discussed shortly, we need to express the propagation constant Kx explicitly in terms of the flow resistance. We use the expression for ρ/ρ ˜ given in connection with Eq. 10.30 and put ρ/ρ ˜ ≈ + ir0 /ωρ ≡ (1 + iξ ),
(10.35)
where ξ = r0 / ωρ and r0 is the steady flow resistance per unit length of the absorber. = 1 + Gs + Gv is the total structure factor, where Gs is the structural induced mass factor and Gv ≈ 0.2 is the low frequency value of the contribution from the viscous interaction. Then, with Q2 = (ρ/ρ)(H ˜ κ/κ) ˜ ≈ H γ ρ/ρ, ˜ we obtain from Eq. 10.34, Kx2 ≈ (1 + γ H 2 d/D1 )
1 + iξ . 1 + (H d/ D1 ) + iξ
(10.36)
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NOISE REDUCTION ANALYSIS
In this context we are interested mainly in the imaginary part Ki of Kx , which determines the spatial decay of the sound pressure amplitude along the duct. It is significant that the geometrical parameters enter as the ratio d/D1 in Eq. 10.36, which means that in the low frequency approximation, the attenuation in a duct with a nonlocally reacting porous liner depends only on this ratio for a given value of the flow resistance r0 . This ratio, in turn, determines the open area fraction of the duct, σ = D1 /(D1 + d). For the locally reacting liner, the situation was quite different (compare Eqs. 10.12 and 10.16). With tan(x)/x ≈= 1 and cos(ky D1 ) ≈ 1 in Eqs. 10.31 and 10.32, it follows that the low frequency approximation for the wave impedance in Eq. 10.33 becomes ρ/ρ ˜ , (10.37) σ ρ/ρ ˜ + (1 − σ ) ˜ for σ = 1 and σ = 0. which goes to the correct limits, 1 and ρ/Qρ ˜ = ρ/Hρ, ζd ≈ (1/Kx )
High Frequency Approximation To determine the high frequency approximation for ky D1 in Eq. 10.31, we first look at the frequency dependence of qy d, the argument of the tangent function on the right-hand side of the equation, (10.38) qy = q 2 − kx2 = q 2 − k 2 + ky2 = k Q2 − 1 + (ky D1 )2 /(kD1 )2 . According to Eqs. 10.27 and 10.30 the normalized propagation constant Q within the porous material is such that Q2 − 1 = (H s + Fv /(1 − Fv )(1 + (γ − 1)Fv ) − 1.
(10.39)
The function Fv depends on the ratio of the boundary layer thickness dv and the equivalent pore width 2a in the porous material, and we can write (10.40) a/dv = 3kd/2H = (3πd/λ)/H = 3π(d/D1 )(D1 /λ)/H . If s H ≈ 1, which is normally the case, we get from Eq. 10.39, Q2 − 1 ≈ γ Fv /(1 − Fv ), and if we use the expression for a/dv in Eq. 10.40 and the high frequency approximation for Fv , we obtain (10.41) qy ≈ (γ /2)(2H /kd)1/2 (1 + i). The argument qy d for the tangent function on the right-hand side of Eq. 10.29 increases with increasing frequency and tan(qy d) → i as ω → ∞. The equation then reduces to ky D1 tan(ky D1 ) ≈ −iqy D1 (ρ/ρ). ˜ (10.42) If |qy D1 | >> 1, the argument ky D1 will lie in the vicinity of π/2, and we express the corresponding approximate high frequency solution for ky D1 as
MATHEMATICAL SUPPLEMENTS AND COMMENTS
341
ky D1 ≈ π/2 + . Insertion into Eq. 10.42 yields tan(ky D1 ) ≈ −1/, and hence ≈ −i(π/2)/qy (ρ/ρ)D ˜ 1 H . Thus, ky D1 ≈
π ˜ 1 ). (1 − i/qy (ρ/ρ)D 2
(10.43)
Thecorresponding high frequency approximation for the propagation constant kx = k 2 − ky2 is then given by kx ≈ k + i
(π/2)2 . 3 qy (ρ/ρ)kD ˜ 1
(10.44)
Then, if we use the expression for qy from Eq. 10.41 in Eq. 10.44 and ρ/ρ ˜ ≈ s we get, with s H ≈ 1, 1/4
kx ≈ k + i
d/D1 1 π 2 ( ) . D1 2 (kD1 )7/4 (H )1/4
(10.45)
Thus, under these conditions, the high frequency attenuation is weakly dependent on the liner resistance, ∝−1/4 , increasing somewhat with decreasing flow resistance. For the locally reacting liner it was independent of the resistance to a first approximation. Two Sides Lined Having established that the normalized admittance to a nonlocally reacting porous layer can be expressed in terms of the propagation constant qy in the porous material, as shown in Eq. 10.29, we can use the result in Eq. 10.9 for the locally reacting liners for determination of the propagation constant in the duct merely by expressing the admittance η1 and η2 in this manner. Thus, with the propagation constants in the
layers denoted by q1 and 2 with the corresponding y-components q 1y =
2 , q12 − q1x
we obtain, with q1x = q2x = kx = k 2 − ky2 , an equation for the determination of the y-component ky of the propagation constant in the air channel, and hence the desired kx =
k 2 − ky2 .
10.3 SUPPLEMENT TO SECTION 8.3, OTHER DUCT TYPES 10.3.1 Rectangular Duct Lined On All Sides The duct under consideration is shown in Figure 8.9. The sides of the rectangular air channel are Dy and Dz . The liners are locally reacting and the normalized admittances are ηy1 , ηy2 , ηz1 , and ηz2 , as shown. We place the origin of the coordinate system at the center of the duct with the y and z being vertical and horizontal, respectively. The x-axis runs into the page along the duct axis.
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NOISE REDUCTION ANALYSIS
The pressure distribution in the duct is expressed as p = A(eiky y + Ry e−iky y )(eikz z + Rz e−ikz z ),
(6.1.21)
and the corresponding velocity field has the components uy = (1/ − iρω)(−∂p/∂y) and yz (1/ − iρω)(−∂p/∂z). Then, by imposing the boundary conditions of the known admittances at the boundaries y = 0, y = Dy , z = 0, and z = Dz , we can express Ry in terms of ky , ηy1 , ηy2 and Rz in terms of kz , ηz1 , ηz2 , and by eliminating Rz and Rz , we obtain equations for ky and kz . Thus, Ky tan(ky Dy ) = −i
ηy1 + ηy2 , 1 + ηy1 ηy2 /Ky2
(10.46)
where Ky = ky /k and k = ω/c. The analogous equation for Kz = kz /k is obtained by replacing y by z. Generally, these equations have to be solved numerically. In the special case when the admittances of opposite walls are the same, η1y = η2y = ηy and η1z = η2z = ηz , the equation for Ky can be expressed as Ky tan(Ky kDy /2) = −iηy
(6.1.21a)
as explained in the discussion following Eq. 10.8 with an analogous equation for Kz . The propagation constant kx then follows from Eq. 10.9, i.e, (10.47) kx = kr + iki = k 2 − ky2 − kz2 . The corresponding attenuation is 20 log 10(e)ki ≈ 8.7ki dB per unit length. A numerical example has already been given and discussed in Figure 8.14. The extension of the analysis to nonlocally reacting liners follows from the treatment of that in Section 10.2.2.
10.3.2 Circular Duct Locally Reacting Liner With the radial component of the propagation constant in the porous liner denoted by qr (in a locally reacting liner there is no other component), the expression for the sound pressure field in the liner is p(r) = AH0(1) (qr r) + BH0(2) (qr r), (1)
(2)
(10.48)
where H0 = J0 + iY0 and H0 = J0 − iY0 are the Hankel functions of zeroth order, being combinations of the Bessel and Neumann functions J0 and Ys . (We could equally well have expressed the total pressure field as a linear combination of J0 and Y0 , which would have been a bit simpler. Try it!) By analogy with plane waves, these relations correspond to exp(±ikx) = cos(kx) ± i sin(kx), H0(1) representing an (2) outgoing and H0 an incoming cylindrical wave. With qr being the only component, we have qr = q, where q is defined in Eq. 10.27.
MATHEMATICAL SUPPLEMENTS AND COMMENTS
343
The corresponding radial velocity amplitude is obtained from the momentum equation −iωρu ˜ r = −∂p/∂r, where, as before, ρ˜ is given in Eq. 10.35, ηw (1) (2) AH1 (qr) + BH1 (qr) . (10.49) ur (r) = i ρc ˜ r /k) (normalized wave admittance), where k = ω/c. The quantity ηw = (ρ/ρ)(q We have used dH0 (z)/dz = −H1 (z), where H1 is the Hankel function of the first order. The boundary condition of zero radial velocity at the outer radius b of the porous layer, which is in contact with the rigid duct wall, yields (1)
(2)
B/A = −H1 (qr b)/H1 (qr b).
(10.50)
Using this relation in combination with Eqs. 10.48 and 10.49, we find for the normalized admittance of the porous liner η = ρc
J1 (qr a)Y1 (qr b) − J1 (qr b)Y1 (qr a) ur (a) = iηw . p(a) J0 (qr a)Y1 (qr b) − J1 (qr b)Y0 (qr a)
(10.51)
For a thin layer, (b − a) << a, we expect the admittance to be the same as for the plane layer, i.e., −iηw qr (b − a) to the first order in qr (b − a). To check if this is consistent with Eq. 10.51, we expand J1 (qr b) = J1 (qr a) + (b − a)J1 (qr a) with an analogous expansion for Y1 (qr b) both in the numerator and the denominator (the prime superscript indicates differentiation with respect to the argument). Next, we use the recursion formula zJ1 (z) = −J1 (z) + zJ0 (z), valid also for Y1 , to find that η ≈ −iηw qr (b − a)/(1 + (b − a)(..)) ≈ −iηw qr (b − a) to the first order in qr (b − a). Having obtained the liner admittance, we can proceed to calculate the attenuation in the duct. We restrict the analysis to the symmetrical mode with no azimuthal variation in sound pressure, i.e., p(r, x) = AJ0 (kr r)eikx x kx = k 2 − kr2
(10.52)
where k = ω/c. The corresponding radial velocity field is obtained from the momentum equation −iωρur = −∂p/∂r ur = i(kr /ωρ)AJ1 (kr r) = i(kr /k)(A/ρc)J1 (kr r).
(10.53)
The boundary condition of a normal admittance η at r = a requires that η = ρcur /p = i(kr /k)J1 (kr a)/J0 (kr a) or (kr a)
J1 (kr a) = −i(ka)η, J0 (kr a)
(10.54)
(10.55)
which takes the place of Eq. 10.9. This equation for the complex quantity kr a generally has to be solved numerically. The propagation constant kx is then obtained from Eq. 10.52 and the attenuation from Eq. 10.12.
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NOISE REDUCTION ANALYSIS
In the low frequency approximation, with the arguments of the Bessel functions much less than unity, we have J0 (z) ≈ 1 and J1 (z) ≈ z/2, so that (kr a)2 ≈ −i2(ka)η.
(10.56)
In the high frequency region ka >> 1 and kr a >> 1, we have J1 /J0 ≈ tan(kr a − π/4), and it follows from Eq. 10.55 that the argument of the tan()-function must be close to π/2, and we put kr a − π/4 = π/2 + . With tan(π/2 + ) ≈ −1/, it follows from the equation that ≈ −i(kr a/kaη) and kr a ≈
3π 1 3π ≈ (1 − i ). 4(1 + i/kaη) 4 kaη
(10.57)
The solution of Eq. 10.55 for kr a over the entire frequency range is obtained numerically and the axial propagation constant kx = kxr +ikxi follows from Eq. 10.52 and the attenuation from p ∝ exp(−kxi x). An example of the computed frequency dependence of the attenuation is shown and discussed in Figure 8.14. Nonlocally Reacting Liner In a liner without transverse partitions (nonlocally reacting), there will be sound propagation within the liner also in the axial direction, and if the corresponding component of the propagation constant is qx , we have qr = q 2 − qx2 , where q is the ‘total’ propagation constant in the porous material as given before in Eq. 10.27. Similarly, in the air channel, we have kr2 = k 2 − kx2 . Since the axial variation in the sound field must be the same in the air channel as in the liner, we have qx = kx . The field matching condition at the boundary of the liner, expressed by Eq. 10.56 in which η is given by Eq. 10.51, is still valid. With kx2 = k 2 − kr2 and qr = q 2 − kx2 ityields an equation for kr from which follows the desired propagation constant kx = k 2 − kr2 in the duct in complete analogy with the procedure for the rectangular duct. The equation for kr has to be solved numerically.
10.3.3 Annular Duct If we add to the circular lined duct a concentric porous core or rigid cylinder with a porous liner, the air channel becomes an annular duct, which is another configuration of practical interest. A sound field with no azimuthal angle dependence, to which the fundamental acoustic mode belongs, will be of the form p = [AJ0 (kr r) + BY0 (kr r)]eikx x = A[J0 (kr r) + RY0 (kr r)]eikx x kx = k 2 − kr2 ,
(10.58)
where k = ω/c and R = B/A. The corresponding radial velocity, obtained from the momentum equation −iωρur = −∂p/∂r, is ur = −
1 A kr kr [AJ1 (kr r) + BY1 (kr r)] = i [J1 (kr r) + RY1 (kr r)]. iωρ ρc k
(10.59)
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345
The inner and outer radii of the annulus are denoted by a and b. Both p and ur are continuous at these boundaries, and we denote their normalized admittances by ηa and ηb . Thus, at r = b we have ηb = i
kr J1 (kr b) + RY1 (kr b) . k J0 (kr b) + RY0 (kr b)
(10.60)
Replacing b by a, and ηb by −ηa ,1 we get the corresponding equation at r = a. From these two relations, we can eliminate R and obtain the following equation for determination of the radial propagation constant kr −
i(kr /k)J1 (kr b) − ηb J0 (kr b) i(kr /k)J1 (kr a) + ηa J0 (kr a) = . Y0 (kr a)ηa + i(kr /k)Y1 (kr a) Y0 (kr b)ηb − i(kr /k)Y1 (kr b)
(10.61)
It is instructive, and also useful as starting points for numerical root-finding routines, to consider the form of these relations in the limits of low and high frequencies corresponding to small and large values of the arguments of the Bessel functions. For z << 1, we have J0 (z) ≈ 1 and J1 (z) ≈ z/2. Both Y0 and Y1 go to infinity, Y0 logarithmically, and Y1 as 1/z. With these approximations, Eq. 10.61 reduces to kr2 (b2 − a 2 ) = −i2k(bηb + aηa ). (10.62) With a = 0, we obtain the result for the circular duct. If the width of the annulus is small compared to its radius, we expect the result to be about the same as for the rectangular duct. Indeed, with b2 − a 2 = (b − a)(b + a) ≈ 2a(b − a) and with b ≈ a on the right-hand side of Eq. 10.62, we get kr2 D 2 ≈ −ikD(ηa +ηb ), where D = b −a. This is consistent with the result in Eq. 10.9 to the first order in kD. In the high frequency limit we use J0 (z) ∝ cos(z − π/4), J1 (z) ∝ sin(z − π/4), Y0 (z) ∝ sin(z − π/4), and Y1 (z) ∝ − cos(z − π/4), the constant of proportional√ ity in each case being 1/ πz. After a little algebra, in which we use the identity tan(z1 − z2 ) = (tan(z1 ) − tan(z2 ))/(1 + tan(z1 ) tan(z2 )), we find the high frequency approximation ηa + ηb kr D tan(kr D) ≈ −ikD , (10.63) 1 + (k/kr )2 ηa ηb where D = b − a is the channel width. This has the same form as Eq. 10.9 for the rectangular duct, and it can be further approximated as shown in and after Eq. 10.22 to yield (10.64) kr D = π − iπ/(ηkD), where η = η1 η2 /(η1 + η2 ). The Boundary Admittances It remains to express the admittances in terms of the properties of the outer and inner porous liners. Actually, for the outer liner this has already been done in the analysis of 1 The admittance is defined as the ratio of the amplitude of the velocity into the boundary, i.e., in the
negative r-direction at the core, hence the minus sign.
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NOISE REDUCTION ANALYSIS
the circular lined duct, Eq. 10.51. We need to change the notation, however, to make it consistent with the present one. Thus, η becomes ηb , a becomes b, and b becomes R = b + d, where d is the thickness of the porous layer. Thus, ηb = iηw
J1 (qR)Y1 (qb) − J1 (qb)Y1 (qR) , J1 (qR)Y0 (qb) − J0 (qb)Y1 (qR)
(10.65)
where, as before, q is the propagation constant in the porous material and ηw the normalized wave admittance. For their definitions, we refer to the previous section on the circular lined duct. As shown in the discussion of Eq. 10.51 when applied to a thin layer, the admittance ηb is approximately −iηw q(R − b), i.e., the value for a plane porous layer as expected. For the inner boundary, consider first the core to be a uniform porous cylinder of radius a. Again, we denote the propagation constant and the wave impedance in the material by q and ηw ; they need not be the same as for the outer liner. The amplitudes of pressure and radial velocity are of the form p = AJ0 (qr) ur = (1/iωρ) ˜ ∂p ∂r = i(A/ρc)ηw J1 (qr).
(10.66)
The admittance, the ratio of the velocity into the core (negative r-direction) and the pressure, is then ηa = (−ur /p)ρc = −iηw
J1 (qa) . J0 (qa)
(10.67)
For low frequencies (small values of the argument), it reduces to −iηw qa/2, i.e., half the value for a plane layer of thickness equal to the core radius. If the core is a rigid cylinder or radius r0 covered with a porous liner with an outer radius a, the calculation of the admittance parallels that for the outer liner and we obtain J1 (qa)Y1 (qr0 ) − J1 (qr0 )Y1 (qa) ηa = −iηw . (10.68) J0 (qa)Y1 (qr0 ) − J1 (qr0 )Y0 (qa) We note that with r0 = 0 (uniform porous core), it reduces to Eq. 10.67, as it should. We refer to Figure 8.15 for an example of the computed attenuation in an annular duct channel.
10.4 SUPPLEMENT TO SECTION 8.6, HIGHER MODES AND FLOW The complex sound pressure amplitude in a rectangular channel with hard walls at y = 0 and z = 0 and with a lining at y = D is of the form p(ω) = A cos(ky y) cos(kz z)eikx x , where kx =
k 2 − ky2 − kz2
(10.69) (10.70)
MATHEMATICAL SUPPLEMENTS AND COMMENTS
347
and k = ω/c. With the two walls perpendicular to the z-axis, at z = 0 and z = Dz being rigid and impervious, the boundary condition of zero normal velocity amplitude requires that kz = mπ/Dz , where m = 0, 2, 3 . . . and Dz the distance between the rigid walls. The boundary conditions at the lined walls, perpendicular to the y-axis, also produce a set of values for ky . They are not as simple as for kz but require the solution to the transcendental equation 10.8 for a complex variable. In the numerical analyses so far, we have considered only the fundamental mode with m = 0 and with the lowest order solution for ky . Modes which correspond to the lowest order for ky but higher order in m, we shall refer to as ‘z-modes,’ and we shall start by considering these. ‘Z-Modes’ The axial velocity amplitude is obtained from −iωux = −∂p/∂x, (where, for harmonic time dependence, we have used ∂/∂t → −iω), and it follows from Eq. 10.69 that A kx cos(ky y) cos(kz z)eikx , (10.71) ux (ω) = ρc k where k = ω/c. Regarding p and ux as rms values, the acoustic intensity in a mode is Ix = {pu∗x } =
|A|2 kmr −2kmi x , | cos(ky y)|2 cos2 (kz z) e ρc k
(10.72)
where {} stands for the ‘real part of’ and kmr and kmi are the real and imaginary parts of kx (for the mth mode). The acoustic power is obtained by integrating over the silencer area. When there are several modes present, we readily find that the total power is the sum of the powers in the individual modes, and that the integral of the cross terms in the expression for the intensity will be zero. In our evaluation of the role of the higher modes, we shall consider only the attenuation (not transmission loss and insertion loss) in the duct. The calculation of TL requires information about the reflection of higher modes from the silencer and, for the IL, we must account also for reflections from the ends of the main duct. Although these are interesting theoretical questions, they will not be considered here. Integration of the intensity over z will yield a factor Dz /2 for all higher modes. Furthermore, the factor | cos(ky y)|2 will be the same for all modes. Thus, for the purpose of calculating the average attenuation it is sufficient to realize that the total power is 2
W (x) = C |A0 | k0r e
−2k0i x
1 2 −2kmi x , + kmr |Am | e 2
(10.73)
m>0
where the constant of proportionality C is the integral of (Dz /k)| cos(ky y)|2 from y = 0 to D. Quantities kmr and kmi are the real and imaginary parts of the propagation constant kx of the mth mode, as obtained from Eq. 10.70.
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NOISE REDUCTION ANALYSIS
The corresponding average attenuation in dB is obtained from the ratio of the powers W (0) and W (L) at the beginning (x = 0) and at the end (x = L) of the duct, or, expressed in decibels, Atten = 10 log[W (0)/W (L)].
(10.74)
In order to what extent this average attenuation might differ from the attenuation of the fundamental mode, we consider a case in which all the amplitudes Am of the first 20 modes are the same and find that there is no significant difference between the average attenuation and the attenuation of the fundamental mode. ‘Y-Modes’ The numerical results in Figures 8.1 and 8.3 apply only to the lowest order solution for ky , corresponding to the fundamental mode. Higher order values can be obtained as additional solutions to Eq. 10.8. The corresponding attenuation will be greater than for the fundamental mode but after they have decayed after a relatively short distance from the entrance of the duct, the fundamental mode attenuation takes over. For a sufficiently long duct, the latter generally determines the overall average attenuation. A more detailed analysis requires not only a determination of the higher mode values of ky but also the amplitude of the various modes at the entrance of the duct. These depend on the nature of the sound field incident sound field. We shall limit ourselves here merely to an estimate of the role of higher order modes in the region of high frequencies where ray acoustics can be used as an approximation. Semi-Empirical Higher Order Mode, TL Correction In order to get an idea of the TL at very high frequencies, we considered a line source placed at the beginning of the duct, as shown in Figure 8.22. The width of the air channel is D and the length L. If the intensity radiated from the source in a direction corresponding to the angle φ (see figure) is I (φ), the total power radiated into the duct is W =
π/2
−π/2
I (φ) dφ.
(10.75)
If the liner is perfectly absorbing, there will be no reflections of the rays that strike the liner, and the only rays that reach the end of the duct are those emitted in the angular range from −φ0 to φ0 , where φ0 = arctan(D/2L). The transmitted power is then We =
φ0
−π0
I (φ) dφ.
(10.76)
In the absence of the liner, the power leaving the duct will be the same as the power W emitted into the left hemisphere of the source. Thus, the transmission loss is T L = 10 log(W/We ). (10.77)
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349
In the special case of an omni-directional source, I (φ) = W/π, We = (2φ0 /π )W , and TL = 10 log(π/(2φ0 ). For small angles, φ0 ≈ D/2L and TL ≈ 10 log(π L/D), as given in the text. To get an idea of the role of directivity, we consider as an example I (φ) = A cos(φ) = (W/2) cos(φ). The power escaping without reflections is then W sin(φ0 ), and the corresponding transmission loss becomes about 2 dB smaller than for the omnidirectional source. In reality, the liner is not totally absorbing, and we can improve the expression for We by accounting for the number of reflections suffered by a ray along its path to the exit of the channel. This number depends on the angle of emission φ of the ray. After each reflection, the intensity is reduced by the factor |R|2 , where R is the pressure reflection coefficient of the liner. Then, if the number of reflections is n(φ), the emitted power from the duct will be We =
π/2
−π/2
I (φ)|R|2n(φ) dφ.
(10.78)
With a given duct liner, the angular dependence of the absorption coefficient can readily be obtained. In our case, there will be one reflection from the acoustic liner in the angular range between −φ0 = arctan(D/2L) and φ1 = arctan(2D/3L). With reference to the discussion of Figure 8.23, refraction of a ray in the nonuniform flow in the duct changes the angles φ0 and φ1 , and hence the output power We . The ray will be curved and, as mentioned, the radius of curvature will be R = dM/dy. It follows from Figure 8.23 that (L + R cos φ0 )2 + (R cos φ0 ) − (D1 /2)2 = R 2 from which it follows D1 /2 L 1 − (L2 + D12 /4)/4R 2 − . (10.79) φ0 = 2R L2 + D12 /4 A positive (negative) value of R corresponds to downstream (upstream) propagation√leading to a decrease (increase) of the critical angle φ0 . At a value R = (1/2) L2 + D 2 , the angle will be zero. In this manner, we can get an idea of the effect of refraction on the transmission loss for higher modes in the high frequency limit. The line source model used here for the purpose of illustration can be replaced by one in which the sound enters the duct from a diffuse field in a plenum chamber.
10.5 SUPPLEMENT TO SECTION 8.7, LIQUID PIPE LINES 10.5.1 Liquid Pipe Line With Slightly Compliant Walls Wall Admittance The sound pressure in the water is denoted by p and the induced radial displacement of the pipe wall by ξ . As indicated earlier, local reaction of the boundary is assumed. If the acoustic wavelength is much larger than the pipe diameter, we can assume the
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NOISE REDUCTION ANALYSIS
displacement to be independent of the azimuthal angle. The strain is ξ/a and the stress in the wall σ = Eξ/a, (10.80) where E is the elastic modulus, and a the unperturbed radius of the pipe. We have assumed that h << a, where h is the wall thickness. With the mass density of the wall material denoted by ρ1 , the mass per unit area of the pipe is m = ρ1 h, and the equation of motion of a mass element of the pipe, referred to in the first sketch in Figure 10.2, becomes m
∂ 2ξ = −Ewξ/a + pa. ∂t 2
(10.81)
We have then neglected the radiation load on the outside of the pipe. For harmonic time dependence, (∂/∂t → −iω), introducing the corresponding complex amplitudes ξ(ω), p(ω) and the velocity amplitude of the wall u1 = −iωξ , we obtain from Eq. 10.81 the wall admittance u1 (ω) i ω2 , ≡Y = p(ω) ωm ω2 − ω02
(10.82)
where we have introduced the resonance frequency of the pipe f0 = ω0 /2π =
1 E/ρ1 = c1 /2π a. 2πa
(10.83)
√ The quantity c1 = E/ρ1 is the longitudinal wave speed in the wall material. The resonance frequency of this symmetrical mode of oscillation is often called the ring frequency. The corresponding period is the time required for a longitudinal wave to make a roundtrip along the circumference of the pipe. With c1 ≈ 5000 m/sec, we get f0 ≈ 2.1 × 104 Hz for a 3 inch steel pipe. To account for damping in the wall material, we let the elastic modulus be complex, and replace E by E(1 − i), where is the loss factor. The resonance frequency ω0 is also affected, and ω02 has to be replaced by ω02 (1 − i). We normalize the admittance with respect to ρ0 c0 , where c0 is the sound speed in water, c02 = 1/κρ0 , and κ is the compressibility. Then, with ω02 = c12 /a 2 , m = ρ1 w, and k0 = ω/c0 , the normalized admittance η = Yρ0 c0 can be written, accounting for the damping, ρ0 c02 a 1 η = −i k0 a . (10.84) w E 1 − i − ω2 /ω02 In this equation E and ω02 are the real parts (no is involved). The approximation of uniform pressure across the pipe is expected to be good only if η << 1. Propagation Constant In the present approximation, we determine the effect of wall compliance on sound propagation by determining an effective complex compressibility of the liquid,
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MATHEMATICAL SUPPLEMENTS AND COMMENTS
Figure 10.2: Left: A radial displacement ξ produces a stress σ in the wall (corresponding pair of forces on the element shown, each σ h). The mean radius of the wall is a, wall thickness, h. Right: Control volume in the calculation of the equivalent compressibility of the liquid in the pipe when the normalized admittance of the wall is η. The velocity u1 of the wall is ηp/ρ0 c0 , where p is the sound pressure. accounting for the compliance of the wall, and we start with the linearized mass conservation equation ∂ρ (10.85) + ρ0 div u + U div ρ = 0, ∂t where ρ is the mass density of water (ρ0 being the unperturbed value), u the (acoustic) velocity perturbation, and U the mean flow velocity in the pipe. The ratio of the third and second term in this equation is of the order of the flow Mach number U/c in the pipe, and with M << 1, we shall neglect this term. Since (1/ρ0 )∂ρ/∂t = κ∂p/∂t, where κ is the compressibility of the liquid, Eq. 10.85 reduces to κ
∂p + div u = 0 ∂t
(10.86)
or, for harmonic time dependence −iωκp + div u = 0.
(10.87)
Integration of Eq. 10.87 over the control volume shown in the second sketch in Figure 10.2, with A = πa 2 being the area of the pipe, yields −iωκAp + 2πa(ηp/ρc) + A
∂u = 0. ∂x
(10.88)
If we express this relation in terms of an equivalent compressibility κe of the liquid, corresponding to the mass conservation equation −iωκe p = −∂u/∂x, it follows that the equivalent (complex) compressibility of the liquid in the pipe, accounting for the flexibility of the wall, will be κe = κ(1 + i2η/k0 a),
(10.89)
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NOISE REDUCTION ANALYSIS
where we have used κ = 1/ρ0 c02 , k0 = ω/c0 , c0 being the free field sound speed in the liquid. √ The complex sound speed then becomes c = 1/ ρκe (see discussions in Chapters 3 and 5), and the complex propagation constant for the wave in the liquid in the hose becomes, with D = 2a, ω k = kr + iki = = k0 1 + i2η/k0 a, (10.90) c √ where k0 = ω/c0 and c0 = 1/ κρ0 . The phase velocity of the wave in the liquid is then cr = ω/kr
(10.91)
and the attenuation in dB per unit length, Atten = 20 log10 (e)ki ≈ 8.7ki .
(10.92)
If we use the complex wall admittance in Eq. 10.92 and introduce the parameter β = (D/ h)(ρ0 c02 )/E, then, if β << 1, we obtain ki ≈ k0 β/2,
(10.93)
which can be used as a first rough estimate. The corresponding attenuation constant per (free field) wavelength λ0 = 2π/k0 , is then ki λ0 = πβ.
(10.94)
The pressure reflection coefficient at the interface between two pipes of different materials will be c − c0 R= . (10.95) c + c0 (For a discussion of numerical results, we refer to Figure 8.27.)
10.5.2 Liquid Pipe Line With Air Layer Wall Treatment The pipe line, illustrated schematically in Figure 8.28, is rectangular with two opposite sides lined with air layers contained by membranes. With the membranes and air layers loss free, this line, strictly speaking, should not be classified as dissipative. The width is D = 2a. With the boundaries at y = ±a, the expression for the complex pressure amplitude in the line is p = A cos(ky y)eikx x kx = (ω/cl )2 − ky2 = (ω/cl ) 1 − (ky c1 /ω)2 .
(10.96)
With a pressure release boundary, p = 0 at y = ±a, the first mode corresponds to ky a = π/2. If we introduce the cut-off frequency fc = ky c1 /(2π ) = c1 /4a the expression for kx can be written kx = (ω/c1 ) 1 − (fc /f )2 fc = πc1 /2a. (10.97)
MATHEMATICAL SUPPLEMENTS AND COMMENTS
353
In other words, the wave will propagate if f > fc and it is evanescent for f < fc . For a pipe with a width D = 2a = 10 cm and a sound speed in water of 1440 m/sec, the cut-on frequency fc for the first mode will be fc ≈ 7500 Hz. In the evanescent regime, kx = iki is imaginary, where ki D = π(f/fc ) (fc /f )2 − 1. (10.98) The attenuation in dB per width D of duct length is 20 log(e)ki D ≈ 8.7ki D. It starts out with 27.3 dB at f/fc = 0 and decreases monotonically to zero at f = fc . Wall Impedance We treat the gas tube liner as a locally reacting surface with an acoustic impedance z. Without losses in the liner, the impedance will be purely reactive involving the contributions from the gas layer and the limp membrane, which contains the air in the liner. However, the results obtained can readily be extended to include a resistive component of the impedance resulting from losses in the membrane and/or in the interior of the air layer if it contains a porous material, for example. If the thickness of the air layer is d, its reactance becomes iρa c cot(kd), where k = ω/c, c, the sound speed in air, and ρa , the mass density of air. If kd << 1, which corresponds to an acoustic wavelength in air which is much longer than the thickness of the air layer, the impedance reduces to iρa c/kd = iρa c2 /dω = iK/ω, where K = ρc2 /d is the ‘spring constant’ of the air layer. Including the mass reactance of the limp wall, we obtain for the total boundary impedance z = −iωm + iρa c cot(kd). (10.99) In terms of the thickness dw of the membrane and the mass density ρw , we have m = ρw dw . The impedance becomes zero at the resonance frequencies given by the solutions to the equation ωm = ρa c cot(ωd/c). (10.100) In the long wavelength approximation ωd/c << 1, the fundamental resonance frequency becomes (10.101) f1 ≈ (c/2π) ρa /(ρw dw d). With c ≈ 340 m, d = 1.27 cm (≈ 0.5 inches), ρw = 2 g/cm3 , ρa = 0.0013 g/cm3 , and dw = .1 cm, we get from Eq. 10.101, f1 ≈ 381 Hz for a gas pressure of 1 atm. The corresponding resonance frequencies at 10 and 100 atm are found to be 1193 and 3390 Hz. It is the normalized wall impedance ζ = z/ρl c1 or the corresponding admittance η = 1/ζ , which will determine the sound propagation characteristics in the pipe, where ρl c1 is the wave impedance of the liquid, ζ = −i(ωm/ρl c1 ) + i(ρc/ρl c1 ) cot(ωd/c).
(10.102)
The wave impedance for water is z1 = ρl c1 ≈ 1.5 × 106 MKS and for air at P = 1 atm, z ≈ 420 MKS. Thus, the magnitude of the normalized impedance (admittance) generally is much less than (much greater than) unity.
354
NOISE REDUCTION ANALYSIS
Propagation Constants As already indicated, we analyze here the propagation in a rectangular pipe with two opposite walls at y = ±a being treated with the type of air liner described above. The other walls, perpendicular to the z-axis, are assumed to be rigid. The circular pipe can be treated in a completely analogous fashion. With the coordinates chosen as described above, the fundamental acoustic mode in the pipe will have a uniform pressure distribution in the z-direction and the pressure field is that in Eq. 10.18 and the corresponding y-component of the velocity amplitude is uy (y, x, ω) = iA(ky /ωρl ) sin(ky y)eikx x . (10.103) The factor ky /ωρl can be written (ky /kl )/ρl c1 , where kl = ω/c1 . The boundary condition at y = D/2 = a then results in −i
kl cot(ky a) = ζ. ky
(10.104)
Introducing the normalized boundary admittance η = 1/ζ , X = kl a, and Y = ky a, we can rewrite Eq. 10.104 as follows, Y tan(Y ) = −iXη. The propagation constant in the axial direction is given by kx = kl2 − ky2
(10.105)
(10.106)
with the corresponding normalized value Kx = kx /kl =
1 − Ky2 ,
(10.107)
where Ky = Y /X = ky /kl . The x-dependence of the sound pressure amplitude is then expressed by p(x, ω) = p(0, ω)eikx x = p(0, ω)eiKx kl x .
(10.108)
With η being purely imaginary, the right-hand side (RHS) of Eq. 10.105 will be real, positive when the boundary is mass controlled and negative when it is stiffness controlled. If the RHS is positive, the root of Eq. 10.105 will lie between 0 and π/2 for the first branch of the function on the left-hand side (LHS). If the RHS is negative, there are two possibilities; Y can be real and lie between π/2 and π or it can be imaginary, Y = iYi , with the magnitude between zero and infinity. In the latter case, the y-dependence of the wave function for the complex sound pressure amplitude is expressed by cosh(Yi y/a) and will be referred to as the hyperbolic branch of the solutions to Eq. 10.107. This mode will propagate in the duct without attenuation and will determine the resulting overall transmission loss in the frequency range where the boundary is stiffness controlled, i.e., below the boundary resonance frequency.
MATHEMATICAL SUPPLEMENTS AND COMMENTS
355
The evanescent mode will be present for both a mass and a stiffness controlled boundary, but will be of consequence only in the mass controlled region, since the unattenuated hyperbolic mode will determine the transmission loss in the stiffness controlled regime. With Y being purely imaginary in the stiffness controlled region, it follows from Eq. 10.107 that the corresponding normalized propagation constant Kx will be larger than 1 and the phase velocity smaller than the free field wave speed. Actually, for sufficiently low frequencies so that the wavelength is large compared to both d and D and RHS << 1, we get Y tan Y √ ≈ Y 2 , and, from Eq. 10.22, Y 2 ≈ −iXη. Then, from Eq. 10.107, it follows that Kx ≈ 1 + iη/X, and with the low frequency approximation η ≈ −ikd, we get κd Kx = 1 + . (10.109) κ1 D The phase velocity along the axis will be cp ≈ c1 /Kx ,
(10.110)
where κ and κ1 are the compressibilities of the gas and the water, respectively. In Figure 8.27 we have shown an example of the frequency dependence of the phase velocity for a channel with D = 2a = 10 cm and with a thickness of the air layer of d = 1.27 cm. The mass of the membrane enclosing the air is such that the resonance frequency of the membrane-air layer combination is ≈ 380 Hz at a gas pressure of 1 atm. The phase velocity decreases monotonically to zero at the resonance frequency and above this frequency, in the mass controlled regime, the wave becomes evanescent. The phase velocity is reduced considerably by the air layer and it is less than 1/60th of the free field sound speed at a pressure of 1 atm. Transmission Loss The transmission loss is T L = 10 log(1/τ ) = 20 log(|pi /pt |),
(10.111)
where pi and pt are the complex sound pressure amplitudes in the incident and transmitted waves, respectively. In calculating the transmission loss, we shall treat the attenuator as a onedimensional transmission line using as field variables the average pressure and velocity amplitudes across the pipe. From Eqs. 10.22 and 10.23 it follows that the wave impedance, the ratio of the complex amplitudes of pressure and axial velocity in a wave traveling in the positive z-direction, is p(y, ω)/u(y, ω) = ρl c1 /Kx ,
(10.112)
where, as before, Kx = kx /kl is the normalized propagation constant. The same expression applies to the quantities averaged over the cross section of the pipe.
356
NOISE REDUCTION ANALYSIS
With reference to Appendix A, the transmission matrix elements of the duct are T11 = cos(kx L) T12 = −i(1/Kx ) sin(kx L) T21 = −iKx sin(kx L) T22 = cos(kx L)
(10.113)
and the transmission loss is T L = 10 log(|T11 + T12 + T21 + T22 |2 /4).
(10.114)
An example of the computed transmission loss has already been given and discussed in Figure 8.29.
10.6 SUPPLEMENT TO SECTION 9.1, UNIFORM DUCT With reference to Appendix B, the matrix elements of a pipe are T11 = T22 = cos(kL) T12 = T21 = −i sin(kL),
(10.115)
where L is the length of the pipe and, neglecting losses in the pipe, k = ω/c. If we wish to include visco-thermal losses we put k ≈ (ω/c)[1 + iδvh /D],
(10.116)
where δvh is the visco-thermal boundary layer thickness and D the pipe diameter. The insertion loss is obtained from the general equation given in Chapter 8. An interesting result is obtained if the pipe is added to a reflection-free source, i.e., with ζs = 1. Then, if the areas of the pipe and the source are the same, the radiation resistances of the source and the end of the pipe are also same, and the insertion loss (and the transmission loss) of the pipe will be zero, independent of its length. For a constant pressure source, the internal impedance is zero, ζs = 0, and for a constant velocity source, ζs = ∞, and the corresponding insertion losses are I L0 = A + 10 log | cos(kL) − i(1/ζt ) sin(kL)|2 I L∞ = A + 20 log | cos(kL) − iζt sin(kL)|2 , where A = 10 log(As θrs /Ap θr ).
(10.117)
They have already been plotted and discussed in Figure 9.1. The expressions in Eq. 10.117 could have been derived by considering the power delivered by the source with and without the pipe present. For a constant pressure (velocity) source, this power is proportional to the real part of the load admittance (impedance). The normalized input impedance is ζi = p1 /ρcu1 = θi + iχi =
ζt cos(kl) − i sin(kl) , −iζ2 sin(kl) + cos(kl)
(10.118)
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MATHEMATICAL SUPPLEMENTS AND COMMENTS
where ζ2 is the normalized termination impedance of the pipe. We note that when the pipe length is an integer number of half wavelengths, sin(kl) = 0, and for an odd number of quarter wavelengths, cos(kl) = 0. Thus, for = nλ/2 ζ2 ζi = (10.119) 1/ζ2 for = (2n − 1)λ/4. At frequencies fn for which the length of the pipe is an integer number of half wavelengths, L/λ = n/2, the input impedance ζi equals the termination impedance ζ2 . We have assumed that ζ2 is also the termination impedance of the source in the absence of a pipe, and it follows that at the frequencies fn , the power delivered by the source will be the same with and without the pipe, i.e., the insertion loss of the pipe will be zero. This is true for both a constant pressure and a constant velocity source. It is noteworthy that for L/λ < 0.5, the insertion loss IL(0) for a constant pressure source increases monotonically with decreasing L/λ, I L(0) → 20 log[1 + (2L/0.61D)],
L/λ → 0.
(10.120)
For example, with L/D = 10, this limiting value becomes approximately 30.6 dB. In this frequency range, the addition of the pipe in essence means an addition of a mass load ρL to the source. In the absence of the pipe, the mass load is ≈ 0.61ρ(D/2), as contributed by the radiation impedance. This explains the expression for the insertion loss. When L is an odd number of quarter wavelengths, L/λ = (2n + 1)/4, the input impedance is 1/ζ2 and provides a (local) maximum impedance mismatch and insertion loss. The maximum decreases with L/λ because the impedance mismatch is reduced since 1/ζ2 decreases.
10.7 SUPPLEMENT TO SECTION 9.6, ATTENUATION IN TURBULENT DUCT FLOW 10.7.1 Friction Factor in Turbulent Duct Flow It is customary to express the average shear stress τ w at the wall in terms of a friction coefficient defined by τ fc = , (10.121) ρU 2 /2 where U is the mean flow velocity in the duct. For a circular duct, the shear stress at the wall is the same around the perimeter of the duct so that the local value τ is the same as the average value, τ = τ . If the area of the duct is A and its perimeter S, the pressure drop per unit length of the duct can be expressed in terms of τ by the relation A(dP /dx) = −τ S, or dP = −τ (S/A) = −τ (4/D) = −(ψ/D)ρ|U |U/2, dx
(10.122)
where D = 4A/S is the hydraulic diameter of the duct and ψ = 4fc , the duct friction factor. For a circular duct, the hydraulic diameter is the same as the actual diameter.
358
NOISE REDUCTION ANALYSIS
In the laminar flow regime, the pressure drop in a circular duct is given by ∂P /∂x = −32μ/D 2 . With the Reynolds number equal to R = U Dρ/μ, it follows from Eq. 10.122 that this is equivalent to ψ = 64/R. For Reynolds numbers in the approximate range 2000 to 4000, the flow undergoes a transition to turbulent flow, and the friction factor then becomes noticeably dependent on the roughness.
10.7.2 Acoustic Perturbations and Dispersion Relation At sufficiently low frequencies, it is justified to treat a sound wave in the duct as a quasi-static perturbation of the mean flow in the duct. Thus, the acoustic perturbations in density and velocity, δ and u, cause a perturbation in the wall stress such that dτ/τ = dfc /fc + 2dU/U0 + dρ/ρ = dfc /d + 2u/U0 + δ/ρ0 ,
(10.123)
and the linearized equations of mass and momentum balance are then ∂ ∂u 0 )δ + ρ0 ∂x + [u − (δ/ρ0 )U ] ∂ρ ( ∂t∂ + U0 ∂x ∂x = 0 ∂ 0 ρ0 ( ∂t∂ + U0 ∂x )u − (ρ0 u + U δ) ∂ρ ∂x ∂p 2 c = − ∂x − (ψ/D)ρ0 |U0 |u(1 + fc−1 ∂f ∂U ) − (ψ/D)(U0 /2)δ,
(10.124)
where we have made use of the relation ∂(ρ0 U0 )/∂x = 0 for the unperturbed flow and ψ = 4fc . We shall neglect the x-dependence of the unperturbed variables and for harmonic time dependence with the spatial dependence of the wave expressed by exp(ikx), Eq. 10.122 reduces to [−(ω/c) + kM]p + k(ρ0 cu) = 0 [−(ω/c) + kM − i(θ/D)]ρ0 u + kp = 0, where θ = ψ|M|[1 + (U/2f )df/dU ] and where we have used p = δc2 . Solving for the propagation constant we obtain θ M (ω/c + i (ω/c)2 + i(ωθ/cD), ) ± k≈− 1 − M2 2D
(10.125)
(10.126)
where the signs of the radical correspond to propagation in the downstream (+) and upstream (−) directions, respectively. If θ/(ωD/c) << 1, expansion of the square root leads to k± =
ω/c θ ± . 1+M 2D(1 + M)
(10.127)
The corresponding decay of the sound pressure is then given by |p(x)| = |p(0)| exp(−βx), where the attenuation constant is β=
θ . D(1 + M)
(10.128)
359
MATHEMATICAL SUPPLEMENTS AND COMMENTS With θ ≈ ψ|M| (Eq. 10.123), the attenuation in dB in a distance x is then 20 log[p(0)/p(x)] = 20 log(e)βx/D ≈ 8.7
ψ|M| x , 1+M D
(10.129)
where, in the last step, we neglected the term containing dfc /dU. As discussed in the previous section, this is valid for sufficiently large Reynolds numbers. With the pressure and the mean density decreasing with x, conservation of mass flux requires that the flow velocity and Mach number will increase with x. If we wish to account for the x-dependence of the Mach number we can do that approximately in the expression for the decay of the sound pressure by replacing Eq. 10.128 by x
|p(x)| = |p(0)| exp − β(x) dx . (10.130) 0
10.7.3 A Comparison With Visco-Thermal Attenuation It was shown in Chapter 2 that the pressure attenuation in a duct due to viscothermal wall effect is given by p(x) = p(0) exp(−βx), where βp = (1/D)k0 δvh
(10.131)
and D is the hydraulic diameter of the tube. The attenuation in dB in a length equal to D is then p(0) ] = 20 log(e)βD = 20 log(e) 2k0 δvh ≈ 5 × 10−4 f , (10.132) p(D) √ where we have used δvh ≈ 0.31/ f (cm) and f is the frequency in Hz. As an example we note that at a frequency of 400 Hz this attenuation is only 0.01 dB per diameter length, and a tube length of 100 diameters is required to get 1 dB of attenuation. It should be emphasized that the visco-thermal attenuation given here is valid only if the acoustic boundary layer thickness is much larger than the duct diameter. √ With √ δv ≈ 0.31/ f cm and k0 = 2πf/c, we get from Eq. 10.132 that βp D ≈ 0.6× −4 10 f . This should be compared to the corresponding flow induced attenuation constant in Eq. 10.126. With θ ≈ ψM and ψ typically ≈ 0.02 we get βD ≈ 0.02 M/ (1 ± M). The visco-thermal and flow induced attenuation then become equal at a frequency 2
M f ≈ 105 Hz. (10.133) 1+M 20 log[
As an example, with M = 0.1, the flow induced attenuation will be larger than the visco-thermal for frequencies below 1000 Hz; for M = 0.2 this frequency will be ≈ 4000 Hz.
Appendix A
Transmission Matrices A.1 INTRODUCTION Most of the matrices we are dealing with here are analogous to those used for linear networks in electrical engineering, i.e., 2 × 2 matrices, which correspond to four-pole networks with two input terminals and two output terminals. Only passive systems are considered. Then, there are no sources within the network so that the values of the output variables depend only on the values of the input variables. For a linear electrical four-pole network shown in Figure A.1 we have then the following relations V1 = A11 V2 + A12 I2 I1 = A21 V2 + A22 I2 ,
(A.1)
where (V1 , I1 ) and (V2 , I2 ) are the input and output values of voltage and current and Aij the elements of the (transmission) matrix of the network. Suppose the source of power is on the left-hand side in Figure A.1. Then, in order for the network to be passive, we must have I1 = 0 if V1 = 0 so that V2 /I2 = −A12 /A11 = −A22 /A21 or A11 A22 − A12 A21 = 1.
Figure A.1: Electric four-pole and its equivalent T-network. 361
(A.2)
362
NOISE REDUCTION ANALYSIS
In other words, the four matrix elements are not independent but must be such that the determinant of the matrix is unity. This relation between the matrix elements can also be seen if we recall that the most general four-pole can be represented in terms of a ‘T-network,’ shown in Figure A.1, with three independent impedances Z1 , Z2 , and Z3 . If we express the relations between V1 , I1 and V2 , I2 in this network and express the four matrix elements Tij in terms of the impedances, we again find the relation in Eq. A.2.
A.1.1 Choice of Variables Another comment should be made in this context. In an acoustical circuit, the variables which correspond to voltage and current can be sound pressure p and the volume flow rate qf , i.e., the product of velocity and cross sectional area of the acoustical element involved. With the choice of qf as the velocity variable, this quantity will be continuous across a discontinuity in a cross sectional area, just like the electrical current. The determinant of an acoustical ‘circuit’ matrix then will be unity. However, if velocity u rather than volume flow rate is chosen as a variable, the determinant of the matrix will not be unity but rather A2 /A1 , where A1 and A2 are the input and exit areas of the acoustical circuit. Despite this lack of elegance, we shall use p and u (rather than qf ) as the acoustical variables; in cases where there are no changes in the cross sectional area, the determinant will be unity even with this choice. Actually, if the temperature is uniform over the acoustical circuit, it is frequently convenient to use the velocity variable ρcu, where ρc is the wave impedance at this temperature, p1 = T11 p2 + T12 ρcu2 ρcu1 = T21 p2 + T22 ρcu2 .
(A.3)
With this choice, the matrix elements Tij become dimensionless, and we shall use this choice unless stated otherwise.
A.2 APPLICATION OF MATRICES Knowledge of the matrix elements of an acoustical ‘barrier’ facilitates the calculation of many quantities of interest in the analysis of its acoustical characteristics such as reflection, absorption, and transmission coefficients, impedance, transmission loss, etc., and we shall now show how these quantities can be computed from known transmission matrices.
A.2.1 Impedance With reference to Figure A.2, we consider an acoustic ‘barrier,’ such as a layer of porous material or a combination of several layers. The x-direction is placed along the normal to the barrier and the front surface is at x = 0.
363
TRANSMISSION MATRICES
Figure A.2: An acoustic barrier can be a combination of any number of elements, in this case four different layers sandwiched between two panels. The complex amplitudes of sound pressure and the fluid velocity component in the x-direction at the front side of the barrier are p1 (ω) and u1 (ω), and the corresponding quantities on the other side are p2 (ω) and u2 (ω). It should be realized that when a sound wave is incident on the barrier, the quantities p1 and u1 are the sums of the contributions from the pressures and velocities in the incident and reflected waves. The normal impedance of the layer is z1 = p1 /u1 and the corresponding normalized value is ζ1 = z1 /ρc. It follows from Eq. A.3 that Input impedance ζ1 = p1 /ρcu1 = (T11 ζ2 + T12 )/(T21 ζ2 + T21 )
(A.4)
where ζ2 = p2 /ρcu2 : Termination impedance. We have assumed the wave impedance ρc to be the same on both sides and that there is no mean flow of the air or fluid on either side of the barrier. In the special case of a barrier backed by a rigid wall, we have ζ2 = ∞, and if it is backed by free space, ζ2 = 1/ cos φ, where φ is the angle of incidence. Thus, we get ζ1 =
T11 /T21 (T11 + T12 cos φ)/(T21 + T22 cos φ)
rigid wall backing free space backing.
(A.5)
A.2.2 Reflection and Absorption Coefficients The reflection and absorption coefficients can be expressed in terms of the impedance ζ1 . For convenience we review the derivation here. The pressure reflection coefficient is defined as the ratio of the reflected and incident pressure amplitudes (plane waves are assumed). We consider an incident wave with an angle of incidence φ with respect to the normal to the boundary. The complex amplitude of the corresponding pressure and fluid velocity fields are then pi = Aeikx x+iky y ≡ pi (0)eikx x ∂p ux = (1/iωρ) ∂x = (pi /ρc) cos φ,
(A.6)
where k = ω/c, kx = k cos φ, ky = k sin φ, and pi (0) is the pressure amplitude at x = 0, the front boundary of the barrier.
364
NOISE REDUCTION ANALYSIS
The reflected pressure wave is pr = R pi (0) exp(−ikx x), where R is the pressure reflection coefficient, R = pr (0)/pi (0), and the corresponding velocity wave is ux = −(pr /ρc) cos φ. The amplitudes of the total pressure pi (0) + pr (0) and velocity at the boundary are given by p1 = pi (0)(1 + R) ρcu1 = pi (0)(1 − R) cos φ.
(A.7)
With ζ1 = p1 /ρcu1 , we get from these equations, ζ1 cos φ = (1 + R)/(1 − R) and the pressure reflection coefficient becomes Pressure reflection coefficient R=
ζ1 cos φ−1 ζ1 cos φ+1
=
(T11 ζ2 +T12 ) cos φ−(T21 ζ2 +T22 ) (T11 ζ2 +T12 ) cos φ+(T21 ζ2 +T22 )
(A.8)
where ζ1 , ζ2 : See Eq. A.4. In the special case of a rigid termination, ζ2 = ∞, R=
T11 cos φ − T21 T11 cos φ + T21
(ζ2 = ∞).
(A.9)
In this case, the absorbed acoustic intensity within the barrier is the difference between the incident and the reflected intensity, i.e., (1/ρc)(|pi |2 | − |pr |2 ), and the absorption coefficient α is the ratio of the absorbed and the incident intensity, α = 1 − |R|2 .
(A.10)
If there is a transmitted wave, the absorption within the barrier corresponds to the coefficient α0 = 1 − |R|2 − |τ |2 , (A.11) where τ is the transmission coefficient to be described next.
A.2.3 Transmission Coefficient and Transmission Loss Again, a plane wave is incident on the barrier at an angle φ. In addition to the reflected wave discussed above, there may also be a transmitted wave and the complex pressure amplitude of it at the back side of the barrier is p2 , which is involved in Eq. A.3. If the front side of the barrier is at x = 0, the backside is at x = L, where L is the thickness of the barrier. The (pressure) transmission coefficient of the barrier is defined as τ = p2 /pi (0), where pi (0) is the incident pressure amplitude at the front side of the barrier. Adding the two equations in Eq. A.7 and making use of Eq. A.3, we get 2pi (0) = p1 + ρcu1 / cos φ = (T11 + T21 / cos φ)p2 + (T12 + T22 / cos φ)ρcu2 . (A.12)
365
TRANSMISSION MATRICES
The barrier is assumed to be located in free space so that the transmitted wave is a plane wave traveling in the same direction as the incident wave so that ρcu2 = p2 cos φ (u2 is the component of the velocity in the x-direction, normal to the boundary), and it follows that 2pi (0) = (T11 + T12 cos φ + T21 / cos φ + T22 )p2
(A.13)
so that τ = p2 /pi (0) =
2 . T11 + T12 cos φ + T21 / cos φ + T22
(A.14)
The corresponding transmission loss is Transmission loss TL =
10 log(1/|τ |2 )
= 10 [log |T11 + T12 cos φ + T21 / cos φ + T22 |2 /4]
(A.15)
Duct: TL vs TL0 It was implied in this derivation that the barrier was an infinite wall in free space with a transmitted plane wave traveling in the direction φ without being reflected. Many applications involve an acoustical element in a duct, as described in Chapter 7. The relation between p2 and u2 on the backside of the barrier is no longer p2 /u2 = ρc/ cos φ but is expressed in terms of an impedance p2 /ρcu2 = ζ2 , which depends on how the duct is terminated. If the termination is reflection-free, ζ2 = 1. If the duct configuration involve many elements, the total transmission matrix is the product of the transmission matrices of the individual elements (including changes in duct area), as will be discussed shortly. Then, the exit area A2 of the duct configuration need not be the same as the area A1 at the source. With reference to the discussion in Chapter 7, two different definitions of transmission loss were introduced, T L and T L0. The former is analogous to the one given above. The latter, however, is based on the net values of the indecent and transmitted powers. To obtain T L, we merely have to put the impedance cos φ = 1 in Eq. A.15 and use the factor A1 /A2 to account for a possible area change, i.e., T L = 10 log[(A1 /A2 )|T11 + T12 + T21 + T22 |2 /4].
(A.16)
As an example, we consider the transmission loss of a rigid resistive screen, normalized flow resistance θ, in a duct with uniform cross section, A1 = A2 . With reference to Eq. A.45, we have T11 = T22 = 1, T12 = θ and T21 = 0. The transmission loss then becomes T L = 20 log(1 + θ/2)
(rigid, resistive screen).
(A.17)
The transmission loss T L0 of an element (barrier) is defined in Section 7.3.2 as the ratio, expressed in dB, of the net incident power into the element and the net
366
NOISE REDUCTION ANALYSIS
transmitted power (the net value being the difference between the primary incident power and the reflected). The net values can be measured by means of intensity probes as explained in the text. The difference between the net incident power and the net transmitted is absorbed by the element. With ζ2 = p2 /ρcu2 , as before, if follows from p1 = T11 p2 + T12 ρcu2 that p1 /p2 = T11 + T12 /ζ2 .
(A.18)
Furthermore, the normalized input impedance of the silencer, as given in Eq. A.4, is ζ1 = [T11 ζ2 + T12 ]/[T21 ζ2 + T22 ]. (A.19) Thus, the net input and exit powers can be expressed as W1 = (|p1 |2 /ρc)(θ1 /|ζ1 |2 )A1 W2 = (|p2 |2 /ρc)(θ2 /|ζ2 |2 )A2 ,
(A.20)
where A1 and A2 are the entrance and exit areas. The transmission loss is then T L0 = 10 log(W1 /W2 ).
(A.21)
It is instructive to determine T L0 in a different manner, based more directly on its definition. Thus, we denote the (primary) incident pressure by pi and the reflected by pr . Then, with pi = ρcui and p2 = −ρcur , we get p1 = pi + pr = T11 p2 + T12 ρcu2 = p2 (T11 + T12 /ζ2 ) ρcu1 = ui + ur = pi − pr = T21 p2 + T22 ρcu2 = p2 (T21 + T22 /ζ2 ), (A.22) where ζ2 is the normalized impedance at the back of the barrier. Addition and subtraction of these equations yields, respectively, 2pi = (T11 + T12 /ζ2 + T21 + T22 /ζ2 )p2 2pr = (T11 + T12 /ζ2 − T21 − T22 /ζ2 )p2 .
(A.23)
With the resistive part of ζ2 denoted by θ2 , the net transmitted power is W2 = A2 (|p2 |2 /ρc)(θ2 /|ζ2 |2 .
(A.24)
The net incident power W1 is the difference between the incident and the reflected W1 = A1 (|pi |2 − |pr |2 )/ρc = A1 (|T11 + T12 /ζ2 + T21 + T22 /ζ2 |2 −|T11 + T12 /ζ2 − T21 − T22 /ζ2 |2 )|p2 |2 /4ρc.
(A.25)
Thus, T L0 = 10 log(W1 /W2 ) = 10 log((A1 /A2 )|(T11 + T21 )/ζ2 + T12 + T22 |2 −|(T11 − T21 )/ζ2 + T12 − T22 )|2 /4θ2 .
(A.26)
367
TRANSMISSION MATRICES For the example above of a resistive screen in a uniform duct, we get T L0 = 10 log(1 + θ)
(rigid, resistive screen),
(A.27)
which should be compared with the result obtained for T L in Eq. A.17. For a purely reactive element, such as an impervious limp wall, with θ replaced by −iωm/ρc, where m is the mass per unit area, we get T L0 = 0. In this case there is no power absorbed by the barrier. The transmission loss T L is not zero, however.
A.2.4 Insertion Loss Transmission loss is based on a comparison of the incident and transmitted acoustic powers, i.e., it involves two different locations, in front of and behind the barrier. Insertion loss, on the other hand, concerns the sound pressure level at a single location and is defined as the change in the sound pressure level at that location resulting from the insertion of a barrier or muffler. Actually, in the case of a muffler in a duct system, it is frequently defined as the change in the acoustic power from the duct as a result of the insertion of the muffler. Unlike the transmission loss, which is always positive, the insertion loss can be either positive or negative. It depends not only on the characteristics of the barrier or muffler but also on the acoustical properties of the rest of the system, including the sound source. In electrical engineering practice for networks, one describes a source in terms of an internal impedance Zi and an electromotive force Ei so that the output voltage and current are related by V1 = Ei −zi I1 . In an analogous manner for an acoustical source, we have p1 (ω) = pi (ω) − zi u1 (ω), where pi , an ‘internal’ pressure, corresponds to the electromotive force, and zi is an analogous internal impedance. If the internal impedance is zero, zi = 0, the pressure delivered by the source at the input of the load will be constant and equal to pi . The source is then referred to as a ‘constant pressure’ source; an ordinary fan is approximately of this type. On the other hand, a source which produces a constant velocity amplitude independent of the acoustic load is a ‘constant velocity’ source, and its internal impedance is zi = ∞. A positive displacement pump is an example which approximates this kind of a source. Here, and in what follows, we use specific acoustic impedances normalized with respect to the wave impedance ρc of air, ρ being the density and c, the sound speed. Furthermore, harmonic time dependence is implied, and the acoustic variables are expressed in terms of the complex amplitudes of pressure p(ω) and velocity u(ω), defined on the basis of a time factor exp(−iωt). We let these amplitudes be rms values to avoid having to repeatedly use a factor of 1/2 in expressions involving the time average acoustic power. In muffler applications, the insertion loss generally refers to the effect of the muffler on the radiated acoustic power and is defined as I L = 10 log(W1 /W2 ),
(A.28)
where W2 and W1 are the radiated powers with and without the muffler present.
368
NOISE REDUCTION ANALYSIS
We consider first the radiation from the bare source. Then, if the average radiation impedance of the source is ζ1 and the internal impedance ζi , the amplitudes of pressure and velocity at the radiating surface of the source are given by p1 = pi − ρcζi u1 = ρcζ1 u1 from which follows
ρcu1 =
pi . ζi + ζ 1
(A.29) (A.30)
With the radiating area of the source denoted by A1 , and the average specific radiation impedance by ζ1 = θ1 + iχ1 , the acoustic power is W1 = A1 |u1 |2 ρcθ1 =
|pi |2 A1 θ1 . ρc |ζi + ζ1 |2
(A.31)
To determine the insertion loss of a muffler applied to the source, we proceed to compute the velocity amplitude u2 at the end of the muffler and the corresponding radiated acoustic power. The muffler, and associated pipe sections, is considered a transmission line, described acoustically by a transmission matrix Tij . Thus, with the field quantities at the beginning and the end of the line labeled by the subscripts 1 and 2, the following relations apply p1 = pi − ζi ρcu1 = T11 p2 + T12 ρcu2 ρcu1 = T21 p2 + T22 ρcu2 .
(A.32)
Multiplying the second equation by ζi and adding it to the first, we obtain pi = (T11 + ζi T21 )p2 + (T12 + ζi T22 )ρcu2 .
(A.33)
The average normalized radiation impedance at the end of the line is denoted by ζ2 = θ2 + iχ2 , so that p2 = ζ2 ρcu2 . It follows then that ρcu2 = pi /(T11 + ζi T21 )ζ2 + (T12 + ζi T22 ).
(A.34)
The radiated power is W2 = A2 |u2 |2 ρcθ2 ,
(A.35)
where A2 is the radiating area of the muffler or pipe line following the muffler. The insertion loss is then
A1 θ1 |(T11 + ζi T21 )ζ2 + T12 + ζi T22 |2 W1 . (A.36) = 10 log I L = 10 log W2 A 2 θ2 |ζi + ζ1 |2 We note that if in this expression for the insertion loss we put ζ1 = ζ2 = ζi = 1, we obtain the transmission loss in Eq. A.15 including the area correction mentioned after that equation. Often the reference system used in the definition of the insertion loss is not the bare source but the source with a duct attached to it. We can readily derive an expression for the insertion loss based on such a reference but in essence it can be obtained
369
TRANSMISSION MATRICES
merely by applying Eq. A.36 to both configurations and then taking the difference between the insertion loss values thus obtained. The radiation impedances ζ2 usually refers to the open end of a pipe in which case we can use the following approximate relations, ζ2 = θ2 + iχ2 ≈
0.25(ka)2 0.61ka −i , 1 + 0.25(ka)2 1 + 0.25(ka)2
(A.37)
where a is the pipe radius, k = ω/c = 2π/λ and c the sound speed. The impedance ζ1 is expressed in the same way, with the appropriate radius inserted. The result refers to a circular pipe, but it can be used also for a square or rectangular cross section if a √ is an equivalent radius, based on the area A of the duct, a ≈ A/π . Thus, in the low frequency approximation, ka << 1, the normalized radiation resistance is (ka)2 /4 and the reactance −i0.61(ka). For a piston source in an infinite rigid baffle, the impedance is calculated with the low frequency approximations for the normalized resistance and reactance being θ2 ≈ (ka)2 /2 and χ2 ≈ −i(8/3π )ka = −i 0.85 (ka). In the presence of a mean flow in the pipe, the interaction of sound with the (turbulent) exit flow results in additional resistance with a normalized value ≈ M, where M is the exit Mach number. This addition to θ2 and θ1 should be applied everywhere except in the factor (A1 θ1 /A2 θ2 ), since it does not contribute to the radiated sound. When several duct elements are connected in series to form a cascade, the total transmission matrix of the cascade is obtained as the product of the matrices of the individual elements. As an example, let us consider the experimental arrangement in Figure 7.2 which, starting from the source, involves an unlined duct section of length L1 , the silencer, and an unlined duct section of length L2 . The insertion loss depends not only on the silencer characteristics but also on the two lengths L1 and L2 , the impedance of the source, and the radiation impedance at the end of the main duct. The total transmission matrix T of the duct silencer system is the product of the individual matrices, (A.38) T = D1 × S × D2 , where the D matrices refer to the two sections of the main duct and the S to the matrix of the silencer (a lined duct including sudden changes in cross section at the two ends). In a laboratory test of the kind illustrated in Figure 7.2, the insertion loss of the silencer is evaluated with respect to an empty duct of length L + L1 + L2 .
A.2.5 Noise Reduction Noise reduction, N R, is defined as the difference between the sound pressure levels on the two sides of the barrier, N R = 10 log |p1 /p2 |2 , and it follows from Eq. A.3, with ρcu2 = p2 /ζ2 , that Noise reduction N R = 10 log(|p1 /p2 |2 ) = 10 log |T11 + T12 /ζ2 |2
(A.39)
370
NOISE REDUCTION ANALYSIS
In large facilities, such as power plants, it is not always practical to measure the insertion loss of a silencer because a meaningful acoustic reference power may not be readily available. In such cases, to test the performance of a silencer in situ, the noise reduction is often measured. Then, if the distance from the silencer to the exit of the stack in which it is installed is L2 , the impedance ζ2 in Eq. A.39 is the input impedance of an open ended pipe of length L2 . The matrix elements of this pipe are P11 = P22 = cos(kL2 ), P12 = P21 = −i sin(kL2 ), where k = 2π/λ = ω/c. Then, the impedance ζ2 in Eq. A.39 is obtained as ζ1 from Eq. A.33 with Tij replaced by Pij and with ζ2 being the radiation impedance at the end of the stack. From Eqs. A.18, A.21, and A.39 it follows that N R = T L0 + 10 log(Y2 /Y1 ) + 10 log(A2 /A1 ),
(A.40)
where Y is the real part of the duct admittance and A the duct area with the subscripts 1 and 2 referring to the entrance and the exit of the duct, respectively.
A.3 COMMONLY USED MATRICES A.3.1 Porous Screen After having discussed some general properties of transmission matrices, we turn to calculations of the matrices of elements, which have been dealt with in this book. We start with the thin limp screen which was used in connection with sheet absorbers. In that context, the equivalent impedance was determined in terms of the interaction impedance z of the screen and its mass m per unit area. This interaction impedance was defined in such a manner that the product of it and the air velocity relative to the screen is the pressure drop across the screen, which is also the acoustically induced force on the screen per unit area. This impedance can be considered to be known from measurements in which the screen is held rigid. It has a resistive and a mass reactive component, which is induced by the distortion of the flow in the screen. Since an acoustical analysis deals with first order perturbations, the absolute velocity amplitude u1 in front of the screen is equal to the velocity amplitude u2 on the other side.1 We shall consider separately two cases. In the first, the screen is not in contact with any other solid object, like a porous layer, for example, and in the second case, it is. Free Screen The sound pressure amplitudes on the front and back sides of the screen are p1 and p2 , and the velocity amplitude of the screen is u . The mass per unit area of the screen is m, and we assume that any stiffness reactance of the screen can be neglected (frequency higher than the resonance frequency of the screen element). Furthermore, we assume that the screen is not in contact with any other structure, 1 Rigorously, it is the mass flux that is continuous, but the difference in the density on the two sides of the screen in the absence of a mean flow is of the first order, and, from conservation of mass flux, it follows that the difference in the velocities will be of the second order.
371
TRANSMISSION MATRICES
such as a flexible porous layer, i.e., it has air on both sides. Under these conditions it follows from the definition of the interaction impedance z ≡ ρcζ that p1 − p2 = z(u2 − u ) −iωmu 2 = z(u2 − u ).
(A.41) (A.42)
For a purely resistive screen with a flow resistance r, we have z = r = ρcθ . Usually, this assumption is satisfactory. In the equation of motion of the screen, it is assumed that there is no contact force acting on the screen. It follows then that u 2 = u2 z/(z − iωm) (A.43) and p1 = p2 + ζ ρcu2 ζ = ζ /(1 + iζρc/ωm).
(A.44)
With u1 = u2 the linear relation between p1 , u1 and p2 , u2 then can be expressed as
p1 ρcu1
1 = 0
ζ
1
p2 . ρcu2
(A.45)
Thus, the screen can be represented by a 2 × 2 matrix. To make the screen rigid (immobile), we put m = ∞. Then, for a purely resistive screen, ζ = θ , where θ is the normalized flow resistance. Screen in Mechanical Contact If the screen is in mechanical contact with some other structure, such as a flexible porous layer, the equation of motion (A.42) has to be modified. For example, if it is applied to a rigid porous structure, the motion of the screen is suppressed and only the resistive part of the screen impedance is included. Under more general conditions, with the contact pressures on the two sides of the screen denoted by p1 and p2 , the equation of motion becomes −iωmu = p1 − p2 + z(u2 − u ) − p2 .
(A.46)
Thus, the complete set of equations expressing the relations between the variables involved are p1 = p2 + zu2 − zu
u1 = u2 p1 = p2 + zm u
u 1 = u 2 = u , where zm = −iωm ≡ ρcζm .
(A.47)
372
NOISE REDUCTION ANALYSIS
These equations can be expressed as ⎛ 4 ×⎞4 matrix ⎛ for a flexible⎞screen ⎞ ⎛ p1 1 ζ 0 −ζ p2 ⎜ ⎟ ⎜ ⎟ ⎟⎜ ⎜ρcu1 ⎟ ⎜0 1 0 0 ⎟ ⎜ρcu2 ⎟ ⎜ ⎟=⎜ ⎟ ⎟⎜ ⎜ p ⎟ ⎜0 0 1 ζ ⎟ ⎜ p ⎟ m ⎠⎝ 2 ⎠ ⎝ 1 ⎠ ⎝ ρcu 1 0 0 0 1 ρcu 2
(A.48)
where ζm = −iωm/ρc, ζ : Interaction impedance, m: Mass per unit area. For a purely resistive screen, z = r = ρcθ is the flow resistance of the screen and zm = −iωm, where m is the mass per unit area of the screen. The screen is now represented by a 4 × 4 matrix. The acoustic-elastic properties of the structure in contact with the screen determine the relation between u , p1 , and p2 .
A.3.2 Area Discontinuities Area Expansion A commonly occurring acoustical element encountered in sound propagation in ducts is a sudden change in cross sectional area. With reference to Figure A.3, the areas of the duct are denoted by As and A . We shall deal only with the plane wave component in the duct and assume that the frequency is low enough so that all higher modes are evanescent. Whether a discontinuity should be classified as an area expansion or contraction is determined by the direction of the incident sound. We shall consider first an area expansion in which the area As is the smaller area and A the larger, the subscripts s and indicating this. Neglecting the difference in mass density, continuity of mass flow for the plane wave leads to the relation As u1 = A u2 for the plane wave components of the velocity amplitudes. In the transition between the small and the large area, the wave field will have a transverse velocity component (which is accounted for by evanescent higher order modes that are excited at the discontinuity). The effect of such a change in direction of the oscillatory flow can be accounted for in terms of plane wave variables by means of an induced mass reactance. The induced mass is distributed over a region which extends from the discontinuity a distance of about one diameter of the small
Figure A.3: Area expansion in a duct.
373
TRANSMISSION MATRICES
duct, but as far as the plane wave equations are concerned, it can be treated as a lumped reactance placed at the discontinuity. We express the induced mass as As ρδ1 , where δ1 is a characteristic length, often called an ‘end correction.’ For a circular opening which connects to free space, this end correction is known to be ≈ 0.61as at low frequency, kas << 1, where as is the radius. As a semi-empirical correction of this result, to make it applicable when the opening connects to a duct rather than free space, we apply a factor (1 − As /A ), which guarantees that the end correction be zero when there is no change in cross section. The corresponding mass reactance is −iωAs ρδ1 , and by dividing by As and ρc, we obtain the normalized specific reactance −ikδ1 , where k√= ω/c. For the circular aperture with the area As , the radius is as = As /π , and the end correction then becomes δ1 ≈ 0.36 As (1 − As /A ).
(A.49)
For engineering purposes, it is a good approximation to use this expression also for noncircular ducts. Actually, in addition to the reactance, there is also a small resistance, which accounts for the viscous dissipation which results from the tangential flow over the transverse surface in the transition between the two duct sections. Formally, this resistive component can be accounted for by letting the end correction be complex, thus replacing δ1 by δ1 (1 + i), where is the ratio of the resistive and reactive components. It can be expressed as the product of the surface resistance Rs , introduced earlier, and a characteristic area, which we express as πd1 δr , where δr is a resistive end correction. The corresponding normalized specific resistance is then (Rs /ρc)π d1 δr /As . It can be shown that δr ≈ δ1 and the ratio of the resistive and reactive parts of the transition impedance is ≈ 4(Rs /ρc)/kd1 . A more important contribution to the resistance results from the presence of mean flow in the duct. It is intimately related to the pressure loss in the mean flow due to turbulence, and this loss depends on the flow direction. Considering flow from the small to the large area duct, flow separation occurs, and the interaction between the sound and the flow then leads to dissipation of sound and a corresponding acoustic resistance. As a semi-empirical expression for the resistance per unit area of the small duct we shall use ≈ρU1 (1 − As /A2 ), where U1 is the mean flow velocity in the small duct. The corresponding normalized value is simply M1 (1 − As /A )2 , where M1 is the Mach number of the mean flow in the small duct. If the flow is in the opposite direction, the resistance is smaller, typically by a factor 0.2 to 0.3. Thus, for the normalized transition impedance between the small and the large duct we shall use ζ1 ≈ θ1 − ikδ1 θ1 = C M1 (1 − As /A )2 ,
(A.50)
where C ≈ 1 for flow in the direction of the larger area and C ≈ 0.3 for flow in the opposite direction.
374
NOISE REDUCTION ANALYSIS
In terms of this impedance, we get p1 = p2 + ζ1 ρcu1 = p2 + (A /As )ζ1 ρcu2 ρcu1 = (A /As )ρcu2 .
(A.51)
The corresponding matrix is Sudden area expansion in a duct(Figure A.3) T =
1
(A /As )ζ1
0
A /As
(A.52)
ζ1 : See Eq. A.50. Area Contraction The matrix for a sudden area decrease is obtained in a similar manner. The sound wave now travels from the larger to the smaller area. With reference to Figure A.4, the areas of the large and small ducts are B and Bs , and the normalized specific transition impedance is obtained as for the area expansion ζ2 ≈ θ2 − ikδ2 √ δ2 ≈ 0.36 Bs (1 − Bs /B ) θ2 = C M2 (1 − Bs /B ),
(A.53)
where M2 is the mean flow Mach number in the small area duct. The constant C is ≈ 1 for flow in the direction of the larger area and ≈ 0.3 for flow in the opposite direction. The relations between the acoustic variables are then p1 = p2 + ζ2 ρcu2 ρcu1 = (Bs /B )ρcu2
(A.54)
and the matrix is Sudden area contraction in a duct (Figure A.4) T =
1
ζ2
0
Bs /B
ζ2 : See Eq. A.53.
Figure A.4: Area contraction.
(A.55)
375
TRANSMISSION MATRICES
A.3.3 Duct Element We consider a uniform duct of length L along the x-axis with rigid, impervious walls. The general expression for a plane wave pressure field in the duct is p(x, ω) = Aeikx + Be−ikx ,
(A.56)
where k = ω/c. The corresponding velocity field is obtained from the equation of motion −iωρu = −∂p/∂x, and hence u(x, ω) = (1/ρc)(Aeikx − Be−ikx ).
(A.57)
We now wish to relate the values of the field variables at the beginning of the duct, p1 and u1 , in terms of the values p2 and u2 at the end of the duct. To do this, we express A and B in terms of p2 and u2 , and by placing x = 0 at the end of the duct (and x = −L at the beginning), we get A + B = p2 A − B = ρcu2
(A.58)
so that A = (p2 + ρcu2 )/2 and B = (p2 − ρcu2 )/2. Using these values in Eqs. A.56 and A.57, we get p1 = cos(kL)p2 − i sin(kL)ρcu2 ρcu1 = −i sin(kL)p2 + cos(kL)ρcu2
(A.59)
and the corresponding transmission matrix Ductsection with loss-free walls cos(kL) −i sin(kL) T = −i sin(kL) cos(kL)
(A.60)
where k = ω/c = 2π/λ, L: Duct length. Frequently, the duct carries a uniform flow and to account for it we need to modify the result above. The basic effect of the flow is to make the wave speed different in the two directions in the duct. If the flow velocity is U in the positive x-direction, the wave speed will be c+ = c + U = c(1 + M) in the positive x-direction and c− = c(1 − M) in the negative direction, where M = U/c is the flow Mach number. Then, with k+ = ω/c+ and k− = ω/c− , the pressure field becomes p = Aeik+ x + Be−ik− x .
(A.61)
Introducing C = exp[(k− − k+ )x/2] and k = (k+ + k− )/2 = k/(1 − M 2 ), we can rewrite this expression as
(A.62) p = C Aeik x + Be−ik x
376
NOISE REDUCTION ANALYSIS
and the corresponding velocity field is
ρcu = C Aeik x − Be−ik x .
(A.63)
Proceeding in the same manner as before we obtain the transmission matrix Duct withmean flow Mach number M −i sin(k L) cos(k L) i T =e −i sin(k L) cos(k L)
(A.64)
where = −k LM, k = k/(1 − M 2 ).
A.3.4 Contracted Duct Section, Perforated Plate From the matrices for a sudden contraction, a duct element, and a sudden expansion, we can obtain the matrix for an orifice in a duct section with a smaller area As than the area A of the main duct. The transmission matrix of this contracted duct can then be expressed as the matrix product T = Tc Td Te ,
(A.65)
where Tc and Te are the matrices for a sudden contraction and sudden expansion, respectively, and Td , the matrix for the duct section in between. A perforated plate can be handled in the same manner. In this case, however, the length of the duct element, i.e., the thickness of the plate, is much smaller than the wavelength, and the matrix can be brought into the same form as for a thin sheet. Actually, using the matrices obtained for area changes and a duct and assuming the thickness L of the plate small compared to a wavelength, the product in Eq. A.65 becomes (keeping only the first order terms in kL = 2π L/λ),
Perforated plate
1 T = −ikL
(1/s)[(ζ1 + ζ2 ) − ikL] 1
(A.66)
where s: Open area fraction, ζ1 : Eq. A.50, ζ2 : Eq. A.53. The element T12 is the impedance of the orifice plate, consisting of the mass reactance −ikL of the air in an orifice and two impedances ζ1 and ζ2 that account for viscous resistance, the mass end corrections and the resistance due to sound-flow interaction. The matrix element T21 = −ikL accounts for the compression of the air in an orifice (or, which amounts to the same thing, that the velocity is not quite constant along the orifice channel). If we neglect this effect, the transmission matrix has the same form as for the thin screen with ζ = (1/s)[−ikL + ζ1 + ζ2 ]
(A.67)
being the normalized (average) impedance of the perforated plate. To account for the acoustic nonlinearity of an orifice plate, we refer to the discussion in Section 7.5.
377
TRANSMISSION MATRICES
A.3.5 ‘Expansion Chamber’ and Elbow An expansion chamber in a duct can be created by combining a sudden expansion, a duct element, and a sudden contraction, and the total transmission matrix is obtained by multiplying the matrices of these elements, in the same way as was done in the previous section. Although not necessary, it is sometimes convenient to carry out the matrix multiplication explicitly to get the transmission matrix for the expansion chamber. Thus, if the length of the chamber is L, the area of the main duct S1 , and the area of the expanded duct section S2 , we obtain for the transmission matrix elements T12
T11 = cos(kL) − i(ζ1 /S1 ) sin(kL) = (ζ2 + (S2 /S1 )ζ1 ) cos(kL) − i(S2 + ζ1 ζ2 /S1 ) sin(kL) T21 = −i sin(kL)/S1 T22 = (S2 /S1 ) cos(kL) − i(ζ2 /S1 ) sin(kL).
(A.68)
If the walls of the chamber are treated with absorption material, we can account for that by using the transmission matrix for a lined duct rather than an unlined duct as used here. This change, of course, is easily accounted for in a computer program. As in the previous section we consider the case when the wavelength is much larger than the chamber dimensions (acoustically compact chamber), in which case the transmission matrix will be approximately independent of the shape of the chamber. Rather than use the general formulas above for small values of kL, it is instructive to start from the linearized equation for the conservation of mass ∂ρ + ρdiv (u) = 0. ∂t
(A.69)
With ∂ρ/∂t = (1/c2 )∂p/∂t and κ = 1/ρc2 integration of this equation over the volume V of the chamber yields ∂p + A 2 u2 − A 1 u1 = 0 ∂t with the corresponding complex amplitude equation κV
−iωκVp + A2 u2 − A1 u1 = 0,
(A.70)
(A.71)
where A1 and A2 are the areas of the entrance and exit ports of the chamber. The transition impedances at these ports, ζ1 and ζ2 , can be approximated by those introduced for the sudden expansion and sudden contraction in Eq. A.4, and we get p1 = p + ζ1 ρc u1 p2 = p − ζ2 ρc u2 .
(A.72)
Combining these equations, and introducing V = Av L, where Av is an average chamber area and L a corresponding average length, we find, with S1 = A1 /Av and S2 = A2 /Av , p1 = (1 − iζ1 kL/S1 )p2 + [ζ2 + ζ1 (A2 /A1 ) − (iζ2 kL/S1 )]ρcu2 ρcu1 = −ikL/S1 p2 + [(A2 /A1 ) − iζ2 kL/S1 ]ρcu2
(A.73)
378
NOISE REDUCTION ANALYSIS
and the corresponding transmission matrix is
Expansion chamber
1 − iζ1 kL T = −ikL
ζ2 + ζ1 (A2 /A1 − iζ2 kL) A2 /A1 − iζ2 kL
(A.74)
where ζ1 : See Eq. A.50, ζ2 : See Eq. A.53. Again, if we wish to include the effect of an absorptive wall treatment in this low frequency approximation, we can readily do so by including in Eq. A.71 a term (p/ρc)ηS, where S is the area of the wall treatment and η, its normalized admittance. This term expresses the sound pressure induced oscillatory flow into the wall treatment. Thus, −iωκVp will the be replaced by −iωκV (1 + iScη/ωV ) = −iωκV (1 + iη/k = −iωκ V˜ ), where we have used κ = 1/ρc2 , = V /S, k = ω/c, and introduced the complex volume V˜ = V (1 + iη/k).
(A.75)
In other words, in order to account for the wall treatment, we merely have to replace V by V˜ in Eq. A.71, which means that the quantity L in the subsequent equations will be replaced by L˜ = L(1 + iη/k). Otherwise, everything else remains the same. Elbow Frequently, an elbow is present in a duct system, as in the exhaust stack of a gas turbine power plant. The elbow is frequently lined on the wall perpendicular to the entrance duct, and the exit and entrance areas are often different. We can treat this approximately as an expansion chamber with the modification that the exit is turned 90 degrees from the entrance. This introduces an additional inertial reactance, which has to be incorporated in the transmission matrix.2
A.3.6 Lined Duct The transmission matrix for a lined duct section is obtained from Section A.3.3 replacing the propagation constant k by the complex propagation constant kx for the fundamental mode and introducing the corresponding normalized wave impedance by ζw . The transmission matrix for the lined duct portion of an attenuator of length L is
Lined duct section
−iζw sin(kx L) cos(kx L) Td = −i(1/ζw ) sin(kx L) cos(kx L)
where ζw = k/kx : See Chapter 8. 2 See, for example, Morse and Ingard, Theoretical Acoustics, 1968, Problem 13 in Chapter 9.
(A.76)
379
TRANSMISSION MATRICES
For a locally reacting liner or baffle, the expression for the wave impedance refers to the air channel alone, and we must combine it with the matrices T1 and T2 for an area contraction and an area expansion, to obtain the total matrix of the silencer T = T 1 Td T2 .
(A.77)
A.3.7 Side-Branch Tube To facilitate the calculation of the transmission and insertion loss we derive its transmission matrix for the side-branch tube. As in all sections in this appendix, the analysis is limited to the wavelengths, which are large compared to the cross sectional dimensions of the duct, and, in this case, the tube. With reference to Figure A.5, which shows both a straight and folded version of the resonator tube, the cross section area of the side branch tube is As . The length Ls includes the end correction, which is approximately 0.85as , where π as2 = As . The area of the main duct is A. We account for the possibility that the entrance to the side-branch tube is covered with a screen or perforate with a normalized impedance ζ . The impedance of the side-branch tube is, with Ls ≈ L + 0.85as , ζt = iζw cot(QkLs ),
(A.78)
where ζw = 1/Q is the normalized wave impedance, Q, the normalized propagation constant in the tube, and k = ω/c. For a tube with loss-free walls, Q = ζw = 1. For an untreated solid pipe, the losses are due to visco-thermal effects at the walls and yield an imaginary part of the propagation constant, which is typically of the order of 0.01 of the real part. If the tube is filled uniformly with a porous material, the input impedance has to be modified accordingly. Providing damping with a porous material in the tube rather than with a screen over the tube opening has the advantage that the speed of sound in the cavity and the tube length required for resonance at a particular frequency will be reduced. If the resonator opening is covered with a perforated plate, the interaction of sound with a mean flow in the duct results in acoustic losses and a corresponding flow induced resistance of the plate. This resistance should be included in the impedance z of the screen, and we refer to Chapter 8 for a discussion of this question and of related nonlinear effects.
Figure A.5: Side-branch tube in a duct with a resistive screen.
380
NOISE REDUCTION ANALYSIS
The total input impedance of the side-branch tube is then ζs = ζt + ζ,
(A.79)
and the corresponding admittance ηs = 1/ζs . With the complex amplitudes of sound pressure and fluid velocity being p1 (ω), u1 (ω) and p2 (ω), u2 (ω) just before and after the side-branch (see figure), we obtain in the long wavelength approximation, p1 = p2 ρcu1 = (As /A)ηs p2 + ρcu2 .
(A.80)
The corresponding transmission matrix is then Side-branch tube (resonator) 1 0 T = (As /A)ηs 1
(A.81)
where ηs = 1/ζs : See Eqs. A.78 and A.81.
A.3.8 Side-Branch Helmholtz Resonator The uniform side-branch tube in the previous section has its first resonance when the length of the tube is one quarter wavelength. In practice this may result in unacceptably long tubes for the attenuation of very low frequencies. The length can be reduced somewhat if the opening is provided with a constriction (orifice plate) or replaced by a Helmholtz resonator (bottle resonator). The constriction introduces an inertial mass contribution, which results in a decrease of the resonance frequency of the tube. If the wavelength is so long that the sound pressure throughout the cavity can be considered to be uniform, most of the kinetic energy in the system is concentrated in the region of the constriction and the potential energy resides in the cavity. The volume element can be of arbitrary shape. In the long wavelength approximation, the pressure amplitude in the cavity can be considered to be constant, and the impedance of the corresponding spring can be obtained directly by integrating the conservation of mass equation ∂ρ + ρdiv u = 0 ∂t over the volume. Thus, with ∂ρ/∂t = (1/c2 )∂p/∂t and κ = 1/ρc2 , we get V κ∂p/∂t + div (u) dV = 0.
(A.82)
(A.83)
With the last term replaced by a surface integral which is −uAs , where u is the velocity in the orifice and As the orifice area, we get for harmonic time dependence −iωκ Vp = As u.
(A.84)
381
TRANSMISSION MATRICES Thus, the specific input impedance to the volume is p/u = i
As . ωκV
(A.85)
The impedance of the neck is −iωρ + r, where is the neck-length (including end corrections) and r = θρc, a resistance which can be due to visco-thermal effects and/or a resistive screen across the orifice. The total specific impedance, normalized with respect to ρc, is then (recall κ = 1/ρc2 )
ω02 cAs ωo ζs = (−ik) + θ + i ωV = −ik 1 − ω2 + iD ω where ω0 = 2πf0 = c VAs , D = θ/k0 . (A.86) Quantity f0 is the resonance frequency, θ = r/ρc, θ − ik, the normalized impedance of the resonator neck, D ≡ 1/Q, the damping factor, k0 = ω0 /c, and Q the ‘Q-value’ (quality) of the resonator. The specific normalized input admittance to the side-branch is then η = 1/ζs and in terms of it, the transmission matrix takes the same form as for the side-branch tube in Eq. A.81. Flow in the duct produces a flow induces contribution to the aperture resistance, as discussed in Section 7.5, typically with the normalized value of the order of 0.1M, where M is the Mach number in the duct.
A.3.9 Parallel Channels Consider the transmission of sound through a duct which is divided into two parallel channels, A and B, by means of an acoustically hard partition along the duct, as illustrated in Figure A.6. One channel may be lined with an absorptive treatment and the other unlined. The phase velocities in the two paths will then be different, and this can be utilized to produce an interference attenuator such that the sound pressures at the exits of the two channels will interfere destructively. In a sense, it is an alternate version of the old Quincke tube, in which the interference is obtained by having different lengths of the two channels with the same wave speeds. Here, the lengths are the same but the wave speeds are different. To study quantitatively the characteristics of such a system, it is useful to determine the transmission matrix for this parallel duct combination. To do that, we start from
Figure A.6: Parallel transmission channels.
382
NOISE REDUCTION ANALYSIS
the equations, which relate the field variables at the two ends of the individual duct sections. The main duct is assumed to be loss-free and the frequency low enough so that only the plane wave component of the incidence wave will propagate. Accordingly, under such conditions, the fundamental modes in the individual parallel duct elements will dominate. The complex amplitude of the sound pressures at the beginning of the duct elements are considered to be the same and equal to p1 . We have then neglected the pressure drops that may be produced by any distortion of the flow at the entrance, which will lead to (small) inertial mass reactances. Similarly, the sound pressure amplitude at the end of the ducts is the same for both and denoted by p2 . The velocity amplitudes at the two ends are u1a , u1b and u2a , u2b . With the matrix elements of the two duct elements denoted by Aij and Bij the following relations then apply for duct A, p1 = A11 p2 + A12 ρcu2a ρcu1a = A21 p2 + A22 ρcu2a .
(A.87) (A.88)
The corresponding set of equations for duct B is p1 = B11 p2 + B12 ρcu2b ρcu1b = B21 p2 + B22 ρcu2b .
(A.89) (A.90)
The areas of the two duct branches are σa S and σb S, where S is the area of the main duct. The total velocity amplitudes in the main duct at the beginning and the end of the attenuator are then u1 = σa u1a + σb u1b and u2 = σb u2a + σb u2b , and we wish to determine the relation between these amplitudes and p1 and p2 . From Eqs. A.88 and A.90, we obtain ρcu2a = ρcu2b =
p1 A12 p1 B12
− −
A11 A12 p2 B11 B12 p2 .
(A.91) (A.92)
Multiplying Eq. A.91 by σa and Eq. A.92 by σb and adding, we obtain ρcu2 in terms of p1 and p2 , or (A.93) p1 = T11 p2 + T12 ρcu2 . In a similar manner, by multiplying Eq. A.88 by σa and Eq. A.90 by σb , and adding, we can express ρcu1 in terms of p1 and p2 . Then, using the expression for p1 in Eq. A.93, we obtain, after some straight-forward algebra, the expressions for T21 and T22 . Thus, Parallel ducts (Figure A.6) T11 = (σa A11 B12 + σb B11 A12 )/(σa B12 + σb A12 ) (A.94) T12 = (A12 B12 )/(σa B12 + σb A12 ) T21 = (T11 T22 − 1)/T12 T22 = (σa A22 B12 + σb B22 A12 )/(σa B12 + σb A12 ) The matrix elements for the individual ducts are obtained from Eq. A.76.
383
TRANSMISSION MATRICES
A.3.10 Rigid Porous Layers The transmission matrix for the rigid porous layer is obtained in much the same way as for the duct element in Eq. A.76. The pressure field is expressed as the sum of an outgoing and a reflected wave and the corresponding velocity wave is obtained from the momentum equation −iωρ˜ = −∂p/∂x, p = Aeiqx x + Be−iqx x ρc u = (1/ζw )(Aeiqx x − Be−iqx x ).
(A.95)
Here ζw = (ρ/ρ)/Q ˜ x is the normalized complex wave impedance and Qx = qx / (ω/c) the normalized complex propagation constant. The field variables are labeled by the indices 1 and 2 at the beginning and end of the layer, respectively. We put x = 0 at the end of the layer and x = −L at the beginning, and it follows that p2 = A + B ρc u2 = (1/ζw )(A − B)
(A.96)
and, consequently, A = (p2 + ρcζw u2 )/2 B = (p2 − ρcζw u2 )/2.
(A.97)
Inserting these values into Eq. A.96 and putting x = −L yields p1 = cos(qx L) p2 − iζw sin(qx L) ρcu2 ρcu1 = −i(1/ζw ) sin(qx L) p2 + cos(qx L),
(A.98)
which defines the transmission matrix
Rigid porous layer
−iζw sin(qx L) cos(qx L) T = −i(1/ζw ) sin(qx L) cos(qx L)
(A.99)
where ζw = (ρ/ρ)(q ˜ x /k) and qx .
A.3.11 Flexible Layer A flexible layer supports both an air wave and a structure wave, and we are consequently dealing with four field variables, the complex amplitudes of pressure and velocity for both these waves. It is natural, therefore, to derive a 4 × 4 transmission matrix as follows. The field variables are denoted by p, u, p , and u , where the prime refers to the structure wave. There are two characteristic wave modes with propagation constants q1x and q2x .
384
NOISE REDUCTION ANALYSIS
Although it is quite cumbersome to write out all the details in the analysis explicitly, we shall nevertheless do so rather than to use some abbreviated form of expression. Thus, p = A1 eiq1x x + B1 e−iq1x x + A2 eiq2x x + B2 e−iq2x x u = C1 eiq1x x + D1 e−iq1x x + C2 eiq2x x + D2 e−iq2x x p = 1 A1 eiq1x x + 1 B1 e−iq1x x + 2 A2 eiq2x x + 2 B2 e−iq2x x u = V1 C1 eiq1x x + V1 D1 e−iq1x x + V2 C2 eiq2x x + V2 D2 e−iq2x x .
(A.100)
The quantity 1 and P i2 are the ratios of the pressure amplitudes of the air-borneand structure-borne wave in the two modes, and V1 and V2 are the corresponding ratios for the velocity amplitudes. We have here made use of the relation between the amplitudes of the air-borne and structure-borne amplitudes expressed by the factor for pressure and V for velocity. As before, we place x = 0 at the end of the layer and x = −L at the beginning, and label the field variables at the beginning and the end by the subscripts 1 and 2. Thus, we obtain p2 = A1 + B1 + A2 + B2 u2 = C1 + D1 + C2 + D2
p2 = 1 A1 + 1 B1 + 2 A2 + 2 B2 u 2 = V1 C1 + V1 D1 + V2 C2 + V2 D2 .
(A.101)
The amplitudes A1 and C1 are related through the momentum equation applied to mode 1 (with u = V1 u1 ) to yield ˜ 1 /ωρ A1 = ρc ρ/ρ−izV C1 ≡ ρcζ1 C1 Q1x
B1 = −ρcζ1 D1
ζ1 =
ρ/ρ−izV ˜ 1 /ωρ , Q1x
(A.102)
and the analogous relations for mode 2 involving A2 , C2 and B2 , D2 and ζ2 are obtained by changing the subscripts from 1 to 2. We refer to Chapter 5, Section 5.5 for the definition of the quantities Q1x and Q2x . Introducing these relations into Eq. A.101 yields p2 = ζ1 (C1 − D1 ) + ζ2 (C2 − D2 ) ρcu2 = (C1 + D1 ) + (C2 + D2 ) p2 = 1 ζ1 (C1 − D1 ) + 2 ζ2 (C2 − D2 ) ρcu 2 = V1 (C1 + D1 ) + V2 (C2 + D2 ).
(A.103)
Solving for the velocity amplitudes in terms of the field variables at the end of the layer, we get 2C1 = α1 p2 + β1 ρcu2 + γ1 p2 + δ1 ρcu 2 2D1 = −α1 p2 + β1 ρcu2 − γ1 p2 + δ1 ρcu 2 2C2 = α2 p2 + β2 ρcu2 + γ2 p2 + δ2 ρcu 2
2D2 = −α2 p2 + β2 ρcu2 − γ2 p2 + δ2 ρcu 2 ,
(A.104)
385
TRANSMISSION MATRICES where 2 ( 2 − 1 )ζ1 , 2 β1 = V2V−V , 1 1 γ1 = − ( 2 − 1 )ζ1 1 δ1 = − V2 −V 1
α1 =
1 ( 1 − 2 )ζ2 1 β2 = V1V−V 2 1 γ2 = − ( 1 − 2 )ζ2 1 δ2 = − V1 −V2 .
α2 =
(A.105)
Using the velocity amplitudes in Eq. A.104 in terms of the field variables at the end of the layer, we return to Eq. A.102 and express the field variables at the beginning of the layer (corresponding to x = −L) in terms of those at the end. The result can be written ⎛ ⎞ ⎛ ⎞ p1 p2 ⎜ρcu1 ⎟ ⎜ ⎟ ⎜ ⎟ = M ⎜ρcu 2 ⎟ , (A.106) ⎝ p ⎠ ⎝ p ⎠ 1 2
ρcu1 ρcu2 where the elements of the 4 × 4 transmission matrix are M11 = ζ1 α1 cos(q1x L) + ζ2 α2 cos(q2x L) M12 = −iζ1 β1 sin(q1x L) − iζ2 β2 sin(q2x L) M13 = ζ1 γ1 cos(q1x L) + ζ2 γ2 cos(q2x L) M14 = −iζ1 δ1 sin(q1x L) − iζ2 δ2 sin(q2x L) M21 = −iα1 sin(q1x L) − iα2 sin(q2x L) M22 = β1 cos(q1x L) + β2 cos(q2x L) M23 = −iγ1 sin(q1x L) − iγ2 sin(q2x L) M24 = δ1 cos(q1x L) + δ2 cos(q2x L) M31 = 1 ζ1 α1 cos(q1x L) + 2 ζ2 α2 cos(q2x L) M32 = −i 1 ζ1 β1 sin(q1x L) − i 2 ζ2 β2 sin(q2x L) M33 = 1 ζ1 γ1 cos(q1x L) + 2 ζ2 γ2 cos(q2x L) M34 = −i 1 ζ1 δ1 sin(q1x L) − i 2 ζ2 δ2 sin(q2x L) M41 = −iV1 α1 sin(q1x L) − iV2 α2 sin(q2x L) M42 = V1 β1 cos(q1x L) + V2 β2 cos(q2x L) M43 = −iV1 γ1 sin(q1x L) − iV2 γ2 sin(q2x L) M44 = V1 δ1 cos(q1x L) + V2 δ2 cos(q2x L).
(A.107)
If we make use of the expressions for α1 . . . δ2 in Eq. A.105, we note that they are not independent and the expressions for the matrix elements can be simplified accordingly. The 4 × 4 matrix of the porous layer can readily be reduced to a 2 × 2 matrix for layers with one or both ends open or closed, and we start with the case of both ends being open.
386
NOISE REDUCTION ANALYSIS
‘Open’ and ‘Closed’ Surfaces In the following discussion, the designations ‘open’ and ‘closed’ refer to the condition of the surface of the porous material. Thus, by ‘open’ is meant that the surface of the porous material does not contain an impervious skin or cover (or any cover, for that matter) in which case the air velocity and the velocity of the structure are not constrained to be the same at the surface. A ‘closed’ surface, on the other hand, does have such a cover in which case the boundary condition requires that the velocity amplitude of the air and the structure be the same at the surface. Open-Open Layer With both ends open we have p1 = p2 = 0 and if these conditions are introduced in Eq. A.106, we can express u 2 in terms of p2 and u2 , ρcu 2 = −(M31 p2 + M32 ρcu2 )/M34 .
(A.108)
Using this result, together with p2 = 0, in the expressions for p1 and ρcu1 we get p1 = T11 p2 + T12 ρcu2 ρcu1 = T21 p2 + T22 ρcu2 ,
(A.109)
where T11 = M11 − M14 M31 /M34 T12 = M12 − M14 M32 /M34 T21 = M21 − M24 M31 /M34 T22 = M22 − M24 M32 /M34 .
(A.110)
Closed-Closed Layer The boundary conditions for a layer with both ends closed are u2 = u 2 , u1 = u 1 , and from the latter we can express p2 in terms of p2 and u2 , p2 = (M21 − M41 )/(M43 − M23 )p2 + (M12 + M14 )ρcu2 .
(A.111)
Making use of this relation and u 2 = u2 , the 2 × 2 matrix elements for the layer are found to be T11 = M11 + M13 (M21 − M42 )/(M43 − M23 ) T12 = M12 + M14 + M13 (M22 + M24 − M42 − M44 )/(M43 − M23 ) T21 = M21 + M23 + M23 (M21 − M41 )/(M43 − M23 ) T22 = M22 + M24 + M23 (M22 + M24 − M42 − M44 )/(M43 − M23 ). (A.112) Open-Closed Layer The boundary conditions are now p1 = 0 and u 2 = u2 . From the first it follows that p2 = −(M31 /M33 )p2 − (M32 + M34 )/M33 ρcu2 .
(A.113)
387
TRANSMISSION MATRICES From this relation and u 2 = u2 it follows that T11 = M11 − M13 M31 /M33 T12 = M12 + M14 − M13 (M32 + M34 )/M33 T21 = M21 − M23 M31 /M33 T22 = M22 + M24 − M23 (M32 + M34 )/M33 .
(A.114)
Closed-Open Layer The boundary conditions are now p2 = 0 and u 1 = u1 . From the latter it follows that ρcu 2 = (M21 − M41 )/(M44 − M24 )p2 + (M22 − M42 )/(M44 − M24 )ρcu2 , (A.115) and we obtain T11 = M11 + M34 (M21 − M41 )/(M44 − M24 ) T12 = M12 + M14 (M22 − M42 )/(M44 − M24 ) T21 = M21 + M24 (M21 − M41 )/(M44 − M24 ) T22 = M22 + M34 (M22 − M42 )/(M44 − M24 ).
(A.116)
Example: Input Impedance of Porous Layer Backed by a Rigid Wall As an illustration of the use of the transmission matrix we calculate the input admittance or impedance of a porous layer backed by a rigid wall. With the boundary conditions u2 = u 2 = 0, the expressions for p1 and u1 are p1 = M11 p2 + M13 p2
u1 = M21 p2 + M23 p2 ,
(A.117)
and the input admittance is obtained as ρcη = u1 /p1 . The boundary condition at the input end of the layer is that the stress in the porous material is zero, p1 = 0, which means p1 = T31 p2 + T33 p2 = 0,
(A.118)
i.e., p2 /p2 = −M31 /M33 . The expression for the admittance is then η = u1 /p1 = (M11 M33 − M23 M31 )/(M11 M33 − M13 M31 ).
(A.119)
Inserting the expression for the matrix elements in Eq. A.119, we find that this admittance indeed is the same as that obtained by other means.
A.3.12 Thin Porous Plate The effect of bending stiffness in the expression for the structural impedance of a screen with a mass m per unit area was shown to be
−iωm ω2 ζs = (A.120) 1 − 2 sin2 φ , ρc ωc
388
NOISE REDUCTION ANALYSIS
√ where ωc = c2 / B and B = v 2 h2 /12, the bending stiffness, and φ the angle of incidence. Quantity v is the longitudinal wave speed in the plate, v 2 = Y /[ρp (1 − σ 2 )], Y , the Young’s modulus, ρp , the mass density of the plate, σ , the Poisson ratio, and φ the angle of incidence. This would be the impedance for a solid, impervious plate. When the plate is porous with a normalized interaction impedance ζ , the equivalent impedance will be ζs ζ ζ = (A.121) ζs + ζ as explained in Chapter 2. The corresponding transmission matrix is then
1 ζ
T = . 0 1
(A.122)
Appendix B
Flow Resistance Measurements B.1 SIMPLE METHOD FOR STEADY FLOW In a conventional flow resistance apparatus, flow is forced through the sample to be tested, and the flow rate and the pressure drop across the sample are measured. The total flow resistance of the sample is then obtained as the ratio of the pressure drop and the flow velocity. Division by the sample thickness then yields the flow resistance per unit length. The pressure drop and the related density change are assumed to be small enough so that there is no significant difference between the upstream and downstream flow velocities. The apparatus requires a flow moving device, typically a blower of some kind, and instruments for measuring the pressure drop and the flow rate. In the apparatus described here, the need for these auxiliary pieces of equipment is eliminated. The flow is produced by a piston of known weight, which falls under the influence of gravity through a vertical (or inclined) tube, covered at one end (or both) with the porous sample to be tested, as shown schematically in Figure B.1. The speed of the piston depends on the flow resistance of the sample. As we shall demonstrate shortly, the piston quickly reaches its terminal velocity, which is determined by measuring the time required for the piston to fall a known distance L in the tube. Apart from the small leakage flow between the piston and the tube wall, the flow velocity through the sample is simply the velocity of the piston if the areas of the tube and the sample, A and S, are the same. Otherwise, we have to multiply the piston velocity by A/S. The tube can be pivoted about a horizontal axis and locked at any desired angle of inclination. If the sample is mounted at the top of the tube, an elastic strap (with negligible flow resistance) across the bottom end of the tube catches the piston as it comes down. When the piston has reached its terminal speed, the pressure drop across the sample is the same as the pressure drop across the piston, which, in turn, is determined by the weight of the piston if the friction force on the piston is negligible. As we shall see, this is the case in the range of flow resistances of interest, and we then find that
389
390
NOISE REDUCTION ANALYSIS
Figure B.1: A simple apparatus for the measurement of the steady flow resistance of a porous material. A piston falls under gravity in a tube and pumps air through the porous sample mounted at the bottom (or at the top). The time of fall over distance L determines the flow resistance. flow resistance is given by the expression Measurement of flow resistance r=C
Mg cos(φ)S ts LA2
(B.1)
where M: The mass of the piston, g: Acceleration of gravity, φ: Angle of tube axis with the vertical, S: Sample area, A: Tube area, L: Travel distance of the piston, ts : Travel time over the distance L when the sample is present. (See also Eq. B.3.) The quantity C is a correction factor accounting for the small leakage between the piston and the tube wall and the friction against the wall. It can usually be set equal to unity. It is given by 1 − t0 /ts C= , (B.2) 1 − ts /t∞ where t0 is the travel time when there is no sample (tube open) and t∞ the time when the tube is closed with an impervious cover plate. Since, generally t0 /ts and ts /t∞ are << 1, we can in most cases put C = 1. In our tube (see Figure B.1) we found t0 ≈ 0.1 s and t∞ ≈ 75 s. With a porous sample in place, the measured travel time typically is ≈ 1 s and with t0 ≈ 0.1 and t∞ ≈ 75 s, the correction factor becomes C ≈ 1.014. If the travel time t is of the order of 10 seconds or larger the influence of the leakage flow should be corrected for. Thus, with t = 10, we get C ≈ 1.15. The relative error r/r in
FLOW RESISTANCE MEASUREMENTS
391
the measured flow resistance r is a minimum, 4ts /t∞ , if ts = t∞ /2. This minimum becomes about half a percent if the error ts in the time measurement is assumed to be ≈ 0.1 sec. In our experimental model of the apparatus, a Lucite tube was used with an inner diameter of 88 mm and a wall thickness of 6 mm. The total length of the tube was 120 cm, but only about half of this length was used as the travel distance L of the piston. The piston, also made from Lucite, had a mass of 264 grams and a length of 10 cm. The annular gap between the cylinder and the tube wall was approximately 0.2 mm. Porous samples were placed at one or both ends of the tube and held in place by specially designed sample holders. The tube was supported by a stand so that it could be pivoted about a horizontal axes through the center of the tube. Thus, with M = 264 g = 981 cm/s2 , S = A = π 8.82 /4 cm2 , and L = 62 cm, we get (from Eq. B.1) for the measured normalized flow resistance, r ≈ 1.64 ts cos(φ). ρc
(B.3)
If the area S of the sample is not equal to the area A of the tube, this expression should be multiplied by S/A. This makes it possible to extend the range of the flow resistances, which conveniently and accurately can be measured with the apparatus. By varying the angle φ or the weight of the piston, the velocity dependence (if any) of the flow resistance can also be determined. It should be noted that the method can be used for measurement in both gases and liquids. If one end of the tube is covered with a solid plate so that there is no air leakage into the tube, r ≈ ∞, the piston moves very slowly down the tube due to the small air leakage between the piston and the tube wall. (As already mentioned t∞ = 75 seconds for the piston to fall L = 62 cm in our apparatus.) Therefore, the piston can essentially be held at rest by closing the top of the tube with a cover plate and then released by removing the plate. For an open tube, r ≈ 0, the travel time was found to be t0 ≈ 0.13 sec. When a sample is in place, the time ts typically is between 1 and 20 seconds for most materials of interest. After a sample has been inserted, the tube is pivoted into the vertical position so that the piston will be at the top of the tube, and held in place by covering the top of the tube with a plate, as just mentioned. Removal of the cover plate starts the motion of the piston, which quickly acquires its terminal constant speed. The time ts required for the piston to fall a distance L (62 cm in our case) is measured with a stop watch. The range of measurable flow resistance can be altered by varying the physical parameters of the apparatus, the piston mass M, the sample area S, and the inclination angle φ. The range is limited on the lower end by the friction force between the piston and the tube wall, expressed in terms of the shortest possible time t0 (tube open), and at the upper end by the air leakage between the piston and the tube wall, expressed by the longest possible time t∞ of travel. With S/A = 1, these limits for the normalized flow resistance in our apparatus were ≈ 0.16 and 60. In summary, the flow resistance can be determined simply by means of a stop watch and does not require any flow moving device, flow rate meter, or pressure gauge. The procedure of measurement is the following: (1) Place the sample in the sample holder at the top of the tube.
392
NOISE REDUCTION ANALYSIS
(2) Pivot the tube so that the piston moves to the top of the tube. Once the piston is at the top, place a rigid cover plate over the top so that the piston can be held at the top (its motion is negligible). (3) Remove the cover plate. The piston then starts to fall. Measure the time of travel between two given marks on the tube (in our case separated by 62 cm).
B.1.1 Equations of Motion In the following discussion, we imagine the sample to be at the bottom of the tube. If the terminal velocity of the piston is v, conservation of mass requires that Avρ = (P − P0 )(Sρ/r + Sa ρ/ra ),
(B.4)
where P is the pressure between the piston and the sample and P0 the ambient pressure outside the tube and on the top of the piston. As before, r is the flow resistance of the sample, S the area of the sample, and ρ the air density. Sa and ra are the corresponding quantities for the annular gap between the piston and the cylinder. Thus, the first and the second term in the equation account for the mass flow rate through the sample and the leakage flow through the annulus between the piston and the cylinder. The latter is not known a priori but can be determined from the measurement of the velocity of the piston when the tube is closed, corresponding to r = ∞. Thus, in terms of corresponding travel time t∞ , we have Sa /ra = A2 v∞ /Mg = A2 L/(Mgt∞ ).
(B.5)
When terminal speed has been reached, the net force on the piston is zero, and we have Mg = (P − P0 )A + αv, (B.6) where we have set the piston area equal to the tube area A. The second term represents the friction force between the piston and the tube wall and is not known a priori. However, it can readily be determined from the travel time t0 when the tube is open, and we get α = Mg/v0 = Mgt0 /L. (B.7) Thus, combining these equations, we obtain the result given in Eq. B.1, having used v = L/ts and neglected a term t0 /t∞ << 1. As an interesting side note, one might wish to calculate the leakage flow and the friction force on the piston under the assumption that there is no contact between the piston and the cylinder so that only the viscous flow in the annulus plays a role. If we treat the annulus locally as a region between two parallel plates, we can use the results already obtained in Chapter 3 for the flow resistance. Accordingly, the characteristic times t0 and t∞ can be computed in terms of the coefficient of shear viscosity and the other parameters of the apparatus. The result is t0 = (μ/Mg)π DLH /d t∞ = (12μ/Mgπ)A2 LH /(Dd 3 ).
(B.8)
FLOW RESISTANCE MEASUREMENTS
393
In practice, no doubt, there could be some additional friction due to contact between the piston and the cylinder; this would result in a characteristic time ts , which is larger than the value t0 given by this equation. The time t∞ , however, is indeed consistent with the measured value. The gap size, which corresponds to t∞ = 75 sec, is ≈ 0.023 cm. Generalization The previous discussion was based on the assumption that the terminal speed had been reached when the measurement of piston velocity was made from the time ts required to travel the distance L. We consider now an alternate procedure and measure the time required to travel a distance L, but this time we start the clock when the piston is released. How can we from this measurement determine the flow resistance? We start with the equation of motion for the piston Mdv/dt = Mg − (P − P0 )A − αv,
(B.9)
where the last term is the friction force from the tube wall. We start by neglecting this term. The same applies then to the second term in Eq. B.4 so that (P − P0 ) = (Aρr/S)v in which case the equation of motion takes the form dv = g − βv, dt
(B.10)
where β = A2 r/SM. In steady state, the solution of v = g/β, and with v = L/ts , we get Eq. B.1. The general solution for velocity is v = (g/β)(1 − exp(−βt)) and for the travel distance gt 1 − e−βt x = (1 − ) (B.11) β βt which, of course, again reduces to Eq. B.1 as βt gets large. We now solve this equation numerically for β, and hence for the flow resistance r in terms of the measured value of x and the corresponding value of t. To improve on this even further, we can introduce the effect of wall friction and leakage in terms of the measured values of t0 and t∞ . However, this will not be pursued here but is saved for one of the problems in Appendix D.
B.1.2 Nonlinearity of Flow Resistance In a conventional apparatus for the measurement of the steady flow resistance, air is forced through a porous sample with a blower or suction device, and the flow rate through and pressure drop across the sample are measured with a flow meter and a (differential) pressure gauge. For a porous material, the flow resistance can be considered to be independent of the flow speed only at sufficiently low speeds but generally increases with increasing speed. It should be realized, however, that the velocity amplitude in a sound wave, typically of the order of 1 cm/sec, is much smaller than typical mean flow velocities, and
394
NOISE REDUCTION ANALYSIS
the relevant flow resistance values of a porous material ideally should be measured at such low flow speeds. However, it is difficult to measure the corresponding small pressure drops accurately, and to obtain the flow resistance in the linear regime of a porous material, the experimental data normally have to be extrapolated down to zero velocity. Examples of the measured velocity dependence of the flow resistance of samples of commercially available fiber metals are shown in Figure B.2. For curve (1) in the left graph, the flow resistance is essentially constant for flow velocities less than 20 cm/sec, and the value extrapolated to zero velocity is approximately 45 CGS (450 MKS). The main reason for the nonlinearity at higher flow velocities is that in addition to the viscous drag force on the material, there is also a turbulent contribution which increases as the square of the velocity. In some very flexible materials, there is also the possibility that an increase in the velocity can cause a deformation of the material with a related change in the resistance. Densely woven wire mesh screens have characteristics similar to those of feltmetal data, and a detailed discussion of the resistance of wire mesh cloth will be given later in connection with the measurements of the acoustic flow resistance, i.e., resistance to oscillatory flow. It should be noted that the nonlinearity of the flow resistance can be considerably different for different materials even though their linear flow resistance may be approximately the same. For flow through an orifice in a rigid plate, the turbulent contribution to the flow resistance is usually dominant, and the steady flow resistance is then found to increase linearly with the velocity, a reflection of the fact that the pressure drop is approximately proportional to the square of the velocity. In a conventional flow resistance apparatus, it is generally not possible to measure accurately the low pressure drops, which are involved in the low flow regime where the viscous resistance is dominant and where the flow resistance is independent of flow speed. This is illustrated in Figure B.2, right
Figure B.2: Left: Typical velocity dependence of the steady flow resistance of some porous materials: (1). Brunsmet, thickness 0.025 inches. (2). Feltmetal, thickness 0.043 inches. (3). Brunsmet, thickness 0.025 inches. (4). Feltmetal, thickness 0.043 inches. Right: Perforated sheet with open area fractions 1.45, 5.5, 9, 14, 22, and 30. Hole diameters and thicknesses: (0.125, 0.05), (0.025, 0.05), 0.085, 0.15), (0.093, 0.015), (0.093, 0.040), and (0.125, 0.015) inches, respectively.
395
FLOW RESISTANCE MEASUREMENTS
graph, which shows the velocity dependence of the measured flow resistance for a number of perforated plates with open area fractions ranging from 1.45 to 30 percent. As can be seen, the flow resistance is approximately proportional to the flow speed in the ranges of flow velocities at which these measurements could be carried out with conventional instrumentation. The pressure drop across a test sample is the product of the flow resistance and the flow speed. The results indicate that this product is very nearly the same for the initial values of all the curves shown in the figure, and the corresponding pressure drop is found to be P ≈ 360 dyne/cm2 or (P )min ≈ 0.14 inches of water. This presumably was the lowest pressure drop that could be measured with the apparatus that was used so that data to the left of the dashed line in the figure could not be obtained. This dashed line corresponds to r = (P )min /U , where U is the flow velocity. As a good approximation, the normalized flow resistance of a perforated plate in the range of open areas and velocities given in the figure can be expressed as 1−s U r ≈ 2 , ρc s c
(B.12)
where U is the average flow velocity over the orifice plate, and s the open area fraction. Effect of a Perforated Facing on a Porous Layer When porous materials such as glass wool are used as sound absorbers or duct liners, they are frequently covered with a perforated facing with an open area of typically 23 percent or more. The effect of the facing is usually negligible when it is combined with a relatively thick porous material, particularly if it is in loose contact with the material. However, if the facing is combined with a thin porous sheet, such as fiber metal, the flow resistance of the combination will be larger than the flow resistance r of the porous sheet by a factor between 1 and 1/σ , where σ is the open area fraction of the perforated facing. This is because the velocity through the screen at each perforation would be larger than the average velocity across the plate by the factor 1/σ , thus generating a larger pressure drop.
B.2 SIMPLE METHOD FOR OSCILLATORY FLOW For oscillatory flow in a porous material, such as produced by sound with harmonic time dependence, the pressure drop across a (thin) layer of the material and the velocity through the layer are generally not in phase. This indicates that the material is not strictly resistive but contains a reactance as well. This is to be expected, of course, since the air in the material has a certain mass which will present an inertial mass reactance. As explained earlier, there is an additional inertial mass contribution, which is caused by two effects, the viscous interaction and the forced deviation of the flow through a tortuous path through the material. These effects can be accounted for by using an effective air mass density in the material, which often is larger than the free field mass density, typically by 50 percent.
396
NOISE REDUCTION ANALYSIS
The steady flow resistance of a porous layer was defined as the ratio of the pressure drop over the layer and the steady flow speed through it. If a layer is strictly resistive (almost true for some very thin materials), the oscillatory flow through the sample is in phase with the oscillatory pressure drop across the sample and the ratio of their amplitudes is the flow resistance of the sample. In general, however, the velocity lags behind the pressure drop by a certain phase angle φ, and a complete description of the porous layer requires that both the ratio |z| of the amplitudes of the pressure drop and velocity and φ be determined. Normally, the resistance r = |z| cos(φ) and the reactance x = |z| sin(φ) are specified rather than |z| and φ. The oscillatory flow resistance increases slowly with frequency because of the frequency dependence of the viscous boundary layer thickness as explained in Chapter 3. The steady flow resistance, if extrapolated down to the linear regime, as discussed above, is found to be the same as the oscillatory flow resistance at sufficiently low frequencies, and in many cases of practical interest, it is a good approximation to neglect the frequency dependence of the resistance and use the measured value for steady flow. An apparatus for the measurement of the flow impedance is shown schematically in Figure B.3. In our prototype model of the apparatus, the driver section consists of a 60 watt horn type loudspeaker to which is attached an extension tube with an inner diameter of 2 inches. The sound source is driven by an ordinary signal generator and amplifier, either with a pure tone or random noise depending on the detection instrumentation that is available. The sample to be tested is placed at the end of the extension tube, as shown, and a microphone, mounted flush with the inner surface of the tube, is placed just ahead of the sample. This microphone measures the (complex) sound pressure amplitude p1 at this location. The receiver tube of length L is terminated with a rigid plug (4-inch long aluminum plug) and contains a flush mounted microphone for measurement of the (complex) sound pressure amplitude p3 at this termination. The length of the tube is L = 85 cm, which is a quarter wavelength at a frequency of 100 Hz. Tubes with other lengths can be used, of course. We shall assume here that the thickness of the sample is much smaller than the wavelength, say less than 1/10 of the wavelength. If the frequency is chosen to be an odd multiple of 100 Hz, i.e., 100, 300, etc., the oscillatory flow resistance of the sample
Figure B.3: Experimental arrangement for the measurement of the flow resistance or impedance of a porous layer for oscillatory flow, i.e., sound.
FLOW RESISTANCE MEASUREMENTS
397
is equal to the ratio between the magnitudes of the sound pressures, p1 and p3 , to an approximation, which is sufficiently accurate for most practical purposes. To be more accurate, the phase difference between the pressures should also be measured, and the ratio just mentioned should be multiplied by cos(φ−π/2), which is a quantity close to 1 since the phase difference differs only slightly from π/2. The phase difference can be measured with a phase meter or by observing the pressure signals on a dual beam oscilloscope. It is then convenient to adjust the amplitudes of the traces to be the same by means of a calibrated attenuator and then read the amplitude ratio on the attenuator. If a two-channel FFT analyzer is available, the measurement can be simplified considerably. Then, with the sound source generating a broad band noise, the frequency dependence of the real and imaginary parts of the complex ratio, p1 (ω)/p3 (ω), can be displayed directly with the analyzer operating in the Transfer Function mode. By moving the frequency cursor of the instrument to the frequencies at which the length L of the receiver tube is an odd multiple of quarter wavelengths (in our case with L = 85 cm, corresponding to 100, 300 Hz, etc.), the numerical values of the real and imaginary parts of p1 (ω)/p3 (ω), and hence the real and imaginary parts of the flow impedance, can be read directly on the instrument screen without any need for further data processing since the normalized resistance is the magnitude of the imaginary part and the reactance is the real part. The sign of the reactance is (−1)n−1 , where n = 1 corresponds to 100 Hz, n = 2 to 300 Hz, etc. As we did in the measurement of the steady flow resistance, it is of interest to explore the limitations of the flow impedance measurement apparatus in terms of the range of flow resistances that can be measured. The primary factors, which determine the smallest and largest flow resistance that can be measured, are the losses in the receiver tube and the flanking transmission at the porous sample. They are determined experimentally by running ‘calibration tests’ with the sample removed and with a sample consisting of a rigid impervious plate. In the absence of a sample, the equivalent flow resistance values obtained at 100, 300, 500, 700, and 900 Hz were 0.021, 0.038, 0.046, 0.052 and 0.058, respectively. These values are consistent with what can be expected from the visco-thermal losses on the walls of the receiver tube, and they are much lower than for the materials commonly of interest in acoustics, and a correction for the effect of the losses is not necessary. It is important, however, that the termination be rigid and without leaks. Our 4 inch aluminum plug was arrived at after some experimentation with other terminations. The upper limit of the measurable flow resistance was determined by using a rigid plate as a sample, and the corresponding normalized resistance was found to be about 20. The frequency at which the receiver tube length L equals one quarter wavelength varies somewhat with the temperature. As a ‘calibration’ preceding a series of measurements, we determined this frequency by varying it until the pressure p1 was minimum when no sample was present (the corresponding pressure ratio p1 /p3 at twice the frequency was then equal to one). In our apparatus, designed for a lowest frequency of 100 Hz, the required variation in this frequency was usually less than 1 Hz.
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NOISE REDUCTION ANALYSIS
B.2.1 Some Experimental Results With a standard two-channel analyzer, as described above, the flow resistance of a number of porous materials has been measured, and some of the results obtained are presented here. Wire Mesh Screens The flow resistance of wire mesh screens is of particular interest. This type of porous material is rather well defined for a particular screen since both the wire diameter and the number of wires per unit area of a screen are known. The experimental results, which we shall discuss here, refer to three sets of screens with 30, 60, and 100 mesh per inch and wire diameters 0.012, 0.0075, and 0.0045 inches, respectively. If the wires form a pattern of squares, with the open squares having a width b, we have b = (1/n) − d and the open area fraction of the screen σ = (1−nd)2 , where n is the number of wires per unit length and d the wire diameter. For the 30, 60, and 100 mesh screen, we get b = 0.021, 0.0092, and 0.0055 inches, respectively, and the corresponding open area fractions are 0.41, 0.30, and 0.30. We can get an idea of the frequency dependence of the acoustic flow resistance of a screen by making use of the results obtained in Chapter 2 for sound propagation in a narrow channel. We found that the resistance is essentially independent of frequency if the channel width is less than the √ viscous boundary layer thickness for oscillatory flow. This boundary layer thickness is 2ν/ω (see Chapter 3), where ν is the kinematic viscosity (dv ≈ 0.15 CGS for air) and ω the angular frequency. At 100 Hz, dv ≈ 0.022 and at 900 Hz, ≈ 0.007 cm. Thus, for the 100 mesh screen, we have dv > b in the entire frequency range of our measurements, 100 to 900 Hz, and the resistance is then expected to be independent of frequency in this range. For the 30 mesh screen, on the other hand, we have dv > b only at frequencies less than 100 Hz, and we expect a frequency dependence in our range of measurements. The experimental results, shown in Figure B.4, indeed confirm these predictions with the resistance for the 100 mesh screen being essentially independent of frequency and increasing with frequency √ for the 30 mesh screen. Theoretically, this frequency dependence should be ∝ ω asymptotically. These results refer to 25 layers of closely packed screens. Figure B.5 shows the flow resistance as a function of the number of screens in a stack. The flow resistance increases linearly with the number of screens, at least when more than approximately five screens are involved. The thickness of a stack of 25 screens of the 100 mesh material was 0.5 cm, and the corresponding normalized flow resistance per cm is then 1.0. For the 60 mesh material the corresponding values are 0.85 cm and 0.35. As already indicated, the open area fractions for the 60 and 100 mesh screens are the same (as are the ratios b/d), and this gives us an opportunity to check the ‘scaling’ law that predicts that, for a given porosity, the resistance per unit length should be inversely proportional to the square of the air gap between the wires (see Chapter 3). The ratio of the flow resistances, 1.0/0.35 ≈ 2.9, is in satisfactory agreement with the ratio of the squares of wire diameters, (7.7/4.5)2 ≈ 2.8.
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0.5
Normalized flow resistance
0.1
100
1000 Frequency, Hz
Figure B.4: Measured frequency dependence of the (normalized) flow resistance of a stack of 25 wire mesh screens. Bottom curve: 30 mesh/inch wire diameter 0.012 inches. Middle curve: 60 mesh/inch wire diameter 0.0075 inches. Top curve: 100 mesh/inch wire diameter 0.00445 inches. Layer thicknesses: 1.4, 0.85, and 0.5 cm, respectively. 0.6
0.6 900 Hz
0.5
700
0.5 0.4
0.4 Normalized flow resistance 0.3
Normalized reactance
0.2
0.2
0.1
0.1
0
0
0
5
10 15 20 25 Number of screens
30
500
0.3 300
100
0
5
20 15 10 Number of screens
25
30
Figure B.5: Left: Measured flow resistance (normalized) of a package of screens described in Figure B.4 vs the number of screens in the package for 30, 60, and 100 mesh/inch screens (bottom, middle, and top curves, respectively). The multiple markers on each curve refer to the frequencies 100, 300, 500, 700, and 900 Hz, and correspond to the frequency dependence shown in Figure B.4. Right: Measured reactance (normalized) for the 30 mesh screens.
For the 30 mesh screens, the open area fraction is larger and the flow resistance per unit length should be smaller than predicted by the scaling law. This also is consistent with the experimental results. The measured reactive part of the flow impedance for the 30 mesh screen is shown in Figure B.5 as a function of the number of screens in the stack with frequency as a parameter. The reactance increases linearly with the thickness of the stack, at least
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NOISE REDUCTION ANALYSIS
when more than approximately 5 screens are involved. With 25 screens, the thickness was 1.4 cm and the normalized reactance per cm at 900 Hz is 0.45. The corresponding reactance of a 1 cm thick air layer is ω/c ≈ 0.14. Thus, the effective inertial mass density in the material is larger than in free field by a factor of 2.7. This factor was found to increase with decreasing frequency toward a value of ≈ 3 at 100 Hz. For the 60 and 100 mesh screens, the factor was found to be about 15 percent larger, and this presumably is related to the fact that the open area of these screens was smaller than for the 30 mesh screen. In Chapter 3 it was shown that the part of the structure factor that is due to the viscous interaction force decreases with increasing frequency, thus, the results obtained here are consistent with this behavior. To check for possible nonlinear effects, the measurements were made at different sound pressure levels up to 130 dB on the front side of the material, corresponding to a velocity amplitude of approximately 200 cm/sec in the material. No nonlinearity was found, however. If the material is flexible, the measured flow impedance will reflect the structural response of the sample, and the frequency dependence can be quite complex and dependent on the size of the sample and the boundary conditions. If one wishes to eliminate these structural effects and obtain an impedance which refers to the relative motion of the air and the sample, one has to mount the sample between rigid, acoustically transparent screens. A porous screen in water. ‘What is the effect of a screen on the flow and sound in water?’ This is a frequently asked question of considerably interest, which deserves a comment at this point. The flow resistance is proportional to the shear viscosity, which for air is μa ≈ 0.0183 and for water μw ≈ 1.0 CGS units (poise), i.e., about μw ≈ 54.6μa . On the other hand, the effect of a screen on sound transmission through the screen is determined by its flow resistance in relation to the wave impedance ρc, which for water is about 3568 times larger than for air. Thus, a flow resistance of 1 (ρc)a in air will be 54.6/3568 ≈ 0.015 (ρc)w in water. In other words, if a screen Sw is to have the same effect acoustically in water as another screen Sa in air, the flow resistance of Sw must be about 67 times larger than the flow resistance of Sa . Assuming the Reynolds number to be low enough so that viscous forces dominate, a steady flow through a screen results in a pressure drop across the screen of P = rU , where r is the flow resistance and U the flow speed. For the same screen and the same velocity, the pressure drop in water will be ≈ 54.6 times that in air, but the screen will be a great deal more acoustically transparent in water than in air. Although the viscosity μ of water is larger than for air, the kinematic viscosity ν = μ/ρ is about 7.1 times smaller. This means that for a given flow velocity and linear size of an object (for example, wire diameter in a wire mesh screen), the Reynolds number in water will be 7.1 times larger than in air. Thus, although the flow in air may be laminar, the corresponding flow in water can be turbulent.
B.2.2 Other Materials With the method described above, the acoustic resistance, reactance, and the corresponding structure factor have been determined for many materials other than wire
FLOW RESISTANCE MEASUREMENTS
401
mesh screens. For example, glass wool, both bonded (boards) and not, have been measured. Thus, a 4.5 lb/ft3 fiberglass board was found to have a normalized flow resistance at 100 Hz of 0.5 per inch and a structure factor of 1.8 for oscillatory flow normal to the board. In the direction parallel with the board the corresponding values were 0.23 and 2. The frequency dependence of the resistance was small. For a fiberglass batt with a density of 6 lb/ft3 , the values were 1.68 and 1.9. Again, the frequency dependence was only slight. In measurements of flexible foam-like materials, the frequency dependence of the flow resistance often was found to be stronger and the structure factor larger. For example, for Solimide with a density of 2 lb/ft3 , the normalized flow resistance at 100 Hz was 1.56 per inch and 1.99 at 500 Hz. The structure factor decreased from 10.7 to 4.5. This is interpreted in part as an effect of the induced motion of the material, the motion being more pronounced at low frequencies, thus making the equivalent mass density of the air and the corresponding structure factor larger.
B.2.3 A Supplementary Note It remains to present in more detail the formal analysis of the measurement apparatus shown in Figure B.3. The impedance of the test sample is defined as zf (ω) = [p1 (ω) − p2 (ω)]/u(ω),
(B.13)
where p1 and p2 are the complex amplitudes of the sound pressures on the two sides of the sample and u(ω) the complex amplitude of the velocity. For a uniform porous layer the impedance z1 per unit length used in Chapters 4 and 5 is obtained simply by dividing by the thickness of the material. With reference to Figure B.13, the sample is placed across a tube with rigid walls and a rigid termination. A sound field is produced in the tube by a sound source located at one end, as shown. It is assumed that the frequency is low enough so that only the plane wave can propagate in the tube. The diameter of the tube in our apparatus was D = 5 cm, which means that measurements are restricted to wavelengths larger than 1.7D ≈ 8.5 cm corresponding to frequencies below 4000 Hz. We shall assume the sample thin enough so that we may put the velocity amplitudes on the two sides of the sample equal, i.e., u1 = u2 (see figure). The sound pressure field between the sample and the rigid termination has the complex amplitude p(x) = p3 cos[k(L − x)], where x = 0 at the back side of the sample and x = L at the rigid wall. The velocity amplitude at x = 0 is obtained from −iωρu2 = −∂p/∂x and we get u2 = −i(p3 /ρc) sin(kL).
(B.14)
The pressure at the back side of the sample is p2 = p3 cos(kL), and if we choose L to be an odd multiple of quarter wavelengths, L = (2n − 1)λ/4 (n = 1, 2, . . .), this pressure amplitude will be zero if the losses in the tube are neglected. Recalling that we may put u1 ≈ u2 ≡ u, the expression for the normalized flow impedance can then be written zf = (p1 −p2 )/u = p1 /u = iρc(p1 /p3 )/ sin[(2n−1)π/2] = (ρc)i(−1)n−1 (p1 /p3 ). (B.15)
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NOISE REDUCTION ANALYSIS
Thus, the real part of the normalized flow impedance ζf ≡ zf /ρc will be θf = | {p1 /p3 }|,
(B.16)
and the corresponding imaginary part is χf = (−1)n−1 {p1 /p3 }. (As before, the sign convention is defined by p(t) = {p(ω) exp(−iωt)}.)
(B.17)
Appendix C
Historical Notes and References, Absorbers C.1 ‘THE ABSORPTION COEFFICIENT PROBLEM’ In the early days of the use of reverberation rooms for the measurement of sound absorption, in the late 1920s and early 1930s, the meaning of the ‘The Absorption Coefficient Problem’ was well-known to most acousticians. The problem had to do with the variability in the results of reverberation room measurements of the absorption coefficient obtained when samples of a material were tested under different conditions, i.e., different rooms, different sample areas, etc. This problem turned out to have a far-reaching influence on the development of acoustics, particularly in this country. As stated by Hunt (1939) at a symposium at the tenth anniversary of the founding of the Acoustical Society of America, the absorption coefficient problem was one of the motivating factors which drew together the group of acousticians who founded the Society. The origin of the reverberation room method for measuring sound absorption goes back to the turn of the century and Wallace Sabine, a physics professor at Harvard University, with x-rays as a specialty. His interest in acoustics came in part from a desire to improve the acoustics of a very bad lecture room at the university. He undertook studies of reverberation in a room using organ pipes as sound sources and his ear as a detector and measured the time the sound could be heard after the source had been turned off. This time was correlated with the number of seat cushions (borrowed from a nearby theater) in the room. These preliminary tests led to further investigations over a period of time during which he also was involved with concert hall acoustics. One of the results that emerged from the studies was the wellknown Sabine formula for the reverberation time in terms of an average absorption coefficient. Detailed and very interesting descriptions of the experiments can be found in Sabine’s collected papers published by the Harvard University Press, Cambridge, 1922. Among other early investigators in this country, intimately involved with the absorption coefficient problem, were Carl Eyring (1929), Paul Sabine (1929), E. C. Wente (1929), V. O. Knudsen, (1929), V. L. Chrisler (1930), and R. F. Norris (1932).
403
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NOISE REDUCTION ANALYSIS
The absorption coefficient problem also inspired experimenters and theorists alike to gain a better understanding of acoustics in rooms in general and the reverberation room method of measurement, in particular. Research groups at M. I. T. under Professor P. M. Morse and at Harvard under Professor F. V. Hunt had a friendly rivalry going in the development of modal theory of room acoustics and the effects of acoustic wall treatments (Morse [1939], Hunt, et al. [1939]). (Some of the basic problems in room acoustics had been discussed already by Morse in his book Vibration and Sound, the first edition of which was published in 1936.) It should be noted in this context that Lord Rayleigh in his Theory of Sound (Vol. II, p 71), i.e., over 100 years ago, expressed the general sound field in a rectangular room with acoustically hard walls as a sum of normal modes. The effect of sample size on the measured absorption coefficient was suspected to be a result of ‘edge effect’ or diffraction, and studies of this aspect were carried out by Morse and his students (Morse and Rubinstein, [1938] and Pellam, [1940]). These and related problems in room acoustics have continued to attract investigators ever since. In the list of references on reverberation measurement given below, the effect of sample size and geometry alone is dealt with in 24 publications. Actually, this list generally does not include papers which deal mainly with sound distribution in rooms and the sound ‘quality’; if it did, the number of references would have vastly increased. The search for explanations of the absorption coefficient problem no doubt also influenced Professor V. O. Knudsen (1931) to carry out his now well-known studies of the effect of humidity on sound absorption in air and on the reverberation in a room. This study and the collaboration with Kneser (1934), who earlier had interpreted the ‘anomalous’ absorption of sound in air and oxygen in terms of molecular collisions and vibrational relaxation (1933), led to a greatly improved understanding of the absorption and attenuation of sound in air. Although the absorption of sound now represents but a small branch of the broad field of acoustics, it is interesting to note from the above remarks that it played a significant role in the early development of the field, and continues to be challenging as evidenced in this book.
C.1.1 Sound Absorption in Porous Materials As in most other areas of acoustics (see comments about room acoustics above and sound attenuation in ducts below), fundamental aspects of the subject were treated over 100 years ago in Lord Rayleigh’s classical book Theory of Sound, the first edition of which was published in 1877; it was revised and enlarged in 1894 and reprinted in 1926 and 1929 in two volumes. A 1945 Dover paperback edition is also available. Sound absorption in porous materials is dealt with in a section of volume 2 (page 327) in connection with the treatment of sound propagation in narrow tubes and between parallel plates, a topic which has been carried further in Chapter 3 of this book. The study of sound absorption in porous materials gradually took form and during the 1930s, considerable advances were made. References to early papers, covering the period from 1911 to 1935, can be found in Lothar Cremer’s book, Die Wissenschaftlischen Grundlagen der Raumakustik, Band III, S. Hirzel Verlag, Leipzig, 1950, which
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contains a thorough treatment of sound absorption by rigid porous materials. A second modified edition, in collaboration with Helmut A. Müller, was published in 1978. It was translated into English by Theodore J. Schultz, under the title Principles and Applications of Room Acoustics, Applied Science Publishers, London and New York (1982). There are, of course, many other books on acoustics and general handbooks which contain sections on sound absorption, but a list of these is not included here. However, there is one remarkable book we wish to mention which contains an almost encyclopedic coverage of essentially all aspects of acoustics, namely E. Skudrzyk, Die Grundlagen der Akustik, Springer, Vienna, 1084 pp (1954). Books devoted solely to sound absorption include the following: C. Zwikker and C. W. Kosten, Sound Absorption Materials, Elsevier Publisher Company, New York, (1949). Although old at this point, this excellent book is still very useful. A detailed mathematical analysis and modeling of porous absorption materials can be found in F. P. Mechel, Schallabsorber, Vol. I, Hirzel Verlag, Stuttgart 672 pp (1989). This book is based in part on Mechel’s many published papers in the field (see reference list) and is an impressively thorough analytical treatment of models of sound absorbers, starting from first principles. A more recent book is J. F. Allard, Propagation of Sound in Porous Media: Modeling Sound Absorbing Materials, Chapman & Hall, London, (1993). It also reflects the many important contributions to the subject by the author and his co-workers (see reference list). Among chapters in handbooks that should be mentioned are: F. P. Mechel and Istvan Ver, “Sound Absorption Materials and Sound Absorbers,” Chapter 8 in Noise and Vibration Control Engineering, edited by Leo L. Beranek and Istvan Ver, Wiley Interscience (1992). David A. Bies, Acoustical Properties of Porous Materials, Chapter 10 in Noise and Vibration Control, Edited by Leo L. Beranek, Revised edition (1988) Institute of Noise Control Engineering, Washington, DC. An important contribution to the field, written by an expert who did his dissertation on sound dissipation in porous media, is K. Attenborough, “Acoustical characteristics of porous materials,” Physics Reports, 82 (1982) p. 179. In addition to an extensive treatment of the subject, this excellent article contains many references. A large collection of experimental data on many different sound absorbers can be found in E. J. Evans and E. N. Bazley, Sound Absorbing Materials, National Physical Laboratory, Teddington, Middlesex (Great Britain) (1978) and in Robert A. Hedeen, Compendium of Materials for Noise Control, National Institute for Occupational Safety and Health (NIOSH), Publication No. 80-116 (1980). Although most of the earlier work dealt with rigid porous materials, the role of flexibility has been considered on and off ever since the mid-1930s (see, for example, Rettinger [1936]) and was treated analytically in some detail by Zwikker and Kosten in their book Sound Absorption Materials (1949) on the simplifying assumption that only dilatational waves were involved. An important generalization was made in a series of papers by Biot (1956–1962) who started from the general stress-strain relation of a porous material and included also transverse waves (‘rotational’ waves). Actually, as indicated by Biot in his first paper, these waves in a porous material were arrived at earlier by Frenkel (1944), based upon essentially the same approach.
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Of interest already in the mid-1930s (for example, Janowski and Spandöck, [1937]), anechoic rooms still receive considerable attention. In part, this is a result of the ever increasing demands on the precision of frequency response measurements, calibrations of loudspeakers and microphones, and various ‘free field’ experiments. The measurement of the absorption coefficient of anechoic wall-treatments, such as wedges, is still a problem, particularly at low and mid frequencies. Furthermore, little is known about the angular dependence of the absorption coefficient of wedges, and methods for testing the quality of anechoic rooms need further attention. Although anechoic wedges are not treated directly in this book, the absorption by a wedge is simulated in Chapter 4 by applying a computer program to multilayered porous absorbers, which provide a gradual transition in flow resistance and impedance from one layer to the next. In contrast to the broad band absorption by porous materials in general and anechoic wedges in particular, resonators are typically narrow band absorbers. The most common is the Helmholtz type resonator but vibrating panels and membranes can also be classified as resonance absorbers. At resonance, even at modest incident sound levels, the velocity amplitude in the neck of a Helmholtz resonator can become large enough for nonlinear effects to play an important role. These effects are associated with acoustically induced steady vorticity and pulsating jets (Ingard and Labate [1950]) and can markedly affect the absorption characteristics.
C.1.2 Regarding the Lists of Publications The lists of publications given in the following sections should be treated as suggestions for further reading or browsing and not as documents reviewed and referenced in the text. The lists include published papers and, only with few exceptions, letters to the editor, abstracts, company reports, and patents have not been included. In these days of electronic retrieval of information, the need for lists, such as those given here, becomes less important than before as databases in specialized areas are more readily available. The present lists could provide at least a start for anyone who is intent on producing a database on everything published in the field of sound absorption and attenuation in ducts. The reference lists have been divided into six major categories, each with the authors, presented in alphabetical order: • Sound absorption concepts and mathematical analysis Deals with sound propagation and absorption in porous materials, both rigid and flexible. Effects of perforated plates and impervious membranes are included. • Measurements: Methods and data This section is divided into the following parts: (a) Flow resistance and other properties. (b) Absorption coefficient reverberation room method and related matters. (c) Tube methods. (d) Free field and other methods. (e) Absorption data and general discussion. The list on the reverberation method contains some references to early papers on room acoustics, but papers on sound distribution in rooms and related topics generally have not been included.
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• Anechoic wedges and rooms • Resonators and related matters Both Helmholtz resonators and ‘plate resonators’ are considered. Included also are studies of both the linear and nonlinear impedance of orifice plates. • ‘Functional’ or ‘volume’ absorbers The absorbers referred to in previous sections represent wall treatments and are referred to in the text as ‘surface’ absorbers. By contrast, functional or volume absorbers are absorptive units such as spheres, cylinders, panels, sheets, etc., which are placed in a room away from the walls and exposed to sound on all sides. Included here are also references to papers on sound absorption by audience and seats in auditoria.
C.2 LISTS OF REFERENCES C.2.1 Sound Absorption, Concepts and Analysis Allard, J. F. Propagation of Sound in Porous Media: Modelling Sound Absorbing Materials, Chapman & Hall, London, 1993. ; and Aknine, A. “Biot theory and acoustical properties of high porosity fibrous materials and plastic foams,” Acustica 56(3), 221-237 (1984) (French; English abstr.). ; Aknine, A.; and Bruneau, A. M. “Acoustical properties of plastic foams with high specific flow resistance,” Acustica 59(2), 142-47 (1985) (French; English abstr.). ; Depollier, Claude; Guignouard, Philippe; and Rebillard, Pascal. “Effect of a resonance of the frame on the surface impedance of glass wool of high density and stiffness,” J. Acoust. Soc. Am. 89(3), 999-1001 (1991). Aso, S.; and Kinoshita, S. “Maximum sound absorption coefficient of fiber assembly,” J. Textile Machinery Soc. Japan 11(3), 81-87 (1965). Attenborough, K. “Acoustical characteristics of porous materials,” Physics Reports, 82, (1982), 179. ; “Ground parameter information for propagation modeling,” J. Acoust. Soc. Am. 92(1), 418-32 (1992). ; “The influence of microstructure on propagation in porous fibrous absorbents,” J. Sound Vib. 16(3), 419-442 (1971). ; “The prediction of oblique-incidence behaviour of fibrous absorbents,” J. Sound Vib. 14(2), 183-191 (1971). Phys. Abstr. 34752 (24 June 1971). Balachandran, C. G.; and Butchers, C. J., “Sound absorption properties of thin fibrous materials,” N. Z. Eng. 31(12), 272-274 (1976). Beranek, Leo L. “Acoustic impedance of porous materials,” J. Acoust. Soc. Am. 13, 248 (1942). “Acoustical properties of homogeneous, isotropic rigid tiles and flexible blankets,” J. Acoust. Soc. Am. 19, 556 (1947). Bliss, Donald D. “Study of bulk reacting porous sound absorbers and a new boundary condition,” J. Acoust. Soc. Am. 71(3), 533-45 (1982). ; and Shawn E. Burke, “Experimental investigation of the bulk reaction boundary condition,” J. Acoust. Soc. Am. 71(3), 546-52 (1982). Bolt, R. H. “Design of perforated facings for acoustic materials,” J. Acoust. Soc. Am. 19 (1947). Bourdier, R.; and Depollier, C. “Biot waves in layered media,” J. Applied Phys. 60(6) 19261929 (1986). ; Biot, M. S. “Theory of propagation of elastic waves in a fluid saturated porous solid. I. Low frequency range,” J. Acoust. Soc. Am. 28, 168 (1956). ; II. Higher frequency range,” J. Acoust. Soc. Am. 28, 179 (1956). ; “Mechanics of deformation and acoustic propagation in porous media,” J. appl. Phys. 33, 1482 (1862). ; “Generalized theory of acoustic propagation in porous dissipative media,” J. Acoust. Soc. Am. 34, 1254 (1962).
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Brillouin, J. “Sound absorption by structures with perforated panels,” Sound Vibration 2(7), 6-22 (1968). Brodhun, D. “Determination of the sound absorption of porous materials from flow resistance measurements,” Exptl. Tech. Phys. (Berlin) 5(2), 68-74 (1957). Brouard, B.; et al. “Acoustical impedance and absorption coefficients of porous layers covered with a facing perforated with parallel slits,” Noise Control Eng. J. 41(1), 289-297 (1993). Byrne, K. P. “Calculation of the specific normal impedance of perforated facing-porous backing constructions,” Applied Acoustics 13, 43-55 (1980). “Calculating the acoustic properties of fabric constructions,” J. Sound Vib. 123(3), 423-435 (1988). Callaway, D. B.; and Ramer, L. G. “Use of perforated facings in designing low frequency resonant absorbers,” J. Acoust. Soc. Am. 24(3), 309 (1951). Caviglia, Giacomo; Morro, Angelo; and Straugham, Brian. “Reflection and refraction at a fluid-porous medium interface,” J. Acoust. Soc. Am. 92(2), 1113-9 (1992). Champioux, Yvan; and Stinson, Michael R. “On acoustical models for sound propagation in rigid frame porous materials and the influence of shape factors,” J. Acoust. Soc. Am. 92(2), 1120-31 (1992). Chandler–Wilde, S. N.; and Horoshenkov, K. V. “Padé approximants for the acoustical characteristics of rigid frame porous media,” J. Acoust. Soc. Am. 98(2), 9-29 (1995). Craggs, A. “A finite element model for rigid porous absorbing materials,” J. Sound Vib. 61(1), 101-111 (1978). ; “Coupling of finite elements acoustic absorption models,” J. Sound Vib. 66(4), 605-613 (1979). Davern, W. A. “Perforated facings backed with porous materials as sound absorbers,” Appl. Acoust. 10(2), 85-112 (1977). Phys. Abstr. 62115 (1 Sept. 1977). Delany, M. E.; and Bazley, E. N. “Acoustical properties of fibrous absorbent materials,” Applied Acoustics, 3, 105 (1970). Depollier, Claude; Allard, Jean F.; and Lauriks, Water. “Biot theory and stress-strain equations in porous sound-absorbing materials,” J. Acoust. Soc. Am. 84(6), 2277-9(L) (1988). Doak, P. E.; and King, M. R. “Mechanical and acoustical properties of porous foam materials,” p. 3. Proc. Br. Acoust. Soc. 1(3), (1972). Phys. Abstr. 69727 (26 Oct. 1972). Dowell, Earl H.; Chao, Chen-Fu; and Bliss, Donald B. “Absorption material mounted on a moving wall–Fluid/wall boundary condition,” J. Acoust. Soc. Am. 70(1), 244-5(L) (1981). Dunlop, J. I. “Acoustic impedance properties of closed front fibre masses,” J. Sound Vib. 34(1), 1-9 (1974). Egorov, N. F. “Scale modeling of sound absorbing layers of fibrous materials,” Sov. Phys.– Acoust. 13(3), 320-323 (1968). Esche, V. V. “Experimentelle Untersuchungen zu Einflussparametern and Grösse des Kanteneffektes,” Acustica 19(6), 301-312 (1968). Evans, E. J.; and Bazley, E. N. Sound Absorbing Materials, National Physical Laboratory, Teddington, Middlesex (Great Britain) (1978). Fand, R. M.; Gogos, C. M.; and Jackson, F. J. “Absorption of sound in liquids by porous metals,” J. Acoust. Soc. Am. 41, 530 (1967). Ferrero, M. A.; and Sacerdote, G. G. “Parameters of sound propagation in granular absorbent materials,” Acustica 1(3), 137-142 (1952). Ford R. D.; Landau, B. V.; and West, M. “Reflection of plane oblique air waves from absorbents,” J. Acoust. Soc. Am. 44, 531 (1968). Frenkel, J. “On the theory of seismic and seismoelectric phenomena in a moist soil,” Journal of Physics, 4, 230-241 (1944). Hamet, J. F. “Parametric analysis of absorption curves of porous materials at normal incidence,” Acustica 52(3), 186-190 (1983).
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Gan, W. S. “Transient response of sound absorbing materials,” Phys. Lett. 48A(1), 71-72 (1974). Hersh, A. S.; and Walker, B. “Acoustical behaviour of homogeneous bulk materials,” AIAA 6th Aeroacoustics Conference, paper AIAA-80-0986, June 4-6, (1980). Hess, H. M.; Attenborough, K.; and Heap, N. W. “Ground characterization by short–range propagation measurements,” J. Acoust. Soc. Am. 87(5), 1975-86 (1990). Ingard, Uno. “Perforated facing and sound absorption,” J. Acoust. Soc. Am. 26, 289-293 (1954). ; “Sound absorption by perforated porous tiles,” J. Acoust. Soc. Am. 26, 289-293 (1954). ; “Locally and nonlocally reacting flexible porous layers; a comparison of acoustical properties,” Journal of Engineering for Industry, 103, 302-12 (1981). ; “Acoustics in physics and engineering,” Acustica 52(3), 127-147 (1983). ; and Bolt, R. H. “Absorption characteristics of acoustic material with perforated facings,” J. Acoust. Soc. Am. 23, 533-540 (1951). Kang, Yeon June; and Bolton, J. Stuart. “Finite element modelling of isotropic elastic porous materials coupled with acoustical finite elements,” J. Acoust. Soc. Am. 98(1), 635-43 (1995). Kemp, G. T.; and Nolle, W. W. “Attenuation of sound in small tubes,” J. Acoust. Soc. Am. 25(6), 1053 (1953). Kinoshita, R. “On the relation between sound absorption coefficient and flow resistance of fibrous materials,” J. Acoust. Soc. Japan 23(6), 415-425 (1967). (Japanese, English abstract). Kitamura, O.; and Sasaki, M. “On the relationship between the absorption coefficient and flow resistance of the cloths,” J. Acoust. Soc. Japan 21(5), 272-280 (1965). (Japanese; English abstr.). Konstantinov, B. P. “On the absorption of sound waves upon reflection from a solid boundary,” Journ. Techn. Phys. 9, 226-28 (1939). (In Russian). Abs. Wireless Engineer 16, 3992 (1939). Korringa, J.; Kronig, R.; and Smit, A. “On the theory of the reflection of sound by a porous medium,” Physics, XI, No. 4, Dec. 209-230, (1945). Kosten, C. W. “Absorption of sound by coated porous rubber wallcovering layers,” J. Acoust. Soc. Am. 18, 457 (1946). ; and Zwikker, C. “Theory of the absorption of sound by compressible walls with a non-porous surface layer,” Physica 8, 251-272 (1941). ; and Zwikker, C.“Extended theory of the absorption of sound by compressible wall-coverings,” Physica 8, 968-978 (1941). ; and Zwikker, C. “Measurements of the absorption of sound by porous rubber wallcovering layers,” Physica 8, 933-967 (1941). ; and J. H. Janssen, “Acoustic properties of flexible and porous materials,” Acustica 7(6), 372-378 (1857). Koyasu, M.; Tate, R.; Ogita, Y.; and Yoshimuta, H. “Note on sound absorption of perforated panel absorbers,” J. Acoust. Soc. Japan 17(1), 31-37 (1961). (In Japanese with English abstract). Kraak, W. “Schallabsorption und Schallisolation poröser Absorber mit sehr leichten elastsischen Skelett,” Hochfrequenz-tech. u. Elektroakust. 71(3), 86-98 (1962). “Die Bestimmung der Schallabsorption geschichteter poröser Absorber bei schrägem und statistischen Schalleinfall mit einem Analogierechner,” Hochfrequenz-tech. u. Elektroakust. 71(5), 155-160 (1962). Kuntz, Herbert L.; and Blackstock, David T. “Attenuation of intense sinusoidal waves in air-saturated, bulk porous materials,” J. Acoust. Soc. Am. 81(6), 1721-31 (1987). Lafarge, Denis; Allard, Jean F.; Brouard, Bruno; Verhaegen, Christine; and Lauriks, Walter. “Characteristic dimensions and prediction at high frequencies of the surface impedance of porous layers,” J. Acoust. Soc. Am. 93(5), 2474-8 (1993). Laeis, W. “Akustisches Verhalten der Kunststoffe,” Kunststoffe 58(4), 305-308 (1968). Lambert, Robert. “Acoustical properties of highly porous fibrous materials,” J. Acoust. Soc. Am. 72(2), 643(T). “Low-frequency acoustic behavior of highly porous, layered, flexible, fine fiber materials,” J. Acoust. Soc. Am. 97(2), 818-21 (1995). ; “Acoustic resonance in highly porous, flexible, layered fine fiber materials,” J. Acoust. Soc. Am. 93(3), 1227-34 (1993). ; “Surface admittance of highly porous foams with finite stiffness,” J. Acoust. Soc. Am. 81(5), 1293-8 (1987). ; “Sound in a layered fine fiber porous material with finite fram stiffness,”
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J. Acoust. Soc. Am. 84(5), 1894-905 (1988). ; and McIntosh, Jason D. “Nonlinear wave propagation through rigid porous materials. II. Approximate analysital solutions,” J. Acoust. Soc. Am. 88(4), 1950-9 (1990). (For part I, see McIntosh and Lambert). Lauriks, Walter; Cops, André; Allard, Jean F.; Depollier, Claude; and Rebillard, Pascal. “Modelization at oblique incidence of layered porous materials with impervious screens,” J. Acoust. Soc. Am. 87(3), 1200-6 (1990). Lebedeva, I. V.; and Nesterov, V. S. “The behavior of a multilayered absorber at oblique incidence and in diffuse field,” Akust. Zh. 11(4), 463-467 (1965). (In Russian). Legeay, V.; and Seznec, R. “On the determination of the acoustical characteristics of absorbent materials,” Acustica 53(4), 171-192 (1983). Maa, Dah-You “Theory and design of microperforated panel sound-absorbing constructions,” Sci. Sin. 18(1) 55-71 (1975). ; “Microperforated-panel wideband absorbers,” Noise Control Eng. J. 29(3), 77-84 (1988). Mariner, T.; and Park, A. D. “Sound–absorbing screens in a marginal industrial noise problem,” Noise Control 2(5), 22-27, 58 (September, 1956). Masiak, J. E. “Effect of surface films on acoustical foam performance,” Sound and Vibration 8(7), 24-29(1974). Mawardi, Osman K. “On the propagation of sound waves in narrow conduits,” J. Acoust. Soc. Am. 21, 482-486 (1949). Matsuzawa, Kiichiro; Hasegawa, Takahi; and Ochi, Masayuki. “Effective density of air in open-cell polyurethane foam at ultrasonic frequencies,” J. Acoust. Soc. Am. 70(6), 1704-6 (1981). Mechel, F. P. “Design charts for sound absorber layers,” J. Acoust. Soc. Am. 83(3), 1002-13 (1988). ; “Extension to low frequencies of the formulae of Delany and Bazley for absorbing materials,” Acustica 35(3), 210-213 (1976) (German, Engl. abstr.). ; and J. Royar, “Experimentalle Untersuchungen zur Theory des porösen Absorbers,” Acustica 26(2), 83-96 (1972) (English abstract). ; and N. Kiesewetter, “A sound absorber made of plastic foils,” Acustica 47(2), 8388 (1981). ; and Istvan Ver, Sound Absorption Materials and Sound Absorbers, Chapter 8 in Noise and Vibration Control Engineering, edited by Leo L. Beranek and Istvan Ver, Wiley Interscience (1992). Melling, T. H. “The acoustic impedance of perforates at medium and high sound pressure levels,” J. Sound Vib. 29(1), 1-66 (1973). Mertens, P. “Nichtlinearität poröser absorber bei Strömungsüberlagerung,” Acustica 15(5), 407-410 (1965). Meyer, E.; and Reipka, R. “Das Reflexions- und Durchlassverhalten von Stosswellen an porösen Absorben (Rayleigh-Modelle),” Acustia 16(3), 149-159 (1965). Mills, Charles A.; and Spickermann, Charles E. “Evaluating acoustic absorption coefficients by comparative analysis–Theory part,” J. Acoust. Soc. Am. 91(2), 696-703 (1992). “Experimental part,” J. Acoust. Soc. Am. 91(2), 704-12 (1992). Mongy, M. “Acoustic properties of porous materials,” Acustica, 28(243) (1973). Monna, A. F. “Absorption of sound by porous walls,” Physica 8, 129-142 (1938). Moore, James A.; and Lyon, Richard H. “Resonant porous material absorbers,” J. Acoust. Soc. Am. 72(6), 1989-99 (1982). Morse, P. M. Vibration and Sound, McGraw-Hill, First ed. (1936). Second ed. (1948). ; Bolt, R. H.; and Brown, R. L. “Acoustic impedance and sound absorption,” J. Acoust. Soc. Am. 12, 217-227 (1940). ; and Ingard, K. U. Theoretical Acoustics, McGraw-Hill (1968). Morse, R. W. “Acoustic propagation in granular media,” J. Acoust. Soc. Am. 24(6), 696 (1952). Nakamura, Aktra. “On the mechanism of sound absorption by flexible and porous vinul foam,” Mem. Inst. Sci. and Ind. Research, Osaka Univ. 17, 13-17 (1960). ;
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“Acoustic properties of porous materials with two kinds of pores,” Mem. Inst. Sci. and Ind. Research Osaka Univ. 18, 1-13 (1961). Nakamura, T.; Nakamura, A.; and Takeuchi, R. “Absorption mechanism of porous material of a sound pulse,” Acustic 45(1), 1-9 (1980). Nelson, A. “Propagation of finite-amplitude sound in air-filled porous materials,” J. Acoust. Soc. Am. 79(6), 2095 (T) (1986). Nilsson, A. C.; and Rasmussen, B. “Sound absorbing properties of a perforated plate and membrane construction,” Acustica 57(3), 139-148 (1985). Northwood, T. D. “Absorption of diffuse sound by a strip or rectangular patch of absorptive material,” J. Acoust. Soc. Am. 35, 1173 (1963). Parrott, T. L.; and Zorumski, W. E. “Nonlinear acoustic theory for rigid porous materials,” p. 27. NASA, Langley Res. Ct., Langley Station, Va. (June 1971) N71-28777. Penman, H. L.; and Richardson, E. G. “Absorption by porous materials at normal incidence– comparison of theory and experiments,” J. Acoust. Soc. Am. 4, 322 (1933). Pizzirusso, J. F. “Flexible Urethan foam–A new look at as versatile acoustical material,” 1973 Nat. Conf. Noise Control Eng., Washington, D.C., 15-17 Oct. 1973. 513-518 (Inst. Noise Control Eng., Pughkeepsie, N. Y., 1973). Phys. Abstr. 39143 (3 June 1974). Purshouse, M. “On the damping of unsteady flow by compliant boundaries,” J. Sound Vib. 49(3), 423-436 (1976). Lord Rayleigh, John William Strutt. The Theory of Sound, Volumes 1 and 2, MacMillan and Co, Ltd., London, (1929). First edition (1878), enlarged second edition, (1894). Dover edition, (1945). Rettinger, Michael. “On the theory of sound absorption of porous materials,” J. Acoust. Soc. Am. 6, 188 (1935). ; “Theory of sound absorption coefficient of porous materials, flexible and nonflexible,” J. Acoust. Soc. Am. 8, 53 (1936). Schwartz, Manual.; and Gohman, Edmund J. “Influence of surface coating on impedance and absorption coefficient of urethane foam,” J. Acoust. Soc. Am. 34, 502 (1962). ; and Buchner, William L. “Effect of light coatings on impedance and absorption of opne-celled foams,” J. Acoust. Soc. Am. 35, 1507 (1963). ; and Bradley, Herbert D. Jr., “Effect of air space on the acoustic characteristics of uncoated and coated foams,” J. Acoust. Soc. Am. 37, 278 (1965). Sides, D. J.; Attenborough, K.; and Mulholland, K. A. “Application of a generalized acoustic propagation theory to fibrous absorbents,” J. Sound Vib. 19(1), 49-64 (1971). Skrudzyk, G. Die Grundlagen der Akustik, Springer, (1954). Slavik, J. B.; Klimes, B.; and Zadrazil, V. “Sound absorption by perforated plates,” Przegl. Telekomm. No. 5, 139-143 (May 1951). (In Polish.) Phys. Abs. 55, 3394 (May, 1952). Stinson, Michael R.; and Champoux, Yvan. “Propagation of sound and the assignment of shape factors in model porous materials having simple pore geometries,” J. Acoust. Soc. Am. 91(2), 685-95 (1992). Taylor, Hawley O.; and Sherwin, Charles W. “Sound absorption and attenuation by the flue-method,” J. Acoust. Soc. Am. 9, 331 (1938). Tichy, J. “Absorptionsgrad eines porösen materials mit Schlitzen,” Hochfrequenz-tech. u. Elektroakust. 67, 149-153 (1959). Tooms, Steven; Taberzadeh, Shahram; and Attenborough, Keith. “Sound propagation in a refracting fluid above a layered fluid-saturated porous elastic material,” J. Acoust. Soc. Am. 93(1), 173-81 (1993). Van den Eijk, J.; and Zwikker, C. “Absorption of sound by porous material,” Physica 8, 149-158 (1941). Van Os, G. J. “Use of foam rubber as sound absorbing material,” Tech. Wetenschap. Tijdschr. 27, 73-77 (April, 1958). (In Dutch).
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Vayssaire, J. C. “Absorption of shock waves by porous homogeneous walls,” Aeronaut. Astronaut. 47, 85-88 (Apr. 1974) (French). Appl. Mech. Rev. 10992 (No.12, 1975). Velizanina, K. A. “Sound absorber with perforated facings,” Akust. Zhur. 7(2), 165-173 (1961). ; English translation of Akust. Zhur. 7(1), 92-94 (1961). ; Voronina, N. N.; and Kodymskaya, E. S. “Impedance investigation of sound–absorbing systems in oblique sound incidence,” Sov. Phys. Acoust. 17(2), 193-197 (1971). Voronina, N. “Improved empirical model of sound propagation through a fibrous material,” Applied Acoustics, 48(2), 121-132 (1996). West, M. “Effective density of flexible polyurethane foam when considered as an acoustic fluid,” J. Acoust. Soc. Am. 46, 859 (1969)(L). Wilson, David K.; McIntosh, Jason D.; and Lambert, Robert F. “Forchheimer–type nonlinearities for high–intensity propagation of pure tones in air–saturated porous media,” J. Acoust. Soc. Am. 84(1), 350-9 (1988). Wirt, L. S. “Sound-absorptive materials to meet special requirements,” J. Acoust. Soc. Am. 57, 126-43 (1975). Yamamoto, Tokuo; and Turgut, Altan. “Acoustic wave propagation through porous media with arbitrary pore size distributions,” J. Acoust. Soc. Am. 83(5), 1744-51 (1988). Zarek, J. H. B. “Sound absorption in flexible porous materials,” J. Sound Vib. 61(2), 205-234 (1978). Zorumski, W. E.; and Parrott, T. L. “Nonlinear acoustic theory for rigid porous materials,” p. 26. NASA Tech. Note D-6196 (June 1971). Appl. Mech. Rev. 8153 (No. 10, 1971). Zwikker, C. and Kosten, C. W. Sound Absorption Materials, Elsevier Publ. Comp., New York (1949).
C.2.2 Measurements, Methods, and Data Flow Resistance and Other Properties Ballagh, K. O. “Acoustical properties of wool,” Applied Acoustics, 48(2), 101-120, 1996. Bies, David A. “Acoustical Properties of Porous Materials,” Chapter 10 in Noise and Vibration Control, Edited by Leo Beranek, Revised edition, (1988), Institute of Noise Control Engineering, Washington, DC. ; and C. H. Hansen, “Flow resistance information for acoustical design,” Appl. Acoust. 13(5), 357-391 (1980). Brown, Richard L.; and Bolt, Richard H. “Measurement of flow resistance of porous acoustic materials,” J. Acoust. Soc. Am. 13, 337 (1942). Bruneau, A. M.; Bruneau, M.; and Delage, P. “An apparatus for fast control of acoustic properties of materials,” Appl. Acoust. 18(4), 257-270 (1985). Champioux, Yvon; and Stinson, Michael R. “Measurement of the characteristic inpedance and propagation constant of materials having high flow resistivity,” J. Acoust. Soc. Am. 90(4), 2182-91 (1991). ; and Gilles A. Daigle, “Air–based system for the measurement of porosity,” J. Acoust. Soc. Am. 89(2), 910-6 (1991). Gerdien, H. “On the flow resistance of porous ceramic materials,” Akust. Zeits. 6, 329 (1943). Huszty, D.; Illenyi, A.; and Van, Gy. “Equipment for measuring the flow–resistance of porous and fibrous materials (Acoustic material for sound-absorbing devices),” Appl. Acoust. 5(1), 1-14 (1972). Ingard, Uno; and Dear, T. A. “Measurement of acoustic flow resistance,” J. Sound Vib. 103(4), 567-572 (1985). Leonard, R. W. “Simplified flow resistance measurements,” J. Acoust. Soc. Am. 17, 240 (1946). ; “Simplified porosity measurements,” J. Acoust. Soc. Am. 20, 39 (1948).
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Mokshantsev, V. V.; and Rosin, G. S. “Instrument for determining the flow resistivity of sound-absorbing materials,” Ind. Lab. 40(10), 1448-1449 (1974). Phys. Abstr. 14302 (3 Mar. 1975). Mongy, M. “Acoustic properties of porous materials,” Acustica, 28, 243 (1973). Müller, L. “Ein einfaches Gerät zur Bestimmung des Strömungswiderstandes von Schallschluckstoffen,” Rdfunktech. Mitt. 3, 153-156 (June 1959). Ann Télecomm. 119174 (AugustSeptember, 1959). Nichols, R. H. Jr. “Flow resistance characteristics of fibrous acoustical materials,” J. Acoust. Soc. Am. 19, 866 (1947). Pritz, T. “Frequency dependence of frame dynamic characteristics of mineral and glass wool materials,” Journal of Sound and Vibration 106(1), 161-169 (1986) Rjevkine, S. N.; and Toumanski, S. S. “Measuring the flow resistance of porous walls for sound,” J. Techn. Phys. USSR, 17, 681-692 (1947). (In Russian). Ann. Télécomun. 3, 17616 (May 1948). Stinson, Michael R.; and Daigle, Gilles A. “Electronic system for the measurement of flow resistance,” J. Acoust. Soc. Am. 83(6), 2422-8 (1988). Tarnow, Viggo. “Measurement of sound propagation in glass wool,” J. Acoust. Soc. Am. 97(4), 2272-81 (1995). Tyzzer, F. G.; and Hardy, H. C. “Properties of felt in the reduction of noise and vibration,” J. Acoust. Soc. Am. 19, 872 (1947). Wigan, E. R. “Transmission parameters of porous sound absorbers (with particular reference to the flow resistance),” Nature 199, No. 4888, 59 (1963). Woehle, W.; and Weber, K. “Method for measuring low flow resistances,” Hochfrequenztech. u. Elektroakust. 68(5), 158-162 (1959). (In German).
Absorption Coefficient, Diffuse Field Method, and Related Matters Andree, C. A. “Effect of position on the absorption of materials for the case of a cubical room,” J. Acoust. Soc. Am. 3, 535 (1932). Awaya, K.; and Ikeda, H. “Investigation on the edge phenomena of multilayer sound absorbing panels,” J. Acoust. Soc. Jpn. 32(11), 683-692 (1976) (Japanese, Engl. abstr.). Bedell, E. H.; and Swartzel, K. D, Jr. “Reverberation time and absorption measurements with the high speed level recorder,” J. Acoust. Soc. Am. 5, 220 (1934). ; and 6, 130 (1935). Chrisler, V. L. “Measurement of sound absorption by oscillograph records,” J. Acoust. Soc. Am. bf 1 168 (1930). ; “Sound absorption coefficients,” J. Acoust. Soc. Am. 6, 115 (1934). ; “Variation of sound absorption with area,” J. Acoust. Soc. Am. 8, 67(A) (1936). ; and Snyder W. F. “Recent advances in sound absorption measurements,” J. Acoust. Soc. Am. 2, 123 (A) (1930). ; and Miller, Catherine E. “Effect of rotating vanes in a reverberation room,” J. Acoust. Soc. Am. 4, 172(A) (1933). ; “Experimental evidence of nonlogarithmic sound decay,” J. Acoust. Soc. Am. 5, 64(A) (1932). ; Snyder, W. F.; and Miller, C. E. “Measurements with a reverberation meter,” J. Acoust. Soc. Am. 3, 12 (A) (1931). ; “Some of the factors which affect the measurement of sound absorption,” J. Acoust. Soc. Am. 4, 8 (1932). Cook, Richard K. “Absorption of sound by patches of absorbent materials,” J. Acoust. Soc. Am. 29, 765 (1057). Cremer, William. “Acoustic absorption coefficient at high frequencies,” J. Acoust. Soc. Am. 22(2), 260 (1950). de Bruijn, A. A. “A mathematical analysis concerning the edge effect of sound absorbing materials,” Acustica 28(1), 33-44 (1971).
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Daniel, Eric D. “On the dependence of absorption coefficients upon the area of the absorbent material,” J. Acoust. Soc. Am. 35, 571 (1963). Dekker, H. “Edge effect measurements in a reverberation room,” J. Sound Vib. 32(2), 19-202 (1974). Eyring, Carl F. “Reverberation time in “dead” rooms,” J. Acoust. Soc. Am. 1(1), 217-241 (1929). “Methods of calculating the average coefficient of sound absorption,” J. Acoust. Soc. Am. 4, 178 (1933). Gilford, C. L. S. “Diffraction and diffusion aspects of sound absorbers,” Insulation (London) 6(9-10), 56-68 (1962). Gomperts, M. C. “On the edge effect in determining the absorption coefficient in a reverberation room,” Acustica 19(2), 116 (1967/1968). Guillermin, J. “Recherche d’une normalisation efficace de la mesure du coefficient d’absorption en chambre réverbérante,” Acustica 13(1), 15-24 (1963). Halliwell, R. E. “Sound absorption variation caused by modifications to a standard mounting,” J. Acoust. Soc. Am. 72(5), 1634-6(L) (1982). Hamet, J. F. “Acoustic absorption coefficient in diffuse field of a rectangular, plane material of finite dimensions, placed on an infinite, perfectly reflecting plane,” Rev. Acoust. 17(71), 204-210 (1984). Harris, Cyril. “Application of the wave theory of room acoustics in the measurement of acoustic impedance,” J. Acoust. Soc. Am. 17, 35 (1945). ; “Effect of position on the acoustical absorption by a patch of material in a room,” J. Acoust. Soc. Am. 17, 242 (1946). ; “Application of the wave theory of room acoustics in the measurement of acoustic impedance,” J. Acoust. Soc. Am. 17, 35 (1945). ; “Effect of position on the acoustical absorption by a patch of material in a room,” J. Acoust. Soc. Am. 17, 242 (1946). ; “Measurement of acoustic impedance by the room transmission-characteristic method,” J. Acoust. Soc. Am. 17, 102 (1945). Hunt, Frederick V. “Investigation of room acoustics by steady-state transmission measurements,” J. Acoust. Soc. Am. 10, 216 (1939). ; Beranek, L. L; and Ma, D. Y. “Analysis of sound decay in rectangular rooms,” J. Acoust. Soc. Am. 11(1), 80-94 (1939). Kath U.; and Kuhl W. “Einfluss von Streufläche und Hallraumdimensionen auf den gemessenen Schallabsorptionsgrad,” Acustica 11(1), 50-64 (1961). Kneser, Hans O. “Interpretation of the anomalous sound absorption in air and oxygen in terms of molecular collisions,” J. Acoust. Soc. Am. 5, 122 (1933). Knudsen, V. O. “Measurement of sound absorption in a room by the intensity method and by the oscillograph method,” J. Acoust. Soc. Am. 1, 27 (1929). ; “Absorption of sound in air, in oxygen, and in nitrogen—effects of humidity and temperature,” J. Acoust. Soc. Am. 5, 112 (1933). ; “Effect of humidity upon the absorption of sound in a room, and a determination of the coefficients of absorption of sound in air,” J. Acoust. Soc. Am. 3, 126 (1931). ; “Resonance in small rooms,” J. Acoust. Soc. Am. 4, 20 (1932). ; and Kneser, H. O. “Absorption of sound in oxygen as influenced by the presence of other gases,” J. Acoust. Soc. Am. 5, 221(A)(1934). ; and Fricke, Edwin F. “Absorption of sound in carbon dioxide and other gases,” J. Acoust. Soc. Am. 10, 89 (1938). Kolmer F.; and Krvnák, M. “Der Einfluss der Fläche des Prüfmaterials auf die Diffusität des Schallfeldes und aut den Schallabsorptionsgrad,” Acustica 11(6), 405-413 (1961). Konstantinov, B. “The damping of sound in room with solid walls and the diffraction coefficient of sound absorption,” J. Techn. Physics, USSR 9, 424-432 (1939). (In Russian). Sci. Abs. B43, 190 (1940). Kosten, C. W. “The problem of the reverberation method solved?,” Acustica 10(2), 126(L) (1960). ; “International comparison measurements in the reverberation room,” Acustica 10 (5-6), 400-411 (1960).
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Koyasu, M. “The effect of shape and volume of the reverberation chamber on the absorption coefficient of acoustic materials,” J. Acoust. Soc. 13, 320-327 (December 1957). (In Japanese with English abstract.). ; “Dependence of the sound absorption coefficient upon the area of acoustic materials,” Bull. Kobayashi Inst. Phys. Research 8, 222-226 (October-December, 1958). (In Japanese with English abstract). ; “On the relation between the reverberant and the normal incidence sound absorption coefficient of acoustic materials,” Bull. Kobayasi Inst. Phys. Research 9(1-2), 57-64 (1958). Kuhl, W. “Der Einfluss der Kanten auf die Schallabsorption poröser Materialien,” Akust. Beih. No. 1, 264-276 (1960). ; and U. Kath, “Bemerkung zur Messung der Schallabsorption im Hallraum bei vollständiger Diffusität,” Acustica 10(2), 125(L) (1960). Lebedeva, I. V. “Reverberation chamber techniques in the measurement of acoustic absorption coefficient,” Soviet Phys. Acoust. 8(3), 258-262 (1963). London, Albert. “Determination of reverberant sound absorption coefficients from acoustic impedance measurements,” J. Acoust. Soc. Am. 22(2), 263 (1950). Makita Y.; and Fujiwara, K. “Effects on precision of a reverberant absorption coefficient of a plane absorber due to anisotropy of sound energy flow in a reverberant room,” Acustica 39(5), 331-335 (1978). Meyer, Erwin. “Reverberation and absorption of sound,” J. Acoust. Soc. Am. 8, 155 and 209 (1937). ; Kuttruff, H.; and Lauterborn, W. “Modellversuche zur Bestimmung des Schallabsorptionsgrades im Hallraum,” Acustica 18(1), 21-32 (1967). Morris, R. M.; Nixon, G. M.; and Parkinson, J. S. “Variations in sound absorption coefficients as obtained by the reverberation chamber method,” J. Acoust. Soc. Am. 9, 76 and 234 (1938). Morse, Philip M. “Some aspects of the theory of room acoustics,” J. Acoust. Soc. Am. 11(1), 56-66 (1939). ; and Rubinstein, P. “The diffraction of sound by ribbons and by slits,” J. Acoust. Soc. Am. 10(3), 258(A) (1939). See also Phys. Rev. 54, 895 (1938). ; and Bolt, R. H. “Sound waves in rooms,” Rev. Mod. Physics 16, 69-150 (1944). Vibration and Sound, McGraw-Hill, First ed. (1936). Second ed. (1948). ; and Ingard, K. U. Theoretical Acoustics, McGraw-Hill (1968). Norris, R. F. “Application of Norris-Andree method of reverberation measurement to measurements of sound absorption,” J. Acoust. Soc. Am. 3, 361 (1932). ; “Discussion of sound absorption coefficients,” J. Acoust. Soc. Am. 6, 43 (1934). Northwood, T. D.; Grisaro, M. T.; and Medcof, M. A. “Absorption of sound by a strip of absorptive material in a diffuse sound field,” J. Acoust. Soc. Am. 31, 595 (1959). Pan, Jie; and Bies, David. “The effect if fluid-structural coupling on acoustical decays in a reverberation room in the high-frequency range,” J. Acoust. Soc. Am. 87(2), 718-27 (1990). Parkinson, John S. “Area and pattern effects in the measurement of sound absorption,” J. Acoust. Soc. Am. 2, 112 (1930). Pellam, John R. “Sound diffraction and absorption by a strip of absorbing material,” J. Acoust. Soc. Am. 11, 396 (1940). Sabine, Paul E. “Measurement of sound absorption coefficients by the reverberation method,” J. Acoust. Soc. Am. 1, 27 (1929). ; “Critical study of the precision of measurement of absorption coefficients by reverberation methods,” J. Acoust. Soc. Am. 3, 139 (1931). ; “Reverberation measurements of sound absorption coefficients,” J. Acoust. Soc. Am. 5, 220 (1934). “What is measured in sound absorption measurements?,” J. Acoust. Soc. Am. 6, 239 (1935). ; “Effects of cylindrical pillars in a reverberation chamber,” J. Acoust. Soc. Am. 10, 1 (1938). Sabine, W. C. Collected Papers on Acoustics, Harvard University Press, Cambridge (1922). Sato K.; and Koyasu, M. “On the measurement of absorption coefficient of acoustic materials by the reverberation method,” J. Acoust. Soc. Japan 13, 249-255 (1957). (In Japanese with English summary.). ; “The effect of the room shape on thesound field in rooms (Studies of the
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measurement of absorption coefficient by the reverberation chamber method I.),” J. Phys. Soc. Japan 14(3), 365-373 (1959). Physics Abstr. 13128 (Decmeber, 1959). Skrudzyk, G. Die Groundlagen der Akustik, Springer (1954). Steffen, E. “Untersuchungen zur Schallabsorptionsgradmessung im Hallraum,” Hochfrequenz-tech. u. Elektroakust. 67, 73-77 (1958). Takahashi, D. “Excess sound absorption due to periodically arranged absorptive material,” J. Acoust. Soc. Am. 86(6), 2215-22 (1989). Tartakovskii B. D.; and Efrussi, M. M. “On the measurement of sound absorbing material in resonance chambers,” Dokl. Akad. Nauk. USSR 82, 373-376 (No. 3, 1952). (In Russian). Phys. Abs. 56, 2296 (1953). Ten, Wolde T. “Measurements on the edge-effect in reverberation rooms,” Acustica 18(4), 207-212 (1967). Thomasson, S. I. “On the Absorption Coefficient,” Acustica, 44, 265 (1980). ; Watson, F. R. “Report of committee on sound absorption measurements,” J. Acoust. Soc. Am. 6, 54 (1934).
Absorption Coefficient and Impedance, Tube Methods Berendt, Raymond B.; and Schmidt, Henry A., Jr. “A portable impedance tube,” J. Acoust. Soc. Am. 35, 1049 (1963). Bruel, P. V. Rormetodens anvendelse i akustiken. (The use of the tube method in acoustics.) IngenVidensk. Skr., 1-184 (No.1, 1945). Ann. Télécom. 5, 32530 (Aug.-Sep. 1950). Cheung, Wan-Sup. “Improved method for the measurement of acoustic properties of a sound absorbent sample in the standing wave tube,” J. Acoust. Soc. Am. 97(5), 2733-9 (1995). Chu, E. T. “Transfer function technique for impedance and absorption measurements in an impedance tube using a single microphone,” J. Acoust. Soc. Am. 80(2), 555-60 (1986). ; “Further experimental studies on the transfer function technique for impedance tube measurements,” J. Acoust. Soc. Am. 83(6), 2255-60 (1988). ; “Extension of the two-microphone transfer function method for impedance tube measurements,” J. Acoust. Soc. Am. 84(1), 347-8(L) (1986). Jones, E.; Edelman, S.; and London, A. “Long-tube method for field determination of soundabsorption coefficients,” J. Research Natl. Bur. Standards 49, 17-20 (July, 1952). Phys. Abs. 56, 2303 (April 1953). Kurze, U. J. “Measurements of the coefficients of absorption and admittance in tubes,” Acustica 50(4), 267-272 (1982). Leedy, H. A. “Theoretical determination of sound absorptivity by the impedance method with experimental verification,” J. Acoust. Soc. Am. 10, 288 (1939). Legouis T.; and Nicolas, J. “Phase gradient method of measuring the acoustic impedance of materials,” J. Acoust. Soc. Am. 81(1), 44-50 (1987). Love, D. P.; and Morgan, R. L. “Small acoustic tube for measuring absorption of acoustical materials in auditoria,” J. Acoust. Soc. Am. 17, 326 (1946). Mawardi, Osman. “Some notes on the measurement of acoustic impedance,” J. Acoust. Soc. Am. 28(3), 351 (1956). Morrical, Keron C. “Modified tube method for the measurement of sound absorption,” J. Acoust. Soc. Am. 8, 67 (1936) and 162 (1937). Northwood, T. D.; and Pettigrew, H. C. “Horn as a coupling element for acoustic impdance measurements,” J. Acoust. Soc. Am. 26(4), 503 (1954). Rivir, R. B.; McErlean, D. P.; and Rabe, D. C. “Shock tube measurement of the weak shock reflection characteristics of acoustic materials,” pp. 72. Air Force Aero Propulsion Lab., Wright–Patterson AFB, Ohio (Aug. (1974). AD/A-003 852/1GA.
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Sabine, Hale J. “Notes on acoustic impedance measurements,” J. Acoust. Soc. Am. 14, 143 (1942). Shaw, E. A. G. “The acoustic waveguide. I. An apparatus for the measurement of acoustic impedance using plane waves and higher order mode waves in tubes,” J. Acoust. Soc. Am. 25(2), 224 (1953). ; “The acoustic waveguide. II. Some specific normal acoustic impedance measurements of typical porous surfaces with respect to normally and obliquely incident waves,” J. Acoust. Soc. Am. 25(2), 231 (1953). Taylor, Hawley O.; and Sherwin, Chalmers W. “Sound absorption and attenuation by the flue-method,” J. Acoust. Soc. Am. 9, 331 (1938). Wente, E. C. “Tube method of measuring sound absorption coefficients,” J. Acoust. Soc. Am. 1, 28 (1929).
Free Field and Other Measurement Methods Allard, J. F.; and Delage, P. “Free field measurements of absorption coefficients on square panels of absorbing materials,” J. Sound Vib. 101(2), 161-170 (1985). Bassery, L.; and Deprez, G. “Contribution to the experimental study of impulse response of acoustically absorbent materials in the audible frequency domain,” C. R. Acad. Sci. Ser. B 287(4), 73-76 (1978). (French, Engl. abstr). Bekkering, D. H.; and Kosten, C. W. “A simple arrangement for measuring sound absorption at low frequencies,” Appl. Sci. Res. B, 105-212 (No.3, 1948). Ann. Télécommun. 4, 23357 (May, 1949). Bolton, J. S.; and Gold, E. “The determination of acoustic reflection coefficients by using Cepstral techniques, I: Experimental procedures and measurements of polyurethane foam,” J. Sound Vib. 110(2), 179-202 (1986). ; “II. Extension of the technique and consideration of accuracy,” J. Sound Vib. 110(2), 203-222 (1986). Champoux, Yvan.; and L’Espérance, A. “Numerical evaluation of errors associated with the measurement of acoustic impedance in a free field using two microphones and a spectrum analyzer,” J. Acoust. Soc. Am. 84(1), 30-38 (1988). ; and Richarz, Werner G. “An aid in the numerical integration for in situ acoustic impedance and absorption coefficient measurements,” J. Acoust. Soc. Am. 87(4), 1809-12 (1990)(L). ; and Stinson, Michael R. “Measurement of the characteristic impedance and propagation constant of materials having high flow resistivity,” J. Acoust. Soc. Am. 90(4), 2182-90. Cho, Alfred C. F.; and Watson, Robert. “Pulse technique applied to acoustical testing,” J. Acoust. Soc. Am. 31, 1322 (1959). Davern, W. A. “Measurement of low frequency absorption,” Appl. Acoustics 21(1), 1-11 (1987). Ernsthausen W.; and Wittern, W. V. “A method of determining the acoustical reflectivity of the ground,” Akustische Zeits. 4, 353-359 (1939). Esmail-Begui, Z.; and Naylor, Thomas K. “Measurement of the propagation of sound in Fiberglass,” J. Acoust. Soc. Am. 25(1), 87 (1953). Ferraro, M. A.; and Sacerdote, G. G. “Measurement of acoustic impedance in a resonant spherical enclosure,” Acustica 8(5), 325-329 (1958). Gigli, A. “Measurement of the acoustic absorption coefficient,” Ingegnere 13, 4-14 (1938); Sci. Abs. B41, 1419 (1938). Hirata, Y. “Measurement of absorption coefficient by an electrical cancelling method,” Austica 27(1), 48 (1972). Hollin, K. A.; and Jones, M. H. “The measurement of sound absorption coefficient in situ by a correlation technique,” Acustica 37(2), 103-111 (1977).
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Hopper, F. L. “Determination of absorption coefficients for frequencies up to 8000 cycles,” J. Acoust. Soc. Am. 3, 415 (1932). Ingard, Uno; and Bolt, R. H. “A free field method of measuring the absorption coefficient of acoustic materials,” J. Acoust. Soc. Am. 23, 509-516 (1951). Kintsl, Z. “Investigation of the sound absorption of wall sections by a pulse technique,” Soviet Phys.–Acoust. 21(1), 30-32 (1975). Klein, C.; and Cope, A. “Angle Dependence of the Impedance of a Porous Layer,” Acustica, 44, 258 (1980). Kosten, C. W. “A new method for measuring sound absorption,” Appl. Sci. Res.B, B1(1), 35-49 (1947). Kuroiwa, K.; Yasuda, T.; and Matsui, M. “Field measurement method of absorption characteristics of acoustic panel by correlation method,” J. Acoust. Soc. Jap. 30(9), 486-491 (1974). (Japanese, Engl. abstr.). Lauriks, Walter.; Cops, André.; Allard, Jean F.; Depollier, Claude; and Rebillard, Pascal. “Modelization of oblique incidence of layered porous materials with impervious screens,” J. Acoust. Soc. Am. 87(3), 1200-6 (1990). Lawrence, D. E. P.; and Don, C. G. “Impulse measurements of impedance and propagation constant compared to rigid-frame and dual-wave predictions for foam,” J. Acoust. Soc. Am. 97(4), 2255-62 (1995). Legeay, V.; and Seznec, R. “On the determination of the acoustical characteristics of absorbent materials,” Acustica 53(4), 171-192 (1983). Lippert, W. K. R. “Method of determining the propagation parameters of wave motion in acoustical materials,” Acustica 9(6), 419-430 (1959). Mariano, S. “Shock tube technique for testing acoustic materials at very high sound amplitudes,” J. Sound Vib. 14(2), 145-150 (1971). ; Phys. Abstr. 34760 (1971). Meyer, Erwin; and Schoch, Arnold. “Measurements of the comparative degree of sound absorption,” Akust. Zeits. 4, 51-61 (1939). Nakamura, A. “A resonance method for measurement of acoustic properties of porous materials,” J. Acoust. Soc. Japan 15, 1-5 (March, 1959). (In Japanese). Powell, John G.; and Van Houlen, John J. “Tone–burst technique of sound absorption measurement,” J. Acoust. Soc. Am. 48, 1299 (1970). ; “Technique for evaluating the sound absorption of materials at high intensities,” Pp. 77. LTV Res. Centre, Arnheim, Calif. Contract NASL–8763 (1971). ; Phys. Abstr. 4045 (20 Jan. 1972). Rivin, A. N. “Determination of the absorption coefficient for various angles of wave incidence,” Sov. Phys.–Acoust., 20(5), 486-487 (1975). Rogers, Charles L.; and Watson, Robert B. “Determination of sound absorption coefficients using a pulse technique,” J. Acoust. Soc. Am. 32, 1555 (1960). Schuster, K. “Determination of the absorption coefficient for oblique sound incidence,” Akustische Zeits. 3, 137-140 (1938). Shields, F. Douglas; Bass, H. E.; and Bolen, L. N. “Tube method of sound-absorption measurement extended to frequencies far above cutoff,” J. Acoust. Soc. Am. 62(2), 346 (1977). Tamura, Masayuki; Allard, J. F.; and Lafarge, Dennis. “Spatial Fourier-transform method for measuring reflection coefficients at oblique incidence. II. Experimental results. J. Acoust. Soc. Am. 97(4), 2255-62 (1995). Tarnov, Viggo. “Measurement of sound propagation in glass wool,” J. Acoust. Soc. Am. 97(4), 2272-81 (1995). Utsuno, Hideo; Tanaka, Toshimitsu; Fujikawa, Takeshi; and Seybert, A. F. “Transfer function method for measuring characteristic impedance and propagation constant of porous materials,” J. Acoust. Soc. Am. 86(2), 637-43 (1989).
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Absorption Data, General Discussion Ando Y.; and Kosaka, K. “Effect of humidity on sound absorption of porous materials,” Appl. Acoust. 3(3), 201-206 (1970). Aso, S.; and Kinoshita, R. “Absorption of sound wave by fabrics–1, 2,” J. Textile Machinery Soc. Japan 9(1), 32-46 (Jan. 1963). ; “Absorption of sound wave by fabrics–3,” J. Textile Machinery Soc. Japan 10(5), 236-241 (1964). ; “Sound absorption characteristics of fiber assemblies,” J. Textile Machinery Soc. Japan 10(5), 209-217 (1964). Balachandran, C. G.; and Murty, T. V. S. “Sound absorption coefficients of Vermiculite Mortars,” J. Sci. Ind. Research (India) 17b, 286-287 (July, 1958). Physics Abs. 1225 (February 1959). Becker, G.; Bobbert, F.; and Brandt, H. “Comparative measurements of acoustic materials,” Akust. Beih. No. 3, AB176-AB180 (1952). Bhandari, P. S.; and Yadav, L. B. “Sound absorption by porous bricks,” Indian J. Tech. 7(10), 337 (1969). Duggar, B. C. “Special sound absorptive materials in noise control,” Am. Ind. Hyg. Assoc. Quart. 20, 447-452 (1959), Dece. Eisenberg, A. “Comparative measurements of sound absorption, 1950,” Akust. Beih. No. 2 AB108-AB114 (1952). Evans, E. J.; and Bazley, E. N. Sound Absorbing Materials, National Physical Laboratory, Teddington, Middlesex (Great Britain) (1978). ; and Parkin, P. H. “Sound absorption of a stone wall,” Acustica 8(2), 117-118 (1958). Delany, M. E.; and Bazley, E. N. “Acoustical properties of fibrous absorbent materials,” Applied Acoustics, (3), 105 (1970). Harris, Cyril M. “Acoustical properties of carpet,” J. Acoust. Soc. Am. 27, 1077 (1955). Hedeen, Robert A. Compendium of Materials for Noise Control, National Institute for Occupational Safety and Health (NIOSH), Publication No. 80-116, 1980. Horoshenkov, K. V.; Hothersall, D. C.; and Attenborough, K. “Porous materials for scale model experiments in outdoor sound propagation,” J. Sound Vib. 194(5) 685-708 (1996). Igarashi, Juichi. “The determination of sound absorption coefficient,” J. Phys. Soc. Japan 5, 249-253 (1950). (In English.) Kaye, G. W. C.; and Evans, E. J. “Sound absorption of snow,” Nature 143, 80 (1939). “Sound absorption properties of common outdoor materials,” Phys. Soc., Proc. 52, 371-379 (1940). Koyasu, M. “Sound absorption characteristics of fibrous materials. Part II. Rockwool and Slagwool,” Bull. Kobayasi Inst. Phys. Res. 12(1-2), 7-17 (1962). Kuroiwa, K.; Yasuda, T.; and Matsui, M. A. “A method of measuring the absorption characteristics of acoustic panels by correlation method,” J. Acoust. Soc. Japan 28(8), 405-413 (1972). (Japanese with English abstr.). ; and R. Lamoral, “On the absorption coefficient of acoustical materials,” Onde Élec. 33, 461-467 (July 1953). (In French). Phys. Abs. 7, 3243 (April, 1954).
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Lane, R. N.; and Botsford, James. “Total sound absorption by upholstered theater chairs with audience,” J. Acoust. Soc. Am. 24(2), 125 (1952). Nyborg, W. L.; Rudnick, I.; and Schilling, H. K. “Experiments on acoustic absorption in sand and soil,” J. Acoust. Soc. Am. 22(4), 422 (1950). O’Keefe, E. “Physical and acoustical properties of urethane foams,” Sound Vib. 12(7), 16-21 (1978). Paffrath, H. W.; and Schmidt, W. “Akustische Eigenschaften von weichen und halbharten Polyurethan-Schaumstoffen,” Kunststoffe 52(10), 599-603, 1-2, 4 (1962). Pancholy, M.; and Atal, B. S. “Sound absorbing properties of Indigenous Mineral Wool,” J. Sci. Ind. Research (India) 17B, 239-241 (July, 1958). Physics Abs. 150 (January, 1959). Parkinson, J. S.; and Jack, W. A. “Re-examination of the noise reduction coefficient,” J. Acoust. Soc. Am. 13, 163 (1941). Rees, W. M. “Forms, properties, and functions of fibrous glass acoustical materials,” Communications 26, 36-38 (1940). Sabine, Hale. “Review of the absorption coefficient problem,” J. Acoust. Soc. Am. 22(3), 387 (1950). ; “Manufacture and distribution of acoustical materials over the past 25 years,” J. Acoust. Soc. Am. 26(5), 657. Seligman, G. “Sound absorption of snow,” Nature 143, 1071 (1939). Slavik, J. B.; and Tichy, J. “An evaluation of the sound absorption coefficients measured by the tube and reverberation methods,” Slaboproudy Obzor. 18(8), 545-548 (1957). Stephens, R. W. B. “Acoustical properties of high polymers,” in Physics of Plastics, pp. 410-439. D. Van Nostrand Co. Inc. Princeton, N.J. (1965). Phys. Abs. 7491 (Mar. 1967). Tyzzer, F. G.; and Leedy, H. G. “Advances since 1929 in methods of testing acoustical performance of acoustical materials,” J. Acoust. Soc. Am. 26(5), (1954). Val, M. “Absorption coefficients of materials,” Appl. Acoust. 2(4), 309-316 (1969). ; and R. Lehman, “Contribution à l’étude des coefficients d’absorption des matériaux,” Rev. Acoust. 3(11), 221-224 (1970). Velizhanina. K. A.; and Yastrebov, V. V. “Small chamber method for investigation of sound absorbing systems at high sound levels,” Akust. Zh. 24(1), 130-132 (1978) (In Russian). Venzke, G. “Die Schallabsorption poröser Kunststoffschäume,” Acustica 8(5), 295-300 (1958). Vogel, T. “Notes on the acoustical properties of materials,” Onde Élect. 36, 428-434 (May 1956). (In French). Phys. Abs. 205 (January 1957). Willig, F. J. “Comparison of sound absorption coefficients obtained by different methods,” J. Acoust. Soc. Am. 10, 293 (1939). Yokoyama, I.; and Awaya, Y. “Measurement of acoustical 4–terminal coefficients by graphical methods,” J. Acoust. Soc. Japan 12, 63 (June 1956). (In Japanese).
C.2.3 Anechoic Wedges and Rooms Anathapadmanabha, T.; et. al. “Design of an anechoic chamber to study acoustic characteristics of contacting surfaces,” J. Inst. Eng. (India) 63(ME5), 189-195 (1983). Phys. Abstr. 10514 (1 Feb. 1984). Anonymous. “The anechoic room at Nanking University,” Acta Phys. Sin 34(4), 252-259 (1975). (Chinese). Phys. Abstr. 83086 (1 Dec. 1975). Ansbro, A. P.; and Arnold, J. M. “Calculation of the Green’s function for the wedge-shaped layer,” J. Acoust. Soc. Am. 90(3), 1539-46 (1991). Araki, K.; et. al. “Acoustic characteristics of reverberation room, anechoic room and listening room in Takenaka Tec. Res. Lab, Regional Office,” Takenaka Tech. Res. Rep. No. 33, 65-72 (May 1985). Phys. Abstr. 118556 (2 Dec. 1985).
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Ballagh, K. O. “Qualification tests in a hemi-anechoic room,” Acustica 54(5), 296-299 (1984). ; “Calibration of an anechoic room,” J. Sound Vib. 105(12), 233-241 (1986). Bell, E. C.; Hulley, L. N.; and Mazumber, N. C. “The steady state evaluation of small anechoic chambers,” Appl. Acoust. 6, 91 (1973). Beranek, Leo L.; and Sleeper, Harvey P., Jr. “Design and construction of anechoic sound chambers,” J. Acoust. Soc. Am. 18, 140 (1946). “Design of anechoic sound chambers,” J. Acoust. Soc. Am. 18, 246 (1946). Berger, L.; and Ackerman, E. “The Penn State Anechoic Chamber,” Noise Control 2(5), 16-21 (1956). Berhault, J. P.; et al. “The design and construction of an anechoic chamber, lined with panels, for investigation of aerodynamic noise,” Acustica 29(2), 69-78 (1973). (French, Engl. abstr.). Bies, D. A. “Uses of anechoic and reverberant rooms for the investigation of noise sources,” Noise Control Engineering 7(3), 154-163 (1976). Bly, S. H. P.; Lightstone, A. F.; Hewlings, T. G.; and Lourador, J. L. “Performance of a large new anechoic chamber at the Canadian Bureau of radiation and medical devices,” J. Acoust. Soc. Am. 93(4), 2344(A) (1993). Brandt, O.; and Hagman, T. “Construction of anechoic rooms,” Tekn. Tidskrift 80, 1087 (1950). (In Swedish). Bruchmüller, H. G. “Measurement of the absorption coefficient of an anechoic chamber wall constructed on the ‘bead’ principle,” Appl. Acoust. 10(3), 101-109 (1977). Burkewitz, Bert L. “Qualification of a hemi-anechoic room for sound power level determination,” J. Acoust. Soc. Am. 74(S1), S113(A) (1983). Burrin, R. H.; Dean, P. D.; and Tanna, H. K. “A new anechoic facility for supersonic hot jet noise research at Lockheed-Georgia,” J. Acoust. Soc. Am. 55, 400(A) (1974). Cohen, R. “Features of a semi-anechoic chamber for vehicles.” Internoise 80. Proc. 1980 Int. Conf. Nose Control Eng. Miami, FL 8-10 Dec. 1980 1097-1102. Phys. Abstr. 16203 (1 Mar. 1982). Crawford, M. I. “Evaluation of reflectivity level of anechoic chambers using isotropic, 3-dimensional probing,” p. 7. Natl. Bur. Stand., Washington, D.C. (June 1974). COM-7550298/9GA. (See also J. Res. Natl. Bur. Stand. Sect. A 79A(4), 590 (1975).) Dagnall, A. “An inexpensive anechoic chamber,” Phys. Educ. 9(4), 247-250 (1974). Phys. Abstr. 39 (2 Jan. 1975). Dämmig, P.; and Deicke, H. “Ein kleiner reflexionsarmer Raum und seine Brauchbarkeit zu vereinfachten akustischen Messungen,” Acustia 34(4), 252-255 (1976). Davy, J. L. “Evaluating the lining of an anechoic room,” J. Sound Vib. 132(3) 411-422 (1989). Delany, M. E.; and Bazley, E. N. “The design and performance of free-field rooms,” 7th Inern. Congr. Acoust., Budapest 1971, Manuscript 24 A6. “The high frequency performance of wedge-like free field rooms,” J. Sound Vib. 55(2), 195-214 (1977). Diestel, H. G. “Zur Schallausbreitung in reflexionsarmen Räumen,” Acustica 12, 113-118 (1962). ; “Messung des mittleren Reflexionsfaktors der Wandenauskleidung in einem reflexionsarmen Raum,” Acustica 20, 101 (1968). Duda, J. “Basic design considerations for anechoic chambers,” Noise Control Engineering 9(2), 60-67 (1977). Dykstra, R. A.; and Baxa, D. E. “Semi-anechoic testing room: Some sound advice,” Sound Vib. 11(5), 35-38 (1977). Easwaran, V.; and Munjal, M. I. “Finite element analysis of wedges used in anechoic chambers,” J. Sound Vib. 160(2) 333-350 (1993). Eckel, Oliver C. “Sound absorbing assembly with an integral cage,” J. Acoust. Soc. Am. 57, 765 (1975).
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Epprecht, G. W.; Kurtze, G.; and Lauber, A. “Construction of an anechoic room for sound waves and 10 cm electromagnetic waves,” Akust. Beih. No. 2, 567-577 (1954). S. Fujii, N. Suzuki, and T. Nagaoka, “Anechoic chamber having column-like sound- absorbing block,” J. Acoust. Soc. Am. 64(S1), S77(A) (1978). Guy, R. W.; and Li, J. “A facility and test procedure designed for sound-power measurement by the point-to-point intensity technique,” Appl. Acoust. (UK) 36(2), 107-121 (1992). Halliwell, R. E. “An acoustical round robin with a difference,” J. Acoust. Soc. Am. 71(S1), S39(A) (1982). Hanna, Y. I. “Design, construction and test of an anechoic chamber,” Indian J. Technol. 22(12), 456-459 (1984). Phys. Abstr. 120030 (2 Dec. 1985). Hardy, H. C.; Tyzzer, F. G.; and Hall, H. H. “Performance of the anechoic room of the Parmly Sound Laboratory,” J. Acoust. Soc. Am. 19, 992 (1947). Head, J. W. “The design of gradual transition (wedge) absorbers for a free-field room,” Brit. J. Appl. Phys. 16(7), 1009-1014 (July 1965). Phys. Abs. 21156 (1 Sept. 1965). Hickey, D.; et.al. “Calibration of the Ames anechoic facility. Phase 1: Short range plan. p. 129 Ames Res. Ctr. NASA Moffett Field CA (Dec. 1980). N82-16091/2 (No. 12, 1982). Hinders, M. K.; Rhodes, B. A.; and Fang, T. M. “Particle loaded composites for acoustic anechoic coatings,” J. Sound Vib. 185(2), 219-246 (1995). Hunter, J. R.; and Klipsch, P. W. “Anechoic chamber with optional boundaries,” J. Audio Eng. Soc. 30(7-8), 528-31 (1982). Illenyi, A.; and Korpassy, P. “Elimination of interfering echoes in acoustic measurements in anechoic chambers,” Magy. Fiz. Foly 28(4), 399-413 (1980) (Hungarian). Phys. Abstr. 59927 (15 July 1981). Ingerslev, F.; Pedersen, O. J.; and M/oller, P. K. “New rooms for acoustic measurements at the Danish Techn. Univ.,” Acustica 1967/68, No. 4, 185-199. Ishida, T.; and Umeda, A. “Acoustic property of an anechoic room,” J. Tex. Mech. Soc. Jpn. 28(3), 145-150 (1975). (In Japanese). Janovsky, W.; and Spandöck, F. “Construction and investigation of a damped room,” Akustische Zeits. 2, 322-331 (1937). (In German). Klimov, B. M.; and Rivin, A. N. “Sound absorbing Phenopolyurethane wedge coating,” Soviet Phys.–Acoust. 8(3), 286-287 (1963). Klipsch, Paul W.; and Hunter, James R. “Anechoic chamber arrangement,” J. Acoust. Soc. Am. 74(5), 1665-6 (1983)(P). Koidan, Walter; and Hruska, Gale R. “Acoustical properties of the National Bureau of Standards anechoic chamber,” J. Acoust. Soc. Am. 64(2), 508 (1978). ; and Pickett, Marshall A. “Wedge design for National Bureau of Standards anechoic chamber,” J. Acoust. Soc. Am. 52(4, part 1), 1071-1076 (1972). Koyasu, M. “Acoustical design of anechoic and semi-anechoic room,” Archit. Acoust. Noise Control 5(3), 251-256 (1976) (In Japanese). Kudrjavcev, F. S.; and Lagunov, L. F. “Instruction for design of anechoic chambers for machine noise measurement”. p. 79. Vses. Central. Nauchno-Issled. Inst. Ohrany. Tr VCSPS, Moscow (1978) (In Russian). Ergon. Abstr. 78493 (No. 1 1980). Kuttruff, H.; and Bruchmüller, H. G. “On measuring techniques for the examination of anechoic rooms,” Acustica 30(6), 342-349 (1974). (German, Engl. abstr.). Lai, J. C. S. “An anechoic chamber at the Australian Defense Force Academy,” Acoust. Aust. 15(1), 14 (1987). Lauchle, G. C.; and Wong, E. “The ARL/FEU (Fluid Engineering Unit) semi-anechoic chamber,” p. 22. Appl. Res. Lab., Pennsylvania State Univ., University Park, PA (Sept. 1975). AD-A021 247/2GA.
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Lorand, P. “Salles anéchoiques,” p. 7. Tech. Ingr. Electron., E 2620 (Sept. 1975). Ann. Télécommun. 273428 (No. 1, 1976). Maekawa, Z.; and Morimoto, M. “New anechoic chamber at the environmental laboratory in Kobe University,” Mem. Fac. Eng., Kobe Univ.– Phys. Abstr. 113576 (15 Nov. 1985). ; “New anechoic chamber with slits between wedges for air conditioning at Kobe University,” J. Acoust. Soc. Am. 64(S1), S77(A) (1978). Meyer, E.; Buchmann, G.; and Schoch, A. “A new sound absorbing arrangement of high efficiency and the construction of an anechoic room,” Akustische Zeits. 5, 352-364 (1940). ; Also in Zeits. f. techn. Physik 21, 372-375 (1940). (In German). Wireless Engineer 18, 2745A (Oct. 1941). Sci. Abs. A, 45, 724 (1942). Mills, Peter J. “Construction and design of Parmly Sound Laboratory and Anechoic Chamber,” J. Acoust. Soc. Am. 19, 988 (1947). Milosevic, M.; and Nicolic, J. “Anechoic chamber of the faculty of Electrical Engineering in Nis.,” Teknika (Belgrad) 38(4), 617-620 (1983) (In Croatian). 90256 (3 Oct. 1983). Moreland, J. B. “Performance of hemi-anechoic rooms,” J. Acoust. Soc. Am. 68(S1), 826(A) (1980). Nakajima, Heitaro. “SONY inaugurates a new acoustical laboratory,” J. Acoust. Soc. Am. 55, 364(T) (1974). Peterson. G. E.; Hellwarth, G. A.; and Dunn, H. K. An anechoic chamber with blanket wedge construction,” J. Audio Eng. Soc. 18, 67 (1967). Pronenko, L. Z.; and Rivin, A. N. “Stable glassfiber sound absorbers or anehoic chambers,” Akust. Zhur. 5(3), 275-281 (1959). (In Russian). Quartararo, L. B.; and Lauchle, G. C. “Inlet wall design for high-volume-flow subsonic anechoic chambers,” Noise Control Engineering 24(3), 86-97 (1985). Reaves, E. R. “The anechoic chamber in the London Test Section of the British Post Office,” Post Off. Elect. Eng. J. 68(2), 91-95 (1975). Ann. Télécommun. 275779 (5, 1976). Robinson, D. W. “Anechoic chamber for acoustic measurements,” Elec. Commun. 28, 70-77 (Mar. 1951). Sato, S.; Fujimori, T.; and Miura, H. “Sound absorbing wedge design using flow resistance of glass wool,” J. Acoust. Soc. Jpn 35(11), 626-636 (1979). Savell, C. T.; and Stringas, E. J. “High velocity jet noise source location and reduction. Task I. Supplement. Certification of the General Electric Jet Noise Anechoic Test Facility,” p. 236. Aircr. Eng. Group, General Electric Company, Cincinnati, OH (Feb. 1977). AD-A042 327/7GA. Schoch, A. “Theory of linings for anechoic rooms, based on the principle of gradual transition,” in Noise and Sound Transmission, Phys. Soc., London, (1949), 167-173. Phys. Abs. 53, 3017 (May 1950). Suetake, K. “Anechoic chamber and microwave absorbing walls,” J. Soc. Instrum. Control Eng. 17(1), 115-120 (1978). (In Japanese). Phys. Abstr. 46861 (15 June 1978). Sunyach, M.; Brunel, B.; and Comte-Bellot, G. “Capabilities of the high-speed anechoic wind tunnel of the Ecole Centrale de Lyon,” Rev. Acoust. 18(73), 316-330 (1985) (In French). Phys. Abstr.113576 (15 Nov. 1985) (See also “Performance of the high-speed anechoic wind tunnel at Lyon University,” p. 23. NASA, Washington, DC (Mar. 1986). N86-28965/9/GAR (No. 24 1986).) Takagi, K.; and Ikegava, K. “Forced deterioration test of glass fiber used for anechoic rooms,” Rept. Electr. Communic. Lab. Japan 6, 220-226 (June, 1958). Ann. Télécomm. 117835 (July, 1959). Tarnoczy, T. “Technical data of MTA acoustic measuring room,” Magy. Fiz. Foly. 28(4), 377-397 (1980). Phys. Abstr. (15 July, 1981).
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Tyzzer, F. G. “Development of a suitable anechoic treatment for the ASD sonic fatigue facility,” Armour Res. Found., Chicago. Final Rept., 138 pp. (March, 1963). Veit, I.; and Sander. H. “Production of spatially limited ‘diffuse’ sound field in an anechoic room,” J. Audio Eng. Soc. 35(3), 138-143 (1987). Velis, A. G.; Giuliano, H. G.; and Mendez, A. M. “The anechoic chamber at the Laboratorio de Acustic y Luminoicnia CIC,” Appl. Acoust. (UK) 44(1), 79-94 (1995). Velizhanina, K. A.; and Rzhevkin, S. N. “The investigation of sound-absorbent structures for the anechoic chamber of the Physics Department of the Moscow State University,” Soviet Phys.–Acoustics 3(1), 21-26. (January-March 1957). (English translation of Akust. Zhur). Viebrock, V. M.; and Crocker, M. J. “Miniature anechoic room design,” J. Sound Vib. 32(1), 77-86 (1974). Vincenti, A.; and Poscetti, I. “The anechoic room set up at the ISPT.,” Note Recens. Not 30(2), 82-91 (9181) (In Italian). Walther, K. “Reflection factor of gradual-transition absorbers for electromagnetic and acoustic waves,” IRE Trans. 8, 608-621 (Nov., 1060). Wang, Ji-qing. “New acoustical test facilities at Tongij University, Shanghai,” Westpac II, Hong Kong, 27-30 Nov (1985), 535-540 (Hong Kong Polytech., Hong Kong, 1985). ; and Cai Biao, “Calculation of free-field deviation in an anechoic room,” J. Acoust. Soc. Am. 85(3), 1206-1212 (1989). Warnaka, Glenn E. “A different look at the anechoic wedge,” J. Acoust. Soc. Am. 75(3), 855-8 (1984). (See also comments on this paper by K. O. Ballagh and C. Wassilieff, J. Acoust. Soc. Am. 77(4), 1612-14(L) (1985). ; “Structure for absorbing acoustic and other wave energy,” J. Acoust. Soc. Am. 77(6), 2208 (P) (1985).) Watters, B. “Design of wedges for anechoic chambers,” Noise Control 4(6), 32-37 (1958). Wuzuniak, J. A.; Shaw, I. M.; and Essary, J. D. “Characteristics of an anechoic chamber for fan noise testing,” p. 32. NASA Tech. Memo. NASA TM X-73555 (Mar. 1977). Wise, R. E.; and Nobile, M. A. “Absorption measurements on anechoic room wedges,” Proc. Int. Conf. Noise Control Eng. Honolulu, 3-5, 409-414 (Dec. 1984). Zhang, Zhong-Peng. “Performance evaluation techniques of anechoic chambers, Acta Phys. Sin. 31(11), 1457-65 (1982) (In Chinese). Phys. Abstr. 42083 (2 May 1983).
C.2.4 Resonators and Related Matters Abramchik, M.; and Maletsky, I. “Three-dimensional multiresonant sound absorber,” Akust. Zhur. 5(3), 275-281 (1959). (In Russian). ; “Multiresonance volume absorber,” Soviet Phys.– Acoustics 5(3), 282-287. ; English translation of Akust. Zhur. “Influence of position on the action of spatial absorbers,” Soviet Phys.–Acoust. 6(4), 491-492(L) (1961). Becker, Carl. “Reactive components in sound absorber construction,” J. Acoust. Soc. Am. 28(6), 1068 (1956). Bruel, P. V. “Panel absorbents of the Helmholtz type,” in Noise and Sound Transmission, Phys. Soc., London (1949), 10-17. ; Phys. Abs. 53, 3012 (May 1950). Choudhury, N. K. D.; and Kanta Rao, M. V. S. S. “Panel absorber for low frequency sound absorption,” J. Inst Telecommun. Engrs. 5, 103-108 (March 1959). Cummings, A.; and Chang, I. J. “The transmission of intense transient and multiple frequency sound waves through orifice plates with mean fluid flow,” Rev. Phys. Appl. 21(2), 151161 (1986). ; Ferrero, M. A.; and Sacerdote, G. G. “Resonant absorbing metallic structures,” Acustica 9(1), 23-26 (1959). Fahy, F. J.; and Schofield, C. “A note on the interaction between a Helmholtz resonator and an acoustic mode of an enclosure,” J. Sound Vib. 74(3), 464 (1981).
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Fukuchi, T.; and Fujiwara, K. “Sound absorption area per seat of upholstered chairs in a hall,” J. Acoust. Soc. Jpn (E)6(4), 271-270 (1985). Gigli, A. “The absorption of sound by means of resonators,” Alta Frequenza 9, 717-744 (1940). Hiraizumi, M.; et. al. “Study of sound absorption by panel vibration–circular panel with clamped boundary,” J. Acoust. Soc. Japan 30(5), 276-184 (1974). Ingard, Uno. “On the radiation of sound into a circular tube with an application to resonators,” J. Acoust. Soc. Am. 20, 665-682 (1948). ; “On the theory and design of acoustic resonators,” J. Acoust. Soc. Am. 25, 1037-1061 (1953). ; “The near field of a Helmholtz resonator exposed to a plane wave,” J. Acoust. Soc. Am. 25, 1062-1067 (1953). ; “Absorption characteristics of nonlinear acoustic resonators,” J. Acoust. Soc. Am. 44, 1155-1156 (1968). ; “Transient response and reverberation of a Helmholtz resonator,” Fundamentals of Waves and Oscillation, p. 108, Cambridge University Press (1988). ; and S. Labate, “Acoustic circulation effects and the nonlinear impedance of orifices,” J. Acoust. Soc. Am. 22, 211-218 (1950). ; and R. H. Lyon, “The impedance of a resistance loaded Helmholtz resonator,” J. Acoust. Soc. Am. 25, 854-857 (1953). ; and Hartmut Ising, “Acoustic nonlinearity of an orifice,” J. Acoust. Soc. Am. 42, 6-17 (1967). Jordan, Wilhelm L. “On sound absorption by resonators,” Akustische Zeits. 5, 77-87 (1940). ; “Application of Helmholtz resonators to sound absorbing structures,” J. Acoust. Soc. Am. 19, 1972 (1947). Kurtze, G. “Improvement of the efficiency of acoustical tiles by resonators,” Acustica 4(4), 447-450 (1954). ; and A. Lauber, “Comparative measurements of sound absorption of acoustic materials by the tube and the reverberation methods,” Tech. Mitt. P T T 32, 249-253 (July 1, 1954). (In German). Lambert, Robert F. “Study of the factors influencing the damping of an acoustical cavity resonator,” J. Acoust. Soc. Am. 25(6), 1068 (1953). Lauber, A. “Akustische Resonatoren als Schallabsorber,” Techn. Mitt. Schweiz. Telegr. Teleph. 30, 209-213 (No. 7, 1952). Lauffer, H. “Sound absorption of panels capable of vibration,” Hochfrequenztech. u. Elektroakustik 49, 9-20 (1937). McGinnis, C. S.; and Albert, V. F. “Multiple Helmholtz resonators,” J. Acoust. Soc. Am. 24(4), 374 (1952). Nesterov, V. S. “The absorption of sound by systems with double resonance,” J. Techn. Physics, USSR, 9, 1727-1739 (1939). ; “The absorption of sound by means of a three-layer resonance system,” J. Techn. Phys. 10, 617-626 (1940). (In Russian). Nolle, A. W. “Small-signal impedance of short tubes,” J. Acoust. Soc. Am. 25(1), 32 (1953). Norris, A. N.; and Wickham, G. “Elastic Helmholtz resonators,” J. Acoust. Soc. Am. 93(2), 617-30 (1992). Ochmann, M. “An asymptotic method for the analysis of non-linear Helmholtz resonators,” Acustica 65(1), 11-20 (1987). (German. Engl. abstr.). Oie, S.; and Takeuchi, R. “Some considerations on acoustical properties of porous plates,” Acustica, 45, 56 (1980). Ormestad, H. “On sound absorption by vibrating plates,” K. Norske Vid. Selsk. Forh. 23, Paper 2, 3-6 (1950). Phys. Abs. 54, 9349 (Dec. 1951). ; “Experiments with sound absorption by damped vibrating plates,” K. Norske Vid. Selsk. Forh. 23, Paper 3, 7-10 (1951). Parkin, P. H.; and Purkis, H. J. “Sound absorption by wood panels for the Royal Festival Hall,” Acustica 1(2), 81-82 (1951). Photiadis, Douglas M. “The effect of wall elasticity on the properties of a Helmholtz resonator,” J. Acoust. Soc. Am. 90(2), 1188-90(L). Ragavan, D. Govinda. “Slit resonators as low frequency sound absorbers,” J. Inst. Telecommun. Engrs, 4, 213-219 (September, 1958).
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Rogers, Robert. “Absorption of sound by vibrating plates backed with an air space,” J. Acoust. Soc. Am. 10, 280 (1939). ; “Theoretical investigation of the absorption of sound by vibrating materials,” J. Acoust. Soc. Am. 10, 85 (1938). Rschevkin, S. N. “High coefficient of sound absorption by using systems of resonators,” Comptes Rendus (Doklady) de l’Acad. des Sciences, USSR 18, 25-30 (1938). ; Sci. Abs. B41, 1778 (1938). “Resonance absorption of sound,” Techn. Phys., USSR 3, 560-576 (1936). ; “Theory and design of the simplest resonance sound absorbing systems,” Comptes Rendus de l’Acad. des Sciences, USSR 22, 564-69 (1939). ; “Resonance sound absorber with yielding wall,” Tech. Mem. NACA No. 1273, 21 pp (May 1951). Phys. Abs. 55, 2573 (April, 1952). ; “Gestaltung von Resonanzschallschluckern und deren Verwendung für die Nachhallregelung und Schallabsorption,” Hochfrequenz-tech. u. Elektroakust. 67, 128-135 (1959). Sabine, Paul E.; Ramer, L. G. “Absorption-frequency characteristics of plywood panels,” J. Acoust. Soc. Am. 20, 267 (1948). Sacerdote, Gino G.; and Gigli, Antonio. “Absorption of sound by resonant panels,” J. Acoust. Soc. Am. 23(3), 349 (1951). Schucardt, Mark E.; Dear, Terrance; and Ingard, Uno. “Air induction for a 4 cylinder engine: Design, noise control, and engine efficiency,” Paper 931317, March 12, 1993, SAE Meeting, Traverse City, Michigan. Takeuchi, R.; and Shindo, T. “The absorption coefficient of a porous plate,” Mem. Res. Inst. Acoust. Sci. Osaka 2, 28-33 (March, 1951). ; Phys. Abs. 55, 2527 (April 1952). Thurston, George B.; and Charles E. Martin, Jr., “Periodic fluid flow through circular orifices,” J. Acoust. Soc. Am. 15(1), 26 (1953). ; and John K. Wood, “End correction for a concentric orifice in a circular tube,” J. Acoust. Soc. Am. 25(5), 861 (1953). Vlˇcek, M. “Die Dämpfung von Resonatoren,” Hochfrequenz–tech. u. Elektroakust. 67, 135-139 (January, 1959). Voelz, Kurt. “Die Dämpfung akustischer Resonatoren,” Z. Angew. Phys. 3, 67-72 (1951). Ward, F. L. “Helmholtz resonators as acoustic treatment at the New Swansea Studios,” B. B. C. Quart. 7, 174-180 (Fall, 1952). Ann. Télécomm. 7, 50847 (Dec., 1952). Willms, Walter. “On sound absorption with the aid of damped resonators,” Akust. Zeits. 4, 29-32 (1939). Wood, John K. “Acoustic resistance of a pipe orifice to steady-state fluid flow,” J. Acoust. Soc. Am. 26(4), 492 (1954). ; and George Thurston, “Acoustic impedance of rectangular tubes,” J. Acoust. Soc. Am. 25(5), 858 (1953). Wöhle, W. “Die Schallabsorption von Einzelresonatoren in der unendlich grossen Wand,” Hochfrequenz–tech. u. Elektroakust. 67, 140-146 (January, 1959). ; “Die Schallabsorption von Einzelresonatoren bei verschiedener Anordnung im geschlossenen Raum (Wandmitte, Kante, Ecke),” Hochfrequenz–tech. u. Elektroakust. 67, 180-187 (March, 1959). ; “Die Schallabsorption con Einzelresonatoren bei linienförmiger Anordnung in der unendlich ausgedehnten Wand and im geschlossenen Raum,” Hochfrequenz–tech. u. Elektroakust. 68, 7-14 (May, 1959). ; “Schallabsorption von Einzelresonatoren bei allseitigem Schalleinfall und by Anordnung in einer Linie, in Raummitte, and der Wand, in der Kante oder Ecke eines Raumes,” Hochfrequenz-tech. u. Elektroakust. 68, 50-56 (July, 1959). Zeller, W. “Sound damping by air resonators in building construction,” Akustische Zeits. 3, 32-35 (1938).
C.2.5 ‘Functional’ or ‘Volume’ Absorbers Abramchik M.; and Maletsky, I. “Three-dimensional multiresonant sound absorber,” Akust. Zhur. 5(3), 275-281 (1959). (In Russian). ; “Multiresonance volume absorber,” Soviet
HISTORICAL NOTES AND REFERENCES, ABSORBERS
427
Phys.–Acoustics 5(3), 282-287. English translation of Akust. Zhur. “Influence of position on the action of spatial absorbers,” Soviet Phys.–Acoust. 6(4), 491-492(L) (1961). Batcheldor, James H.; Thayer, William S.; and Schultz, Theodore J. “Sound absorption of draperies,” J. Acoust. Soc. Am. 42, 573 (1967). Beranek, Leo L. “Audience and seat absorption in large halls,” J. Acoust. Soc. Am. 32, 661 (1960) and 45, 13 (1969). Borisov, L. A.; and Velizhanina, K. A. “A method of study of a single volume sound absorber in a small chamber,” Akust. Zh. 13(2), 287-288 (1967). (In Russian). ; “Calculation of sound absorption coefficient of a volume absorber in the form of a sphere,” Akust. Zh. 13(2), 289-291 (1967). (In Russian). Bradley, J. S. “Predicting theater chair absorption from reverberation chamber measurements,” J. Acoust. Soc. Am. 91(3), 1514-24 (1992). Cook, Richard K.; and Chrzanowski, Peter. “Absorption and scattering by sound absorbent cylinders,” J. Acoust. Soc. Am. 17, 315 (1946). ; “Absorption of sound by absorbent spheres,” J. Acoust. Soc. Am. 21, 167-170 (1949). Davies, W. J.; Orlowski, R. J.; and Lang, Y. W. “Measuring auditorium seat absorption,” J. Acoust. Soc. Am. 96(2), 879-88 (1994). Fujiwara, K.; and Makita, Y. “Reverberant sound absorption coefficient of a plane space absorber,” Acustica 39(5), 340-344 (1978). ; “Experimental study on a method of estimation of the reverberant sound absorption of plane porous material under any mounting condition,” J. Acoust. Soc. Jpn. 34(1), 21-28 (1978). (Japanese, Engl. abstr.). Fukuchi, T.; and Fujiwara, K. “Sound absorption area per seat of upholstered chairs in a hall,” J. Acoust. Soc. Jpn (E)6(4), 271-270 (1985). Furduev, V. V. “Audience sound absorption: Research methods and results,” Sov. Phys.– Acoust. 16(3), 282-291 (1971). Hodgson, Murray; Eldad, Hazen Victor; and Simard; Louis-Philippe. “Evaluation of the performance of suspended baffle arrays in typical industrial sound fields. Part I. 1:8 scale model experiments”. J. Acoust. Soc. Am. 97(1), 339-48 (1995). Part II: Prediction. 349-53. Kath, U. “Der Einfluss der Bekleidung auf die Schallabsorption von Einzelpersonen,” Acustica 17(4), 27-31 (1966). ; and Kuhl, W. “Messungen zur Schallabsorption von Personen auf ungepolsterten Stühlen,” Acustica 14(1), 50-55 (1964). ; and Kuhl, W. “Messungen zur Schallabsorption von Polsterstühlen mit und ohne Personen,” Acustica 15(2), 127-131 (1965). Koyasu, Masaru. “On the sound absorption of persons and theater chairs,” Bull. Kobayashi Inst. Phys. Research 8, 148-154 (April-July, 1958). (In Japanese with English abstract). Lane, R. N. “Absorption characteristics of upholstered theater chairs and carpet as measured in two auditoriums,” J. Acoust. Soc. Am. 28(1), 101-105 (1956). Lax, M.; and Feshbach, Herman. “Absorption and scattering for impedance boundary conditions on spheres and circular cylinders,” J. Acoust. Soc. Am. 20, 108 (1948). Malecki, I. “Die akustische Eigenschaften der Raumabsorber,” Hochfrequenz-tech. u. Elektroakust. 67, 124-127 (January, 1959). Meyer, E.; Kunstmann, D.; and Kuttruff, H. “Über einige Messungen zur Schallabsorption von Publikum,” Acustica 14(2), 119-124 (1964). Olsen, Harry F. “Functional sound absorbers,” RCA Rev. 7, 503-521 (Dec. 1946). Sato, K.; and Kovasu, M. “On the reverberation chamber measurements of absorption by persons and theater chairs,” J. Acoust. Soc. Japan 14, 227-234 (September 1958). Takeuchi, R.; and Shindo, T. “The absorption coefficient of a porous plate,” Mem. Res. Inst. Acoust. Sci. Osaka 2, 28-33 (March, 1951). Phys. Abs. 55, 2527 (April 1952).
Appendix D
Historical Notes and References, Ducts D.1 BRIEF HISTORICAL NOTE Systematic studies of the attenuation of sound in ducts lined with sound absorbing material have been carried out for at least 60 years. The early investigators generally were more familiar with the characteristics of lossy telegraph cables than with lined ducts, and in many of their attempts to understand sound attenuation in ducts, electrical analogies were frequently used (for example, Sivian, [1937]). However, the corresponding description of sound attenuation in ducts turned out to be satisfactory only at wavelengths much larger than the cross sectional dimensions of the duct, and it was not until Morse (1939) and Brillouin (1939) developed the wave theory of sound transmission in ducts that the theoretical understanding of the subject was put on a firm basis. Actually, even these studies can be said to have been inspired by analogy with electromagnetism since the theory of electromagnetic wave guides had been well established at that time (Barrow, [1936]) and higher order sound waves in a hard-walled duct were discussed in 1938 by Hartig and Swanson. Actually, as can be expected, these waves were anticipated and formally expressed in Rayleigh’s Theory of Sound, (volume II, page 73), i.e., at the turn of the century. However, this fact does not seem to have been generally known. For example, in the book Schallabwehr edited by Lübcke (1940), page 4, it is stated that only plane waves can propagate in channels. Intimately related to sound transmission in a duct is the sound field in an enclosure, and it should be mentioned that Rayleigh presented the general expression of the sound field in a rectangular room with hard walls as a superposition of normal modes in his Theory of Sound, (volume II, page 71). To make his 1939 analysis of sound attenuation in lined duct useful for practical applications, Morse included charts from which the attenuation of the fundamental and the first higher mode could be determined from parameters involving the frequency, the duct dimensions, and the impedance of the duct liner. It was implied that the liner was locally reacting so that the normal impedance was known a priori. The ‘Morse charts’ were used extensively for a long time in the design of attenuating ducts and enlarged copies of the charts were quite popular and made available to members of the Acoustics Laboratory at M. I. T. in the late 1940s. 429
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With due respect to the Morse charts, they were quite cumbersome to use and various simple empirical formulas were often tried to express the attenuation in terms of the absorption coefficient of the liner (Sabine [1940], Piazza [1963]). After Morse’s 1939 paper, numerous studies on the subject have been published, both experimental and theoretical, as evidenced by the list of references. For example, the analysis of sound propagation between two infinitely thick parallel nonlocally reacting porous layers was carried out by Willms [1941]), and numerous studies of the various effects of flow on the sound transmission characteristics of ducts were made in the 1950s and beyond. These studies of ducted sound transmission were to a great extent motivated by the importance of fan noise in aircraft engines. Another topic concerned the flow noise generated within a duct and its effect on the insertion loss of duct liners or silencers (Ingard, [1959]). It was frequently found, for example, in jet engine test facilities, that extending the acoustic treatment within the exhaust stack did not produce any significant change in the insertion loss, and it was suspectedthattheflownoisegeneratedatthedischargeorbythetreatmentitselfwasthe culprit. These studies initiated the development of the first standardized test procedure for the measurement of the self-noise and the ‘dynamic’ insertion loss of silencers. This book should not be regarded as a scholarly historical account of the acoustics of lined ducts but as notes of interest and use to practicing engineers. Therefore, publications listed here should be treated as suggestions for further reading or browsing and not as documents reviewed and referenced in the text.
D.2 REFERENCES Abrahamsson, L.; and Kreiss, H. O. “Numerical solution of the coupled mode equations in duct acoustics,” J. Computational Phys. 111(1), 1-14 (1994). Ahrens, C.; and Ronnenberger, D. “Luftschalldämpfung in turbulent durchströmten, schallharten Rohren bei vershiedenen Wandrauhighkeiten,” Acustica, 25, 1971, 150-157. Alfredson, R. J.; and Davies, P. O. A. L. “Performance of exhaust silencer components,” J. Sound Vib. 15(2), 175-190 (1971). Astley, R. J. “A comparative note on the effect of local versus bulk reaction models for air moving ducts lined on all sides,” J. Sound Vib. 117(1), 191-197 (1987). ; and Cummings, A.; and Sormaz, N. A. “A finite element scheme for acoustic propagation in flexible-walled ducts with bulk-reacting liners and comparison with experiment,” J. Sound Vib. 150(1), 119-138 (1991). ; and Eversman, W. “A finite element formulation of the eigenvalue problem in lined ducts with flow,” J. Sound Vib. 65(1), 61-74 (1979). ; and Eversman, W. “The finite element duct eigenvalue problem: An improved formulation with Hermitian elements and no-flow condensation,” J. Sound Vib. 69(11), 13-25 (1980). ; and Eversman, W. “The application of finite element techniques to acoustic transmission in lined ducts with flow,” p. 12. AIAA Pap. No. 660 (1979). ; and Eversman, W. “Acoustic transmission in non-uniform ducts with mean flow, Part II: Finite element method,” J. Sound Vib. 74(1), 103-121 (1981). ; Part I: See Eversman and Astley.; and Eversman, W. “A note on the utility of a wave envelope approach in finite element duct transmssion studies,” J. Sound Vib. 76(4), 595-601 (1981). ; and Walkington, N. J.; and Eversman, W. “Accuracy and stability of finite element schemes for the duct transmission problem,” p. 11, AIAA Pap. No. 2015 (1981). Barrow, W. L. “Transmission of electromagnetic waves in hollow tubes of metal,” Proc. I. R. E., October (1936).
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Baumeister, K. J. “Analysis of sound propagation in ducts using the wave envelope concept,” NASA TN D-7719 (1974). ; “Evaluation of optimized multisectional acoustic liners,” AIAA J. 17(11), 1185-1192 (1979). ; “Applications of velocity potential function to acoustic duct propagation using finite elements,” AIAA J. 18(5), 509-514 (1980). ; “Time-dependent difference theory for noise propagation in a two-dimensional duct,” AIAA J. 18(12), 14701476 (1980). ; “Influence of exit impedance on finite element and finite difference comparison to experiment,” J. Eng. Ind. 104(1), 113-120 (1983). “Acoustics in variable area duct: Finite element and finite difference comparisons to experiment,” AIAA J. 21(2), 193-199 (1983). ; “Numerical techniques in linear duct acoustics–a status report,” J. Eng. Ind. 103(3), 270-281 (1981). ; “Time dependent wave envelope finite difference analysis of sound propagation,” AIAA J. 24(1), 32-38 (1986). ; “Numerical spatial marching techniques in duct acoustics,” J. Acous. Soc. Am. 65(2), 297 (1979). ; and Majjigi, R. K. “Applications of velocity potential function to acoustic duct propagation and radiation from inlets using finite element theory,” p. 7 AIAA Pap. No. 680 (1979). ; and Dahl, M. D. “Finite Element Model for Wave Propagation in an Inhomogeneous Material Including Experimental Validation,” Lewis Res. Ctr., NASA, Cleveland, N87-25821/6/GAR (No. 21, 1987). Beatty, R. J. Jr. “Boundary layer attenuation of higher order modes in rectangular and circular tubes,” J. Acous. Soc. Am. 22, 850 (1950). Benade, A. H. “Equivalent circuits for conical waveguides,” J. Acous. Soc. Am. 83(5), 1764-9 (1988). Benzakein, M. J.; Kraft, R. E.; and Smith, E. B. “Sound attenuation in acoustically treated turbo-machinery Ducts,” ASME Paper 69-WA/GT-11, (1969). Beranek, L. L. “Sound absorption in rectangular ducts,” J. Acous. Soc. Am. 12, 228 (1940). Bies, D. A.; Hansen, C. H.; and Bridges, G. E. “Sound attenuation in rectangular and circular cross-section ducts with flow and bulk-reacting liner,” J. Sound Vib. 146(1), 43-80 (1991). Bokor, A. “Attenuation of sound in lined ducts,” J. Sound Vib. 10, 390 (1969). ; “A comparison of some acoustic duct lining materials according to Scott’s theory,” J. Sound Vib. 14(3), 367 (1971). Boldman, Donald R.; and Rinich, Paul. “Skin friction on a flat perforated acoustic liner,” AIAA Journal, 14(11), 1656-1659 (1976). Brandstatt, P.; Frommhold, W.; and Fisher, M. J. “Program for the computation of absorptive silencers in straight ducts,” Appl. Acoustics (UK) 43(1), 19-38 (1994). Brillouin, L. “Acoustical wave propagation in pipes,” J. Acous. Soc. Am. 11, 10 (1939). Wave Propagation in Periodic Structures, McGraw-Hill, New York (1946). Brillouin, J. “Form and propagation of sound waves in a space limited by absorbing surfaces,” J. de Physique et le Radium 10, 497 (1939). Bruggeman, Jan C. “The propagation of low-frequency sound in a two-dimensional duct system with T-joints and right angle bends. Theory and experiments,” J. Acous. Soc. Am. 82(3), 1045-51 (1987). Cabelli, A. “The acoustic characteristic of duct bends,” J. Sound Vib. 68(3), 369-388. ; “The influence of flow on the acoustic characteristics of a duct bend for higher order modes–a numerical study,” J. Sound Vib. 82(1), 131-149 (1982). ; “Duct acoustics–a time dependent difference approach for steady state solutions,” J. Sound Vib. 85(3), 423-434 (1982). ; “Application of the time dependent finite difference theory to the study of sound and vibration interactions in a duct,” J. Sound Vib. 103(1), 13-23 (1985). ; “The propagation of sound in a rectangular duct with a nonrigid side wall,” J. Sound Vib. 103(3), 379-394 (1985). ; and Shepherd, I. C. “The influence of geometry on the acoustic characteristics of duct bends for higher order modes,” J. Sound Vib. 78, 119-129 (1981). ; and Fontaine, R. F.; and Shepherd, I. C. “The time-dependent finite difference procedure for propagation of sound in a non-uniform lined duct–a comparison with
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experiments,” J. Sound Vib. 100(1), 35-40 (1985). ; and Shepherd, I. C.; and Fontaine, R. F. “Modal filters in rectangular ducts,” J. Sound Vib. 99(2), 285-292 (1985). Cederfeldt, L. “On the use of the finite element method on some acoustical problems,” J. Eng. Ind. 104(1), 108-112 (1982). Chapman, C. J. “Sound radiation from a cylindrical duct, I. Ray structure of the duct modes and of the external field,” J. Fluid Mech (UK) 281, 293-311 (1994). Cho, Y. C. ; and Ingard, U. “Attenuation of sound in lined ducts,” M. I. T. Gas Turbine Lab. Reports No. 119 (1974) and 120 (1975). Christie, D. R. A. “Theoretical attenuation of sound in a lined duct: Some computer calculations,” J. Sound Vib. 17(2), 283-286 (1971). Gogate, G. R.; and Munjal, M. L. “Analytical solution of sound propagation in lined or unlined circular ducts with laminar mean flow,” J. Sound Vib. 160(3), 465-484 (1993). Craggs, A. “A finite element method for the free vibration of air in ducts and rooms with absorbing walls,” J. Sound Vib. 173(4), 568-576 (1994). ; “The application of the transfer matrix and matrix condensation methods with finite elements in duct acoustics,” J. Sound Vib. 132(3), 393-402 (1989). ; “A Comparison of finite element solutions with those from the simple theory of duct acoustics,” Finite Elem. News (UK) (6), 14-17 (1987). Cremer, L. “Theory of sound attenuation in a rectangular duct with an absorbing wall and the resultant maximum coefficient,” Acoustica 3, 249 (1953). Cummings, A. “Sound transmission in curved duct bends,” J. Sound Vib. 35, 451 (1974). ; “Sound transmission at sudden area expansions in circular ducts with superimposed mean flow,” J. Sound Vib. 38(1), 149-155 (1975). ; “Sound attenuation in ducts lined on two opposite walls with porous material, with some application to splitters,” J. Sound Vib. 40(1), 9-35 (1976). ; “The attenuation of sound in unlined ducts with flexible walls,” J. Sound Vib. 174(4), 433450 (1994). ; “Low frequency acoustic radiation from duct walls,” J. Sound Vib. 71(2), 201226 (1980). ; “Stiffness control of low frequency acoustic transmission through the walls of rectangular ducts,” J. Sound Vib. 74(3), 351-380 (1981). ; “Design charts for low frequency acoustic transmission through the walls of rectangular ducts,” J. Sound Vib. 78(2), 269-289 (1981). ; “The effects of flanking transmission on sound attenuation in lined ducts,” J. Sound Vib. 179(4), 617-646 (1995). ; “Sound propagation in narrow tubes of arbitrary cross-section,” J. Sound Vib. 162(1), 27-42 (1993). ; and Chang, I. J. “Sound attenuation of a finite length dissipative flow duct silencer with internal mean flow in the Absorbent,” J. Sound Vib. 127(1), 1-17 (1988). ; “A finite difference scheme for acoustic transmission through the walls of distorted circular ducts,” J. Sound Vib. 104(3), 377-393 (1986). ; and Parrett, A. V.; and Astley, R. J. “A comparison of measured and computed sound pressure levels in a non-uniform acoustically lined duct,” J. Sound Vib. 85(3), 407-414 (1982). Davies, P. O. A. L. “Realistic models for predicting sound propagation in flow duct systems,” Noise Control Eng. J. 40(1), 135-141 (1993). ; “Practical flow duct acoustics,” J. Sound Vib. 124(1), 91-115 (1988). ; and Yaseen, E. A. A. “High Intensity Sound Propagation in Flow Ducts,” J. Sound Vib. 114(1), 153-157 (1987). Doak, P. E. “Fundamentals of aerodynamic sound theory and flow duct acoustics,” J. Sound Vib. 28(3), 527 (1973). ; and Vaidya, P. G. “Attenuation of plane wave and higher order mode sound propagation in lined ducts,” J. Sound Vib. 12(2), 201 (1970). El-Raheb, Michael; and Wagner, Paul. “Acoustic propagation in rigid sharp bend and branches,” J. Acous. Soc. Am. 67(6), 1914 (1980). El-Sharakwy, A. I.; and El-Chazley, N. M. “Critical survey of basic theories used in muffler design and analysis,” Appl. Acous. 20(3), 195-218 (1987). Eversman, W. “The effect of Mach number on the tuning of an acoustic lining in a flow duct,” J. Acous. Soc. Am. 48, 425 (1970). ; “Effect of boundary layer on the transmission and attenuation of sound in an acoustically treated circular duct,” J. Acous. Soc. Am. 49,
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1372 (1971). ; and Beckemeyer, R. J. “Transmission of sound in ducts with thin shear layers– convergence to the uniform flow case,” J. Acous. Soc. Am. 52, 216 (1972). ; and Astley, R. J. “Acoustic transmission in non-uniform ducts with mean flow, Part I: The method of weighted residuals,” J. Sound Vib. 77(2), 301 (1981). Feder, E.; and Dean, L. W. III. “Analytical and experimental studies for predicting noise attenuation in acoustically treated ducts for turbofan engines,” NASA CR-1373. Firth, D.; and Fahy, F. J. “Acoustic characteristics of circular bends in pipe,” J. Sound Vib. 97(2), 287-303 (1984). Fisher, E. “Attenuation of sound in circular ducts,” J. Acous. Soc. Am. 17, 121 (1945). Fuller, Christoffer R.; and Bies, David A. “Propagation of sound in a curved bend containing a curved axial partition,” J. Acous. Soc. Am. 63(3), 681 (1978). Furnell, G. D.; and Bies, D. A. “Matrix analysis of acoustic wave propagation within curved ducting systems,” J. Sound Vib. 132(2), 245-263 (1989). ; “Characteristics of modal wave propagation within longitudinally curved acoustic waveguides,” J. Sound Vib. 130(3), 405-423 (1989). Gopalkrishnan, S.; and Doyle, J. F. “Wave propagation in connected waveguides of varying cross-section,” J. Sound Vib. 175(3), 347-363 (1994). Gogate, G. R.; and Munjal, M. L. “Analytical solution of the laminar mean flow wave equation in a lined or unlined duct two-dimensional rectangular duct,” J. Acous. Soc. Am. 92(5), 2915-23 (1992). Goldstein, M.; and Rice, E. “Effect of shear on duct wall impedance,” J. Sound Vib. 30(1), 79 (1973). Hartig, H. E.; and Swanson, C. E. “Transverse” acoustic waves in rigid tubes,” Phys. Rev. 54, 618 (1938). Hine, M. J.; and Fahy, F. J. “A membrane analogy to an acoustic duct,” J. Sound Vib. 18(1), 1-7 (1971). Hover, Robert M.; Laird, Donald T.; and Miller, Layman N. “Acoustic filter for water-filled pipes,” J. Acous. Soc. Am. 22(1), 38-41 (1950). Howe, M. S. “The interaction of sound with low Mach number wall turbulence with application to sound propagation in turbulent pipe flow,” J. Fluid Mechanics 94(4), 729-744 (1979). Hurst, C. J. “Sound transmission between absorbing parallel plates,” J. Acous. Soc. Am. 67(1), 206 (1980). Ingard, U. “Attenuation and regeneration of sound in ducts and jet diffusers,” J. Acous. Soc. Am. 31, 1202 (1959). ; “Nonlinear attenuation of sound in ducts,” J. Acous. Soc. Am. 43, 167 (1968). ; and Singhal, V. K. “Sound attenuation in turbulent pipe flow,” J. Acous. Soc. Am. 55, 535 (1974). ; and Pridmore-Brown, D. “Propagation of sound in a duct with constrictions,” J. Acous. Soc. Am. 23, 689-694 (1951). ; and Pridore-Brown, D. “Effect of partitions in the absorptive lining of sound attenuating ducts,” J. Acous. Soc. Am. 23(5), 589 (1951). Johnston, G. W.; and Ogimoto, K. “Sound radiation from a finite length unflanged circular duct with uniform axial flow. I. Theoretical analysis,” J. Acous. Soc. Am. 68(6), 1858 (1980). ; “II. Computed radiation characteristics,” J. Acous. Soc. Am. 68(6), 1871 (1980). Kapur, A.; and Mungur, O. “On the propagation of sound in a rectangular duct with gradients of mean flow and temperature in both transverse directions,” J. Sound Vib. 23(3), 401-404 (1972). Keefe, Douglas H.; and Benade, Arthur H. “Wave propagation in strongly curved ducts,” J. Acous. Soc. Am. 74(1), 320-332 (1983). Keller, Joseph. “Nonlinear forced and free vibrations in acoustic waveguides,” J. Acous. Soc. Am. 55, 524 (1974). King, A. J. “Attenuation of sound in lined air ducts,” J. Acous. Soc. Am. 30, 505 (1958).
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Ko, S. H. “Sound attenuation in lined rectangular ducts with flow and its application to the reduction of aircraft engine noise,” J. Acous. Soc. Am. 50, 1418 (1971). ; “Sound attenuation in acoustically lined ducts in the presence of uniform flow and shear flow,” J. Sound Vib. 22(2), 193–210 (1972). ; “Theoretical analysis of sound attenuation in acousticaslly lined flow ducts separated by porous splitters (rectangular, annular, and circular ducts”), J. Sound Vib. 39(4), 471 (1975). ; “Sound wave propagation in a two-dimensional flexible duct in the presence of an inviscid flow,” J. Sound Vib. 175(2), 279-287 (1994). Kurze, U. “Untersuchungen an Kammerdämpfern,” Acustica, 15, 149-150 (1965). ; and Allen, C. H. “Influence of flow and high sound level on the attenuation in a lined duct,” J. Acoust. Soc. Am. 49, 1643 (1971). ; and Ver, I. L. “Sound attenuation in ducts lined with non-isotropic material,” J. Sound Vib. 24(2), 177 (1972). Lansing, D. L.; and Zorumski, W. E. “Effects of wall admittance changes on duct transmission and radiation of sound,” J. Sound Vib. 27(1), 85 (1973). Lambert, R. F.; and Steinbrueck, E. A. “Acoustic synthesis of a flowduct area discontinuity,” J. Acous. Soc. Am. 67(1), 59 (1980). Lester, Harold C.; and Posey, Joe W. “Duct liner optimization for turbomachinery noise sources,” NASA TM X-72789, Nov. (1975). Leventhal, H. G. “Low frequency noise in ventilation systems–Criteria and control,” J. Low Frequency Noise Vib., 13(4), 123-131 (1994). Levitskij, L. A. “Propagation of sound waves in a plane waveguide with thin elastic walls in a fluid medium,” Sov. Phys.-Acous. 26(1), 59-63 (1980). Lübcke, E. Editor, Schallabwehr, Springer Verlag, Berlin, (1940). Majjigi, R. K.; Sigman, R. K.; and Zinn, B. T. “Wave propagation in ducts using the finite element method,” p. 8 AIAA Pap. No. 659 (1979). Malloy, Charles. “The lined tube as an element of acoustic circuits,” J. Acous. Soc. Am. 21, 413-418 (1949). Mariano, S. “Effect of wall shear layers on the sound attenuation in acoustically lined rectangular ducts, J. Sound Vib. 19, 261 (1971). ; “Optimization of acoustic linings in the presence of wall shear layers,” J. Sound Vib. 23(2), 229-235 (1972). Marsh, A. H. “Application of duct-lining technology to jet aircraft,” J. Acous. Soc. Am. 48, 826 (1970). McCormick, M. A. “The attenuation of sound in lined rectangular ducts containing uniform flow,” J. Sound Vib. 39(1), 35 (1975). Mechel, F. P. “Modal solutions in rectangular ducts lined with locally reacting absorbers,” Acustica 73(5), 223-239 (1991). Meyer, E.; Mechel, F.; and Kurtze, G. “Experiments on the influence of flow and sound attenuation in absorbing ducts,” J. Acous. Soc. Am. 30, 165 (1958). Miles, J. “The reflection of sound due to a change in cross section of a circular tube,” J. Acoust. Soc. Am. 16, 14 (1944). ; “The analysis of plane discontinuities in cross section in circular tubes. Part I,” J. Acous. Soc. Am. 17, 259 (1946); Part II: J. Acous. Soc. Am. 17, 272 (1946). Miles, J. H. “Acoustic transmission matrix of a variable area duct or nozzle carrying a compressible subsonic flow,” J. Acous. Soc. Am. 69(6), 1577 (1981). ; “Verification of a onedimensional analysis of sound propagation in a variable area duct without flow,” J. Acous. Soc. Am. 72(2), 621-4 (1982). Morse, P. M. “The transmission of sound inside pipes,” J. Acous. Soc. Am. 11, 205 (1939).; and Ingard, K. U. Theoretical Acoustics, McGraw-Hill, New York (1968). Möring, W. M. “Acoustic energy flux in nonhomogeneous ducts,” J. Acous. Soc. Am. 64(4), 1186 (1978).
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Index Absorption spectra of a perforated plate-cavity resonator, 120 Absorption spectra of a rigid sheet, 61 Absorption spectra, infinite sheet, 83–85 Acoustic barriers, 363 Acoustic nonlinearity, 116–121, 276–277 Acoustic nonlinearity, perforated plate, 139–141 Acoustic perturbations and dispersion relation, 358–359 Acoustic supercharge, 328–330 Acoustic viscous boundary layer, 35 Acoustically equivalent silencers, 251–252 Additional duct shapes, 268–274 Air induction silencer for an automobile engine, 267 Airlet assembly for an automobile engine, 312 Alternate choice of unit cell, 98 Amplitude dependence of wave speed, 214 Angle dependence of wave and input impedance, 184–185 Angle of refraction, 190 Angle of refraction vs angle of incidence for a slot absorber, 153 Angle of refraction vs angle of incidence for a sound incident in porous layer, 189 Angular dependence of absorption coefficient of a locally reacting boundary, 180
1/3 and 1/1 octave band average absorption, 60–61, 90 100% absorption at the lowest resonance, 68–69 4 × 4 matrix for a flexible screen, 372 A Absorption and scattering, 106–116 Absorption and scattering cross sections, 105, 108, 133–136 Absorption area per unit area of sheet material, 85 Absorption coefficient of a nonlocally reacting porous layer, 171 Absorption coefficient of a rigid porous layer covered with a thin resistive screen, 168 Absorption coefficient problem, 403–406 Sound absorption in porous materials, 404–406 Absorption coefficient vs flow resistance, 165, 204 Absorption coefficient vs sheet resistance, 76–77 Absorption cross section of a resonator, 134 Absorption cross section, low frequency approximation, 86 Absorption materials, 7 Absorption peaks but not at resonances, 205–206 Absorption spectra, 59–60, 69–70, 161–173, 204–211 Absorption spectra of a perforated plate-cavity absorber, 121 439
440 Angular dependence of absorption coefficient of an infinite porous layer, 162 Anisotropic layer, 192 Annular duct, 272–274, 344–346 Boundary admittances, 345–346 Apparatus, 212–213, 217–218 Apparatus for measurement of complex compliance, 217–218 Apparatus for measurement of steady flow resistance of a porous material, 390 Apparatus for measuring complex compliance, 218 Application of matrices, 362–370 Approximate complex compressibility in a channel, 51 Area contraction in a duct, 374 Area discontinuities, 372–374 Area expansion in a duct, 372–374 Arrangement for measurement of the flow resistance or impedance of a porous layer, 396 Arrangement for measuring attenuation, 281 Asymmetric cell, 94 Attenuation, 240–241 Attenuation in a circular duct lined with a rigid, locally reacting porous liner, 272 Attenuation in 10 ft long rectangular duct, 268 Attenuation in turbulent flow in ducts, 317–319 Static pressure drop, 318 Sound attenuation, 318–319 Proposed aeroacoustic instability, 319 Attenuation in turbulent duct flow, 357–359 Friction factor in turbulent duct flow, 357–358
NOISE REDUCTION ANALYSIS Acoustic perturbations and dispersion relation, 358–359 Comparison with visco-thermal attenuation, 359 Attenuation mechanisms, 255–257 Attenuation of fundamental acoustic mode in a rectangular duct, 280 Attenuation of fundamental mode in a rectangular duct, 258, 262, 276 Attenuation per unit length, 240 Attenuation spectrum for an annular lined duct, 273 Attenuation vs flow resistance of liner, 266–267 Average complex compressibility in a channel, 50 Average compressibility and propagation constant, resonator lattice, 114 Axial pressure gradient in turbulent pipe flow, 290 Axial propagation constant, 15, 46, 334, 338 Axial velocity amplitude, viscous mode, 43 Axial velocity distribution in a channel, 44 B Background noise, 245–246 Bandwidth of low frequency resonance, 75 Boundary admittances, 345–346 Bragg reflection, 79 C Channel impedance per unit length, 22, 46 Choice of variables (in matrices), 362 Circular and annular lined ducts with locally reacting porous liners, 270 Circular duct, 270–271
INDEX Circular vs square lined duct, 272 Closed surface, 229, 386 Closed-closed layer, 386 Closed-open layer, 387 Collection of sheet data, 72 Commonly used matrices, 370–388 Comparison of attenuation vs normalized flow resistance for nonlocally reacting liners, 266 Compressibility, 19, 157 Complex compressibility, 19–24, 38, 49–51 Complex density and wave impedance, 23–24 Computed absorption spectra of a flexible porous layer, 205, 207–208, 210 Contracted dust section, perforated plate, 376 Contraction chamber, 305–307 Convection, 237, 284–285 Coupled waves, 198–199 Cut-off frequency, 239 Cut-on frequency, 239 D Data acquisition system, 213 Data analysis, 218–219 Data points, 129 Decay rate, 111–112 Definition of view angle and emission angle, 175 Demonstration of amplitude dependence of wave speed, 214 Demonstration of response of a resonator to an incident pulse, 115 Diffuse field absorption coefficient for an absorber, 154 Diffuse field absorption coefficient, anisotropic layer, 185 Diffuse field absorption spectra of a uniform nonlocally reacting porous layer, 167
441 Dispersion relation, 199–201, 224 Dispersion relation, flexible porous material, 200 Dissipation functions, 203–204, 227–228 Dissipation in duct liners, 255–256 Double sheet absorber, 64 Duct acoustics, 235–253 Wave modes, 238–240 Simple illustration, 238–240 Measures of silencer performance, 240–249 Attenuation, 240–241 Transmission loss, TL and TL0, 241 Two-room method, 241 Standard method, 241–244 Insertion loss, IL, 244–245 Multi-source environment effect of background noise, 245–246 Noise reduction, NR, 246 Numerical examples, 246–248 Pressure drop and flow noise (self-noise, SN), 248–249 Lined ducts, 249–251 Reactive silencers, 251 Acoustically equivalent silencers, 251–252 Duct lined on one side, 337–338 Duct liner configurations, 275–282 Duct section with loss-free walls, 375 Duct with mean flow Mach number M, 376 Duct: TL vs TL0, 365–367 Ducts in series and in parallel, 274–275 E Effect of a bonded perforated facing, 206–208 Effect of a cover screen and perforated facing, 144–145 Effect of a perforated facing, its nonlinearity and induced motion, 166–167
442 Effect of a perforated plate, 208–209 Effect of a screen cover, 198 Effect of a screen on a porous layer, 167–169 Effect of absorption, 178–179 Effect of acoustically induced motion, 30 Effect of an air layer, 145 Effect of bending stiffness and structural resonances, 68, 91–92 Effect of cell size in a partitioned air backing, 63 Effect of diffraction, 85–87, 101–103 Effect of duct liner flexibility, 278 Effect of flow resistance, 150 Effect of grazing flow, 192–194 Effect of heat conduction, 151–152 Effect of honeycomb cell size, 62–63 Effect of induced plate motion, 140–141 Effect of internal damping of flexible wall, 19 Effect of mean flow on acoustic resistance, 312–315 Effect of nonlinearity and induced motion of a perforated plate, 277 Effect of partition spacing, 280–282 Effect of perforated facing on a porous layer, 395 Effect of porosity, 150–151 Effect of refraction in a duct, 286 Effect of refraction in grazing flow, 173–181 Effect of temperature and nonlinearity on a sheet absorber, 65–66 Effect of temperature on attenuation in a lined duct, 289 Effects of flow, 122–132 Effects of higher modes and flow, 282–293 Effects of perforated facing, 276–277 Elbow in a duct, 378 Elementary kinetic theory of gases, 9
NOISE REDUCTION ANALYSIS Elimination of orifice/pipe tones, 125–126 Equation for velocity amplitude in an orifice at resonance, 118 Equations for coupled waves, 221–225 Equations of motion, 392–393 Equivalent duct configurations (silencers), 250 Equivalent impedance, 66–68 Example showing angular region in a wind tunnel, 181 Examples of resonators, 110 Expansion chamber, 303–305 Expansion chamber and elbow, 377–378 F Field distribution within a cell, 97–98 Field distributions, 201–204 Finite layer, 162–164 Finite sheet, effect of diffraction, 85–87 Flexible layer, 383–385 Flexible layer with a perforated plate/screen facing, 209–210 Flexible porous materials, 197–232 Coupled waves, 198–199 Dispersion relation, 199–201 Field distributions, 201–204 Pressure and velocity fields, 201–203 Dissipation functions, 203–204 Absorption spectra, 204–211 Absorption peaks but not at resonances, 205–206 Effect of a bonded perforated facing, 206–208 Porous material with closed cells, 210–211 Nonlinear effects and shock wave reflection, 211–216 Apparatus, 212–213
443
INDEX Amplitude dependence of wave speed, 214 Reflection from a flexible porous layer, 214–216 Measurement of complex elastic modulus, 217–219 Apparatus, 217–218 Data analysis, 218–219 Limp material, 219–232 Equations for coupled waves, 221–225 Pressure and velocity fields, 225–227 Absorption coefficients, 228–232 Flexible porous sheet with cavity backing, 66–70 Flexible sheet absorber, equivalent sheet impedance, 66 Flexible sheet cavity absorber, 90–94 Flow excitation of a side-branch resonator in a duct, 128–130 Flow excitation of a slanted side-branch resonator in a duct, 131 Flow excitation of a tube resonator in free field, 128 Flow excitation of orifice and pipe tones, 124 Flow excitation of pipes and orifices, 123–126 Flow generated tone in a film dryer facility, 126 Flow induced acoustic resonance, 122–123 Flow resistance and impedances, 56–57, 155–156 Flow resistance measurements, 389–401 Simple method for steady flow, 389–392 Equations of motion, 392–393 Nonlinearity of flow resistance, 393–395
Simple method for oscillatory flow, 395–397 Wire mesh screens, 398–400 Other materials, 400–401 Free screen, 370–371 Friction factor in pipe flow, 291 Friction factor in turbulent duct flow, 357–358 G Grazing flow, 123, 127, 173–174, 178–180, 192 H Helmholtz resonator, 109–110, 136–137, 380 High frequency approximation, 336–337, 340–341 High frequency attenuation of fundamental mode in lined duct, average compressibility, 331–332 High impedance source, 309–311 Higher modes, 282–284 Higher modes and flow, 346–349 Z-modes, 347–348 Y-modes, 348 Semi-empirical higher order mode, TL correction, 348–349 Higher order mode, 239 Honeycombs, 54 I Illustration of the effect of flow direction on transmission loss, 286 Impedance of a tube resonator, 132–133 Impedance per unit length, 22–23 Impedances, 87–88, 362–363 Industrial dryer, 126 Infinite layer, 161–162 Infinite sheet, diffuse field, effect of induced motion, 100–101 Influence of boundary layer flow on absorption coefficient, 178
444 Influence of porosity on absorption coefficient of slot absorber, 151 Input impedance and absorption coefficient, 99 Input impedance and admittance, absorption coefficient, 190–191 Input impedance of periodic lattice absorber, 78 Input impedance of porous layer backed by a rigid wall, 387 Input impedance of slot absorber, normal incidence, 148 Input impedance, absorption spectra, 148–154 Input impedance, anisotropic material, 185 Insertion loss, IL, 244–245, 304–307, 309, 367–369 Insertion loss of a contraction chamber, 307 Insertion loss of a pipe, 302 Insertion loss of a side-branch resonator, 313 Insertion loss of an expansion chamber, 304 Insertion loss of side-branch tube, 310 Interaction impedance, impedance per unit length, and complex density, 187–188 Interference, 256–257 Interpretation of steady flow resistance data, 30–31 Isotropic porous layer, 154–157, 186–187 Isotropic porous layer, physical parameters, 154–157 K Kirchhoff attenuation, 17 L Labyrinth resonator assembly, 311 Lattice absorbers, 77–82 Law of refraction, 176
NOISE REDUCTION ANALYSIS Limp material, 219–232 Limp sheet, 67–76 Limp sheet absorber for maximum NRC, 70–72 Linearized acoustic equations, 187 Lined duct section, 378 Lined ducts, 249–251, 255–297, 378–379 Attenuation mechanisms, 255–257 Dissipation in duct liners, 255–256 Interference, 256–257 Rectangular ducts, 257–267 Locally reacting liner, 257–261 Nonlocally reacting liner, 261–264 Locally vs nonlocally reacting liner, 264–265 Attenuation vs flow resistance of liner, 266–267 Additional duct shapes, 268–274 Rectangular duct with all sides lined, 268–270 Circular duct, 270–271 Circular vs square lined duct, 272 Annular duct, 272–274 Ducts in series and in parallel, 274–275 Ducts in series, 274 Parallel ducts, interference filter, 274–275 Duct liner configurations, 275–282 Effects of perforated facing, 276–277 Effect of duct liner flexibility, 278 Mulitlayer liners, 278–279 Slotted liner, 279–280 Effect of partition spacing, 280–282 Effects of higher modes and flow, 282–293
INDEX Higher modes, 282–284 Convection, 284–285 Refraction, 285–287 Scaling laws, 287–288 Static pressure drop in ducts, 290–293 Liquid pipe lines, elementary aspects, 293–297 Liquid pipe line with slightly compliant walls, 293–295 Liquid pipeline with air layer wall treatment, 295–297 Liquid pipe line with air layer wall treatment, 295–297, 352 Liquid pipe line with slightly compliant walls, 293–295, 349–352 Liquid pipe lines, 349–356 Liquid pipe line with slightly compliant walls, 349–352 Wall admittance, 349 Propagation constant, 350–352 Liquid pipe line with air layer wall treatment, 352–355 Wall impedance, 353 Propagation constants, 354–355 Transmission loss, 355–356 Liquid pipe lines, elementary aspects, 293–297 Liquid pipe line with slightly compliant walls, 293–295 Liquid pipeline with air layer wall treatment, 295–297 Locally reacting liners, 257–261, 332–337, 342–344 Wave impedance of the duct, 334 Low frequency approximation, 334–336 High frequency approximation, 336–337 Locally vs nonlocally reacting liner, 264–265 Loudspeaker experiment, 238–239
445 Low frequency approximation, 334–336, 339–340 Low frequency resonance, 68–69, 93–94 Low frequency resonance, limp sheet absorber, 74–75 Low impedance source, 311–312 M Macroscopic physical parameters of a porous material, 7 Mass balance, 222 Mass density of a porous material, 156 Material with closed cells, 198 Material with open cells, 197–198 Maximum absorption cross section at resonance, 135 Maximum q-value, 135–136 Measured absorption spectra of a resistive sheet, 62 Measured diffuse field absorption coefficients, 164 Measured frequency dependence of flow resistance of a stack of wire mesh screens, 399 Measurement of complex elastic modulus, 217–219 Measurement of flow resistance, 390 Measures of silencer performance, 240–249 Method of measuring compression, 216 Micro-perforates, 25–27 Mode coupling, 130–131 Momentum balance, 223–224 Mulitlayer liners, 278–279 Multisheet absorber, 55, 82, 144 Multisource environment effect of background noise, 245–246 N Net transmitted power, 242 Noise reduction, NR, 246, 369–370 Noise reduction coefficient, 6, 63–64
446 Nonlinear absorption characteristics, 120–121 Nonlinear attenuation, 319–320 Nonlinear effects and shock wave reflection, 211–216 Nonlinear reflection, transmission, absorption, 315–316 Nonlinearity of flow resistance, 393–395 Nonlocally reacting liner, 261–264, 337–341, 344 Duct lined on one side, 337–338 Wave impedance, 338–339 Low frequency approximation, 339–340 High frequency approximation, 340–341 Two sides lined, 341 Nonperiodic lattice, 80–82, 98–99 Nonuniform porous absorbers, 169–170 Normal incidence absorption coefficient of a slot absorber, 150 Normal incidence absorption spectra, 70, 79, 149–150 Normal incidence and diffuse field absorption coefficients of a nonuniform lattice absorber of limp sheets, 80 Normal incidence and diffuse field absorption coefficients of a nonuniform lattice absorber, 81 Normal incidence and diffuse field absorption coefficients of two absorbers, 169–170 Normal incidence and diffuse field average absorption coefficients of a nonlocally reacting rigid porous layer, 161 Normalized acoustic resistance of a perforated plate-cavity resonator, 119 Normalized dissipation functions, 204
NOISE REDUCTION ANALYSIS Normalized flow resistance, 289 Normalized input impedance, flexible porous layer, closed surface, 229 Normalized input admittance, flexible porous layer, open surface, 228 Normalized input impedance admittance, 190 Normalized input impedance of a tube, 132 Normalized specific impedance of a slot absorber, 149 Normalized wave impedance in a channel, 48 Numerical examples (silencers), 246–248 O Oblique incidence, 48, 60, 152, 183–184 Oblique incidence of a sound wave on a boundary, 184 Oblique incidence, refraction, and diffuse field absorption, 152–154 Octave band average transmission loss, 279 On air induction acoustics, 320–330 Sound pressure and radiated power, 321–326 Pipe impedance, 327–328 Radiated power, 328 Acoustic supercharge, 328–330 Open surface, 228–229, 386 Open-closed layer, 386–387 Open-open layer, 386 Optimization of the absorption spectrum, 64–65 Optimum radius for maximum resonance absorption cross section, 135 P Parallel channels, 381–382 Parallel ducts, 382
INDEX Parallel ducts, interference filter, 274–275 Parameter relations, 181–182 Penetration depth, 17, 159–160 Perforated facing, its nonlinearity and induced motion, 191 Perforated facing, normalized impedance, semi-empirical, 25 Perforated plate, 24–27, 312–317, 376 Perforated plate with (porous) cavity backing, 117–120 Perforated plate-screen combination, laminates, 29–30 Perforated plate-sheet combinations, 73–74 Periodic lattice, 77–80 Periodic lattice absorber, 78 Physical parameter of a slot absorber, 147 Pipe impedance, 327–328 Plane wave reflection, transmission, and absorption, infinite sheet, 99–100 Porosity, 154–155 Porous layer backed by an air cavity, 170–171 Porous material with closed cells, 210–211 Porous plug, 261–264 Porous screen, 370–372 Porous screen in water, 400 Porous sheet cavity absorber, 54 Power dissipation, visco-thermal boundary layer, 13 Pressure amplitude distribution of air-borne and structure-borne waves, 203 Pressure and velocity fields, 201–203, 225–227 Pressure drop and flow noise (self-noise, SN), 248–249 Pressure loss, exit, 292 Pressure loss, inlet, 293
447 Pressure reflection coefficient, 364 Pressure reflection coefficient, absorption coefficient, 59 Profiles of axial velocity in a channel, 18 Propagation constant, 15–18, 157–159, 350–352, 354–355 Propagation constant and wave impedance, 95–97, 188–189 Propagation constant, anisotropic material, oblique incidence, 184 Propagation constant, impedance per unit length, wave impedance, 49 Propagation constant, locally reacting liner, 257 Propagation constant, nonlocally reacting liner, 261 Propagation constant, phase velocity, and attenuation, 44–46 Propagational mode, 41–42 Proposed aeroacoustic instability, 319 Push-pull wave, 239 Q Q-value, 109 Q-value of a cavity resonator, 14 Q-value of a cavity resonator, 39–41 Q-value of a circular tube resonator, 40 R Radiated power, 301–302, 321, 328–330 Ratio of viscous and thermal relaxation frequencies, 51 Reactive duct elements, 299–330 Uniform duct section, 299–303 Role of source impedance, 300–303 Expansion chamber, 303–305 Transmission loss, 304 Insertion loss, 304–305 Contraction chamber, 305–307 Transmission loss, 305–306
448 Insertion loss, 306–307 Side-branch resonator in a duct, 307–312 Transmission loss, 308–309 Insertion loss, 309 Perforated plate, 312–317 Effect of mean flow on the acoustic resistance, 312–315 Nonlinear reflection, transmission, absorption, 315–316 Shock wave interaction with an orifice plate, 316–317 Reactive silencers, 251 Real and imaginary parts of normalized propagation constants in flexible porous material, 200 Rectangular duct lined on all sides, 268–270, 341–342 Rectangular duct with two opposite walls lined with a locally reacting material, 332 Rectangular duct, one side lined with a porous layer, 260 Rectangular ducts, 257–267 Rectangular water line, 296 Reflection and absorption coefficients, 183, 363–364 Reflection from a flexible porous layer, 214–216 Reflection, transmission, and absorption, 83 Refraction, 160, 184, 285–287 Region of total reflection, 179–181 Relaxation times and a note on complex compressibility in a channel, 19–21 Resonance frequency, Helmholtz resonator, 110, 137 Resonances and anti-resonances, 57–59 Resonator absorber in a diffuse sound field, 110–112
NOISE REDUCTION ANALYSIS Resonator in free field with grazing flow, 127–128 Resonators, 106–141 Absorption and scattering, 106–116 Q-value, 109 Helmholtz resonator, 109–110 Resonator absorber in a diffuse sound field, 110–112 Two-dimensional arrays of resonators, 112–113 Three-dimensional lattice of resonators, 113–114 Transient response and reverberation, 115–116 Acoustic nonlinearity, 116–121 Perforated plate with (porous) cavity backing, 117–120 Effects of flow, 122–132 Flow induced acoustic resonance, 122–123 Flow excitation of pipes and orifices, 123–126 Resonator in free field with grazing flow, 127–128 Flow excitation of a side-branch resonator in a duct, 128–130 Impedance of a tube resonator, 132–133 Absorption and scattering cross sections, 133–136 Optimum radius for maximum resonance absorption cross section, 135 Absorption q-value, 135–136 Helmholtz resonator, 136–137 Three-dimensional array of resonators, 137–139 Acoustic nonlinearity, perforated plate, 139–141 Effect of induced plate motion, 140–141 Rigid porous layers, 383 Rigid porous materials 143–194 Slot Absorber, 146
INDEX Isotropic porous layer, physical parameters, 154 Wave motion, 157 Absorption spectra, 161 Effect of refraction in grazing flow, 173 Rigid sheet absorber for maximum NRC, 63 Rigid single sheet cavity absorber, 87–90 Rigid single sheet with cavity backing, 56–63 Role of source impedance, 300–303 S Scaling laws, 287–288 Density, 288 Sound speed, 288 Wave impedance, 288 Shear viscosity, 288 Kinematic viscosity, 288 Reynolds number, 288 Screen in mechanical contact, 371–372 Semi-empirical higher order mode, TL correction, 348–349 Separation of acoustically driven oscillatory flow, 117 Sheet absorbers, 53–103 Single sheet surface absorber, 53–55 Multisheet absorber, 55 Single sheet as a volume absorber, 55–56 Rigid single sheet with cavity backing, 56–63 Flow resistance and impedances, 56–57 Resonances and anti-resonances, 57–59 Absorption spectra, 59–60 Wire screens, 62 Effect of honeycomb cell size, 62–63 Flexible porous sheet with cavity backing, 66–70
449 Equivalent impedance, 66–68 Low frequency resonance, 68–69 Absorption spectra, 69–70 Lattice absorbers, 77–82 Periodic lattice, 77–80 Nonperiodic lattice, 80–82 Volume absorbers, 82–87 Reflection, transmission, and absorption, 83 Absorption spectra, infinite sheet, 83–85 Finite sheet, effect of diffraction, 85–87 Rigid single sheet cavity absorber, 87–90 Impedances, 87–88 Absorption coefficient, 88–90 1/3 and 1/1 octave band average absorption, 90 Flexible sheet cavity absorber, 90–94 Effect of bending stiffness and structural resonances, 91–92 Low frequency resonance, 93–94 Absorption coefficient, 94 Uniform (periodic) lattice, 94–98 Unit cells, 94–95 Propagation constant and wave impedance, 95–97 Field distribution within a cell, 97–98 Alternate choice of unit cell, 98 Nonuniform lattice, 98–99 Transmission matrix, 98–99 Input impedance and absorption coefficient, 99 Sheet as a volume absorber, 99–103 Plane wave reflection, transmission, and
450 absorption, infinite sheet, 99–100 Infinite sheet, diffuse field, effect of induced motion, 100–101 Effect of diffraction, 101–103 Sheet absorber vs uniform porous layer, 171–173 Sheet absorber, input impedance, local reaction, 58 Sheet absorber, input impedance, nonlocal reaction, 59 Sheet absorbers vs uniform porous layers, 145 Sheet as a volume absorber, 99–103 Shock tube experiment, 212–213 Shock wave interaction with an orifice plate, 316–317 Shock wave reflection, 198 Shock wave reflection from an orifice plate, 317 Shock wave reflections from a flexible porous layer, 215–216 Side-branch Helmholtz resonator, 380–381 Side-branch resonator in a duct, 307–312 Side-branch tube, 379–380 Side-branch tube in a duct with a resistive screen, 379 Simple method for oscillatory flow, 395–397 Simple method for steady flow, 389–392 Simulation of boundary layer by a single step change, 177 Single sheet as volume absorber, 55–56 Single sheet surface absorber, 53–55 Slanted resonator in a duct, effect of flow direction, 131–132 Slot absorber, 146–154, 181–185 Slotted liner, 279–280 Sound absorption in porous materials, 404–406 Sound absorption mechanisms, 7–51
NOISE REDUCTION ANALYSIS Steady flow through a (narrow) channel, 9–10 Flow resistance, 9–10 Acoustic boundary layers, 10–14 Viscous boundary layer, 10–12 Surface impedance for shear flow, 11–12 Thermal boundary layer, 12–13 Power dissipation, visco-thermal boundary layer, 13 Sound propagation in a narrow channel, 14–21 Propagation constant, 15–18 Kirchhoff attenuation, 17 Penetration depth, 17 Wave propagation vs diffusion, 17–18 Velocity and temperature profiles, 18–19 Effect of internal damping of flexible wall, 19 Relaxation times and a note on complex compressibility in a channel, 19–21 Impedances, 21–31 Impedance per unit length, 22–23 Complex density and wave impedance, 23–24 Perforated plate, 24–27 Micro-perforates, 25–27 Wire mesh screen, 28–29 Perforated plate-screen combination, laminates, 29–30 Effect of acoustically induced motion, 30 Interpretation of steady flow resistance data, 30–31 Visco-thermal admittance and absorption coefficient of a rigid wall, 31–34 Absorption coefficient, 33–34 Steady flow through a narrow channel, 34–35
INDEX Steady flow resistance, 34–35 Oscillatory flow and viscous boundary layer, 35–37 Acoustic viscous boundary layer, 35 Surface impedance for shear flow, 35–37 Thermal boundary layer, 37–38 Complex compressibility, 38 Power dissipation in the boundary layer, 39–41 Q-value of a cavity resonator, 39–41 Sound propagation in a narrow channel, 41–46 Propagational mode, 41–42 Thermal mode, 42–43 Viscous mode and the total velocity profile, 43–44 Propagation constant, phase velocity, and attenuation, 44–46 Impedances, 46–51 Wall stress and viscous interaction impedance, 47 Wave impedance, 48 Comparison with a one-dimensional transmission line, 48–49 Sound attenuation, 318–319 Sound pressure and radiated power, 321–326 Sound pressure of cavity tones, 131 Static pressure drop, 318 Static pressure drop in ducts, 290–293 Steady flow channel resistance per unit length, 9 Sudden area contraction in a duct, 374 Sudden area expansion in a duct, 374 Surface impedance for shear flow, 11–12, 35–37 Surface with a resistive screen, 229–231 Symmetric cell, 94
451 T Temperature profiles, 18 Theory of Sound book, 8, 143 Thermal boundary layer, 12–13 Thermal boundary layer thickness, 12 Thermal mode, 42–43 Thin porous plate, 387–388 Three ducts in series, 274 Three-dimensional array of resonators, 137–139 Three-dimensional lattice of resonators, 113–114 Total flow resistance of absorbers, 172 Transient response and reverberation, 115–116 Transmission coefficient and transmission loss, 364–367 Transmission loss and insertion loss of a circular duct, 247 Transmission loss of a contraction chamber, 306 Transmission loss of a side-branch pipe, 309 Transmission loss, TL, 241, 304–306, 308–309, 355–356, 365 Transmission matrices, 361–388 Choice of variables, 362 Application of matrices, 362–370 Impedance, 362–363 Reflection and absorption coefficients, 363–364 Transmission coefficient and transmission loss, 364–367 Insertion loss, 367–369 Noise reduction, 369–370 Commonly used matrices, 370–388 Porous screen, 370–372 Area discontinuities, 372–374 Duct element, 375–376 Contracted dust section, perforated plate, 376 Expansion chamber and elbow, 377–378 Lined duct, 378–379 Side-branch tube, 379–380
452 Side-branch Helmholtz resonator, 380–381 Parallel channels, 381–382 Rigid porous layers, 383 Flexible layer, 383–385 Thin porous plate, 387–388 Transmission matrix, 98–99 Tranverse velocity amplitude, thermal mode, 43 Tranverse velocity component, viscous mode, 44 Traverse propagation constant, 45 Tube resonator in a wall, 106–107 Two-dimensional arrays of resonators, 112–113 Two-room method, 241 Two-sided absorption cross absorption per unit area of circular disc, 87 Typical velocity dependence of steady flow resistance of some porous materials, 394 U Uniform (periodic) lattice, 94–98 Uniform duct, 356–357 Uniform duct section, 299–303 Unit cells, 94–95 Universal curves for diffuse field average absorption coefficients of a nonolocally reacting rigid porous layer, 164 Universal normal incidence and diffuse field absorption spectra of a locally reacting rigid porous layer, 163 Use of 4 × 4 matrices, 231–232 V Velocity amplitude distribution in flexible porous layer, 202 Velocity amplitude, propagational mode, 42 Velocity and temperature profiles, 18–19
NOISE REDUCTION ANALYSIS Velocity dependence of normalized acoustic resistance in porous material, 263 View angle vs emission angle, 174–175 Visco-thermal acoustic losses per unit area of a boundary, 13 Visco-thermal attenuation in a duct, 318 Viscous boundary layer, 10–12 Viscous interaction impedance per unit length, 47 Viscous mode and the total velocity profile, 43–44 Volume absorbers, 82–87 Vortex sheet, 177 W Wall admittance, 349 Wall impedance, 353 Wall stress and viscous interaction impedance, 47 Wave admittance and impedance, 189 Wave impedance, 48, 288, 338–339 Wave impedance of a duct, 334 Wave impedance, anisotropic material, 185 Wave modes, 238–240 Wave motion, 157–160 Wave propagation vs diffusion, 17–18 Whistle efficiency, 125 Wire mesh screen impedance, semi-empirical, 28 Wire screens, 62 Wire mesh screens, 28–29, 398–400 X x-component of propagation constant, 189 x-component of propagation constant, wave impedance, 159 Y Y-modes, 282–284, 348 Z Z-modes, 282, 347–348
