Calvin H. Wilcox Applied Mathematical Sciences 50
Sound Propagation in Stratified Fluids
Applied Mathematical Sciences I Volume 50
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Calvin H. Wilcox
Sound Propagation in Stratified Fluids
Springer-Verlag New York Berlin Heidelberg Tokyo
Calvin H. Wilcox University of Utah Department of Mathematics Salt Lake City, Utah 84112 U.S.A.
AMS Classification: 46NO5, 76005
Library of Congress Cataloging in Publication Data Wilcox, Calvin H. (Calvin Hayden) Sound propagation in stratified fluids. (Applied mathematical sciences ; v. 50) Bibliography: p. Includes index. 1. Sound-waves-Transmission. 2. Fluids-Acoustic properties. 3. Stratified flow. I. Title. II. Title: Stratified fluids. III. Series: Applied mathematical sciences (Springer-Verlag New York Inc.) ; v. 50. QA1.A647 vol. 50 [QC233] 510s [534'.23] 84-1447
© 1984 by Springer-Verlag New York Inc. All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer-Verlag, 175 Fifth Avenue, New York, N.Y. 10010, U.S.A. Printed and bound by R.R. Donnelley & Sons, Harrisonburg, Virginia. Printed in the United States of America.
987654321 ISBN 0-387-90986-9 Springer-Verlag New York Berlin Heidelberg Tokyo ISBN 3-540-90986-9 Springer-Verlag Berlin Heidelberg New York Tokyo
Preface
This monograph was begun during my sabbatical year in 1980, when I was a visiting professor at the University of Bonn, and completed at the University of Utah in 1981.
Preliminary studies were carried out during the
period 1972-79 at Utah and while I held visiting professorships at the University of Liege (1973-74), the University of Stuttgart (1974 and 197677) and the Ecole Polytechnique Federale of Lausanne (1979).
Throughout
this period my research was supported by the U.S. Office of Naval Research. I should like to express here my appreciation for the support of the
Universities of Bonn, Liege, Stuttgart and Utah, the Ecole Polytechnique Federale, the Alexander von Humboldt Foundation and the Office of Naval Research which made the work possible.
My special thanks are expressed to
Professors S. Chatterji (Lausanne), H. G. Garnir (Liege), R. Leis (Bonn) and P. Werner (Stuttgart) for arranging my visits to their universities.
I also
want to thank Professor Jean Claude Guillot of the University of Paris who collaborated with me during the period 1975-78 on the spectral theory of the Epstein operator.
That work, and our many discussions during that
period of wave propagation in stratified media, contributed importantly to the final form of the work presented here.
Calvin H. Wilcox Bonn
August, 1982
Introduction
Stratified fluids whose densities, sound speeds and other parameters are functions of a single depth coordinate occur widely in nature.
Indeed,
the earth's gravitational field imposes a stratification on its atmosphere, oceans and lakes.
It is well known that their stratification has a profound
effect on the propagation of sound in these fluids.
The most striking
effect is probably the occurrence of acoustic ducts, due to minima of the sound speed, that can trap sound waves and cause them to propagate horizontally.
The reflection, transmission and distortion of sonar signals by
acoustic ducts is important in interpreting sonar echoes.
Signal scattering
by layers of microscopic marine organisms is important to both sonar engineers and marine biologists.
Again, reflection of signals from bottom
sediment layers overlying a penetrable bottom are of interest both as sources of unwanted echoes and in the acoustic probing of such layers.
Many
other examples could be given. The purpose of this monograph is to develop from first principles a theory of sound propagation in stratified fluids whose densities and sound speeds are essentially arbitrary functions of the depth.
In physical terms,
the propagation of both time-harmonic and transient fields is analyzed. The corresponding mathematical model leads to the study of boundary value problems for a scalar wave equation whose coefficients contain the prescribed density and sound speed functions.
In the formalism adopted here
these problems are intimately related to the spectral analysis of a partial
differential operator, acting in a Hilbert space of functions defined in the domain occupied by the fluid. The intended audience for this monograph includes both those applied
physicists and engineers who are concerned with sound propagation in stratified fluids and those mathematicians who are interested in spectral analysis and boundary value problems for partial differential operators.
vii
viii
INTRODUCTION
An attempt to address simultaneously two such disparate groups must raise the question:
is there a common domain of discourse?
this question is no!
The honest answer to
Current mathematical literature on spectral analysis
and boundary value problems is based squarely on functional analysis, particularly the theory of linear transformations in Hilbert spaces.
This
theory has been readily accessible ever since the publication of M. H. Stone's AMS Colloquium volume in 1932.
Nevertheless, the theory has not
become a part of the curricula of applied physics and engineering and it is seldom seen in applied science literature on wave propagation.
Instead,
that literature is characterized by, on the one hand, the use of heuristic
non-rigorous arguments and, on the other, by formal manipulations that typically involve divergent series and integrals, generalized functions of unspecified types and the like. The differences in style and method outlined above pose a dilemma.
Can an exposition of our subject be written that is accessible and useful to both applied scientists and mathematicians?
An attempt is made to do
this below by beginning each chapter with a substantial summary.
Taken
together, the summaries present the basic physical concepts and results of
the theory, formulated in the simplest and most concise form consistent with their nature.
The purpose of the summaries is twofold.
First, they can be inter-
preted in the heuristic way favored by applied physicists and engineers.
When read in this way they are independent of the rest of the text and present a complete statement of the physical content of the theory.
Second,
readers conversant with Hilbert space theory can interpret the summaries as concise statements of the principal concepts and results of the rigorous mathematical theory.
When read in this second way, the summaries serve as
an introduction to and overview of the complete theory.
Contents Page
Preface CHAPTER 1.
INTRODUCTION
CHAPTER 2.
THE PROPAGATION PROBLEMS AND THEIR SOLUTIONS
Summary
13
2.
The Acoustic Propagator Solutions with Finite Energy
17 18
SPECTRAL ANALYSIS OF SOUND PROPAGATION IN STRATIFIED FLUIDS
1. 2. 3.
4. 5. 6. 7.
8.
9.
10.
CHAPTER 4.
Summary The Reduced Propagator AU Solutions of the Equation Auq Spectral Properties of Au Generalized Eigenfunctions of Au The Spectral Family of Au The Dispersion Relations The Spectral Family of A Normal Mode Expansions for A Semi-Infinite and Finite Layers
TRANSIENT SOUND FIELD STRUCTURE IN STRATIFIED FLUIDS
21
21 32 33 51 65 71 85 90
104 117 125
1.
Summary
125
2.
Normal Mode Expansions of Transient Sound Fields Transient Free Waves Transient Guided Waves Asymptotic Energy Distributions Semi-Infinite and Finite Layers
133 134 150 154 156
3. 4.
5. 6.
CHAPTER 5.
13
1. 3.
CHAPTER 3.
1
SCATTERING OF SIGNALS BY INHOMOGENEOUS LAYERS
1. 2. 3.
4.
5.
Summary Signals in Homogeneous Fluids The Reflected and Transmitted Signals Construction of the Scattering Operator The Scattering Operator and Signal Structure
161
161 167 170 178 184
APPENDIX 1.
THE WEYL-KODAIRA-TITCHMARSH THEORY
189
APPENDIX 2.
STATIONARY PHASE ESTIMATES OF OSCILLATORY INTEGRALS WITH PARAMETERS
193
REFERENCES
197
INDEX
198
ix
Chapter 1
Introduction
This monograph presents a theory of the propagation of transient sound waves in plane stratified fluids whose densities p(y) and sound speeds c(y) are functions of the depth y.
The main goal of the theory is to calculate
the signals produced by prescribed localized sources and to determine their asymptotic behavior for large times.
The results include criteria for the
occurrence of acoustic ducts that trap a portion of the signal and cause it to propagate horizontally.
The mathematical theory is based on a real valued function u(t,X), the acoustic pressure or potential at the time t and the spatial point with Cartesian coordinates X = (x1,x2,y), which satisfies a scalar wave equation A derivation of
that contains the fluid density p(y) and sound speed c(y).
this wave equation from the laws of fluid dynamics is presented here in order to clarify the hypotheses needed for its validity.
Analogous deri-
vations may be found in the monographs of F. G. Friedlander [6]* and L. M. Brekhovskihk [2] and in an article of I. Tolstoy [19]. A compressible, non-viscous, heat conducting fluid is considered.
The
states of the fluid are characterized by the variables
p = fluid pressure p = fluid density v = fluid velocity T = fluid temperature (absolute scale) S = fluid entropy per unit mass
Each is a function of the variables t and X.
These variables satisfy the
*Numbers in square brackets denote references from the list at the end of the monograph.
1
2
1.
INTRODUCTION
equations
v= 0,
(1.2)
Dt+p V
(1.3)
P Dt+Vp =-G,
(1.4)
p Dt
T
V
(kVT) = 0,
f(p,p,T) = 0 and (p,p,S) = 0,
(1.5)
where Du =
(1.6)
Dt
denotes the material derivative.
au
at + v
Vu
Equations (1.2) and (1.3) express the
conservation of mass and linear momentum, respectively.
The field
G = G(t,X) describes the external forces that act on the fluid.
Equation
(1.4) relates the variation of S to the diffusion of heat in a medium with Equations (1.5) are alternative forms of
thermal conductivity k = k(X).
the thermodynamic equation of state.
Sound waves will be studied in a static inhomogeneous fluid in which external forces are negligible.
Thus if the fluid parameters (1.1) for the
static fluid are denoted by a subscript zero then one has
vo
G
at
=
0,
Ot
= 0, Go = G>
a = 0 or po = Po(X) Opo = 0 or po = p0(t)
and
(1.10)
V
(kVT0) = 0.
The last equation, together with suitable boundary conditions, determines a unique
1.
Introduction
3
(1.11)
To = T0(X).
Then (1.5) implies that
(1.12)
f(P0(t),P0(X),T0(X)) = 0,
whence
po = const.
(1.13)
and po(X) may be determined by solving (1.12) with known po and To(X). Finally, S0(X) is determined by
(1.14)
(P0,Pa(X),S0(X)) = 0.
The equations governing sound waves in a static inhomogeneous fluid will be obtained by a formal perturbation method applied to the static solution just discussed.
The sound waves will be assumed to be excited by
body forces with force density
-G = E G1r
(1.15)
where E is a parameter.
Moreover, c and G1 will be assumed to be so small
that the resulting disturbance is adequately described by the first order terms in a Taylor series expansion in C.
The perturbation equations are
obtained by substituting (1.15) and
P=Po +EPi P=Po+EP1 v = E V1
(1.16)
T = To + ET1 1
S=So +ES1
in equations (1.2)-(1.5) and dropping terms containing powers of E higher than the first.
Together with the small displacement assumption (1.16) the additional hypothesis will be made that the acoustic disturbances are so rapid that heat diffusion is negligible during passage of the sound waves.
This
assumption, which is called the adiabatic hypothesis, is equivalent to dropping the diffusion term in (1.4) by setting k = 0.
Thus for sound
4
1.
INTRODUCTION
waves one replaces (1.4) by DS = 0. Dt
(1.17)
Instead of linearizing (1.17) directly one may take the material derivative of the equation of state 0(p,p,S) = 0 and use (1.17) to get
DP+ PDp =0 Dt
(1.18)
where 4P =
p Dt and P = a4/3p.
Substituting (1.16) in (1.18) and retain-
ing only linear terms in t gives (since Opo = 0)
(1.19)
o
aPi + ON, + at
P
at
-'°1)
V P0
= 0
po
where
0 =
p(P" Po(X),So(X)),
p =
P(PO1Po(X),So(X))
(1.20)
This can be written
(1.21)
Rl
at = c2(X)
jatl + Vpo - V1
where O
(1.22)
c2(x) p
is the local speed of sound at X.
a
l
(aPIS
Linearizing equations (1.2) and (1.3)
yields
(1.23) (1.24)
apl + V PO
(Pwl) = 0,
atl + 7P1 = Po 1
Combining (1.21) and (1.23) gives the alternative equation
(1.25)
+ czpo V
vl = 0.
1.
Introduction
5
Finally, differentiating (1.25) with respect to t and using (1.24) to eliminate vl gives a scalar wave equation for pl:
(1.26)
at21 - cZpo V
.
Vp1
(p 0 l
I
-F
JJ
where
(1.27)
G1
F = -cZpo V
is determined by the prescribed source density G1. If Pool = VG then one may introduce an acoustic potential u(t,X) such that
V1 =
(1.28)
PO
at + G.
Vu, P1
Then (1.24) and (1.25) are satisfied if
(1.29)
ate - cZpo V
with F = 2G/Pt.
.
I 1 DuJ = F Po
It will be convenient to use this formulation below.
A static fluid will be said to be plane stratified if the fluid parameters depend on the single depth coordinate y.
In this case,
writing c = c(y), po = p(y) in (1.29) one gets the partial differential equation
(1.30)
92u 2 at2 - c (Y)
a2u
+
32U
2
+ P(Y)
1
aY IP(Y)
2u
8y)
= F(t,Xi'x2.Y)
Sound waves governed by (1.30) are studied in the remainder of this monograph.
The fluid media are assumed to be unlimited in horizontal
planes, so that -°° < x1,x2 < . The principal case treated is that of a completely unlimited fluid for which -°° < y <
This idealized problem
provides a model for the analysis of sound propagation in portions of a finite stratified fluid that are far from the boundaries.
The cases of a
semi-infinite layer, with 0 < y < -, and a finite layer, with 0 < y < h < are also discussed.
These cases provide models for the study of sound
propagation near one and two horizontal plane boundaries, respectively. The fluid parameters p(y) and c(y) are inherently positive. be assumed here that they are bounded and bounded away from zero. suffices to show the solvability of the propagation problem.
It will This
Additional
6
1.
INTRODUCTION
conditions that restrict the behavior of p(y) and c(y) at y = ±w are needed to obtain more detailed information concerning the structure of the acoustic field.
Such conditions are imposed as they are needed.
The method of solving the wave equation (1.30) developed below is based on the spectral theory of selfadjoint operators in a Hilbert space.
The possibility of using this method is suggested by the observation that the partial differential operator A defined by z
2
Au = _C2 (Y) xz +
(1.31)
+ 1) (Y)
P (Y)
aYl)]
is formally selfadjoint with respect to the scalar product defined by the integral
(1.32)
(u,v) =
u(x1.x2,Y) v(x1.x2,Y) c-2(Y) P-1(Y)dx1dx2dy, 1JJ
where c-2(y) = 1/c2(y), p-1(y) = 1/p(y) and the integration is over the domain Q occupied by the fluid.
Indeed, partial integration using the
divergence theorem yields
(Au,v) _ (u,Av) + boundary terms
(1.33)
and if u and v satisfy suitable boundary conditions then one has (Au,v) _ (u,Av).
These facts are used below to construct a selfadjoint realization
of A in a Hilbert space 2C of functions with scalar product (1.32).
This
makes it possible to solve the wave equation (1.30) by means of the functional calculus based on the spectral theorem for selfadjoint operators.
A
concise discussion of the principal concepts and results of this theory will be presented here in order to motivate and clarify the analysis that follows.
Detailed developments of the spectral theory of operators in Hilbert spaces can be found in a number of excellent texts, including those of N. Dunford and J.
T. Schwartz [5] and T. Kato [11].
The set of all complex-valued functions u on a domain 9 in threedimensional space that satisfy
(1.34)
lu(x1,x2,Y)l2 c-2(y) P-1(Y)dxldx2dy < w
I1u112 = (u,u) = 11!
form a Hilbert space x; i.e., a vector space over the complex field C that has a scalar product (1.32) and is complete in the norm hull defined by (1.34).
The completeness property plays the same fundamental role in
1.
Introduction
7
functional analysis that the completeness of the real number field plays To obtain it one must include in x the large class of all
in calculus.
Lebesgue square-integrable functions.
The completeness of such spaces
was discovered and proved by F. Riesz and E. Fischer, working independently, in 1907.
The differential operator A defined by (1.31) can be used to define various linear operators in X.
Of course, the domain of definition D(A)
of such an operator cannot be all of 7C.
Instead, functions u E D(A) must
have the necessary derivatives and perhaps satisfy integrability and Operators with different domains are considered to be
boundary conditions.
different operators, even if the actions of both are defined by the same operator (1.31).
The operators considered here have domains D(A) that, although they are rather "thin" subsets of 7, nevertheless are large enough to be dense in 7f; i.e., every u E x is a limit in the norm (1.34) of a sequence un E D(A).
Such operators are said to be densely defined.
Every such
operator A has a well-defined adjoint operator A* whose domain D(A*) is the set of all u E JC such that
(Av,u) = (v,u*) for all v E D(A).
(1.35)
The action of A* is defined by A*u = u*. The selfadjoint operators A are those for which A* = A; i.e., D(A*) = D(A) and A*u = Au for all u E D(A).
The importance of this rather
special class of operators is due to the discovery of J. von Neumann in 1929
that they are precisely the operators that have a spectral representation
(1.36)
A
< '} is a family of operators in 3C, called the spectral
family of A.
To explain the properties of spectral families and the meaning of (1.36), the notions of subspace and orthogonal projection in a Hilbert space must be defined.
A subspace M C Jf is a subset that is closed under
linear operations in 3C and is complete.
Hilbert space.
Thus a subspace is itself a
To each subspace M there corresponds a second subspace M
called the orthogonal complement of M, which consists of all u E If such that (u,v) = 0 for all v E M. decomposition
Moreover, every u C
has a unique
INTRODUCTION
1.
8
u = ul + u2, u1 E M, u2 E M
(1.37)
i
.
Given a subspace M, the correspondence that assigns to each u E JC its component ul E M defines a linear operator II in If.
(1.38)
It has the properties
11u = ul E M for all u E JC,
112
(1.39)
= 11
(1.40)
Conversely, any operator H with properties (1.39), (1.40) defines a subspace M = INC.
Such operators are called orthogonal projections.
Thus there is
a one-to-one correspondence between the set of subspaces of JC and the set of all orthogonal projections in X. A spectral family {11(A)
< A < co} in JC is a family of orthogonal
projections 11(X) in JC with the properties
11(X) 11(p) = 11(p) for all X > p
(1.41)
for u E X.
(1.42)
lim
11(A)u = U
A->+m
It follows that {11(A)} is a monotone non-decreasing family in the sense that
(1.43)
and all u E X.
(11(p)u,u) < (11(A)u,u) for all A > p
It is also customary to normalize {11(A)} by requiring
continuity from the right:
(1.44)
11(X+0)u = 11(A)u for all A E R and u E JC,
but this is not essential. The spectral integral
(1.45)
J
f(A) d11(A),
1.
Introduction
9
where f(X) is a continuous complex-valued function of A E R, will be interpreted as a linear operator in K. First, if
It will be defined in two steps.
< a < b < - then n
(b
f(a) dl(A)u = lim ITrH-0 a
(1.46)
for all u E JC, where Tr
[a,b] and
ITrI
The existence in JC of the limit in (1.46)
onotonicity property (1.43), just as in the case of
scalar Stieltjes integrals.
Finally M
r°O
(1.47)
f(aj)[ll(A.) - ll(aj-1)]u
a = A0 < A, < . < an = b is a partition of
= Max (aj - aj-1).
follows from the
j=l
J-m f(A) dl(A)u = lim
M,N- f-N
where convergence is in JC.
f(A) dl(A)u,
The domain of the operator so defined is the
set of u E JC for which the limit in (1.47) exists.
It can be shown to
coincide with the set of u E JC such that
(1.48)
Jw I f (a) l2 dlln(A)uII2 <
In particular, the domain of the operator A characterized by (1.36) is the
set of all u C JC such that A2 dlln(A)u(l2 < -.
(1.49)
1W m
As examples the spectral families associated with two special cases of the operator (1.31) will be described. and A = -A is the negative Laplacian.
In both cases p(y) = c(y) = 1
In the first example Sd is a bounded
domain in three-dimensional space and D(A) is the subset of all u E JC for which Au E JC and u(x) = 0 on the boundary of Q.
Then A is selfadjoint and
has a pure point spectrum consisting of real eigenvalues 0 < A, < A2 <
and corresponding eigenfunctions ul(x), u2(x), such that Aun = Anun (i.e. un E D(A) and Aun = -Aun = anun). normalized; i.e.,
If the eigenfunctions are then the spectral family of A
lunll = 1 for n =
is given by
(1.50)
n(A)u =
an(un,u)un A
where the notation indicates a sum over all indices n such that A < A. n Thus in this case (H(A)} is a step function that is constant for A between
10
INTRODUCTION
1.
consecutive eigenvalues.
Note that the jump H(Xn) - "(An - 0) is precisely
the orthogonal projection onto the eigenspace of A for the eigenvalue A
.
n For the second example consider A = -A, acting on functions defined
in the whole space 1 = R3.
If D(A) is the subset of all u E JC for which
Au E JC then A is selfadjoint and has a pure continuous spectrum consisting of all real A > 0. analysis.
The spectral family {11(X)} may be constructed by Fourier
Indeed, if u E JC then the limit 1
u(P) = lim (2n)32
(1.51)
1
X I
exists in JC and one has u(X)
RJJ liimm
(2T )3/2
J P
u(P)dP
IIuM _ Hull. These results are Plancherel's theory of the Fourier transform.
With this
notation the spectral family of A = -A in R3 is given by 1
f
2
(21
u(P)dP,
A > 0,
,
X < 0.
II(X)u(x) _
(1.54)
0
It is clear that in this case 11(X) varies continuously with A.
The differential operator (1.31), acting in the whole space R3 or in a semi-infinite or finite layer, is much closer to the second example than the first.
In particular, it always has a pure continuous spectrum.
However,
when p(y) and c(y) are variable the structure of {11(A)} may be complicated
by the existence of acoustic ducts and corresponding families of subspaces of JC.
Of course, the spectral resolution (1.34) is useful only when an
explicit construction of {11(X)}, such as (1.50) or (1.54), can be found.
This is achieved below for sound waves in stratified fluids by means of the physically motivated concept of normal modes.
The organization of the remainder of the monograph will now be described.
Chapter 2 presents a construction of a selfadjoint realization
of the acoustic propagator A defined by (1.31), followed by a solution of
1.
Introduction
11
the wave equation (1.30) by means of the functional calculus (1.47), (1.48) for A.
The principal results of the monograph are developed in the long Chapter 3 which presents an explicit construction of the spectral family of
A, under the condition that p(y) and c(y) have limits in a suitable sense at y = ±-.
The construction is based on two families of generalized
eigenfunctions, or normal mode functions.
Physically, the first family
describes the response of the fluid to an incident plane acoustic wave. The second family exists only when the fluid has an acoustic duct and represents guided waves that are localized near the duct. Chapter 4 presents a detailed analysis of the large-time behavior of transient sound fields in stratified fluids. normal mode expansion of the fields.
The calculations begin with a
Then, under suitable restrictions on
the behavior of p(y) and c(y) at y = ±W, the large-time behavior of the fields is determined by the method of stationary phase.
The convergence of
u and its first-order derivatives in C to the corresponding asymptotic wave functions is demonstrated and used to determine the asymptotic distribution of energy for large times.
Chapter 5 presents an analysis of the scattering of acoustic signals
by an inhomogeneous stratified layer of finite thickness that separates two homogeneous fluids.
A scattering operator is constructed and shown to
determine completely the asymptotic form of the reflected and transmitted signals that are produced by a prescribed source localized in one of the homogeneous fluids.
The monograph concludes with two appendices.
Appendix 1 presents in
a concise form, with references to the literature, the Weyl-KodairaTitchmarsh theory of singular Sturm-Liouville operators.
These results
provide the foundation for the proof in Chapter 3 of the completeness and orthogonality of the normal mode functions.
Appendix 2 presents a version
of the stationary phase estimate for oscillatory integrals containing parameters.
The uniformity with respect to the parameters of these
estimates is essential for the applications in Chapter 4. The physical literature on sound propagation in stratified fluids is very large.
The modern developments date from the pioneering work of
C. L. Pekeris [17], published in 1948, in which p(y) and c(y) are piecewise constant.
In the intervening years hundreds of papers on the subject have
appeared in the Journal of the Acoustical Society of America alone. of the material has been reviewed in the well known books of L. M. Brekhovskikh [2] and I. Tolstoy and C. S. Clay [20].
Some
12
1.
INTRODUCTION
The work presented here differs from the earlier literature in being based on concepts and results from the theory of linear transformations in Hilbert spaces.
Some of the many advantages of this approach will be
mentioned here.
First, the use of the Weyl-Kodaira-Titchmarsh theory of
singular Sturm-Liouville operators permits a straightforward derivation of
the completeness and orthogonality of the normal mode functions, without ad hoc hypotheses concerning p(y) and c(y). a Hilbert space theory.
This theory is intrinsically
Second, the use of the notions of convergence in
the mean-square and energy norms leads to results on the asymptotic behavior for t - w of transient waves that are not true in the conventional sense of pointwise convergence.
Finally, the introduction of the unitary scattering
operator in Chapter 4 gives a particularly lucid form to the relationships among the incident, reflected and transmitted waveforms in the scattering of signals by an inhomogeneous layer.
Earlier applications of Hilbert space methods to wave propagation in stratified media were made by the author to the special case of the Pekeris profile [23,26,27,28] and by J. C. Guillot and Wilcox [7,8] and Y. Dermenjian, Guillot and Wilcox [4] to the special case of the Epstein profile.
For simplicity, displayed equations are numbered separately in each chapter.
The notation (j.k) always means equation k of section j of the
current chapter.
The notation (i.j.k) is used to refer to equation k of
section j of chapter i.
Chapter 2
The Propagation Problems and Their Solutions
The purpose of this chapter is to present a mathematical formulation and solution of the propagation problems that are studied in the subsequent chapters.
The basic physical concepts and results are summarized in §1.
The remaining sections develop the mathematical theory, together with rigorous proofs or references to the literature. fluid is treated here.
The case of an unlimited
The changes needed to treat semi-infinite and
finite layers are indicated at the end of Chapter 3.
§1.
SUMMARY
The acoustic field in an unlimited stratified fluid will be characterized by a real-valued acoustic potential u(t,X) where x = (x,y) = (x1,x2,y) E R3 denote the rectangular spatial coordinates of Chapter 1. The partial derivatives of u will be denoted by Dku, k = 0, 1, 2, 3, where
(1.1)
Do = B/Bt, D1 = 8/BX1, D2 = a/Bx2, D3 = 3/3y.
With this notation the wave equation of Chapter 1 takes the form
Dou - c2(Y) P(Y) V
Vu)
= F(t,X)
where V = (D1,D2,D3) denotes the spatial gradient operator and F(t,X) is a source density function.
The acoustic pressure p and velocity field v in
the fluid are given by [61
(1.3)
p = -Dou and v = p( y) Vu
respectively.
13
14
THE PROPAGATION PROBLEMS AND THEIR SOLUTIONS
2.
As yet, no conditions other than positivity have been placed on the variations of p(y) and c(y).
In the remainder of this work it is assumed
as a minimal hypothesis that
0 < Pm < P(Y) < PM < °° and 0 < cm < c(y) < cM <
(1.4)
for all y, where pm, pM, cm and cM are suitable constants.
In this chapter
no additional hypotheses are needed (other than the technical hypothesis of In particular, p(y) and c(y) need not be continuous.
measurability).
The
meaning of the wave equation (1.2) when p(y) and c(y) are discontinuous is explained below.
Other conditions than (1.4) that would permit p(y) and
c(y) to be unbounded or to tend to zero can be treated by the methods developed here, but (1.4) is sufficiently general for most applications. The wave sources are assumed to be localized in both space and time.
Without loss of generality it can be assumed that the source density in (1.2) satisfies
(1.5)
supp F C {(t,X)
:
T < t < 0, xi+x2+(Y-Yo)2 < 621
where supp F denotes the support of F (i.e., the smallest closed set outside which F is zero).
Then u(t,X) can be characterized as the solution of
(1.2) for t E R, X G R3 that satisfies the condition u(t,X) = 0 for all t < T.
It is known that (1.4) implies that the supports of solutions of
(1.2) propagate with speed not exceeding cM [221.
Hence, (1.5) implies
that if u(0,X) = f(X), Dou(0,X) = g(X) for X E R3
(1.6)
then
(1.7)
supp f U suPP g C {X : xi+x2+(y-Yo)2 < 621
where 6 = 60 + cMITI.
It follows that u(t,X) can be characterized for
t > 0 as the solution of the homogeneous wave equation
Dou - c2(y) p(y) V '
Ip-y) Du' = 0 for t > 0, X E R3
with initial values f, g that satisfy (1.6), (1.7).
The reduction of the
initial value problem for (1.2) to that for (1.8) by means of Duhamel's principle is shown below.
Summary
2.1.
15
The integral
E(u,K,t) =
(1.9)
{IDu(t,X)I2 + c-2(y) ID0u(t,X)12} P-1(y) dX, J
where dX = dxldx2dy,maybe interpreted as the energy of the acoustic field u in the set K C R3 at time t.
It is known that solutions of (1.8) satisfy
the conservation law
E(u,R3,t) = E(u,R3,0)
(1.10)
where E(u,R3,O) < 4°O.
dered below.
Only solutions with finite total energy are consi-
A necessary and sufficient condition for u to have this
property is that the initial state have finite energy [22]
(1.11)
1
3
{IOf(X)IZ + c -2(y) Ig(X)12} P=1(y) dX < -.
R
The initial value problem in its classical formulation (1.6), (1.8)
will have a solution only if p(y), c(y), f(X) and g(X) are sufficiently smooth.
However, for arbitrary p(y), c(y) satisfying (1.4) the problem may
be shown to have a unique solution with finite energy whenever the initial state f,g has this property.
A proof is outlined in §3 below.
A formal
construction of the solution may be based on the operational calculus for the linear operator A, defined by the differential operator
(1.12)
Au = -c2(y) P(y) V
.
IP(Y) Vul
acting in the Hilbert space 1C with scalar product
(1.13)
u(X) v(X) c-2(y) P_'(y) dX.
(u,v) = J R3
If the domain of A is defined to be the set of all u E X such that Vu E dC and V
(1.14)
(p Vu) E 3C then A is a selfadjoint non-negative operator.
(cos t A1'2 )f + (A-'/' sin t
is the solution of the initial value problem
Dou+Au=0, t > 0, (1.15) u(O) = f, Dou(0) = g.
A1/2 )g
Moreover,
16
2.
THE PROPAGATION PROBLEMS AND THEIR SOLUTIONS
It can be shown that (1.14) is the solution with finite energy whenever the initial values have finite energy.
The initial value problem for (1.2) where F satisfies (1.5) and u(t,X) F 0 for t < T can be reformulated as
Dou + Au =
t E R,
u(t) = 0 for t < T.
The solution is given by the Duhamel integral
t J{A 1/2 sin (t-T)A1/2}
(1.17)
dT, t > T.
T
It is clear that for t > 0 this solution satisfies (1.15) with
f = u(0)
0
{A 1/2 sin T A1/2}
dT,
T
(1.18)
g = Dou(0) = Jo {cos T A1/2}
dT.
T
The remainder of the analysis will be based on the representation It will be convenient to rewrite it in the form
(1.14).
u(t,X) = Re {v(t,X)}
(1.19)
where v(t,X) is the complex-valued wave function defined by
e
(1.20)
-i to 1/2 h
and
h= f + i A1/2g
(1.21)
This representation is valid if f and g are real-valued and A1/2 f, f, g and A 1/2 g are in 3C.
A rigorous interpretation of (1.14)-(1.21) may be based
on the calculus of selfadjoint operators in Hilbert spaces.
It is noteworthy that the same formalism is applicable to the cases of semi-infinite and finite layers. R+ = {X
:
For the semi-infinite layer
y > 0}, IC is the Hilbert space with scalar product
The Acoustic Propagator
2.2.
(1.22)
17
u( X) v(X) c-2(y) p-1(y) dX
(u,v) = J R3
and the domain of A is the set of all u E 3C such that Vu E 3C,
V.(p Vu) E IC and u satisfies the Dirichlet or Neumann condition at y = 0.
The relations (1.14)-(1.21) then hold as before.
hold for the finite layer Rh = {X
§2.
:
The analogous statements
0 < y < h}.
THE ACOUSTIC PROPAGATOR In this section a precise definition of the acoustic propagator A is
given and a proof of its selfadjointness in 3C is outlined.
Here and
throughout this work p(y) and c(y) are assumed to be Lebesgue measurable functions that satisfy the boundedness conditions (1.4).
(2.1)
It follows that
c-2(Y) P_'(y) dX
m(K) = J K
defines a measure in R3 that is equivalent to Lebesgue measure (i.e. m is absolutely continuous with respect to dX and dm/dX = c-2(y) p-1(y)).
Hence
the Lebesgue space
3f = L2(R3,c-2(y) p-' (y) dX)
(2.2)
is a Hilbert space with scalar product (1.13).
Note that (1.4) implies
that 3C is equivalent as a normed space to the usual Lebesgue space
L2(R3,dX), although they are distinct as Hilbert spaces. A selfadjoint realization of the differential operator A in 3C, also to
be denoted by A, is obtained by defining the domain of A to be
D(A) = L2(R3) n {u
(2.3)
:
(p -'Vu) E L2(R3)}
V
where L2(R3) = L2(R3,dX) and
L2'(R3) = L2(R3) n {u
(2.4)
is the usual first Sobolev space [1].
:
Vu E L2(R3)}
All the differential operations in
(2.3), (2.4) are to be understood in the sense of the theory of distributions.
It follows that the linear operator A in 3C defined by (1.12),
(2.3) satisfies
(2.5)
A = A* > 0
2.
18
THE PROIAGATION PROBLEMS AND THEIR SOLUTIONS
where A* is the adjoint of A with respect to the scalar product (1.13).
proof of (2.5) may be given by the method employed in [26].
A
Alternatively,
(2.5) may be derived from Kato's theory of sesquilinear forms in Hilbert space [11, p. 322].
Indeed, if one defines a sesquilinear form A in JC by
D(A) = LZ(R3) C X
(2.6)
and
A(u,v) = JR
(2.7)
Vv p-1(y) dX
Du 3
then it is easy to verify that A is closed and non-negative, and that A is the unique selfadjoint non-negative operator in JC associated with A.
As an
additional dividend it follows from Kato's second representation theorem [11, p. 331] that D(A1/2) = L2(R3) and for all u E D(A1/2) (
(2.8)
(A1/2 ul12 = A(u,u) =
3
J
IDu1 2
one has
P-1(y) dX
R
where 11-11,X is the norm in JC. §3.
SOLUTIONS WITH FINITE ENERGY It was shown in [22] that the initial value problem (1.6), (1.8) has
a unique generalized solution with finite energy (= solution wFE) whenever the initial state f, g has finite energy; i.e., (1.11) holds.
These
conditions may be written, by (2.7),
A(f,f) + IgIl2 = IIA1/2 fll2 + IIgMi2 <
f E LZ(R3), g E JC. It follows that the function u(t,X) defined by (1.14) is the unique solution wFE and
(3.3)
u E C1(R,3C) n C(R,L(R3)),
k (Here C (I,JC) is the set of k times continuously differentiable JC-valued function on I.)
2.3.
Solutions with Finite Energy
19
The acoustic fields studied below are defined by (1.14) where f, g satisfy (3.2).
It will be convenient to specialize further by writing u
in the form (1.19)-(1.21).
This is correct provided f and g are real
valued, (3.2) holds and g E D(A 1'2).
However, D(A
1/'2)
Of course, D(A 1"2) # x, in general.
is dense in 3C, by the spectral theorem.
This fact can
often be used to extend results derived from (1.19)-(1.21) to general solutions wFE.
Chapter 3 Spectral Analysis of Sound Propagation in Stratified Fluids
The purpose of this chapter is to construct families of normal mode functions and to derive corresponding normal mode expansions for the The basic physical concepts and results are sum-
acoustic propagator A.
A more detailed formulation and complete proofs of all the
marized in §1.
results are developed in the remaining sections.
This rather long chapter
provides the analytical foundation for the solutions of the propagation problems that are developed in Chapters 4 and 5.
§1.
SUMMARY
Throughout this chapter it is assumed that p(y) and c(y) satisfy the boundedness conditions (2.1.4) and the four conditions
IP(y) - P(± )I dy < m, ±jp
Ic(y) - c(±-)I dy <
±[
where p(+°°), p(-W), c(+-) and c(-oo) are suitable constants.
Under these
conditions the spectrum of A fills the interval [0,m) and is continuous with no embedded eigenvalues.
These facts, which are verified below, imply that
the normal mode functions of A are generalized eigenfunctions; that is, solutions p of the differential equation
(1.2) that are not in I.
-c2 (y) p(y) V , (Y)
Ate
A V (x,y)
Solution of (1.2) by separation of variables leads to
solutions of the form
(1.3)
i(x,y) =
eip*x
21
(y), p = (p1,p2) E R2,
22
3.
SPECTRAL ANALYSIS OF SOUND PROPAGATION
where p- x = plx1 + p2x2 and tp(y) is a solution of the equation _C2 (y) [P(y)
(1.4)
IPI2P] dy 1
j
1
-
_ AP
with IP12 = pl + P2 The operator A. defined by
Auk
_C2 (y) [P(Y)
(P(Y)
dy
dy)
will be called the reduced acoustic propagator. Liouville operator on -w < y < w.
- U2 ]
It is a singular Sturm-
Its proper and improper eigenfunctions
will be used to generate the normal mode functions of A.
Note that (2.1.4) and (1.1) imply that
PM < P(±-) < PM, cm < c(+°°) < cm.
(1.6)
Moreover, in the presence of (2.1.4), conditions (1.1) are restrictions on the behavior of p(y) and c(y) near y = ±a' only.
Thus it is natural to
compare solutions of Auk = a4 with solutions of its limiting forms for
y -r±w 2 day
+ (A c-2(±w) - U2) = 0.
In what follows it will be convenient to assume that
(1.8)
c(`°) < c(-`°).
This condition does not limit the generality since it can always be achieved by proper choice of the depth coordinate y. The general solution of (1.7) for A # c2(±0°)p2 can be written iq+(p,X)y
cl e - cl e
+ c2 e
-iq+(p,A)y
-
,
or
+ c2 e
where q+(U (1.10)
(ac 2(±w) - U2)1/2 > 0 for A > c2(±w)112
3.1.
Summary
23
and
q+(u,A) = (u2 - xc
2
(t_) ) 1/2 > 0 1
for A < c2(+°°)j12 q+(u,A) _ -iq+(u,A)
In particular, the solutions are oscillatory when A > c2(tm)u2 and non-oscillatory when A < c2(t°)u2.
The first step in the analysis of Au is
to construct special solutions of Auk = a that have the asymptotic behaviors q+(u,A)Y [1 + 0(1)], y - +-
1(Y,u,l) = e
A # c2(")u2,
q+(u,X)Y [1 + o(1)], Y + I
e-
(1.12)
q'(U,A)Y [1 + 0(1)], y
e
a # c2(-°')u2
-q (u,l)Y 4(Y,u,a) = e
[1 + 0(1)], y i -°°
It follows from an asymptotic calculation of the Wronskians that 1 and 2
are a solution basis for AuO = a when A # c2(°D)u2, while 3 and 4 are a basis when A #
c2(-W)u2.
The nature of the spectrum and eigenfunctions of Au can be inferred from (1.12).
It follows that if A < c2(o)u2 then A 0 = a¢ has bounded P
solutions only if 02 and 03 are linearly dependent.
(1.13)
Thus
F(u,A) = P-1 W(02103) = 0
is an equation for the eigenvalues of Au, where W denotes the Wronskian and p _1W is independent of y.
(1.14)
The corresponding solutions
k(Y,u) = ak(u) 02(Y,u,ak(u)) = ak(u) 03(Y,u,Ak(u)),
where A = Ak(u) is a root of (1.13), are square integrable on R and hence are eigenfunctions of A A > c2(o)p2, by (1.12). interval [cmu2,c2(m)u2].
Moreover, Au can have no point eigenvalues Thus 00(A11 ), the point spectrum of Au, lies in the
Criteria for ao(A
to be empty, finite or
countably infinite are given in §4 below.
It will be shown that the continuous spectrum of Au is [c2(m)u2,-) and corresponding generalized eigenfunctions will be determined from (1.12).
24
3.
For c2(_)}12 < A <
SPECTRAL ANALYSIS OF SOUND PROPAGATION
c2(_W)u2 there is a single family of generalized eigen-
functions of the form
(1.15)
= ao(u,A)
For A > c2(- )j12 there are two families defined by
a+(u,A) it
(1.16)
l P_(Y,u,A) = a_(u,X) 1(Y,u,A)
It will be shown that these functions have the following asymptotic forms.
+
ck(u) e ck(u) e
-q+(u,ak(u))Y ,
y + +-,
q'(u,Ak(u))Y ,
-iq+(u,X)Y
e
Y
+ R0(u,A) e
iq+(u,A)Y ,
Y + {m,
q'(U,X)Y
TO (u,A) e -iq+(u,A)Y
+ R+(u,X) e
e
+(u,a) e
iq+(u,X)Y
,y++
-iq(u,X)Y
iq+(u,X)Y
T_(u,A) e (1.20)
y + 4°O,
,
i_(Y,u,A) - c_(u,A) iq_(11,A)Y
e
+ R_(u,A) e
-iq_(u,A)Y ,
y + -°°
Here ak(u), ak'(u), ao(u,X), a+(u,A), ck(u), co(u,A), c+(u,X), Ro(u,A),
R+(u,A), T0(u,A) and T+(u,A) are functions of u and A that will be calculated below.
Families of normal mode functions for A may be constructed from those for AIpI by the rule (1.3).
The following notation will be used.
3.1.
Summary
25
(1.21)
+(x,y,P,X) _
(1.22)
i,(x,Y,P,A) _
(1.23)
(2Tr)-'
P+(Y,lpl,x), (P,X) C Q,
(2Tr)-i
1k(x,y,P) _ (21r)_'
eiP'x
eip*x
o(y,IPI,a), (P,X) E Qa,
p E 52k, k > 1,
k(Y'IPI),
The parameter domains Q, PQ, Qk are defined by
2 = {(P,X) E R3
(1.24)
o = {(P,X) E R3
(1.25)
I
I
c2(°°)
c2(-°°) IPI2 < X}, IPI2 < X < c2(-m)
IPI2}
Qk={pCR2 I IPI C0k}, k > 1,
(1.26)
where 0k is the set of p > 0 for which AP has a kth eigenvalue. The three families have different wave-theoretic interpretations that are characterized by their asymptotic behaviors.
Thus for (p,A) E P one has
+R+e
e
i(P'x+qy) ,
y 14-,
c+(IPI,A) (1.27) 2Tr
i(P'x-q_y) T+ e
e
y
c-(IPI,X) (1.28) 2Tr
i(p,x+q-y) e
where q+ = q+(IPI,X), R+ = R+(IPI,X), etc. y
i(P'x-q_Y)
+ R
e
,
y -- -
,
Hence t+(x,y,p,X) behaves for
like an incident plane wave with propagation vector ki = (p,-q+)
plus a specularly reflected wave with propagation vector kr = (p,q+), while for y -; - it behaves like a pure transmitted plane wave with propagation vector kt = (p,-q_).
The incident and transmitted plane waves can be shown
to satisfy Snell's law n(m) sin 8(°) = n(-°) sin 0(-) where e(-) and 6(--') are the angles between the y-axis and ki and kt, respectively, and n(±-) = c-'(± ).
(x,y,p,X) has a similar interpretation.
26
SPECTRAL ANALYSIS OF SOUND PROPAGATION
3.
For (p,A) E S20 one has
y ' +-,
+ Ra e
e co(IPI,A) (1.29)
Vio(x,Y,P,A) "
4'Y
2T r
e
To
Hence for y ->
,
o(x,y,p,A) behaves like an incident plane wave plus a
specularly reflected wave while for y -. -
it is exponentially damped.
This is analogous to the phenomenon of total reflection of a plane wave in a homogeneous medium of refractive index n(-) = c-1(oo) at an interface with
a medium of index n(-) = c-1(-0) < n(-). A < c2(- )
Indeed, the condition
p12 is equivalent to the condition for total reflection:
n(co) sine(-) >n(-). For p E 0 k' k > 1, one has e-q+Y
ck(IPI) 2?r
(1.30)
V1k(x,y,P) -
i
ck(IPI) Il
2r
ei px e q'y ,
Y
Hence the functions lpk(x,y,p) can be interpreted as guided waves that are trapped by total reflection in the acoustic duct where c(y) < c(±-).
They
propagate in the direction it = (p,0) parallel to the duct and decrease exponentially with distance from it.
The coefficients R+, Rp and T+, To in (1.27), (1.28), (1.29) may be interpreted as reflection and transmission coefficients, respectively, for the scattering of plane waves by the stratified fluid.
They will be shown
to satisfy the conservation laws q+
(1.31)
P(± )
q+
IT+I2 =
IR+IZ + P(+W)
The three families J+, _ and i4i
-
q+
,
IR0I = 1.
P(+°°)
represent, collectively, the response (p,q) E R3.
of the stratified fluid to incident plane waves exp To see this consider the mappings
(1.32)
(p,q) = x+(P,),) = (P,q+(IPI,A)),
(p,X) E Q,
(p,q) = X0(P,A) = (P,q+(IPI,A)),
(p,X) E 00,
(p,q) = X_(P,A) = (p,-q_(IPI,A)),
(P,A) C 0.
3.1.
Summary
27
X+ is an analytic transformation of Q onto the cone
(1.33)
C+ = {(p,q)
q > a IpI}
I
where
a = ((c(-°°)/c(°°))2 - 1)172 > 0.
(1.34)
Similarly, X0 is an analytic transformation of 0o onto the cone
(1.35)
Co = {(p,q)
I
0 < q < a IpI}
and X_ is an analytic transformation of 0 onto the cone
(1.36)
C_ = {(p,q)
I
q < 0}.
Thus, the asymptotic forms of p+ and io for y + ±' show that 4+(x,y,p,X) with (p,A) E S2 is the response of the fluid to a plane wave exp
qy)}
with (p,q) E C+, ip(x,y,p,a) is the response to a plane wave with (p,q) E CO and i)_(x,y,p,A) is the response to a plane wave with (p,q) E C_.
Note that
R3 = C+ U Co U C_ U N
(1.37)
where N = DC+ U ac-, aC+ = {(p,q)
:
q = a IpI} and aC_ = {(p,q)
:
q = 01.
Physically, vectors (p,-q) with (p,q) E DC+ separate the plane waves incident from y = 4
that have a transmitted plane wave at y = -- from those
that are totally reflected.
Vectors (p,0) E DC_ correspond to plane waves
that are incident parallel to the stratification.
The momenta (p,q) E N
form a Lebesgue null set in the momentum space R3 and are not needed in the theory developed here.
The interpretation of
+, y_ and ip given above suggests the intro-
duction of a composite normal mode function
(1.38)
where
+(x,y,p,q) _ (2'rr)-1
0+(y,p,q), (p,q) E R3,
28
SPECTRAL ANALYSIS OF SOUND PROPAGATION
3.
(2q)1/2 c(_) $+(y,IPl,x), (P,A) = X+1(P,q), (p,q) E C+.,
o(y,IPI,X), (p,A) = X01(P,q), (p,q) E Co,
(2q) 112 c(°°) (1.39)
$+(y,P,q) _
c(-m) $_ (y,IPl,A), (P,X) = X1(P,q), (p,q) E C_,
(2IgI)112
0
,
The normalizing factors (2q) of the Jacobians of X+1,
X01
112
(P,q) E N.
c(am) and (2q)h/'2 c(-°°) are the square roots
and X-
The function $+(x,y,p,q) is a
.
solution of the differential equation
A
(1.40)
A(P,q) $+(,,P,q)
where
c2(°O)(IPI2 + q2),
(1.41)
A(p,q) =
(p,q) E C+ U Co,
c2(-°°)(IPI2 + q2), (p,q) E C_, 0
,
(p,q) E N.
Its asymptotic behavior is described by
+ R+ e
,
(p,q) E C+1
el(P'x-qy) + Ro ei(p-x+qy),
(p,q) E Co,
e
(1.42)
(x,y,p,q) - c(p,q)
ei(P'x+q+(Ipl,A)y)
T
,
(p,q) E C_,
for 5i(p'x_q-(IPI,X)y)
T
(1.43)
$+(x,y,p,q) ` c(p,q)
a
T0
(p,q) E C+1
eip'x eq'(Ipl,X)y
ei(P'x-qy) + R
el(p-x+qy),
(p,q) E Co,
(p,q) E C
Summary
3.1.
for y - --.
29
In §9 it is shown that one may take
(2n)-3/2
c(°°) p1I2 (°D) , (p,q) E C+ U Cp
,
c(p,q) =
(1.44)
(2n)-312 c(-°°) P1I2
,
(p,q) E C_.
Another family of normal mode functions for A is defined by
(1.45)
4_(x,y,p,q) = +(x,y,-P,q),
(P,q) E R3.
It is clear that A 0_ = A(p,q) ¢_ and $_(y,P,q)
(2n)-i
(1.46)
_(x,y,P,q) =
where
(1.47)
4_(y,p,q) = +(y,p,q)
The asymptotic behavior of _ for y (1.44).
may be derived from (1.42), (1.43),
It is given by
R{
(1.48)
c_(x,y,P,q) ` c(p,q)
ei(P'x+gy) + Ra
el(P'x-q+(IPI,X)y)
T
T
_(x,y,P,q) - c(p,q)
Ta
ei(P'x+q-(IPI,X)y),
eq!(IPI,A)y
ei(P'x+qy) + R- ei(p'x-qy),
for y -
(p,q) E C+1
(p,q) E Co,
(p,q) E C_,
and
for y
(1.49)
ei(p-x-qy),
-°°.
(p,q) E C
(p,q) E CO,
(p,q) E C_,
These relations clearly imply that _(x,y,p,q) is not simply a
multiple of 0+(x,y,p,q). symmetry property
By contrast the guided mode functions have the
30
SPECTRAL ANALYSIS OF SOUND PROPAGATION
3.
(1.50)
k > 1,
Yx.y.-P),
Yx'y.p) =
because the functions ,yk(y,Ipl) are real-valued and depend on p only through Ipl.
is derived from that of
The completeness of the family ...} in §9.
The existence of the two families 0+ and 0_ is a
consequence of the invariance of the wave equation under time reversal. In Chapter 4 the family
is used to construct asymptotic solutions for
t -* +oe of the propagation problem.
The normal mode expansions which are the main results of this chapter
For clarity, the case in which A has no guided
will now be formulated. modes is described first.
The general case is described at the end of the
section.
The normal mode expansions for A are in essence Hilbert space expansions; that is, they converge in a mean square, or Hilbert space, sense. The space JC in which A acts was defined by (2.1.13). n
L2(Q) of square integrable functions on Q C R
The Lebesgue spaces
are also needed.
L2(Q) is
a Hilbert space with scalar product
(1.51)
u(P) v(P) dP
(u,v) = J Q
where dP = dpl...dpn is the volume element in Rn. P e R3} is a
If A has no guided modes then the family complete family of normal mode functions for A.
This statement means
that every h E JC has a unique generalized Fourier transform h+ E L2(R3) such that
(1.52)
h(X) c-2(y) P-'(y) dX,
h+(P) = L2(R3)-lim r R
(1.53)
IIhIIX = IIh+lIL2(R3)
and
(1.54)
h(X) = JC-lim JR3 +(X,P) h+(P) dP.
Equation (1.53) generalizes the Parseval relation of Fourier analysis.
Equation (1.52) is a condensed notation for the assertion that if {Km} is a
nested sequence of compact sets in R3 such that UKm = R3, and if
3.1.
Summary
31
h(X), X E Km,
hm(X) _
(1.55)
X E R3 - Km,
0 ,
then the integrals
(1.56)
1R3
(X.P) h(X) c-2(y)P-1(y)dX
+.(X.P) hm(X) c-2(y) P-'(Y)dX = 1IK
m are finite for every P E R3 and define functions hm+ E L2(R3) such that the sequence (hm+} is convergent in L2(R3).
h+ is by definition the limit of
this sequence.
Equation (1.54) has the analogous interpretation.
Thus it represents
an arbitrary h E JC as a superposition of normal mode functions.
useful because it provides a spectral representation of A.
(1.54) is
This means that
if h is in the domain of A, so that Ah E JC, then X(P) h+(P) E L2(R3) and
(1.57)
(Ah)+(P) = A(P) h+(P)
where X(P) is defined by (1.41).
This property is used in Chapter 4 to
solve the wave equation for A.
In what follows the K-lim notation is sometimes suppressed for the sake of simplicity.
However, integrals such as (1.52) and (1.54) are always to
be interpreted as Hilbert space limits.
Of course, for special choices of
h it may happen that the integrals also converge pointwise.
However, even
in these cases it is the mean square convergence that is the more relevant to problems of wave propagation.
To describe the general case where A has guided modes let No - 1 denote
the number of distinct guided normal mode functions k for A where 2 < No < +m.
Then (1.52) still holds for every h E JC and in addition for
every k such that 1 < k < N. there exists a unique function hk E L2(Qk) such that
(1.58)
k(X,p) h(X) c 2(Y) P-1(Y)dX.
hk(p) = L2(Slk)-lim 1R3
Moreover, the Parseval relation (1.53) is replaced by No-1 (1.59)
112
= IIh+IIL2(Ra) +
k=l
IIhk1IL2(Qk)
and the normal mode expansions asserts that the limits
32
SPECTRAL ANALYSIS OF SOUND PROPAGATION
3.
(1.60)
3 yX,P)
hf(X) = JC-lim 1
h+(P)dP
R
Uk(X.P) hk(p)dp, 1 < k < No,
hk(X) = C-lim J
Qk
h=hf+
No-1 hk
k=1
The representation
where if No = i' then the last sum is convergent in X.
(1.52), (1.58)-(1.62) is a spectral representation for A in the sense that if h is in the domain of A then (1.57) holds and in addition
(Ah)k(P) = Ak(IPI) hk(p) for 1 < k < No
(1.63)
where
Ak(IPI)
YX,p)
form a second complete set of normal mode
The functions functions for A.
The normal mode expansion for this family is defined by
(1.52), (1.58)-(1.62) with + and h+ replaced by - and h_.
} form still another complete set.
The functions
The normal mode
expansion for this family is derived in §9.
§2.
THE REDUCED PROPAGATOR Au The Sturm-Liouville operator Au defined by (1.5) has a selfadjoint
realization, also to be denoted by A, in the Hilbert space
JC(R) = L2(R,c z(y) P-'(y)dy)
(2.1)
The domain of A is the set u (
(2.2)
Y IP
D(AU) = LZ(R) n li
l
l
1(y) dy) E L2(R)}.
l
))1
The properties
(2.3)
All = AA > cmuz,
where A* is the adjoint of A
in JC(R), can be verified by showing that AP
P
is the operator in JC(R) associated with the sesquilinear form Au in JC(R)
3.3.
Solutions of the Equation APB = ;0
33
D(AP) = L2' (R) C C(R)
P-1(y)dy
ja dy + R l
The spectral analysis of A will be derived below from that of A main steps of the analysis are the following.
The
First, conditions (2.1.4)
and (1.1) are used to construct the special solutions of A 0 = a0 defined P
by conditions (1.12).
Second, these solutions are used to construct an
eigenfunction expansion for A.
The construction is based on the Weyl-
Kodaira-Titchmarsh theory of singular Sturm-Liouville operators.
Finally,
the expansion for A1pI and Fourier analysis in the variables x1, x2 are used to construct a spectral representation for A.
This method has been
applied to the special cases of the Pekeris and Epstein profiles [8,26]
where explicit representations of the solutions of APB = a$ by means of elementary functions are available.
Thus the main technical advance in
the work presented in this chapter is the construction, for the class of density and sound speed profiles defined by (2.1.4) and (1.1), of solutions of APB = aO that have prescribed asymptotic behavior for y
'_W and
sufficient regularity in the parameters A and p to permit application of the methods of [8,26].
§3.
SOLUTIONS OF THE EQUATION APB _ 0 The special solutions 0i(y,P,A) (j = 1,2,3,4) described in §1 are
constructed in this section.
Analytic continuations of these functions to
complex values of A are used in §6 for the calculation of the spectral family of Au.
Hence the more general case of solutions of APB
with
E C will be treated.
The equation APB = 0 cannot have solutions in the classical sense unless c(y) and p(y) are continuous and continuously differentiable, respectively.
A suitable class of solutions is described by the following
definition in which AC(I) denotes the set of all functions that are absolutely continuous with respect to Lebesgue measure in the interval I = (a,b) C R. Definition. (3.1)
A function
:
I = (a,b) i C is said to be a solution of '(Y)),
AP0(y) a -c2(y){P(Y)(P-1(y)
-
CO(Y)
34
3.
SPECTRAL ANALYSIS OF SOUND PROPAGATION
#/dy) if and only if
in the interval I (where
'
E AC(I)
E AC(I), P-'
(3.2)
and (3.1) holds for almost all y E I.
The following notation will be used in the definition and construction of the special solutions 0j(y,u,?).
For each K > 0
L(K) _ {c
I
< K2},
Re
(3.3)
R(K) _ { I Re C > K2}, R±(K) = R(K) n {C
,
±Im C > 0}.
The definitions (1.10), (1.11) will be extended as follows.
C-2(±w) - 1u2)1/2 -Tr/4 < arg
(3.4)
Tr/4
r
E R(c(±°°)u)
r
E L(c(1o')i )
and
(p2 -
C-2(±_))112 1
-Tr/4 < arg q+(pl,C) < TT/4
(3.5)
q+(u,4) _ -i q+(u,4)
The results of this section will now be formulated. Theorem 3.1.
Under hypotheses (2.1.4), (1.1) on p(y), c(y) there
exist functions
j:
(3.6) (where R+ _
1
R x R+ x (L(c(`°)p) U R(c(°°)u)) - C, j = 1,2,
u > 0}) such that for every fixed (u,r) E R+ x
x (L(c("')u) U R(c(-)u)), 0i(y,u,4) is a solution of (3.1) for y E R and j = 1,2 and
Solutions of the Equation AO = 4
3.3.
35
= exp {q+(p,4)Y}[1 + o(l)] (3.7) = p-1(°')
P-1(Y)
exp {q+(p,C)y}[l + o(l)]
and
2(Y,U10 = exp {-q+(p,C)Y}[l + o(l)] (3.8)
y -P-1(")
P 1(Y)
}
exp {-q+(u,C)y}[1 + o(l)]
Similarly, there exist functions
(3.9)
¢j
:
R x R+ X (L(c(--)p) U R(c(--)p)) '' C, j = 3,4,
such that for every fixed (p,C) E R+ X (L(c(--)p) U R(c(-0O)p),
(y,u,l) is
a solution of (3.1) for y E R and j = 3,4 and
03(y,p,C) = exp {q'(p,C)Y}[1 + o(l)]
y--,
(3.10)
o(1)]
P-1(y) m3(Y,p,4) = P-1(-°°) q'(p,c) exp
and
exp {-q'(11,4)Y}[l + o(l)]
y -y
(3.11)
exp {-q'(p,c)Y}[l + o(1)]
4
j
The following three corollaries describe the dependence of the soluon the parameters p and ?.
tions
Corollary 3.2.
j, P-E C[R
(3.12)
for j = 1,
satisfy
The functions
X
U {(1p,4)
E L(c(°°)p)
p>0
2 and
(3.13)
$j,
E CIR X U {(p,4) ll
for j = 3, 4.
Moreover
p>0
I
E L(c(-0°)p)}/
36
3.
SPECTRAL ANALYSIS OF SOUND PROPAGATION
P-lei
1,
E CIR x U {(u,4) p>0
4 E
(lll
lull
(3.14)
2, PE CIR X U {(u,4) l
111111
03, pE CIR X U {(u,4) I
04, PE
C{R x
4 E R (c(m)u)}I,
u>0
4 E R (c(--)u)}
u>0
Jll
U {(u,4)
4 E R+(c(-m)u)}I
u>0
Corollary 3.3.
For each fixed (y,p) E R x R+ the mappings
4 i j(Y,u,4), 4 i P-1(Y) V.(y,u,4)
(3.15)
are analytic for
j= 1, 4 E L(c(°)u) U R+(c(m)u)int (3.16)
j = 2, 4 E L(c(°)u) U R
(c(w)u)int
(c(--)p)int
j= 3, 4 E L(c(--)u) U R
j= 4, 4 E L(c(--)u) U R+(c(-)o) where R±(K)int = R(K) n {4
I
int
±Im 4 > 0). p-'
Corollary 3.4.
The asymptotic estimates for
and
of Theorem
3.1 hold uniformly for (u,4) in any compact set F. such that for
j= 1, Fl C U {(11,4)
4 E L(c(-)u) U R+(c(W)u)),
u>o (3.17)
4 E L(c(m)u) U R(c(-)u)},
j= 2, F2 C U {(u,4) u>0
j= 3, F3 C U {(u,4)
1
4 E L(c(--)u) U R (c
u>0
j= 4, r4 c U {(u,4) 14 E L(c(--)u) U R+(c(-o)u)}. U>0
The special solutions 0i(y,u,4) are not, in general, uniquely determined by the asymptotic conditions (3.7), (3.8), (3.10), (3.11).
Indeed,
if Re q+(u,X) > 0 (resp., Re q+(u,X) < 0) it is clear that any multiple of
3.3.
Solutions of the Equation A 0 = 0
37
U
02 (resp., 01) can be added to 1 (resp., 2). 03 and 04.
However, for each
A similar remark holds for
c C a sub-dominant solution (one with
minimal growth at y = - or y = -W) is unique.
Re q,' (11,C) > 0 for
In particular, since
E L(c(±-)u)
(3.18)
0 for c E R (c(±-)u)
Re
Re q+(}1,0) < 0 for c E R+(c(±w)u)
one can prove Corollary 3.5.
The solution 02 is uniquely determined by (3.8) for
E L(c(W)p) U R (c(oo)p).
all
Similarly, 03 is uniquely determined by
(3.10) for 4 E L(c(-W)p) U R (c(-oo)p), 01 is uniquely determined by (3.7)
in R+(c(W)p) and ,, is uniquely determined by (3.11) in R+(c(-)p). When Re
= c2(± )}12 Theorem 3.1 provides no information about the
asymptotic behavior for y -+ ±" of solutions of Auq = Co.
However, positive
results can be obtained by strengthening hypothesis (1.1).
The following
extension of a known result [14, p. 209] will be used in §4. Theorem 3.6.
(3.19)
Assume that p(y) and c(y) satisfy hypothesis (2.1.4) and
f u IP(y) - p(°°)
Then there exist functions (3.20)
dy <
to y2
c(Y) - c(W)
dy < cc.
1
R x R+ - R, j = 1,2,
such that for every p E R+ the pair 01(y,p), 2(Y,u) is a solution basis for
A40 = c2(°°)u24, 01(Y,U) = 1 + o(1) (3.21)
P-1 (Y) j (Y,p) = 0(1) J and N
(Y,u) = P(00)Y[l + 0(1)] (3.22) P-1(Y) q;(Y,p) = 1 + o(l) J
38
3.
SPECTRAL ANALYSIS OF SOUND PROPAGATION
Lagrange's formula for Au may be written c_2(y)P-1(Y)dy
r Y
(3.23)
= ["](y2) - [W(Y1)
{,PAUL -
2 J
yl
where
101p] (y) = 0(y) (P-1(Y) W' (Y)) - (y) (P-1(Y) '(y)).
(3.24)
In particular, if 0 and
are solutions of Au4
Auto = 4$ on an
interval I then [4fl(y) = const. on I and [cty](y) = 0 on I if and only if 0 and 1 are linearly dependent there.
By combining these facts and
Theorem 3.1 one can show that
W2(',11,01 = -2 p-1(°°) q+(P,0, (3.25)
4(.,p,0] = -2 which imply The pair
Corollary 3.7. Auk
is a solution basis for
for all (U,c) E R+ x (L(c(-)}1) U R(c(m)U)).
Similarly, the pair
g4(y,u,C) is a solution basis for all (p,C) E R+ x (L(c(-')U) U R(c(-')p)).
This completes the formulation of the results of §3 and the proofs will now be given.
The method of proof involves replacing Au0 = 0 by an
equivalent first order system.
The latter can be regarded as a perturba-
tion of the corresponding limit systems for y -> ±m.
In this way integral
equations are established for solutions with prescribed asymptotic behavior for y - - or y -> -- and these equations are solved by classical Banach space methods.
This technique for constructing solutions with
prescribed asymptotic behavior is well known - see for example [5, p. 1408] and [16, Ch. VII].
A first order system equivalent to A _ of (3.1) on an interval I and if
(3.26)
i
1 (Y) _ (y)
then 1, IP2 G AC(I) (cf. (3.2)) and
If m(y) is any solution
Solutions of the Equation A 3 = m
3.3.
39
u
(Y) = P (y) Wz (Y) (3.27)
Z (y) = P for almost every y E I.
c
1 (Y) [112
z
(Y) l 1 (Y)
Thus the column vector (y) with components 1(y),
>V2(y) is a solution of the first order linear system
(3.28)
V(Y) = M(Y,P,0 BU (y)
where
(3.29)
0
P(Y)
P-1(Y)[.Z - C c-2(Y)1
0
M(y,11,0 =
Conversely, if 1P E AC(I) is a solution of (3.28), (3.29) and if (y) = 01(y) then 0 is a solution of (3.1).
The solutions of Theorem 3.1 will be
constructed by integrating (3.28), (3.29). The limit system for y - +°° and its solutions.
By replacing p(y),
c(y) in (3.28), (3.29) by P(-), c(-) one obtains the system
V(Y) = M0(u,O (Y)
(3.30)
where
0
(3.31)
P (°o)
M0(p,0 =
P-
1(_)[p2 - C
c-Z
(°°) l
0
Ma(p1,C) has distinct eigenvalues q+(p,C), -q+(p,C) for C E L(c(°°)p)
U R(c(m)u)
(3.32)
The columns of
B(L,C) _ p-1(°°) I
are corresponding eigenvectors.
(3.33)
q+(U,C)
Hence
M0(u,C) B(p,C) = B(p,C)
-p 1(°°) q+(v,C)
40
3.
SPECTRAL ANALYSIS OF SOUND PROPAGATION
where
0
(3.34)
D(u,0) = -q+(u,0)
0
System (3.30), (3.31) may be integrated by the substitution
= B(11,0)z
(3.35)
It follows that z'(y) = D(u,0) z(y), whence
z1(y) = c1 exp (3.36)
Z2(y) = c2 exp {-q+(u,oy}
and therefore
$1(y) = c1 exp {q+(u,0)y} + c2 exp {-q+(u,0)y} (3.37)
V2(Y) = P 1(OD) q+(u,2)(cl exp {q+(u,0)y) - c2 exp
where c1, c2 are constants of integration. Application of perturbation theory.
System (3.28) may be regarded as Thus if N(y,u,C) is defined by
a perturbation of the limit system (3.30).
(3.38)
M(y,u,0) = M0(11,0) + N(y,u,C)
then
0
(3.39)
al(y)
N(y,11,0) =
u2 a2(y) + 0 a3(y)
0
where
al(y) = P(y) - P(OD) (3.40)
a2 (y) = P-1 (y) a3 (y) _
_[P-1(y) C-2(y) - P-1 (O°) C-2(_)].
3.3.
Solutions of the Equation App _ 0
41
Note that each of these functions is in L1(yo,00) for every yo E R.
a1(y) this is part of hypothesis (1.1). from (2.1.4) and (1.1).
(3.41)
For
For a2(y) and a3(y) it follows
For example, one can write
a2 (y) _ (-P-1 (°°) P-1 (y)) (P(y) - p(°D))
which exhibits a2 as a product of a bounded measurable function and a function in L1(yo,').
On combining (3.28), (3.38) and making the substitution (3.35), one finds that (3.28) is equivalent to the system
z'(y) = D(p.c) z(y) +
(3.42)
z(y)
where
B-1(p,4) N(y,P,4) B(p,Q
(3.43)
has components that are in L1(yo,O°).
Solutions of (3.42) will be
constructed which are asymptotically equal, for y -{
-t
,
to the solutions
(3.36) of z' = Proof of Theorem 3.1.
The proof will be given for the function 01 only.
The remaining cases can be proved by the same method.
Solutions of (3.42)
are related to the corresponding solutions of (3.1) by
(3.44)
P-1 (°°) q+(z1
- z2), q+ = q+(p,c)
Thus d will be a solution of (3.1) that satisfies (3.7) if z is a solution of (3.42) that satisfies
(3.45)
z1 = exp {q+y}n1, z2 = exp {q+y}n2
and
(3.46)
nl(y) = 1 + o(1), n2(y) = o(l) for y--.
Equations (3.42) and (3.45) imply
42
SPECTRAL ANALYSIS OF SOUND PROPAGATION
3.
nl =
E11 n1 + E12 n2
(3.47) fZ = -2 q+ n2 + E21 n1 + E22 n2,
and hence by integration (
Y
T11(y) = c1 +
E1j(Y') nj(y')dy' J
(3.48)
Y° rY
n2(Y) = exp {-2 q+y} c2 +
exp {-2 q+(y - Y')} E2j(Y') nj(Y')dy' J
YI
where c1, c2, y0, y1 are constants and the summation convention has been
used (j is summed over j = 1,2). Construction of 01 for r c- L(c(W)u)
E L(c(o)p).
By (3.18), Re
0 for all
Thus to construct a solution of (3.47) that satisfies (3.46)
it is natural to choose c1 = 1, c2 = 0, y0 = 4w and y1 finite in (3.48). This gives the system of integral equations
n,(y) = 1 - Jm Elj(Y') nj(Y')dY' Y
(3.49)
y > y1. rY
n2(y) =
I
exp {-2 q+(Y - y')} EZ.(Y') nj(Y')dY'
JJJyl
It is natural to study system (3.49) in the space
X = CB([y1,m),C2)
(3.50)
of two-component vector functions of y whose components are continuous and bounded on y1 < y < -.
X is a Banach space with norm
sup
(3.51)
( H1 (Y) I + I f2 (Y) )
Y?y1 The system has the form
(3.52)
T1 (y) = n° + F K(Y,Y') n(Y')dy', y > y1, YI
where f(y) and n° are column vectors with components (nl(Y),n2(Y)) and (1,0), respectively, and the matrix kernel K(y,y') is defined by
_ 0
Solutions of the Equation A
3.3.
43
1
0,
(3.53)
Y' < Y,
K1j(Y,Y') =
exp {-2 q+(Y - y')} EZ.(Y'), 1 < Y' < Y, (3.54)
K2j(y,Y') = y1 < y < y',
0,
and j = 1,2.
The conditions Ejk E L1(y1,w) and Re q+ > 0 imply that the
operator K defined by (3.52), (3.53) and (3.54) maps X into itself.
To
show that K is a bounded operator in X and estimate its norm note that
(3.55)
(Kn) j (Y) I
<
IITill
( I Ej
Jw
I (y) I + I Ej 2 (Y) I) dy
YI
for j = 1,2 and all y > y1.
It follows that 2
(3.56)
IIKII
<
J
IEjk(Y)I dy.
y1 j,k=l In particular, since Ejk E L1(yo,°°) for every y0E R, (3.56) implies that IIKII
< 1 for every sufficiently large y1.
For such a value of y1 the
equation
n = no + Kn
(3.57)
has a unique solution n E X given by
(3.58)
Km n°
n = m=0
Moreover, (3.57), or equivalently (3.49), implies that n1(y), n2(y) satisfy (3.47) for y > y1.
These functions then have unique continuations to
solutions of (3.47) for all y E R, by the classical existence and uniqueness theory for linear systems.
Of course, n1 and n2 are functions of p and q+ and the Ejk depend on these variables.
as well as y because
The solution I of Theorem 3.1
will be defined by
(3.59)
41(Y,U,4) = exp {q+(p,S)Y}
To complete the proof that 1 is the desired function on R x R+ x L(c(oo)p)
44
SPECTRAL ANALYSIS OF SOUND PROPAGATION
3.
it is only necessary to verify that (3.46) holds for each (p,4) E R+
x L(c(')u)
It is clear from (3.49) that
12 (3.60)
n1(y) - II <
IIn1I
W
dy' = o(l), y 1 4o.
X
Y1 j=l
For n2, (3.49) implies that z
r
I n 2 ( Y )
I
<
J
(3.61)
exp {-2q+(y-y')}
Y
I l n l l
Y1
for every y2 > y1 and every y > y2.
2
C
L IE5.(Y')IdY' + J E IEzi (Y,)Idy, j=l Y2 ]=l
Hence for any fixed y2 one has 2
(
(3.62)
lim sup Inz(Y)I <
llnll 1
Y
because Re q+ > 0.
-
J=l
IE2.(Y')I dy'
y2
Since Y2 in (3.62) is arbitrary it follows that
n2(Y) = 0(1)
Construction of 1 for
E R(c(oo)p1).
By (3.3) R(c(°o)U) has the
decomposition R(c(°°)u) = R+(c(°°)u) U R
(3.63)
(c(°°)u)int.
Moreover, for C E R (c(-)U)int one has Re q+(p,C) > 0 and hence the construction of the preceding case is valid.
In the complementary case
where C E R+(c(oo)p) one has Re q+(U,C) < 0 and it is permissible to take c1 = 1, c2 = 0, yo = y1 = oo in (3.48).
The resulting system of integral
equations
n1(Y) = 1 -
T
ni (Y')dy'
y (3.64)
exp {-2q+(Y-y')} E2(Y') ni (Y')dY'
n2(Y) Y
again defines an equation (3.57) in the Banach space X.
Moreover,
lexp {-2q+(y-y'))I< 1 for y < y' < °° and (3.56) is again valid.
It follows
that for y1 large enough (3.64) has a unique solution given by (3.58).
The
solution has a unique continuation to a solution of (3.47) on the interval y E R.
The validity of the asymptotic condition (3.46) is obvious from
(3.64); cf. (3.60). Proof of Corollary 3.2. Note that by (3.39)
Again the proof will be given for 1 only.
Solutions of the Equation App _
3.3.
(3.65)
45
N(y,p,t) = N1(y) + p2 N2(y) +
N3(y)
where the components of N.(y) are in L1(yo,m) for j = 1,2,3 and every yo E R.
Thus by (3.43)
B-1(p,C) N1(y) B(p,C) + p2 B 1(p,C) N2(y) B(p,c) (3.66)
+ C B 1(p,C) N3(y) B(p,C)
Proof of (3.12) for 41.
Note that
and hence also B(p,C) and
B-1(p,c) are continuous functions on the set
(3.67) in X one can
Thus by using the estimate (3.56) for the operator K =
show that for each compact subset r of the set (3.67) and each 6 < 1 there is a constant y1 = yl(r,6) such that, taking y1 = yl(r,6) in the definition of K(p,4), one has
(3.68)
IIK(p,4)11 < 6 for all (p,1) E r.
Hence the series (3.58) converges uniformly in X for (p,0) E r which implies the continuity of 01 and P-101 on the set [y1,w) x r.
Their continuity on
R x r then follows from the classical theorem on the continuous dependence of solutions of initial value problems on parameters.
This implies the
result (3.12) for 01 because r was an arbitrary compact subset of the set (3.67).
Proof of (3.14) for 01.
The method used in the preceding case is
applicable to the operator K(p,C) in X defined by (3.64). Remark on Corollary 3.2.
The argument given above can also be used
to show that
(3.69)
P-lc E CIR x U
E R
I
(c(o)p)int}
p>0
However, the continuity of ¢1 and
(3.70)
R x U
p-1
1
on the set
I
C E R(c(m)p)}
p>0
cannot be asserted since the constructions for
E R+(c(=)p) and
46
3.
C E R (c(')u)int are different.
SPECTRAL ANALYSIS OF SOUND PROPAGATION
Indeed, continuity of 1 on the set (3.70)
is not to be expected since 1 is not uniquely determined when E R (c(-)u)int Proof of Corollary 3.3.
The components of the matrix-valued function R+(c(m)u)int
E(y,u,4) are analytic functions of C E L(c(°°)p) U
for fixed
Hence the uniform convergence of the Neumann series (3.58)
values of y, u.
on compact subsets of this set, which follows from the proof of Corollary 3.2, implies the validity of Corollary 3.3 for 01.
The remaining cases can
be proved by the same method. Proof of Corollary 3.4.
The proof will be given for the case of 01
in a compact subset r of the set (3.67).
and
be proved similarly.
The remaining cases can
01(y,p,4) was defined by (3.59) and the functions
nj(y,u,C) satisfy 1y
Elj(Y,u,O nj(y',u,C)dY',
1 = (3.71) -Joo
n2(Y,u,4) _
exp {-2q+(u,C)(y-y')} E2.(Y',u,4) Ti .(y',u,4)dy' y
It must be shown that these integrals tend to zero when y -> -, uniformly for (u,4) E r.
Now (3.58) and the estimate (3.68) from the proof of Corollary
3.2 imply that IIn(',u,C)II < (1 - d)-1 for all (u,4) E r. for fixed y'
It follows that
> y > y1(r,d) one has
IEkj(Y',u,O nj(Y',p,oI < IIE(Y',p,4)II IIn(Y',p,0II (3.72)
< (1 - 6)-1 IIE(y',u,C)II for (u,c) E r.
Now the continuity of B(u,C) implies that there is a
y = Y(r) such that
(3.73)
E r.
IB(u,C)II IIB-'(u,d)II (1 + u2 + ICI) < Y for
It follows from (3.66) that 3
(3.74)
E r.
I IIN.(Y')II, j=l
Combining (3.71), (3.72) and (3.74) gives 3
(3.75)
In1(y,u,c) - 11 < y(l - d)-' J
E
y j=l
IINj(y')II dy'
Solutions of the Equation AU4 = 0
3.3.
for all y > yl(r,d) and (U,0) E r.
47
Since each N. E L2(yo,°°), (3.75)
implies that ril(y,U,C) - 1 = o(l) uniformly for (11,0) E r.
The case of n2(y,U,r) is more complicated.
Note that 3
(3.76)
exp {-2 Re q+(P,0)(y-y')}
Ir12(y,V,O)I < Y(1-d)-l y
C
IIN.(y')Il dy'. j=1 1
Now the continuity of q+(U,0) and the definition of L(c(°°)U) imply that there is a K = K(r) > 0 such that
(3.77)
2 Re q+(U,C) > K > 0 for all (U,0) E F.
Combining (3.76), (3.77) one has, if Yl =
I n2 ( y , u , 0 )
I
< Yl
((Y2 exp {-K(y-y')}
N fw L y2 j=1 J
L
y
j=1
J
(3.78) 3
< yl 11
3
v exp {-K(y-y2) Jw IINj(Y')Ildy' + Yl j=1
IINi (y')Ildy'
Y2 j-1
for all Y2 > yl(F,d), y > y2 and (U,0) E
Now let E > 0 be given and choose Y2 = Y2(E,r,d) > yl(r,d) such that
Yl F
E
IIN
y2 j=1
.(y')Ildy' < E/2, J
y1
3
In2(y,U,O)I < Yl exp {-K(Y-Y2)} 1W
IIN
I
j=1
for all y > y2(c,r,6) > yl(r,d) and (U,0) E r.
.(y')Ildy' + 6/2 J
Finally, choose a
y3(c,r,d) > '2(E'r'S) such that °°
(3.81)
yl exp {-K(y-y2)} j
3 G
IIN1(Y')IldY' < c/2
yj j=1 for all y > y3(e,r,6).
It follows from (3.80) and (3.81) that
In2(y,U,C)I < e for ally > y3(c,r,6) and (p,C) E F; i.e., n2(y,1J,C) = o(l) uniformly for (U,0) E r. Proof of Corollary 3.5.
It will be shown that l(Y,U,C) is uniquely
determined by (3.1) and (3.7) when 0 E R+(c(=)p). proved similarly.
The other cases are
'1
48
3.
SPECTRAL ANALYSIS OF SOUND PROPAGATION
Assume that for some t E R+(c(-)u) there are two solutions of (3.1), (3.7).
Then their difference m(y) would satisfy (3.1) and (y) = o(1),
p-1(y) 4'(y) = o(l) because Re q+ < 0 for C E R+(c(W)u).
It follows that
the corresponding pair n1(y), rjZ(y), defined by (3.44) and (3.45), would necessarily satisfy -Jm
nI(Y) =
E1i (Y') ni (y')dY'
Y (3.82)
-I' exp {-2q+(Y-y')}
n2(Y) _
ni (Y')dy'
Y
since Iexp {-2q+(y-y')I < 1 for y < y'. equation n = Kn in X.
But (3.82) is equivalent to the
If yI is chosen so large that 114 < 1 then Ti = Kn
has the unique solution n(y) = 0 for y > y1.
The unique continuation of
this solution of (3.47) is then zero for all y E R.
Thus (y) - 0 for
y E R, which proves the uniqueness. Proof of Theorem 3.6.
The equation Auk = c2(m)u20 is equivalent under
the mapping (3.26) with the system (see (3.31), (3.38))
,' = p(o°)i2 + (3.83)
1
Z = where B1(y) = a1(Y) = p(y) - p(-) and
B2(Y) = p2 p-1(Y)(1 - c2(o°)c 2(Y)) (3.84)
= u2 p-1(y) c-2(Y)[c(Y) + c(°°)][c(Y) - c(W)].
It follows from hypotheses (2.1.4) and (3.19) that
B1(y), B2(Y), y B2(Y), y2 B2(Y) E L1(y0,oo)
(3.85)
for every yo E R and every u > 0.
Construction of 1 Application of the variation of constants formula to the system (3.83) gives the integrated form
g1(Y) = cl + p(oi) c2Y + J (3.86)
y
{p(-)(Y-Y') B2(Y')ii1(y') +BI(y')
Yo
y B2(y') w1(y')dy'.
1p2(Y) = c2 + J YO
ip2(y')}dy'
3.3.
= C
Solutions of the Equation A u
Now 1 will satisfy
4
c2(w)u2o1 and the asymptotic condition (3.21) p-lei
satisfies (3.86) and
provided that 1 = ¢1, 2 =
1 (Y) = 1 + 0(1)1 2 (y) = 0(1)1 y - w,
(3.87)
in (3.86).
To construct such a solution take c1 = 1, c2 = 0 and y° This gives the system
$1 (Y) = 1 -
{ P (°°) (Y-Y') B2 (Y') P1 (Y') + B1 (Y') 2 (y')) dy' J
Y
(3.88)
z (Y) = -
y
B2 (Y') 1 (Y') dy'
Y
or
(3.89)
V(Y)
K(y,y') U(Y') dy',
+ Jw
y > Y1,
Y1
where (y) and i° have components1(y), iV2(y) and 1, 0 respectively, K(y,y') = 0 for y > y' and
K11(Y,Y') = P(°°)(Y'-Y) B2(y') (3.90) K12(Y,Y') = -B1(Y')
K21(Y,y') _ -B2(y')
K22(y,y') = 0
for y < y'.
(3.91)
As in the proof of Theorem 3.1, one has
I(Kv)i (Y)I
(IKj(Y,Y')I + IKjz(Y,Y')I) dy'
IWI Jw Y1
and hence
(K)1 (Y) I
<
III II
P(-) Jw (y+y') I B2 (Y') I dY' + Jw (B1 (Y') I dY' Y
Y
(3.92) Jw
< III
Y
and
IB1(Y,)dY,
2P(°°) Jw Y' IB2(Y')Idy' + Y
50
SPECTRAL ANALYSIS OF SOUND PROPAGATION
3.
(3.93)
I
2 (y) I< III II
l Yy
I B2 (y') I
dy'
In particular,
(3.94)
j K 1 1
< 2P(°°) fm
y '
IB,(Y')IdY'+ Jm
IB2(Y')Idy'+f
Y1
Y1
IB2(Y')Idy'.
Y1
Thus (3.85) implies that K is contractive in X for y, large enough.
Hence
(3.88) has a unique solution on [y,,') which can be continued as a solution of (3.83) to all y E R.
Moreover, (3.88), (3.92) and (3.93) imply that
I0,(y) - 11 = I(K'P)1(Y)I = o(l) and %(Y)I = I(Kq)2(Y)I = o(1).
In fact
o(y_2).
Thus (3.87) is satisfied.
(3.85) implies that Construction of 02.
02 will satisfy Au02 = c2(°')312o2 and the asymptotic
condition (3.22) provided
(Y) _ $2 (y) = P (°°) Y n, (y) (3.95)
2(Y) = P-1(y)m2(y) = n2(Y)
where
n,(y) = 1 + 0(1), n2(y) = 1 + 0(1), y + °°
(3.96)
Substituting in (3.86) with cl = 0, c2 = 1 and yo = m gives, after simplification,
n,(y)=1-1, {P(°D)(Y'-Y-1y'2)E2(y')n,(y')+P-1(°°)y-'B,(y')n2(y')}dy' (3.97)
TI, (Y) = 1- Jy p(°O)Y'B2(y')n1(Y')dy'. Y
or
(3.98)
n(y) = no + J
K(y,y') n(y') dy', y > y1 Y1
where q(y) and no have components n1(y), n2(y) and 1, 1 respectively, K(y,y') - 0 for y > y' and
3.4.
Spectral Properties of A11
51
K11(Y,Y') = P(-)(-Y' + Y-1Yt2)B2(Y') (3.99)
K12 (Y>Y') = -P-1 (cc)Y-1 81(y ' ) K21(Y,y') = -P(°) Y' B2(Y')
K22(y,y') = 0
for y < y'.
It follows from (3.91) that r
I(K1)1(Y)
<
InI
{P(=) f (Y'+Y-1Y,2)IB2(Y )IdY'+p 1(°°)Y 1r J
Y
y
IB1(Y') dy'j
(3.100)
(Kn)2(Y)I <
IIKII
IIn1I
P(°°>
Y'
IB2(y')I dy'.
Y' IB2(Y')Idy' + P(°')Y11 1- yr2IB2(y') Idy'
< P(cc) 1 Y1
Y1
(3.102)
+ P-1(m)Y11
Y'IB2(y')Idy'
IB1(Y')IdY' + P(-) l J
v1
Y1
Hence (3.85) implies that K is contractive in X for y1 large enough and a unique solution is obtained as in the preceding case.
Finally, (3.100)
and (3.101) together with (3.85) imply that (3.96) is satisfied.
The existence of the special solutions 1, 2 has thus been proved. Their linear independence follows directly from (3.21), (3.22). Proof of Corollary 3.7.
§4.
This was verified by (3.25).
SPECTRAL PROPERTIES OF Au The results of §3 are used in this section to derive precise results
concerning the location and nature of the spectrum of A.
The notations
a(All),cyp(A1), ac(A) and Qe(AU) will be used to denote the spectrum, point
spectrum, continuous spectrum and essential spectrum of A The definitions of [11, Ch. X] will be used.
respectively.
In particular, ac(AU) is a
closed set and ae(AU) is the set of all non-isolated points of a(A).
Note
that the properties of Au described by (2.3) imply that a(AU) C [cmp2,cc).
52
3.
The Point Spectrum of A
SPECTRAL ANALYSIS OF SOUND PROPAGATION
Theorem 3.1 and its corollaries imply the
following three lemmas concerning Lemma 4.1.
For all p > 0,
ap (AU) C [c2 2, c2 (°D)u2 ] .
Lemma 4.2.
For all p > 0, a(A.) fl [cmu2,c2(°°)u2) C ao(AU).
(4.2)
Moreover, OD(AU) is either a finite set (possibly empty) or a countable set with unique limit point c2(°O)p2. Lemma 4.3.
The eigenvalues of A
that lie in the interval u
[cmu 2,C2 (-)j?) are all simple.
The possibility that c2(°°)p2 E a0(AU) is not excluded by the hypotheses (2.1.4), (1.1) alone.
Criteria for c2(40)u2 q ap(AU) are given below.
It will be convenient to use a notation that permits a unified
discussion of the cases of finite and infinite point spectra G0(AU). The number of eigenvalues in [c2p2,c2(°°)u2) will be denoted by N(u) - 1.
Thus
N(p) is an extended integer-valued function of p > 0 (1 < N(u)
The
eigenvalues of AP in [cmp2,c2(oo)u2), arranged in ascending order will be denoted by Ak(u), 1 < j < N(u).
Thus
cmu2 < X J(4) < X2 (}1) < ... < C2 (°O)u2
(4.3)
The corresponding eigenfunctions are
k(y,u) = ak(u) 2(y,u,Ak(u)), k = 1,2,...
(4.4)
where ak(u) > 0 is chosen to make
1.
The Continuous and Essential Spectra of A
Lemma 4.2 implies that
ae(Ap) C [c2(o')u2,=).
Moreover, ac(AU) and Ge(AU) are closed and ac(A1
C ae(A0) [11, Ch. X].
The characterization of these sets will be completed
in §6 by showing that (c2(°°)p2,_) C ac(A
).
These facts imply
u
Theorem 4.4.
(4.5)
For all p > 0,
ac (AU) = ae(AU) = [c2(m)p2,-)
A direct proof of Theorem 4.4 can be given by using the special solutions of §3 and a criterion of Weyl; see [5, p. 1435].
3.4.
Spectral Properties of A11
53
It is known that the bottom point in the essential spectrum of a Sturm-Liouville operator A can be characterized by the oscillation properties of the solutions of A = A$ [5, p. 1469].
For the operator AP
the characterization is described by Corollary 4.5.
The equation Auk = a4 is oscillatory (every real
solution has infinitely many zeros) for every A > c2(W)u2
The equation
is non-oscillatory (every real solution has finitely many zeros) for every
< c2(O0)p2. These results for Au follow directly from Theorem 3.1. The Point Spectrum of Au
(continued).
may not be oscillatory for A = c2(°°)p2.
provide a criterion for op(A
The equation A11
= a may or
This property is shown below to
to be finite.
The basic tool in
establishing such criteria is the classical oscillation theorem of Sturm. A version suitable for application to Au may be formulated as follows.
Let I = (a,b) be an arbitrary interval (- < a < b <) and consider a pair of equations
(4.6)
L.
_ (P_1(yW)' + Qj(y) = 0, j = 1,2,
where P.(y) and Q.(y) are defined and real valued for almost every y E I,
Pj(y) > 0 for almost every y E I and Pj, Qj are Lebesgue integrable on compact subsets of I (j = 1,2).
A solution of (4.6) on I is a function
E AC(I) such that P-1D' E AC(I) and (4.6) holds for almost all y E I. J
Such solutions are uniquely determined by the values c(yo) = cp, P-1(yo)¢'(yo) = cl at any point yo E I.
Pairs of equations (4.6) such that
P1(y) < P2(y), Q1(y) < Q2(y) for almost all y E I
(4.7)
When (4.7) holds the operator I. is said to be a Sturm
will be considered.
majorant of operator L1, and the operator L1 is said to be a Sturm minorant of operator L2, on I. Theorem 4.6.
Sturm's theorem may now be formulated as follows.
Let j(y) # 0 be solutions of L.
.
= 0 on I (j = 1,2)
and assume that y1 and y2 are successive zeros of q1(y) in I, with y1 < y2. Moreover, let L2 be a Sturm majorant of L1 on (y1,y2). least one zero in [y1,y2).
Then 02(y) has at
In addition, if either Q1(y) < Q2(y) or
P1(y) < P2(y) and Q2(y) # 0 on a subset of (yl,y2) having positive measure
then 2(y) has a zero in (y1,y2). The special solution 3(y,p,A) is real valued, tends to zero exponentially when y i -- and has finitely many zeros when A < c2(m)p2. 4.6 will be shown to imply
Theorem
54
SPECTRAL ANALYSIS OF SOUND PROPAGATION
3.
If Al < A2 < c2(m)p2 then 3(y,u,k2) has at least as
Corollary 4.7.
many zeros as 3(y,p,Y. It will be convenient following [5, p. 1473] to introduce the sets
(4.8)
1
k
= Ik(u) _ {a
I
3(y,u,A) has exactly k zeros}, k =
Note that by Corollary 4.5 each Ik C (-co,c2(W)p2].
or may not be in one of the sets Ik.
The point C2 (_)p? may
Corollary 4.7 implies that each Ik
is an interval and Ik lies to the left of Ik+1 for k =
It is
important for the analysis of O (Ap) to know that the intervals Ik # 0 for k = 1,2," ,N(u) - 1.
This is a corollary of the following fundamental
oscillation theorem. If oo(&u)
Theorem 4.8.
then for k = l,2,---,N(P)- 1 the eigen-
function Pk(y,p) has precisely k- 1 zeros. Corollary 4.9.
(4.9)
Ik = (Xk(11)>ak+1(11)], k = 0,1,...,N(u)- 2
Moreover, if N(p) < m then (a N(p)_1(p),c2(_)u2)
(where A0(p)
C
If a0(Ap ) # 0 then
IN(u)-1.
Corollary 4.10.
The number of eigenvalues that satisfy
k(p) < A < c2(°°)p2 is equal to the number of zeros of 3(y,u,A) Criteria for the Finiteness of a0(Ap).
The principal criterion for
a0(A) to be finite is described by a0(AU) is finite if and only if the equation App
Theorem 4.11.
= c2(W)p2 is non-oscillatory on R.
Hence a0(Ap) is infinite if and only
if App = c2(_)p2 is oscillatory on R. It is shown below that Theorem 4.11 is a consequence of Theorem 4.8 and Sturm's comparison theorem. Corollary 4.12.
is finite if and only if
If c(-) < c(-m) then ao(A
3(y,p,c2(-)p2) has only a finite number of zeros. Specific criteria for the finiteness of a0(Ap) will now be obtained by
deriving criteria for App = c2(_)p2 to be non-oscillatory and using It will be assumed that c(m) < c(-W) so that App = c2(oo)p2 Theorem 4.11.
is non-oscillatory in neighborhoods of y = -
.
Cases for which c(m) = c(--)
may be treated by applying non-oscillation criteria at both y = - and y = --. A criterion for ao(AA) to be finite is provided by Theorem 3.6.
For
under the conditions of the theorem App = c2(o')p2c has a solution basis 01, 02 satisfying (3.21), (3.22). non-oscillatory.
This implies
It follows that the equation is
Spectral Properties of Au
3.4.
Theorem 4.13.
55
If p(y), c(y) satisfy (2.1.4), (1.1), c(-) < c(-') and
(4.10)
JD
Y2 jc(Y) - c(oo)l dy < 00
then oo(AU) is finite for every p > 0.
Alternative criteria for the finiteness of oo(A
can be derived by
constructing Sturm majorants of A4 = c2(o)p2Q that are non-oscillatory and using Theorems 4.6 and 4.11.
Similarly, criteria for Go(AU) to be
infinite can be derived by constructing Sturm minorants that are oscillatory.
Several criteria of this type will be given.
The equation Auo = A
can be written
(4.11)
P-1(Y)(X C -2(y) - u2)$ = 0.
In particular, for A = c2(m).12 one has
(4.12)
P-1(Y)u2(c2(°o)c-2(Y) - 1) = 0.
The first factor p-1(y) in (4.12) is unimportant for the oscillation properties of the equation.
Replacing it by pM' gives the majorant
V' + PM 0-1 (Y)P2 (c2 (°°) c 2 (Y) - 1) 4 = 0.
(4.13)
Each non-oscillatory Sturm majorant of (4.13) gives a criterion for the finiteness of GO(Ap).
Since solutions to (4.13) are non-oscillatory on any
interval (--,yo) when c(W) < c(-W), it is enough to construct majorants of (4.13) on intervals (yo,-). is $" = 0.
An obvious non-oscillatory majorant for (4.13)
Thus oo(A.) is finite for every p > 0 if there is a yo such
that c2(°)c-2(y) - 1 < 0 for all y > yo; that is,
(4.14)
c(y) > c(W) for y > yo.
This means the graph of c = c(y) lies above or on the limit line c = c(W) in a neighborhood of y =
Weaker hypotheses that include this case can
be derived by comparing (4.13) with
(4.15)
" + a y 2 $ = 0
which is oscillatory on (yo,-) if a > 1/4 and non-oscillatory if a < 1/4.
56
3.
SPECTRAL ANALYSIS OF SOUND PROPAGATION
Oscillation theorems based on (4.15) were first given by A. Kneser [12]; see [5, p. 1463]. Theorem 4.14.
Comparison of (4.12) with (4.15) gives If c(°°) < c(-) and
lim sup YZ(c 2(Y) - c-2(°°)) < 0
(4.16)
Y-1-
then O0(AJ) if finite for all p > 0.
Conversely, if
lim inf y2 (c2 (y) - C-2(_)) > 0
(4.17)
y-,
then there exists a PO > 0 such that ao(AU) is infinite for every p > uo. Note that the criterion (4.16) includes (4.14) as a special case.
Note also that sufficient conditions for (4.16) or (4.17) to hold are the existence of constants yo, K and E > 0 such that
c(y) > c(°°) - K y
(4.18)
2-E
for y > yo
or (4.19)
c(y) < c(°°) - K
respectively.
y-2
for y > yo,
In particular, ao(AU) is finite for all i1 > 0 if c(y)
approaches c(°°) from below sufficiently rapidly.
Criteria that Guarantee ao(Aµ) # 0.
Such criteria may be derived by
constructing Sturm minorants for Aµ = c2(°°)p2$ whose solutions have zeros. If the minorant has solutions with infinitely many zeros then GO(AU) is infinite.
If the minorant has a solution with finitely many zeros then it
can be shown that 03(y,p,c2(°°)p2) has at least as many zeros and one may
use the following refinement of Theorem 4.11. Theorem 4.15.
If AWO = c2(-)120 has a solution having a finite number
k of zeros on R then the part of a(A11 ) below c2(°)p2 is finite and has at
least k - 1 and at most k + 2 points. To apply the method in cases where ao(A
is finite consider first the
case p(y) = const. so that (4.12) becomes
(4.20)
V, + p2(c2(00)c 2(Y) - 1)m = 0.
Note that c2(°°) c-2(y) - 1 > c2(w)c02(y) - 1 for all y E R if and only if
(4.21)
c(y) < co(y) for all y E R.
3.4.
Spectral Properties of A11
57
If co(y) can be chosen in such a way that
$
(4.22)
+ pz(cz(°°)co2(Y) - 1) = 0
has a solution on R with k zeros then a0(Ap) will have at least k - 1 points by Theorem 4.15. Theorem 4.16.
In this way one can prove Let p(y) = const. for all y E R and assume that there
is a constant co > cm and an interval I = [a,b] with b > a such that
c(y) < co < c(-) < c(-W) for all y E I.
(4.23)
In fact, N(p) + - when p - -.
Then o0(Ap) # ¢ for all sufficiently large p.
Theorem 4.16 can be proved by comparing c(y) with a suitable piece-wise constant function co(y) that satisfies (4.21).
An analogue of Theorem 4.16
can be proved in the general case where p(y) # const. by making the change of variable y -* n in (4.12), where Y (4.24)
p(Y') dy'.
n = J 0
The details, which are elementary but lengthy, are omitted. This completes the formulation of the results of §4 and the proofs will now be given.
Note that Lemma 4.1 is an immediate consequence of
Theorem 3.1 which implies that for X > c2(_)p2 the equation App = Xc has no solutions in JC(R).
Proof of Lemma 4.2. JC(R)
The resolvent of Ap is an integral operator in
[5, XIII.3]:
(4.25)
(Ap - 4) -1 f(Y) = JI
Gp(Y,Y',4)
f(Y,)
c-2(y,) p-1(y') dy'.
R
is known to have the form
G(y,y',4), the Green's function of A [5, p. 1329]
(4.26)
Gp(Y,Y,,4) _
W(Y) _(Y'),
y _ Y',
_(Y) L_(Y'),
y >_ Y, ,
1
where 0_ and , ,_ are non-trivial solutions of A0 = CO that are in L2(0,W) and L2(-W,0), respectively.
_m = 3 and one has
Thus for 4 E L(c(oo)p) C L(c(-)p),
58
3.
(4.27)
SPECTRAL ANALYSIS OF SOUND PROPAGATION
43(Y,p,0
Y <_ Y',
,2(Y,p,4) c3(Y',u,4),
y > y'.
Gp(Y,Y',O _
It follows from Corollary 3.3 that G11 (y,y',4) is meromorphic in L(c(m)p)
with poles at the zeros of
(4.28)
As remarked in §1, these are precisely the eigenvalues of A. that are less than c2(_)p2.
Their only possible limit point is c2(co)p2 since F(p,C) is
analytic in L(c(W)p) by Corollary 3.3.
These results imply the two state-
ments of Lemma 4.2. Proof of Lemma 4.3.
This follows from Theorem 3.1 which implies that
Apd = a always has at least one solution that is not in C(R). It was remarked above that ac( u) C
Proof of Theorem 4.4. C [c2(W)p2,m).
oe(Au)
Hence to prove (4.5) it is enough to show that
(c2(°°)p2,°) C a,(Ap).
This may be done by constructing a characteristic
sequence for each A E (c2(°)p2,'); i.e., a bounded sequence
X(R) such that each n C D(Ap) and (Ap convergent subsequences.
in
0 in C(R) but {on} has no
Indeed, a suitable sequence has the form
n(Y) = n(Y) 3(Y,H,A) where En C D(A0), n(Y) = 1 for lyl < n, supp En C [-n- l,n + 1] and an(y) and (p-'(y)En(y))' are bounded for all y and n.
Such a sequence {fin} can be constructed but the details are
lengthy.
They will not be given here since the inclusion (c2(W)112,")
C ac(Ap) is proved in §6. Proof of Corollary 4.5.
For A # c2(W)p2 every solution of Apc
is a linear combination of 01(y,p,A) and 2(y,p,A) (Corollary 3.7).
= a It
follows from Theorem 3.1 that every real solution with A > c2(W)p2 has infinitely many zeros in any interval (yo,').
On the other hand for
A < c2(-)p2 Theorem 3.1 implies that every real solution of Apo = X$ is either exponentially large or exponentially small for y - ±W.
In every
case (y) has constant sign outside of some interval [-yo,yo] and hence can have only finitely many zeros. Proof of Theorem 4.6.
Results equivalent to Theorem 4.6 are proved in
[9, Ch. XI] under the additional hypothesis that the P. and Q. are continuous.
The same method will be shown to be applicable under the hypotheses
of Theorem 4.6.
The method is to study the phase plane curves
Spectral Properties of A
3.4.
59 V
(4.29)
(E,n) =
Y E 1, j = 1,2,
defined by the solutions 0j(y) and to transform to polar coordinates (Prufer transformation).
(4.30)
Thus (4.29) can be written
(C,n) = (rj(Y) cos 9.(y),rj(Y) sin B.(y)), y E I, j = 1,2.
Moreover, the curves (4.29) cannot pass through the origin because
.(y) J
Thus rj(y) > 0 and 0j(y) is uniquely defined by continuity and its value at Finally, 0j E AC(I) and (4.6) implies that 0j is a
the point y1 E I.
solution of the first order equation
(4.31)
0 (y) = P.(Y) cos20.(Y) + Qj(y) sin20.(Y), Y E I.
To prove the first statement of Theorem 4.6 note that one can assume
without loss of generality that 1(y) > 0 for y1 < y < y2 and 2(y1) > 0. Thus 0j(y) (j = 1,2) may be defined as the unique solutions of (4.31) such that 01(y1) = 0 and 0 < 02(y1) < ii. y, < y < y2 and 01(y2) = it. [y1,y2).
It follows that 0 < 01(y) < it for
It must be shown that 02(y) has a zero in
If 02(y1) = 0 there is nothing to prove.
If 02(y1) > 0 then
0 < 02(y1) < r and it follows from (4.31) and (4.7) that 02(y) > 01(y) for all y > y1 (see [9, p. 335]).
In particular, 02(y2) > 01(y2) = iT whence by
continuity 02(yo) = it and therefore 02(yo) = 0 for some yo E (yl,y2).
To prove the second statement of theorem 4.6 it is only necessary to remark that if Q1(y) < Q2(y) or P1(y) < P2(y) and Q2(y)
0 on a subset of
(y1,y2) having positive measure then 02(y2) > Ti even if 02(y1) = 0; see [9, p. 335].
This completes the proof.
Proof of Corollary 4.7.
(4.32)
The function 3(y,u,X) is a solution of
Thus if .(y) = 3(y,11,Xj), j = 1,2, then
equation (4.11).
(P 1(yW-)' + P-1(Y)(X.c 2(Y) -
0, j = 1,2.
These equations have the form (4.6) with P.(y) = p(y) and Q.(y) P-1(Y)(A J.c 2(y) - u2).
Hence P1(y) = P2(y) and Q1(y) < Q2(y) for all
y E R, since p(y) and c(y) are always positive, and the second part of Theorem 4.6 is applicable.
It follows that if y1 < y2 <
< Yk are the
zeros of 1(y) = 03(y,u,X1) then 02(y) _ 03(y,u,A2) has k - 1 zeros in the interval (yl,yk). a zero in (--,yl].
Hence it will be enough to show that 03(y,u,X2) also has To verify this apply Lagrange's formula (3.23) to 01(y)
0.
60
3.
and 2(y) in (-°°,y1].
SPECTRAL ANALYSIS OF SOUND PROPAGATION
This is possible because q1(y) and 2(y) are
exponentially small at y = -
.
The result can be written
ryl
(X2 - A1) l-oo $1(Y) 2(y) c-2(Y) P-1(y)dy (4.33) Y1
c-2 p-1 dy
{01
_ 02(YI) {P-1(Y1) 0i(y1)}
since 1(y1) = 0 and 1 and 02 vanish at y = -w. has no zero in (-oo,y1].
by Theorem 3.1.
Moreover, 01(y) > 0 for -m < y < y1 and p-1(YI) i(y1) < 0
because y1 is the first zero of 01. negative.
Now suppose that 02(y)
Then g2(yl) > 0 because c2(y) > 0 near y =
Thus the right hand side of (4.33) is
But the left hand side is clearly positive.
This contradiction
completes the proof. Proof of Theorem 4.8.
For regular Sturm-Liouville operators
A _ -d" + q(y)4 on finite intervals the oscillation theorem goes back to Sturm.
For singular operators in
it was first proved by H. Weyl. B. M. Levitan and I.
such that q(y)
+w when y - +o
More recently the result was proved by
S. Sargsjan [14, p. 201] under more general conditions
on Q(y) that guarantee that the solutions of A0 = a0 are non-oscillatory on 0 < y < m for all A E R (and hence Q(A) is discrete).
It will be shown
here that the method of Levitan and Sargsjan is applicable to the case of Theorem 4.8.
The method of [14] is to regard the Sturm-Liouville problem for A on
0 < y < - as a limit of regular problems for A on 0 < y < b < - and to study the behavior of the eigenvalues and eigenfunctions as b i m.
Here the
operator AP in C(R) will be regarded as a limit of the regular operator in C(a,b) = L2(a,b;c-2(y) p I(y)dy), -o < a < b < m, defined by AU and the
boundary conditions (a) _ 0(b) = 0. will be denoted by Ap,a,b.
The corresponding operator in ((a,b)
The limit a -- -- will be studied first.
The operator Au is more general than the operator studied in [14].
However, the examination of the proofs in [14] reveals that nothing is used but the Sturm comparison theorem, the convergence of the eigenvalues when b i -, the continuity and asymptotic properties of the solution (y,A) of
A = a that satisfies the boundary condition at y = 0 and the nonoscillatory character of A = X spectrum.
in a A-interval containing the point
All of these properties have been established for AU.
3.4.
Spectral Properties of Au
61
The solution of Ac = a that satisfies O(b) = 0, p-1(b) '(b) = 1 will be denoted by Ob(y,A).
For A # c2(-°°)p2, 4b(y,X) is a linear
combination of 3(y,p,X) and 04(y,p,X) and hence has the regularity properties of Corollaries 3.2 and 3.3. roots of the equation 0b(a,A) = 0. k =
The eigenvalues of Au,a,b are the
They will be denoted by Ak,a,b'
with the convention that Ak,a,b < Ak+i,a,b.
The corresponding
eigenfunctions k,a,b(y) - Ob(y,xk,a,b) have precisely k - 1 zeros by the classical oscillation theorem. For the class of operators considered here this result can be proved by the method of [14, p. 17].
The zeros of
k,a,b(y) will be denoted by yjka,b' 1 < j < k - 1. The operator in 1C(-°,b) defined by Au and the boundary condition fi(b) = 0 will be denoted by All b.
The methods used to study a(AU) above
can be used to show that a(A, b) n [cmp2,c2(.o)p2) (C ao(Apb)) is finite or countably infinite with unique limit point c2(-°)u2.
The number of
eigenvalues in [cmu2,c2(-°)p2) will be denoted by N(u,b) - 1 (< +o) in analogy with the notation for Au, and the eigenvalues will be denoted by are The eigenfunctions for b(y'Xk,b) k,b(y) Ak,b (Ak,b < Ak+i,b). The proof of the oscillation theorem for Au,b by the method of [14]
will now be outlined.
(4.34)
First
lim
Ak,a,b = Xk,b for 1 < k < N(p,b).
a-+-This follows, for example, from the convergence of the Green's functions. It follows that
(4.35)
lim a->-W
IP k,a,b(y) = k,b(y) for
uniformly on bounded subsets of (-°°,b].
< y < b,
The proof of the oscillation
theorem given in [14] is based on the following three lemmas. Lemma 4.17.
,N(p,b) - 1 and each fixed b E R one
For each k =
has
sup
(4.36)
a
Lemma 4.18.
For each k =
(k)
3'k-la'b < ,N(}i,b) - 1, each j = 1,2,,,,,k - 1
and each fixed b e R one has
(4.37)
inf a
yj(k),b
(yj+i(k),a,b
-
,a1 > 0.
62
3.
Lemma 4.19.
SPECTRAL ANALYSIS OF SOUND PROPAGATION
1 and each fixed b E R one
For each k =
has
inf (Xk+i,a,b - Ak,a,b) > 0. a
(4.38)
The proofs of these lemmas and the oscillation theorem for Ap, b are
the same as those given in [14, pp. 202-4] and will not be repeated here. The proof of Theorem 4.8 may now be completed by regarding Au as a limit of AP b for b ->
and repeating the argument given above.
The solu-
tion of Auk = a4 that satisfies the condition of square integrability at
y = -- is 3(y,p,A) and is non-oscillatory for all y G R when A < (Corollary 4.5).
The remainder of the proof follows as before.
Proof of Corollary 4.9.
Theorem 4.8 implies that Xk+1(u) E Ik for
Moreover, a continuity argument based on Corollary
,N(u) - 2.
k =
C2(_)P2
3.2 shows that the intervals Ik have the form Ik = (ak,ak+l] where ao < a1 <
< aN(u)-1 (see [5, p. 1475]).
Thus to prove (4.9) it will
suffice to prove that
(4.39)
F(p,ak) =
for k =
1.
0
This proof will be based on the following two
lemmas.
Lemma 4.20.
Let a0 <
c2(-w)p2 and let y0 be a zero of 03(y,}.,)0)
Then to each sufficiently small E > 0 there corresponds a d > 0 such that for la - X01 < 6 the function 03(y,p,X) has exactly one zero in the
interval ly - yd < E. This result follows from Corollary 3.2 and the fact that p 1(y) '(y) cannot vanish at a zero of a non-trivial solution of Apd = X0.
For a proof
see [14, p. 16].
For X E Ik let y1(X) < y2(X) < ... < yk(a) denote the zeros of 0?3(y,p,X).
Then each y.(X) is uniquely defined for A E Uk>j Ik and one has
Lemma 4.21.
Each of the functions ye(a) is continuous and strictly
monotone increasing. The continuity follows immediately from Lemma 4.20.
The strict
monotonicity follows from the proof of Corollary 4.7. Proof of Corollary 4.9 (concluded). contradiction.
(4.39) will be proved by
Assume that F(U,ak) ¢ 0 and note that for A < c2(W)j1Z
one has
(4.40)
$3(Y,u,X) = c(u,A) $1(y,U,A) + c'(p,X)
3.4.
Spectral Properties of Au
Moreover, Theorem 3.1 implies that
by Corollary 3.7.
(4.41)
63
c(u,X) =
P(°') F(u,X)/2iq+(U,A).
Thus c(u,ak) # 0 and by continuity (Corollary 3.2) there is an interval IA - aki < 6 in which c(p,A) # 0.
It follows from (4.40) and the uniformity
of the asymptotic estimates of Theorem 3.1 (Corollary 3.4) that there is an M > 0 such that
(4.42)
1 for all y > M and lA - akj < 6.
Note that Lemma 4.21 implies
lim
(4.43)
ye(a) = y (ak), j = l,2,...,k.
A+ak Now consider yk+1(X) which is defined for X> ak. has yk+1(a) < M by (4.42).
For ak < A < ak + d one
It follows that the limit
y = lim yk+ (A) A+ak
(4.44)
exists and y > yk(ak). = yk(ak).
But 3(y,li,ak) = 0 by continuity and hence y
But this implies that for 6 > 0 small enough and ak < A < ak + d
every neighborhood ly - yk(ak)l < e contains two zeros of 3(y,u,A) in contradiction to Lemma 4.20.
This completes the proof of (4.9).
The last statement of Corollary 4.9 follows from the proof of Theorem 4.8.
Indeed, if N(u) < m and N(u) < N(p,b) then N(u) < N(u,b) - 1 and
AN(u),b is defined.
In this case
lim inf AN(u)
(4.45)
b-'
since otherwise aN
b(k)
'
,b > c2 (°°)u2
< c2(°°)u2 for some subsequence {b(k)},
which would imply that a° was an additional eigenvalue of A.
If N(u)
N(p,b) < m the same argument can be applied to the operator Au
Proof of Theorem 4.11. of the theorem.
It will suffice to prove the second statement
To this end let
b(y,u,A) be the solution of App = a that
satisfies Ob(b,u,A) = 0, p-1(b) Ob(b,p,A) = 1.
Then Ob(y,p,X) and
p-1(y) 0b(y,u,A) are continuous functions of (y,A) E R2. Apq = c2(_)p2O is oscillatory. zeros.
b'
This follows immediately from Corollary 4.9.
Proof of Corollary 4.10.
Now assume that
Then Ob(y,p,c2(-)i12) has infinitely many
It follows by the method used to prove Lemma 4.20 that the number of
3.
64
SPECTRAL ANALYSIS OF SOUND PROPAGATION
zeros of b(y,P,A) tends to infinity as A ; c2(m)p2.
But then the same is
by Theorem 4.6, and it follows from Corollary 4.10 that
true of
To prove the converse note that if O0(AP) is infinite
ao(AP) is infinite.
then Theorem 4.6, applied to the kth eigenfunction and any solution of
APB = c2(m)u2 implies that
has k - 2 zeros.
Since k is arbitrary it
follows that APB = c2(m)P2P is oscillatory. Proof of Corollary 4.12.
This follows immediately from Theorem 4.11.
The hypothesis c(m) < c(-m) is needed only to ensure that 3(y,P,c2(m)P2) is defined.
Proof of Theorem 4.13.
This follows immediately from Corollary 4.12
and Theorem 3.6.
Proof of Theorem 4.14.
To prove the first half of the theorem it will
be shown that condition (4.16) implies the existence of a non-oscillatory majorant for equation (4.13) for every P > 0.
This implies that (4.12),
is non-oscillatory for every p > 0 and the finiteness
i.e., AP¢ =
of ao(A) follows from Corollary 4.12. To construct a majorant for (4.13) note that (4.16) implies that for
every e > 0 there is a yo = yo(e) such that
y2(c2(m)c-2(y) - 1)+ < e for all y > yo(e),
(4.46)
where a+ = Max (a,0).
It follows that for every p > 0 there is a
Ya = Yo(P) such that
PM P-1(Y) Y2(c2(m)c 2(Y) - 1) < PM P -1(y) Y2(C2(m)c 2(y) - 1)+ (4.47)
< PM Pmt y2(c2(m)c 2(Y) -
+
< 4 P2 for all y > Yo(P)
Hence for any p > 0 one has
(4.48)
PM
P_1(y) p2(C2(m)c-2(Y) - 1) < 4 y2 for all y > Yo(P)
It follows on comparing (4.13) with (4.15) with a = 1/4 that (4.13) is non-oscillatory on yo(p) < y < m. for any p > 0 because c(m) < c(-m). Theorem 4.14.
It is non-oscillatory on -
< y < yo(p)
This proves the first half of
Generalized Eigenfunctions of Au
3.5.
65
To prove the second half it will be shown that (4.17) implies the
existence of a PD > 0 such that APq = c2(o)u2 is oscillatory for every u > Po.
The result then follows from Theorem 4.11.
To this end note that
if e satisfies
0 < e < lim inf y2(c2(w)c-2(Y) - 1)
(4.49)
then there is a yo = yo(E) such that
Y2(c2(°°)c-Z(y) - 1) > E for all y > yo(E).
(4.50)
In particular, given any a > 1/4 there is a PD > 0 such that
0 < PM Pmt a/P2 < lim inf y2(c2(_)c-z(y) - 1).
(4.51)
y-*
It follows that there is a yo = yo(a) such that
(4.52)
Y2 (c2 (W)
C-2
(Y) - 1) > PM p
1
a/P2 for all y > yo (a) .
This implies that
(4.53)
Pm PM-1 Po(c2(°')c-2(y) - 1) > a/y2 for all y > yo(a).
Hence, comparison of
(4.54)
$'
+ P. PM-1 pp(c2(oo)c-2(y) - 1)$ = 0
and (4.15) with a > 1/4 implies that (4.54) is oscillatory. a Sturm minorant of (4.12); i.e., APB =
But (4.54) is
provided P > Po.
Hence
the latter is oscillatory for all p > Po. Proof of Theorem 4.15.
This result is proved in [5, p. 1481] for
Sturm-Liouville operators with smooth coefficients.
The proof is based on
the oscillation theorem (Theorem 4.8), Sturm's comparison theorem and the
continuous dependence of the zeros of solutions of Auk = 4 on X (Lemma 4.20).
§5.
Hence it extends immediately to the operator A.
GENERALIZED EIGENFUNCTIONS OF AP The eigenfunctions 1P(Y111) corresponding to the point spectrum of AP
were constructed in the preceding section.
In this section the special
66
3.
SPECTRAL ANALYSIS OF SOUND PROPACATION
(j = 1,2,3,4) of §3 are used to construct generalized eigensolutions $, pL functions of corresponding to the points of the continuous spectrum. These functions will be used in §6 to construct the spectral family {f(a)} of AU and to prove that ac(AU) = De (AU) _ [c2(00)p2,_).
To construct the generalized eigenfunctions ji0(y,u,A), +(y,u,A) described in §1 recall that the special solutions $j(y,P,A) are defined for all real A # c2(±')112 and the pairs $1, $2 and $3, $4 are solution bases for A $ = a$ (Corollary 3.7). U
)
It follows that
$j = Cj3$3 + cj4$4, j = 1,2,
(5.1) j = cjl$1 + cj2$2, j = 3,4. l
The coefficients c.k = cjk (U,A) can be calculated by means of the bracket
operation
(5.2)
of Lagrange's formula.
$k(',U>k))
Indeed, by forming the brackets of equations (5.1)
with $4, $3' $2 and $1 in succession and using the asymptotic forms of Theorem 3.1 one finds
(-2iq_)cj3 = P(-°°)[$j$4]
j = 1,2,
(5.3) (2iq_)cj4
= P(-=)[$. 3]
(-2iq+)cj1 = P(W)[$j$2] r
(2iq+)cj2
j = 3,4.
= P(W)1$j$11
In particular, Corollary 3.2 implies that each cjk(U,a) is a continuous function for A # c2(±_)U2.
These relations will be used to determine the
generalized eigenfunctions of AU.
(5.4)
The notation
A = A(p) _ {A
I
c2(-W)112 < A},
3.5.
67
Generalized Eigenfunctions of Au
(5.4 cont.)
c2(m)p2 < A < c2(-°)p2}
AO = A,(11) _ {A
will be used.
Note that AO 0
The Spectral Interval A.
I
only if c(W) < c(-°°).
The generalized eigenfunctions of Ap are the For A E A,
bounded solutions of the differential equation Ap0 = A.
Theorem 3.1 and the relations (5.1) imply that all the solutions are bounded It will be shown that the functions
+(Y,p,X) = a+(p,A) 4(Y,p,)') (5.5)
= a-(p,A) 1(y,p,A)
have the asymptotic forms described in §1. of these functions will be proved in §6.
The completeness in IIp(A) 3C(R)
which provides
The pair
an alternative basis, will not be treated explicitly here. to correspond to the second family
I
It may be shown
q > 0} described at the
end of §1.
It follows from (5.5), (5.1) and Theorem 3.1 that the asymptotic
behavior of + is given by
c41 e (5.6)
lq+y
-iq+Y
+ C42 e
+ o(1) , y -> -fo ,
t+(Y,p,A) = a+ -iq_y
y
+ o(1),
e
The equivalence of (5.6) and the asymptotic form (1.19) of §1 follows from Lemma 5.1.
For all p > 0 and A E A the coefficients c41(p,A),
c42(p,A) satisfy
(5.7)
IC42I2
p-1(-) q+
= p-1(m) q+ IC41I2 +
p-1(-O°)
q_.
In particular, c42(p,A) # 0.
The proofs of Lemma 5.1 and subsequent lemmas are given at the end of the section.
The asymptotic forms (1.19) and (5.6) coincide if the coefficients satisfy
(5.8)
c+ = a+ C421 c+ T+ - a+, c+ R+ _ a+ c41-
In particular, the first relation and (5.3) imply that
68
SPECTRAL ANALYSIS OF SOUND PROPAGATION
3.
+ (u,a)
(5.9)
= a+(u,a)
(u,A) 2iq+(}i,1)
P(°D)
The normalizing factor a+(p,X) will be calculated in §6.
The factors Indeed,
R+(P,A), T+(p,A) of (1.19) are independent of the normalization. on combining (5.8), (5.9) and (5.3) one finds
2iq+(u,a)
(5.10)
R (U,X)
(5.11)
[M11(u,X) '
Note that the denominator [Ml] is not zero by Lemma 5.1 and relations (5.3).
Equations
may be discussed similarly.
The asymptotic behavior of (5.5), (5.1) and Theorem 3.1 imply
iq+y
+ 0(1),
e
(5.12)
= a-
c13 e
lq_y
y
+ e14 e
-iq_y
+ o(1), y
and one has Lemma 5.2.
For all p > 0 and A G A the coefficients c13(u,a),
c14(U,A) satisfy
(5.13)
p-1
(-°°) q_ 1c13 1 2 = p-1 (-°°) q_ l c14 12 + p-'(-) q+.
Comparison of (1.20) and (5.12) gives
(5.14)
c_ = a_ c13, c_ T_ = a_, c_ R_ = a_ c14.
Solving these equations for c_, T_ and R_ and using (5.3) gives
(5.15)
(5.16)
(5.17)
c (u,A) = a -(u,A) T_
P(-°°) [4
2iq-(p,A) -
R (11,X) = -
i ] (p,A)
2iq (u,A)
[131(11,A) [1441(u,A)
3.5.
Generalized Eigenfunctions of A
69
u
The Spectral Interval Ao.
For X E Ao, Theorem 3.1 and the relations
(5.1) imply that the only bounded solutions of A114 = a¢ are multiples of 4,3(y,u,A).
It will be shown that
so (Y.V,X) = ao (p,X) 3 (y.V,X)
(5.18)
Indeed, (5.18), (5.1) and Theorem 3.1
has the asymptotic form (1.18). imply that
c31 e (5.19)
o(y.V.a) = as
iq+y
+ c32 e
-iq+Y
+ o(1) , y
+°°,
q'y [1 + o(1)],
e
The equivalence of (5.19) and (1.18) follows from Lemma 5.3.
For all V > 0 and X E Ao one has
c32(u,X) = c31(11,X) # 0.
(5.20)
Comparison of (1.18) and (5.19) gives
co = ao c32, co To = ao, co R. - ao c31
(5.21)
Solving for co, Tp and Ro and using (5.3) gives
(5.22)
2iq+(V,A)
(5.23)
T, (11, X) =
2iq+(u,X)
o
421 (11,)')
Ro(11.),)
(5.24)
0 by Lemma 5.3 and relations (5.3).
The denominator
Finally, note that the conservation laws (1.31) hold; i.e., q,
(5.25)
P (±°°)
(5.26)
q
q+
IR+I2 +
for all X E A,
IT+I2 =
P (j°)
P (±m)
IR0I = 1 for all X E A0.
In fact, relations (5.25) are equivalent to relations (5.7) and (5.13) of
SPECTRAL ANALYSIS OF SOUND PROPAGATION
3.
70
Lemmas 5.1 and 5.2, as may be seen by combining (5.7), (5.8) and (5.13), (5.14).
Similarly, relation (5.26) follows from Lemma 5.3 because (5.21)
implies that R0 = C31/C32. Proof of Lemma 5.1.
Relation (5.7) can be verified by calculating
[044] in two ways, using the asymptotic forms of 4 as y - W and y Note that for A E A one has T4 = 03 by the uniqueness theorem (Corollary 3.5) and Theorem 3.1.
at y = - gives
Hence, calculating
(5.27)
[4,$31 = 2i p-'(-")q_,
by (3.25).
Next, relations (5.1) and Theorem 3.1 give (with the notation
c.c. for complex conjugate)
[q,;
]
_ 4{p-1Ty} - C.C.
(5.28) C42Y'2){C41
P-12} - c.c.
+ C42e^l q+)
(C41e lq
-lq+y
_1
x {c41P
c42
(o)(-iq+e
) + C42P
-1
(°O)(iq+e
_
2iq+y y P
1 (°°) (-iq+) { I C4 1
1 2 - C 4 1 C4 2 e
lq+y )} - c.c. + o(l)
+ C4 1 C4 2 e
-2iq y
+
-
I C4 2
12 } - c . c . + o (1)
-2iq+p-1(m){1'4112
- IC42I2} Combining (5.27) and (5.28) gives (5.7). Proof of Lemma 5.2.
(5.13) can be verified by calculating
two ways, in analogy with the proof of Lemma 5.1.
in
It can also be derived
directly from (5.7) and the relations (5.3). Proof of Lemma 5.3.
Note that for A E A0 the uniqueness theorem
(Corollary 3.5) and Theorem 3.1 imply that 3 is real valued and 1 = dm2' Hence relations (5.3) imply
(5.29)
C31 = p(°O)
C32.
Moreover, if c32 = 0 then c31 = 0 by (5.29) and hence by (5.3) one has
3.6.
The Spectral Family of A
71
u (5.30)
0.
But this would imply that 1 and 2 are linearly dependent which contradicts (3.25).
§6.
Hence c32 # 0.
THE SPECTRAL FAMILY OF Au The eigenfunctions and generalized eigenfunctions of §4 and §5 are used
in this section to construct the spectral family {ll (X)} of Ap.
The
construction is based on the Weyl-Kodaira-Titchmarsh theory as presented in Appendix 1.
Note that the operator A.
has the form (A.1) with I = R,
p(y) = P(Y), q(y) = u2 P-1(Y) and w(y) = c -2(y) P-1(y).
It is clear that
p, q and w satisfy (A.2), (A.3), (A.4) when P(y) and c(y) are Lebesgue measurable and satisfy (2.1.4). It will be convenient to decompose R into the disjoint union
(6.1)
R = Ad U {c2(oo)p2} U Ao U {c2(-a0)p2} U A
if c(o°) < c(--) and (6.2)
R = Ad U {c2(oo)p2} U A
if c(°°) = c(-m) where (6.3)
Ad = Ad(u) _ (-°°,c2(°°)u2),
and Ao and A are defined by (5.4).
The spectral measures of the components
of (6.1) and (6.2) will be studied separately. The Spectral Family in A.
The spectral measure fl (A) of intervals 13
A = (a,b) C A will be calculated by applying the Weyl-Kodaira theorem to A. in A.
The solution pair
(6.4)
'P2 (Y, X) _ 01 (Y,lp,X) will be used to obtain a spectral representation in terms of the generalized
eigenfunctions +(y,p,A) defined by (5.5).
The normalizing factors a+(p,X)
will be chosen after the matrix measure for (p1,y2) has been determined. Note that the pair
satisfies the hypotheses of the Weyl-Kodaira
72
SPECTRAL ANALYSIS OF SOUND PROPAGATION
3.
theorem.
Indeed, (A.11) follows from Corollary 3.2 and (A.12) from Lemma
5.1.
The Weyl-Kodaira theorem implies that
(6.5)
Ilnu(A)fII2 = JQ f.(A) fk(1) mi.k(dX), A C A,
for all f E 3C(R) where (mik(A)) is the spectral measure on A associated
with the basis (6.4) and rM
(6.6)
i.(y,A) f(y)w(y)dy,
f.(A) = lim f M- °'
))) -M
the integrals converging in L2(A,m).
Thus to complete the determination of
(A) for A C A it is only necessary to calculate {mjk(A)}.
fl
Now TT(A) can
11
be calculated from the resolvent
(6.7)
R11 (C) = (A
- 7;)-i
by means of Stone's theorem (see, e.g., [25, p. 79]).
For A C A the
theorem takes the form
(6.8)
IIT' (A)f1I2 = lim
C-0+ because oo(AU) n A
a- J
by Lemma 4.1.
A
Moreover, Ru(b) is an integral
operator in K(R) whose kernel, the Green's function of Au, can be represented by the analytic continuation of the basis (6.4) into the 4-plane.
This procedure, whose details are presented in the proofs at the end of the section, leads to Theorem 6.1.
For all f E 3C(R) and all A C A the spectral measure
IIu(A) satisfies
(6.9)
IIn (A)f112 = JA {A*(U,A) Ifl(A)I2 + A2(u,A) If2(A)I2}d1
where q+(p,A) (6.10)
Corollary 6.2. given by
A+(u,a) _
The matrix measure (mjk(A)) for the basis (6.4) is
The Spectral Family of Au
3.6.
(6.11)
73
m11(A) = JA A+(p,X)dA, m22(A) =
A' (1j, A) dX
1A
and m12(A) = m21(A) = 0 for all A C A. These results suggest an appropriate choice of the normalization factors a+(p,A) of (5.5).
Note that if instead of the basis (6.4) one
takes (5.5) then (6.9) becomes
(6.12)
11RU(A)fll2 = J4 {A+ Ia+I-2
If+I2
+ A2 la_ I-2 I f_I2}da
where (M
(6.13)
f+(U,X) = lim f
f(y)w(y)dy -M
M-*
This suggests the
converge in the space L2(A,m') corresponding to choice Ia+I2 = A+ or
i0+(u,A) (6.14)
a+(U,A) = e
A+(11,A)
where 6+(p,A) is an arbitrary real valued continuous function.
The matrix
is independent of the choice of the phase factors
measure {m'k} for
exp {i0+(p,a)} and one could take 0+(p,X)
--
0.
However, it will be more
convenient to choose 6+(p,A) in §9 in a way that simplifies the asymptotic
form of the normal mode functions +(y,p,q).
Theorem 6.1 and the above
remarks imply Corollary 6.3.
If the basis (tP+(y,p,a), (y,p,A)) is normalized by
(6.14) then for all A C A one has
(6.15)
III' (A)f112 = j
(If+(11 ,A)12 + If_(u,a) l2)da
A
and the matrix measure
(6.16)
for (W+, ) is given by
mil (A) = m22(A) = IAI
and m12(A) = m21(A) = 0 where IAl The Spectral Family in Ao.
is the Lebesgue measure of A.
The spectral measure of intervals A C AO
will be calculated by applying the Weyl-Kodaira theorem to Au, Ao and the solution pair
74
SPECTRAL ANALYSIS OF SOUND PROPAGATION
3.
l)1(y,A) _ 3 (y,u ,A), (6.17)
2(y,A) _ 41(y,u,A) The function 3 is chosen to obtain a representation in terms of the generalized eigenfunction 0o defined by (5.18).
The second function could
be replaced by any independent solution of Auk = X
.
The pair (6.17)
satisfies (A.11) by Corollary 3.2 and (A.13) by Lemma 5.3. Calculation of the spectral measure in A0 by the method described above leads to Theorem 6.4.
(6.18)
For all f E W(R) and all A C A. one has
A2(u,A) I11(A)J2 dA
IIn,1(A)f112 = J A
where
g+(u,A) Ao (u,A) =
(6.19)
ip (°D)
I
[
]12 1
3
Hence the matrix measure (mjk(A)) associated with the basis (6.17) is given by
(6.20)
A2(u,A)dA
m11(A) = J A
and m12(A) = m21(A) = m22(A) = 0 for all A C A0. On replacing (6.17) by the basis
and defining
the normalizing factor by
(6.21)
ao(u,A) =
ei00(u,A)
Ao(u,A)
where 0o(p,A) is an arbitrary real valued continuous function one obtains Corollary 6.5.
If ao(pi,A) is defined by (6.21) then for all A C A0
one has
(6.22)
Mll
Ifa(u,A)I2 dA
(A) f112 = J A
where
(6.23)
ipo(y,u,A) f(y)w(y)dy.
fo(u,A) = lim J M-,W
-M
The Spectral Family of Au
3.6.
75
In particular, the matrix measure
for the pair
(6.24)
is given by
m11(A) = JAI
and m12(A) = m21(A) = m22(A) = 0 for all A C A0, and the integral in (6.23) converges in L2(A0). The Spectral Family in Ad.
The portion of 0(A,,,) in Ad was shown in §4
to be the set of eigenvalues {Xk(P)
I
1 < k < N(P)).
Moreover, each ak(P)
is a simple eigenvalue with normalized eigenfunction k(y,p) defined by (4.4), and corresponding orthogonal projection Ppk defined by
(6.25)
ppk f(Y) _
Hence, recalling that by convention IIp(A) = 1I11 (A + 0), one has
(6.26)
II
p
(A) f(Y) =
( k(-"'f)
I
k
(Y'P)' A E n
d
.
kk(p)
The notation in (6.26) denotes a summation over all indices k such that Ak(0) < A.
The sum in (6.26) is finite for all k E Ad. When
The Spectral Measure of the Points c2(_)p2 and c2(-W)p2.
0 because in this case the
c(-) < c(--) one always has
02(y,p,X) are defined for A =
special solutions
c2(-')p2 and
Theorem 3.1 implies that Ap0 = c2(-o,)p20 has no solutions in JC(R).
point A = c2(W)p2 may be an eigenvalue of Ap.
The
In each case this question
must be decided by determining the behavior of solutions of App = c2(o)p2 for y -> ±-.
Theorem 3.6 gives simple sufficient conditions for
fl ({c2(_)p2}) = 0.
For simplicity, it will be assumed in the remainder of
the report that c2(°)p2 jZ ao(Ap).
In cases where c2(co)p2 is an eigenvalue
a corresponding term must be added to the eigenfunction expansion. The Eigenfunction Expansion for Ap.
Combining the representations of
IIp obtained above, one finds the representation
N(p)-1
H(A - Ak(p)) I?k(p) I2
III, p(A)fIJ2 =
k=1
(6.27)
H(A - A')
+ J
Ifo(p,A')I2 dA'
AO
+
1
H(), - x')(If+(p,A')I2 + If_(1, ,a')I2)da'
76
SPECTRAL ANALYSIS OF SOUND PROPAGATION
3.
where H(a) = 1 for A > 0, H(a) = 0 for A < 0,
(6.28)
C-2 (y) P-'(y)dy,
k(y,U) f(y)
k(U) = J
1 < k < N(U),
R (M
(6.29)
0(y,u,A) f(y) c -"(y) P-'(y)dy,
f0(U,A) = L2(A0)-lim 1 -M
and (M
(6.30)
f+(U,A) = Lz(n)-lim f
M
f(y) c-2(y) P-'(y)dy -M
In particular, on making A -* - one obtains the Parseval relation for Au: N(11)-1 IfHI2 =
X
(11'X) 12 dA +
fk(u)I2 +
k=l
no
(6.31)
If_(u,a)I2)da.
+ JA Thus the correspondence
(6.32)
f -+ TP f = (f+(U,'),f_(U,'),fo(U,'),fl(U),fz(U),...)
defines an isometric mapping Y'U of JC(R) into the direct sum space
(6.33)
L2(A) + L2(A) + L2(no) + CN(U)-1
and one has Theorem 6.6.
T11 is a unitary operator from JC(R) to the space (6.33).
This result will be shown to follow from the Weyl-Kodaira theorem and the corresponding properties of the partially isometric operators
YU+
X(R) -* L2 (n) ,
Y' U0
X(R) 4 L2 (no) ,
`Y
X(R) -- C, 1 < k < N(U),
(6.34)
Uk
defined by
The Spectral Family of Au
3.6.
77
(6.35)
Y'uof = fo(u,'),
Tpkf=fk(u)' 1
The operators (6.34) are partially isometric and if
orthogonal projections in 3C(R) are defined by
Puk = Y`uk `YUk, 0 < k < NO),
Pu+ = Y' u+ %+1
(6.36)
then
Pu+ + Pu_ = 1I (A) , (6.37)
Puo = nu(A,),
N(uC)-1
Puk = II 11 (Ad)
k=1 Moreover,
'Y + 'Y*+ = 1 in L2 (A(u)) ,
(6.38) 'Y
Uo Tuo = 1 in L2(Ao(u)),
1 in C, 1 < k < N(u)
T
Corollary 6.8.
The inverse isometries ''* p-
,
T*
uk
,
0 < k < N(p), are -
given by (M
(f*+f+)(Y) = X(R)-lim
+(y,p,a) f+(X)da,
I
&-0,M- 1c2(-m)u2+6
(6.39)
yo(y,u,A) fo(A)da,
(YUofo)(Y) = 3f(R)-lim J
(YUkfk)(Y) = fk k("
,
1 < k < N(u)
78
3.
SPECTRAL ANALYSIS OF SOUND PROPAGATION
The spectral property of the unitary operator Tu is described by Corollary 6.9.
(6.40)
For every f E D(AU) one has
Apf =
'P
This completes the formulation of the results of §6 and the proofs will now be given. Proof of Theorem 6.1.
The integral representation of the resolvent
will be used.
Thus, as in the proof of Theorem 4.2,
(6.41)
% W f(Y) =
G11(Y,Y',c) f(y')w(y')dy',
J
0 a(A,,),
R
where w(y) = c -?-(Y) p-1(y), and the Green's function Gµ has the form
(6.42)
Gu(Y,y',r) =
$_(y<,4) '00(y>,4)
where y< = Y<(Y,Y') = Min (y,y') , y> = Y>(Y,Y') = Max (y,y') and +_(Y,4)
are non-trivial solutions of Auk = 4 such that E L2(0,-).
E L2(-',0) and
To identify these solutions note that by (3.4)
Im q+(p,4) > 0 for 4 E
R+(c(--)p) int
(6.43)
Im q+(p,4) < 0 for 4 E R (c(--)u)int
and
(6.44)
e
± y q+'
=e
±i y q+
=e
y Im q+±i y Re q±
It follows from Theorem 3.1 that
0W(Y,4) _ 1(Y,L,4) for 4 E
(6.45)
R+(c(-m)u)int
0_.(Y,0 _ $y(Y,U,4)
and
for C E R (c(-m)u)int
(6.46)
0__(y,4) _ g3(Y,u,4)
The Spectral Family of Au
3.6.
79
Now the functions appearing in (6.45) and (6.46) have continuous extensions to R+(c(-W)p) and R (c(-oo)p), respectively, by Corollary 3.2.
Indeed,
(6.42), (6.45), (6.46) and Corollary 3.2 imply Lemma 6.10.
For all A E A one has
(6.47)
Gu(y,y',A+i0) _
(6.48)
Gp(y,y',A-i0) =
c2(Y,,u,A),
(A)
and the limits are uniform on compact subsets of R X R x A. To prove Theorem 6.1 it is clearly sufficient to verify (6.9) for the functions f of a dense subset of JC(R).
It will be convenient to use the
subset
(6.49)
J(com(R) = JC(R) n (f
I
supp f is compact).
Note that for f E JCcom(R) the integrand in Stone's formula (6.8) can be
written
(f,[R11 (A+iE) - Ru(A-ie)]f) (6.50)
2Tti
1{c (Y,Y',A+ie) - Gu(Y,Y',A-iE)} f(Y) f(y') w(y) w(y') dydy'. R JR
Lemma 6.10 implies that for all f E JCcom(R) this expression tends to a limit
(6.51)
(f,HU(A)f) = 1
Hp(Y,y',A) f(y) f(y') w(y) w(y') dydy',
J
R R uniformly on compact subsets A C A, where
(6.52)
H(y,y',A) =
27Ti
{G(Y,Y',A+I0) - G(Y,Y',A-i0)}.
It follows that
(6.53)
(f,H. (A)f)dA
IIlip (A)fII2 = f A
for all f E JC °m(R).
The proof of Theorem 6.1 will be completed by
calculating H11 (y,y',A).
implies that Gu(y',y,C) =
Note that the well-known property Ru(C)* = Ru(C) and hence
80
3.
(6.54)
SPECTRAL ANALYSIS OF SOUND PROPAGATION
GP(Y,Y',A-i0) = GC(Y',Y,X+i0).
Equations (6.52) and (6.54) imply that Hu(y,y',X) is Hermitian symmetric:
(6.55)
(Y,Y"a)-
HV(Y',Y,A) = 1
Hence, it will be enough to calculate H,(y,y',A) for y < y'. Calculation of Hu(y,y',X).
Definition (3.4) of q+(ll,C) implies that
Q+(u _ -q+(11,
(6.56)
)
for
E R(c(-©)U)
It follows from Theorem 3.1 and Corollary 3.5 that
for C E R+(c(-0O)V)
(6.57)
4 (Y,31
= 3 (Y,u,T)
)
and
02(Y,u,0 =
for C E R (c(-°°)p).
(6.58)
3(Y'l1'0 = 4(Y'P'Z) It follows that
(6.59)
1(Y',u,A) = T2(7',U,X) = o21(u,X) 1(Y'.u,X) + C24(U,A) j4(Y',u,X)
where
(6.60)
c21
=
c
24
Similarly
(6.61)
Q1(Y,u,A)
and
(6.62)
C31(p,X) 1(Y,U,A) + C34(ll,X) 04(Y,U,A)
3.6.
The Spectral Family of A
81
u
where
__
(6.63)
_[¢341t
C31
,
C34
Combining these relations and Lemma 6.10, one finds for A E A, y < y'
Gu(Y,Y',A+i0) = (6.64)
T1(y',u,A) + C24 4(Y,u,A) y (Y',u
and
Gu(Y,Y',a-i0) =
W2(Y'1111)')
(6.65)
01(y',u,A) + C34 04(Y,u,A) 01(Y',u,A)}. Combining (6.52), (6.64) and (6.65) gives
Hu(Y,Y',a) =
14 27ii
4]
4(Y,u,X) 4(Y',11,A)
(6.66)
-2l
C
3]} 04 (Y
(Y',u ,X))
To calculate the coefficients in (6.66) recall that 01(Y,u,A) = 2(Y,u,a),
3(y,u,A) = 4(y,11,A) for A E A.
It follows from (6.60), (6.63) and (3.25)
that (6.67)
(6.68)
(6.69)
C24
=
-C31
=
[203]
C21_ _
Thus (6.66) can be written
2iP_1(m)q
-[D3c41
_
2ip-1(_')q_
Ti2
= 2Tri A+,
= 2Tli A2_,
C34
0.
[04011[203]
82
3.
SPECTRAL ANALYSIS OF SOUND PROPAGATION
HH(Y,v',A) = A+ ,(v,u,x) 4(Y',u,A) + A2 1(Y,u,X) 1(Y',u,X) (6.70) 2
= A+ 1(Y,a) 1(Y',a) + A This was proved for y < y'.
z(Y,a)
However, both sides of (6.70) are Hermitian
symmetric and it therefore holds for all (y,y') E R2.
Multiplying (6.70)
by f(y) f(y') w(y) w(y') and integrating over R2 gives
(f,HU(X) f) = A+ If1(A)I2 + A 2 Jf2(X)12
(6.71)
and combining (6.53) and (6.71) gives (6.9).
This completes the proof of
Theorem 6.1.
Proof of Corollary 6.2.
It follows from (6.5) and (6.9) that
(f,RU(A)g) = J f (X) Br (a) A
(6.72)
jA {A+ f1(X) g1(X) + A2 f2(X) 82(X)}da
for all f,g E C(R).
Moreover, the Weyl-Kodaira theorem implies that
f = (fl,fz) and g = (gJ'gz) can be arbitrary vectors in L2(A,m).
Now all
vectors f with bounded Borel functions as components are in L2(A,m).
Thus
the first of equations (6.11) can be obtained from (6.72) by taking f1 = g1 = XA, the characteristic function of A, and fz = gz = 0.
The remaining
equations of Corollary 6.2 are obtained similarly. Proof of Theorem 6.4.
The proof follows that of Theorem 6.1.
Equations (6.41), (6.42) for the resolvent are still valid.
However,
instead of (6.43) one has
Im q+(u,4) > 0 for 4 E
R+(c(m)u)int
(6.73)
Im q+(u,4) < 0 for 4 E R (c(m)p)int
Im q (u,4) < 0 for 4 E R(c(-)u) n L(c(--)u).
Thus by Theorem 3.1
The Spectral Family of Au
3.6.
(6.74)
83
for c E R+(c(00)o)int n L(c(--)p),
h
--(Y,4) = 3(Y,11,0
and
(6.75)
E R (c(m)u)int n L(c(--)p)
for
_m(Y,4) _ 3(Y,u,S) Hence Corollary 3.2 implies Lemma 6.11.
(6.76)
For all X E AO and y < y' one has
Cu(Y,Y"A+iO) =
(6.77)
Cu(Y,Y',X-iO)
3(Y,p,A)
=
03(Y,p,A) 2(Y',u,X),
and the limits are uniform on compact subsets of R X R x A0. Proceeding to the calculation of Hu(y,y',A) one has
94.(p,T) = -4'(p,C) (6.78)
h
for C E R(c(m)u) n L(c(-oo)p)
9'(p,T) = J
whence
for C E R+(c(°°)u) n L(c(-°)p)
(6.79)
and
for
(6.80)
E R(c(°°)p) n L(c(-°°)p).
It follows that (6.81)
where
$1(Y',p,X) _
2(Y',u,X) = c21(p,A) T1(7',o,X) + c23(p,X) 3(Y,p,X)
84
SPECTRAL ANALYSIS OF SOUND PROPAGATION
3.
(6.82)
c21
and c23 - [_2_11
[01031
[0301]
These relations and Lemma 6.11 imply that for X E A0 and y < y',
GP(y,Y',X+iO) =
[
103]-I1c
c2303(Y,U,X)03(Y',u,X)}
(6.83)
and
(6.84)
03(Y,u,a) 1(y',U,A).
=
G}1(y,Y',X-+O)
Hence
HU(Y,Y',),)
(6.85)
2Tr3'
1101 3103(Y,U,A)03(Y',}3,X) +
From the relations 1(y,p,A) = 2(Y,u,A), 3(Y,U,A) = 3(y,u,a), together with (6.82) and (3.25) it follows that
(6.86)
C23
[
1
2i
2l 3]
2
1(m)q
- 2Tri A20,
and
(6.87)
c21
Thus (6.85) can be written
(6.88)
Hu(Y,Y',A) = A0 3(Y,U,A) 43A) = A0 01(Y,A) ii1(Y',A)
for y < y' and hence for all (y,y') C R2.
It follows by integration that
for all f C Jt om(R) one has (6.89)
(f,HU(X)f) = A2
Combining (6.89) and (6.53) gives (6.18).
A C A0.
Finally, (6.18) implies (6.20)
by the argument used to prove Corollary 6.2. Theorem 6.4.
This completes the proof of
3.7.
The Dispersion Relations
Proof of Theorem 6.6.
85
Y11 is isometric by (6.31), (6.32).
Hence to
prove that Y'p is unitary it is only necessary to prove that it is surjective.
But this is an immediate consequence of Theorem 6.7, equations (6.38).
The
latter are implied by the Weyl-Kodaira theorem. Proof of Theorem 6.7 and Corollaries 6.8 and 6.9.
These results are
direct consequences of the Weyl-Kodaira theorem.
V.
THE DISPERSION RELATIONS FOR THE GUIDED MODES The eigenvalues of Ap determine the relation between the wave number
IpI and frequency w = functions
k(x,y,p).
(4.4) of k(y,p).
Xk( p ), or dispersion relation, for the guided mode The functions Xk(p) also appear in the definition
The purpose of this section is to provide the information
concerning the p-dependence of Xk(p) and
'Pk(y,p)
that is needed for the
spectral analysis of A in §8 and §9. The domain of definition of the function Ak(p) is the set
0k = {p
I
N(p) > k + 1}, k = 1,2,....
Note that 0k is not empty if and only if 1 < k < N. where
No = sup N(p) < +m. p>O
(7.2)
Clearly, if No < +w then No - 1 is the maximum number of eigenvalues of Ap for p> 0.
If No = -I° then either G0(A1) is infinite for some p > 0 or
Go(A.) is finite for all p > 0 and N(p) -> W when p - -. implies that both cases occur. Theorem 7.1.
Theorem 4.14
The principal result of this section is
For 1 < k < No the set 0k is open and X
k
:
0k ± R is an
analytic function.
The proof of this result given below is based on analytic perturbation theory as developed in [11].
The curves A = Xk(p), P E Ok' can never meet or cross because each eigenvalue is simple and the corresponding eigenfunction lPk(y,p) has
exactly k zeros (Theorem 4.8).
(7.3)
cmp2
Thus for 1 < k < No - 1 one has
< Ak(P) < Xk+i(p) < c2(m)p2,
' E Ok+i
Moreover, if 0k is unbounded then (7.3) implies
(7.4)
c2 < lim inf p-2 Ak(p) < lim sup P-2 Ak(p) < c2(°').
86
SPECTRAL ANALYSIS OF SOUND PROPAGATION
3.
In particular, if 0k is unbounded then
lim ak(u) _ +°°,
(7.5)
A related property is given by For 1 < k < No the function Ak(u) is strictly monotone
Theorem 7.2.
increasing; i.e., for all pl, u2 E 0k one has
(7.6)
Xk(ul) < Ak(u2) when Pl < u2.
The proof of (7.6) given below is based on a variational characterization of ak(u).
By Theorem 7.1, 0k is open and is therefore a union of disjoint open intervals.
Hence the curve A = Ak(u) consists of one or more disjoint
analytic arcs.
It is interesting that these arcs can terminate only on the
curve A = c2(°°)U2.
More precisely, one has
Corollary 7.3.
Let Po be a boundary point of 0k.
Then
lim Ak(u) = c2(°O)up.
(7.7)
P±Pa It is clear from Theorem 7.2 and (7.3) that the limit in (7.7) exists and does not exceed c2(O0)Uo.
The equality (7.7) is proved below.
The result (7.4) can be improved by strengthening the hypotheses concerning c(y).
A result of this type is
Theorem 7.4.
Let cm < c(°) and assume that for each e > 0 there is an
interval I(c) C R such that
c(y) < cm + c for all y E I(c).
(7.8)
Then No = +W, 0k is unbounded for each k > 1 and Ak(u) - c2p2 when u -* in °k; i.e.,
lim u-2 Ak(u) = c2.
(7.9)
The analyticity of Ak(u) and Corollary 3.2 imply the continuity of the eigenfunctions l,k(y,u) Corollary 7.5.
(7.10)
More precisely, one has
For 1 < k < No the function lPk(y,p) satisfies
k' P-1 k E C(R x 0 k),
87
The Dispersion Relations
3.7.
This completes the formulation of the results of V. Proof of Theorem 7.1.
VII] will be used.
The analytic perturbation theory of [11, Ch.
Note that the operator AP may be defined for all p E C
by (1.5), (2.2) and is a closed operator in JC(R).
Moreover, the domain
D(AP) is independent of p and is a Hilbert space with respect to the norm defined by
(7.11)
IIm' II3C(R) + II
II;C(R)
It follows that p } AP is holomorphic in the generalized sense.
Indeed, in
the definition of [11, p. 366] one may take Z = D(AP) (independent of P) and define U(P)
:
JC(R) to be the identification map.
Z
Then U(P) is bounded
holomorphic (in fact, constant) and
V(P)$ = AP U(P)O _ -cep [(P-10')' - P-1p2
(7.12)
E Z.
is holomorphic for all
Thus AP is holomorphic.
]
It follows by [11,
p. 370] that each Ak(p) has a Puiseux expansion at each point PO
Ok.
But
each Ak(po) is simple (Lemma 4.3) and hence the Puiseux series can contain Thus Ak is in fact analytic at
no fractional powers of P - po [11, p. 71]. each Po E 0k.
This proves both statements of Theorem 7.1. The eigenvalue Ak(P) can be
Proof of Theorem 7.2 and Corollary 7.3.
characterized by the variational principle [5, pp. 1543-4]
k (P)
(7.13)
= inf MESk
(A 010)
sup Q>`hflD(AP) II
II =1
where Sk denotes the set of all k-dimensional subspaces of JC(R).
Moreover,
D(AP) is independent of p and
(7.14)
III
=
I0(y)I2) P-1(Y)dy
J
for all 0 E D(AP) [11, p. 322].
Hence if pl < PZ then
(7.15)
for all
E D(A
) = D(A P1
).
In particular, (7.13) and (7.15) imply that
P2
if P1, PZ E Ok then
(7.16)
ak(P1) < llk(PZ)
88
3.
SPECTRAL ANALYSIS OF SOUND PROPAGATION
which proves the weak monotonicity of Ak.
It will be convenient to use
(7.16) to prove Corollary 7.3 before proving the strong monotonicity. To prove Corollary 7.3 note that (7.16) and (7.3) imply that the limit in (7.7) exists and does not exceed c2(_)u0.
But if lim
ak(u)
k
IJ'P
< c2(-)uo then corollary 3.2 implies that u°, ak satisfy
F(p°,ak) = 0;
(7.17)
see (4.28).
Moreover, F(u,A) is analytic at u°, ak by Corollary 3.3.
follows that ak is an eigenvalue of Au
It
and hence u° E 0k by Theorem 7.1.
This contradicts the assumption that 11° is a boundary point of Ok.
To prove that each Ak(u) is strictly monotonic in Ok two cases will be considered.
First, if A1, A2 are in the same component of 0k' say
(a,b) C 0k' then Xk(u1) = Ak(u2) would imply that Ak(u) = const. in [u1,u2] and hence in (a,b), since Ak(u) by Theorem 7.1.
But this contradicts
Corollary 7.3 since
(7.18)
c2(W)a2 = lim Ak(u) < lim Ak(u) = c2(W)b2. uib
In the second case ul and u2 lie in different components of 0 U1 E (al,b1) C Ok and u2 E (a2,b2) C Ok with b1 < a2.
k'
say
In this case, by the
preceding argument one has
(7.19)
ak(in) < c2(co)b2 < c2(-)a2 < ak(u2)
which completes the proof. Proof of Theorem 7.4 (sketch).
The proof is based on the method
proposed for the proof of Theorem 4.16 and the variational principle (7.13).
Note that the hypothesis (7.8) and Theorem 4.16, generalized to non-constant p(y), imply that N(u) - - for u +
Hence No = +- and each 0k is unbounded.
To prove (7.9) choose piece-wise constant functions c°(y) and c°(y) such that
(7.20)
c°(y) < c(y) < c°(y) for all y E R,
(7.21)
c°(y) = cm < c°(m) on an interval 1°, and
(7.22)
cm + e = c°(y) < c°(-) on an interval I(e).
The notation
The Dispersion Relations
3.7.
89
JC°(R) = L2(R,c22(y) P-1(y)dy) (7.23)
7C°(R) = L2 (R, c°-2 (y) P-1 (y) dy) will be used.
The three spaces JC(R), JC°(R) and 7C°(R) have equivalent norms.
In particular, if 110110 and 11$II ° denote the norms in 7C°(R) and JC° (R) , respectively, then by (7.20)
(7.24)
11$4I°
<_
II$II
< II$II°.
Now note that the variational principle can be formulated in the
homogeneous form (A11
(7.25)
X (u) = inf
,$)
sup
heck $EWD(A1)
110112
$#0 where
IR(1$,12 + 1,21$I1)P-1(y)dy (7.26) 11$112
110112
Both the numerator in (7.26) and D(A.) are independent of c(y). A2
Hence, if
and Au denote the operators corresponding to p(y), c2(y) and p(y),
c ° (y) respectively, then (7.24) and (7.26) imply
(AU$,$)
(Aou$,$) (7.27)
< 110112
(AU$,$) <
110112
11$1102
for all 0 a D(A°u) = D(AU) = D(A°) such that 0 # 0.
The variational
principle (7.25) and (7.27) imply that
(7.28)
a°k(P) < ak(u) < ak(u)
for all sufficiently large p, where {X2k(u)} and {X (u)} are the eigenvalues of A°u and Au, respectively.
The proof of (7.9) can now be completed by showing by direct calculation that (see [26])
(7.29)
lim U_2 a°k(u) = cm
P-
90
3.
jj
lim
(7.29 cont.)
P-2
SPECTRAL ANALYSIS OF SOUND PROPAGATION
ak(u) _ (cm + E)2
It follows from (7.28), (7.29) that
(7.30)
c2 < lim inf 11-2 Ak(u) < 1im sup u-2 Xk(u) < (cm + W. Pu-1-
Equation (7.9) follows because, by hypothesis, e > 0 is arbitrary. The result (7.10) is immediate from Corollary
Proof of Corollary 7.5.
3.2, the relation
3(y,u,ak(u)) k (y,11)
(7.31)
_
and Theorem 7.1.
§8.
THE SPECTRAL FAMILY OF A
The acoustic propagator A was defined in Chapter 2 and shown to be a selfadjoint non-negative operator in the Hilbert space K. the spectral family {ll(X)
I
In this section
X > 0} of A is constructed by means of the
normal mode functions
The method of construction is to
use Fourier analysis in the variables x E R2 to reduce A to the operator Alp, and then to use the spectral representation of IIIPI(a) developed in §6.
The construction is given in Theorems 8.1-8.4.
In the remainder of the
section the proofs of Theorems 8.1-8.4 are developed in a series of lemmas and auxiliary theorems.
The formal definitions of the normal mode functions +(x,y,p,a), o(x,y,p,X) and 1Pk(x,y,p) were given in §1, equations (1.14)-(1.16) and (1.21)-(1.26).
The definitions were completed by the construction of the
special solutions 0i(y,u,X) in §3 and of the normalizing factors a+(u>a), ao(u,X), ak(j1) in §6.
The construction of 11(A) will be based on these
normal mode functions and the corresponding generalized Fourier transforms. Formally the latter are the scalar products of functions f E 3C with the normal mode functions.
(8.1)
(8.2)
The following notation will be used.
f(x,y) c-2(y) P-1(y) dxdy,
f+(P,A) = J
R3
J
R3
f(x,y) c-2 (y) P-1(y) dxdy,
fo(P,A) =
The Spectral Family of A
3.8.
(8.3)
fk(P) _
3
91
f(x,y) c -2(y) P-'(Y) dxdy, k > 1.
1Pk(x'Y'P)
R Of course these integrals need not converge since the normal mode functions are not in C.
Instead, they will be interpreted as Hilbert space limits as
in the Plancherel theory of the Fourier transform.
This interpretation will
be based on the following three theorems. Theorem 8.1.
If f E L1(R3) then the integrals in (8.1), (8.2), (8.3)
are absolutely convergent for (p,X) E St, (p,X) E Q. and p E S2 k,
respectively, and
f+ E C(O), fo e C(00), fk E C(Ok), k > 1.
(8.4)
For each f E JC and M > 0 define
f(x,y) if IxI < M and IyI < M, (8.5)
fM(x,Y) = i
l
0
if IxI > M or IyI > M.
It is clear that fM -). f in JC when M
Moreover, fM E JC n L1 (R3) and
one has Theorem 8.2.
(8.6)
For every f E JC and M > 0,
fM± E L2(0), fMo E L2(0(,), fMk E L2(Qk), k > 1,
and the Parseval relation holds: _
(8.7)
IIfMIIx = IjM+112
NoC-1
+ IIfM-IIL2(0) +
k=O
IIfMkIIL2(ok)'
The relation (8.7) suggests the introduction of the direct sum space
No-1 (8.8)
Jf = L2 (O) + L2 (O) +
L2 (Q ) . k=O
JC is a Hilbert space with norm defined by
No-1 (8.9)
see [5, p. 1783].
sequence fM =
(8.10)
IhIIjC = Ih}I22(S2) + IIh-IL2(S2) + kIO IIhklIL2(Stk) Theorem 8.2 implies that for each f E JC and M > 0, the E JC and
IIfMIIx = IIfMIlk.
92
3.
SPECTRAL ANALYSIS OF SOUND PROPAGATION
For arbitrary f E JC the generalized Fourier transforms associated with A are defined by Theorem 8.3.
M
For all f E JC, ifM} is a Cauchy sequence in Jf, for
-, and hence
(8.11)
lim fM = f = M4.
exists in C.
In particular, each of the limits
f+ = L2(Q)-lim fM± M->w
(8.12)
fo = L2 (S2o)-lim fMo M-K-
fk = L2(Qk)-lim fMk, k > 1, M-
exists and the Parseval relation _
No-l
Ilf_IIL2(2)
lfljf = f ic = If+ 11L2(Q)
(8.13)
_ 112
+
+
k=O
k)
holds for every f E X. Theorem 8.3 associates with each f E JC a family of generalized Fourier transforms f =
E Jf such that rM_
(8.14)
f+(p,X) = L2(S2)-lim
i+(x,Y,P,X) f(x,Y) C -2(y) P-1(Y)dxdy,
J
M-
J
Ixl<M
M
rM
(8.15)
fo(P,X) = L2(20)-lim
', (x,Y,p,A) f(x,Y)c-2(Y) P-1(Y)dxdy,
M
1
-M
-M
J
(M
(8.16)
1
IxI<M
r
fk(p) = L2(Qk)-lim J M-+-
k(x,Y,P) f(x,Y) c-2(Y) P 1(Y)dxdy,
Ixl<M
k > 1.
It is easy to verify that if f E JC n L1(R3) then the functions
defined by Theorems 8.1 and 8.3 are equivalent and hence the notation is unambiguous.
A construction of the spectral family {R(U)}
based on these functions is described by Theorem 8.4.
relation
For all f,g E JC and all real U > 0, 1(p) satisfies the
3.8.
The Spectral Family of A
93
(f,R(U)g) = fH(P - A) (f+(P,A) 8+(P,A) + f_(P,A) g_(P,A)dpda O (8.17)
+J
H(P - A) fo (p,X) go (p,A)dpda 0a
No-1
+
H(u - ak(IPI)) fk(P) gk(P)dP
1
k=
=1
SZk
where H(p) = 1 for p > 0 and H(p) = 0 for p < 0. The remainder of §8 presents the proofs of these theorems.
The proof
of Theorem 8.1 will be based on Lemma 8.5.
The normal mode functions satisfy
]P+(x,Y,p,a) E C(R3 x 0), (8.18)
0p(x,Y,p,A) E C(R3 X Qo),
k(x,Y,P) E C(R3 X SZk), k > 1.
Moreover, for each compact set K C 0 there exists a constant MK such that
MK for all (x,y) E R3 and (p,A) E K.
(8.19)
Similarly, for each compact K C S20 there exists a constant MK such that
MK for all (x,y) E R3 and (p,X) E K,
(8.20)
and for each k > 1 and compact K C SZk there exists a constant MK such that
(8.21)
I'k(x,y,p)I < MK for all (x,y) E R3 and p E K.
Proof of Lemma 8.5.
(8.22)
To prove (8.18) note that, by (1.16), (1.21)
Ip+(x,y,P,A) = (2Tr)-3 elp-x a+(Ipl,a) $,,(Y,IpI,a)
where a+(u,X) is defined by (6.14). follows from Corollary 3.2.
The continuity of q,,(y,Ipl,A) on R x 0
The continuity of a+(Ipl,A) on 0 follows from
(6.10), (6.14) and the assumed continuity of the phase function 0+(}1,a). Thus the continuity of ip+ on R3 x 0 follows from (8.22).
The proofs for i4-
94
and
3.
,
SPECTRAL ANALYSIS OF SOUND PROPAGATION
are similar and will not be given.
The continuity of k, k > 1,
follows from Corollary 7.5.
To prove (8.19) for + note that (8.22) and the continuity of a+(IPI,X) imply that it is enough to prove the existence of a constant MK such that
IP4(Y,IPI,A)I < MK for all y E R and (p,A) E K.
(8.23)
Now the uniformity of the asymptotic estimates (3.11) on the compact sets r4 of Corollary 3.4 implies that
(8.24)
4(y,IPI,X) = exp {-iy 4_(IPI,A)}[l + o(1)] , y
uniformly for (p,A) E K.
Hence, there exists a constant yK such that
IP4(y,IPj,a)j < 2 for all y < -yK and (p,A) E K.
(8.25)
Similarly, using the relation
(8.26)
$,(y,u,X) = c41(u,)) $1(y,p,X) + c42(u,X) 42(y,u,A)
from (5.1), (5.2), (5.3), the continuity of c41(p,A) and c42(}i,X) and the
uniformity of the asymptotic estimates for 41, P2 when y - +°, one finds that there exist constants yK, MK such that
I04(y,1PI,A)I < M. for all y > yK and (p,A) E K.
(8.27)
Finally, the continuity of 4(y,Ipl,A) on R x 0, which follows from Corollary 3.2, implies the existence of a constant MK such that
IP,(y,IPI,X)I < MK for -yK < y < yK and (p,A) E K.
(8.28)
Combining (8.25), (8.27) and (8.28) gives the estimate (8.23) with MK = Max (2,MK,MK).
The proofs of (8.19) for 4_ and of (8.20) and (8.21) can be given by the same method.
This completes the discussion of Lemma 8.5.
Proof of Theorem 8.1.
Consider the function ff(p,X).
The absolute
convergence of the integral in (8.1) for each (p,A) E H follows from (8.19).
To prove that f+ E C(H) let (po,ao) E H and let K C H be compact
and contain (po,ao) in its interior.
Then by Lemma 8.5
3.8.
The Spectral Family of A
(8.29)
95
I4+(x,y,p,X) f(x,Y)I < MK If(x,y)I for (x,y) E R3, (p,X) E K.
Hence the continuity of f+ at (pa,A0) follows from (8.18) and (8.29) by Lebesgue's dominated convergence theorem. follows by the same argument.
The continuity of f_, fo and fk
This completes the proof of Theorem 8.1.
Relationship of A to Ap.
As a preparation for the proofs of Theorems
8.2, 8.3 and 8.4 the operator A will be related to AIpI by Fourier analysis in the variables x E R2.
To this end note that if u E JC then Fubini's
theorem implies that F
:
E L2(R2) for almost every y E R.
Thus if
L2(R2) - L2(R2) denotes the Fourier transform in L2(R2) then the
Plancherel theory implies that
(8.30)
G(p,y) _ (Fu)(p,y) = L2(R2)-lim (27r)-' M-+-
e-'p 'x u(x,y)dx
J
I x I <M
exists for almost every y E R and
(8.31)
JR2 Iu(p,y)12 dp = JR2 Iu(x,Y)I2 dx for a.e. y E R.
Another application of Fubini's theorem gives
Lemma 8.6.
u E JC if and only if 0 = Fu E JC and the mapping F
:
JC - JC
is unitary. In particular (8.32)
II0IIx =
IIulI,t for all u E
C.
The Fourier transform of the acoustic propagator A will be denoted by A.
Thus
A= F A F-' , D(A) = F D(A).
(8.33)
A more detailed characterization of D(A) is needed to relate A to AIpI. will be based on
Lemma 8.7.
Let u E JC.
Then Dju e JC (j = 1,2) if and only if
pju(p,y) E K and (8.34)
F Dju = pj F u, j = 1,2.
Similarly, Dyu E JC if and only if Dyu E JC and
(8.35)
F Dyu = Dy F u.
It
96
3.
Proof of Lemma 8.7.
SPECTRAL ANALYSIS OF SOUND PROPAGATION
The distributional derivatives DJu, Dyu may be
characterized as temperate distributions on the Schwartz space S(R3) of rapidly decreasing testing functions [10]. F.
S(R3) is mapped onto itself by
The proof of (8.34) is essentially the same as in the standard
Plancherel theory.
To verify (8.35) note that the distribution-theoretic
definition of D u E JC is y (8.36)
JR3 Dyu(x,y) $(x,y)dxdy = -JR3 u(x,y) Dy$(x,y)dxdy for all $ E S(R3).
Application of Parseval's relation gives
(8.37)
JR3 (F Dyu)$ dpdy = -JR3 (Fu)(F Dy$)dpdy for all $ E S(R3).
Now for $ E S(R3) it is easy to verify that
(8.38)
D
y
D.
0 = Dy F0 = F Dy$
Thus (8.37) is equivalent to
(8.39)
JR3
(F Dyu)$ dpdy = -JR3 (Fu) D $ dpdy for all $ E S(R3) y
which in turn is equivalent to (8.35). Application of Lemma 8.7 to A gives Lemma 8.8.
The operator A is characterized by the relations
F L2'(R3) _ {u
(8.40)
(8.41)
D(A) = F L2'(R3) n {u
(8.42)
plu, p2u and Dyu are in NJ,
Dy (P-'D yu) - 1p!2 u E JC}, and
u = -c2 {PDy(P-'Dyu) - IpV2 u}, u E D(A).
Proof of Lemma 8.8.
These results follow from application of Lemma
8.7 to the definition of LZ(R3), D(A) and A - equations (2.1.12), (2.2.3) and (2.2.4). Corollary 8.9.
(8.43)
For all u E D(A) one has
E D(AIp1) and
3.8.
The Spectral Family of A
(8.44)
97
(Au)(P,-) = AIPI u(P,-)
for almost every p E R2.
Corollary 8.9 is an immediate consequence of Lemma 8.6, Lemma 8.8 and Fubini's theorem.
The Sets JCcom, JC' , JC' and JC" . The following subsets of JC will be used in the proofs of Theorems 8.2, 8.3 and 8.4.
(8.45)
JCCO1° = JC n {f
(8.46)
I
supp f is compact},
= F-'D(R3) = {f
J('
I
f = Ff E D(R3)},
(8.47) JC'={(f(x,y)=f1(x)f2(Y) I f1EF-1D(R2),f2EJC(R),supp f2 compact},
3C" = span JC' =
(8.48)
as fa
f =
I
as E C,fa E JC'
a=1
In (8.46) and (8.47), D(Rn) denotes the Schwartz space of testing functions with compact support [101.
The sets
folds of JC which are dense in JC.
dense in L2(R3).
JCcom,
JC' and JC" are linear submani-
Indeed, it is well known that D(R3) is
This fact implies that J(com is dense in JC.
of JC in JC follows from that of D(R3) and the unitarity of F.
The denseness The
denseness of JC"' follows from the fact that JC is the tensor product of
L2(R2) and X(R) . It is clear that each of the sets JCcom, JC' and JC" is a subset of JC n L1(R3).
Hence, for f in one of these sets, the transforms f+, fo and
fk defined by (8.1)-(8.3) are continuous functions by Theorem 8.1. alternative characterization is given by
Lemma 8.10.
If f E JCcom U JC' U JC" then I+(Y, P ,a) f(P,Y) c
(8.49)
2(Y)
P 1(Y)dy,
J( R
(8.50)
o(Y, P ,A) f(P,Y) c 2(Y) P(Y)dy,
J
R
(8.51)
fk(P) =
Tk(Y, P ) f(P,Y) c -2(y) P-1(Y)dy, k > 1.
1
R
An
98
3.
Proof of Lemma 8.10.
SPECTRAL ANALYSIS OF SOUND PROPAGATION
Equations (8.49)-(8.51) follow from (8.1)-(8.3)
on substituting the definitions (1.21)-(1.23) of the normal mode functions and carrying out the x-integration.
These operations are justified for
f E L1(R3) by Lemma 8.5 and Fubini's theorem. Corollary 8.11.
If f E JC' and f(x,y) = fl(x) f2(y) then
(8.52)
f+(P,X) = fl(P)fz±(IPI'X) = (F ID Ipl+fl(P.X) = 10IPI±FfI(P.x).
(8.53)
fo(P,X) = f1(P)fzo(IPI,A) = [F Dlplofl(p,A) = I01ploFfl(P,A),
(8.54)
fk(P) = fl(P)f2k(IPI) _ (F D plkfl(P) = I1DIplkFfl(P)
These results follow immediately from Lemma 8.10 and the results of §8. The notation
R(T,4) _ (T - 0-1
(8.55)
will be used for the resolvent of T.
The proofs of Theorems 8.2, 8.3 and
8.4 will be based on Stone's theorem relating R(A,4) and the spectral family of A, together with the following three lemmas relating A and Alpl. Let f E IC" and let uC = R(A,4)f or, equivalently,
Lemma 8.12.
u =
Then
(8.56)
0 (p,y) = j GIpI(y,y'.4) f(p,y') c-2(Y') P-1(Y')dy' R
for almost every (p,y) E R3 where G"I
Proof of Lemma 8.12. £ E JC'.
Let f(x,y) = f1(x)fz(y) be such a function so that f(p,y)
= f1(p)f2(y).
(8.57)
Now
E D(A) and hence by Corollary 8.9
((A - ouC)(P.Y) = ((AIPI - )u4)(p,y) = f1(P)fz(Y)
for almost every (p,y) E R3.
(8.58)
is the Green's function for A
It is enough to verify (8.56) for functions
It follows that
aC(p,y) _ [R(Alpl.4)f1(P)f2l(Y) = f1(P)(R(Alpl.C)fzl(Y)
which is equivalent to (8.56) because
3.8.
The Spectral Family of A
(8.59)
99
f2(y') c -'(y') P-1(y')dy'.
J
R Lemma 8.13.
Let f(x,y) = f1(x)f2(y) and g(x,y) = g1(x)g2(y) be
elements of JC' and let C = A + it with A E R and e > 0.
Then for all p E R
one has u
1-1 (8.60) (u
_
f1(P) g1(P) J1
= 1R2
Proof of Lemma 8.13.
The Plancherel theory implies that
(f[R(A,4) - R(A,c)Jg) = (f,[R(A,4)
(8.61)
Combining this and (8.58) gives, by Fubini's theorem
(f,[R(A,C) - R(A,4)]g) (8.62)
R(AIpI,4]g2)dp.
JR 2
Now a standard estimate for the resolvent of a selfadjoint operator [11, p. 272] implies that
(8.63)
E Ifl(P)81(P)IIIf2fl1g21I
Thus integrating (8.62) over -1 < A < u gives (8.60) by Fubini's theorem. Lemma 8.14.
The spectral family {II(u)) satisfies relation (8.17) of
Theorem 8.4 for all f,g E JC". Proof of Lemma 8.14.
Stone's theorem in its general form is [25,
p. 791 (b
1im 1 J e-O+
(f,[R(A,A+ie) - R(A,A-iE)Jg)da
a
(8.64)
_ (f,[R(b) + 11(b-) - II(a) - II(a-)]g).
In the present case 6(A) C [0,00) and (8.64) implies
100
SPECTRAL ANALYSIS OF SOUND PROPAGATION
3.
V1
(8.65)
Z(f,[IT(}i) + II(u-)]f) = 2Ii lim
J-
e-*O+
(f,[R(A,X+iE) - R(A,A-iE)]f)da. 1
Moreover, since R(u-) = lira 11(u - 6) and lira 11((}i- 6)-) = II(u-), (8.65) 6-*0+ 6-*0+
implies
('u (8.66)
lim
(f,ll(p-)f) = 2Li lim
6-0+ t-'O+
(f,[R(A,A+iE) - R(A,X-ie)]f)da.
J
-1
If f(x,y) = f1(x)f2(y) E JC' then combining (8.60) with g = f and (8.66) gives
(f,R(V-)f)
(8.67) u-6
(
lim
21
lim E-*0+
1
R2
(f2,[R(Alpl,A+it)
If1(p)12 I _1
- R(Alpl,X-it)]f2)dadp.
Now application of the spectral theorem to Alpl gives
(8.68)
2ic
(f2,[R(A,A+ie) - R(A,A-iE)]f2) = IPI
IPI
R
(f2,II
(a-a) +E
IPI
(da')f2).
It follows that U-o
- R(Alpl,A-ic)]f2)da
(f2,[R(AIPI,A+it)
I-1
(8.69)
u-6 I
R
I
-1
l
(A-a
(f2,111P1(da')f2).
dal J
Moreover, u-6
(A-A')2+c2 da
(8.70) I-1
for all A' E R and all E > 0.
2
- IR (a-A
)2+E2 dA = 2
Combining (8.69) and (8.70) gives
W-6 (8.71)
(f2,[R(Alpl,A+it) - R(AIPI,A-ic)]f2)dAl < 2,r
J_
11f211
1
for all p E R2 and all u > 0, 6 > 0 and e > 0.
In addition Stone's theorem
applied to Alpl gives 1 2-r1
u-6
lim
lim
6-0+ c-'O+
(8.72)
(f2,R(Alpl,A+ie) - R(Alpl,A-iE)1f2)da = (f2,nlpl(p-)f2). I
3.8.
The Spectral Family of A
101
Equations (8.67), (8.72) and the estimate (8.71) imply, by Lebesgue's dominated convergence theorem,
(8.73)
(f,n(u-)f) =
If1(P)I2 (f2+"IPI(u-)f2)dp
2 J
R
It follows by polarization that
(8.74)
(f,n(u-)g) = JR 2 f1 (P) g1(P)(f2,1IPI(u-)g2)dp
for all f,g E 3C'.
The same argument applied to (8.65) gives the relation
(8.75)
(f,(n(u) + 11(1i-)1g) = 1 2 f1 (P) B1(P)(f2,(nlpI(p) + nlpl(u-)192)dp.
R Subtracting (8.74) from (8.75) gives
(8.76)
(f,n(p)g) =
f1(P) 81(P)(f2,11IPI(p)g2)dp
12
R for all f,g E 3C'.
To prove the relation (8.17) for f,g E 3C' note that the
construction of nu given by (6.27) implies, by polarization
(f2,11Ip1(11)g2)
(8.77)
H(u - A)(f2+(IPI,A) g2+(IPI,A) + 12_(IPI,a) g2_(IPI,A)}da
A(IPI) +
H(11- A) f20(Ipl,A) g20(lpl,a)dA
I
A0(IPI) N(IPI)-l
+
L
H(u - Ak(IPI)) f2k(lpl) g2k(Ipl)
k=l
for all p E R2 such that IpI > 0, where
A(IPI) = {a
I
c2(-oo)IPI2 < X}, and
(8.78)
A0(IPI) _ {A
I
c2(°O)IPI2 < A < c2(-')IPI2}.
Substituting this into (8.76) and recalling the definitions of 0, Q S2k (k > 1) gives (8.19) for f,g c= 3C'.
to all f,g E 3C" by linearity.
and
The relation extends immediately
This completes the proof of Lemma 8.14.
102
3.
Proof of Theorem 8.2.
SPECTRAL ANALYSIS OF SOUND PROPAGATION
Let f E Jf and M > 0 be given.
Then since JC"
is dense in Jf there exists a sequence {gn} in JC" such that gn - fM in X.
Note that since
M IIfM - gn112 = 1_
f
M
If(x,Y) - gn(x,Y)IZ c-2(y) P-1(Y)dxdy
R2
(8.79)
Ign(x,Y)I2 C -2(y) P-1(Y)dxdy IYI>M
RZ
it may be assumed that gn(x,y) = 0 for IyI > M.
Now Lemma 8.14 implies
that Parseval's relation
No-1
(8.80)
IlglI
=
Ilgll
= IIg+II2 + IlgIIZ +
k=
IIgkII2
=0 holds for all g E JC" , where g = Cg F,g_,go,gl. " ')
the differences gn - gm it is found that in Cauchy sequence in
.
On applying (8.80) to
= (gn+'gn-'gno'gn1' ) is a
Hence there exists a limit
lim in = h = (h+,h_,ho,hl,...) E 3f, n-'m
(8.81)
since 3C is a Hilbert space.
To complete the proof of Theorem 8.2 it will be
enough to show that
fM±(p,a) = h+(p,A) for a.e. (p,A) E 2, (8.82)
fMo(p,X) = ho(p,X) for a.e. (p,A) E 0o,
fMk(p) = hk(p) for a.e. p E SZk' k > 1.
This clearly implies (8.6) since h E C.
Moreover, since Hilbert space
convergence implies convergence of the norms, the relation (8.80) for gn E or" implies N
(8.83)
IlfMlix =
Ilgnll;C =
n
n
Ilgnll
=
Ilhllit = IIh+II2 + Ilh_IIZ +
o-1
kIo
IIhkII2
which is equivalent to (8.7) when (8.82) holds. Relation (8.82) will be proved for fM+. cases are entirely similar.
The proofs for the remaining
To prove that fM+(p,X) = h+(p,X) for a.e.
(p,X) in Q note that if K is any compact subset of 0 then fM+ E C(K) C LZ(K) by Theorem 8.1 and
3.8.
The Spectral Family of A
(8.84)
IIfM+
103
n
h+IIL2(K)
-
IIfM+ - gn+IIL2(K)
Now by Lemma 8.10
(8.85) fM+(P'
lU+(Y, P ,a) [fM' y) - gn(P,y)]c-2(Y)P-1(y)dy
-
J
M
Hence by Lemma 8.5 and Schwarz's inequality
_
M
2(Y)P-1(Y)dY
IfM(P,Y) - gn(P,Y)I c
IfM+(P,A) - gn+(P,X)I < MK j-
M (8.86)
M
1/2
_2
MK J_M c
(y)P-1(Y)dY
M
lI
{JMMP.Y)
1/2
- Sn(P,Y)I2c-2(y)P-1(Y)dy )
for all (p,A) E K.
(8.87)
It follows that there is a constant C = C(K,M) such that
IIfM+ -
Since gn - fM in 3C,
gn+IlL2(K) < C IIfM - gnllK = C IIfM - gnllJX.
(8.87) implies that the limit in (8.84) is zero and
hence fM+(p,X) = h+(p,A) for a.e. (p,X) E K.
This completes the proof
since K C 2 was an arbitrary compact set. Proof of Theorem 8.3.
for M
To prove that {fM} is a Cauchy sequence in 3C
let M > 0 and N > 0 be arbitrary numbers and let {gM},
sequences in 3C" such that gMn - fM , 9N n - f N in X. M N in
n
( N} be n
Then, as proved above, N
M
fM and in - fN in X and Parseval's relation (8.80) holds for gn
- gn-
Passage to the limit n -> m gives
11fm - fNli;' = IIfM - fNII;Z
(8.88)
which implies that {fm) is a Cauchy sequence because fm -r f in JC when M -r w.
Finally, one gets (8.13) for arbitrary f c 3C by passage to the
limit in (8.9).
Proof of Theorem 8.4.
It will be enough to prove (8.17) for f = g E JC
since the general case then follows by polarization.
Now by Lemma 8.14
H(11- A)(Ig+(P,A)I2 + Ig_(P,X)I2)dpol
(g,IT (U)g) = J
(8.89)
+J
H(u - ),) I8o (P,),) I2 dpda
No-1 +
H(u - Xk(lpl)) Igk(P)IZ dP k=1
J Qk
104
3.
for all g C JC".
Let f e JC, M > 0 and let {gn} be a sequence in SC" such
that gn - fM in If.
Then it follows from the proof of Theorem 8.2 that
Replacing g by gn in (8.89) and making n 4 1 gives (8.89)
gn - fM in K. with g = fM.
SPECTRAL ANALYSIS OF SOUND PROPAGATION
If No = +' then passage to the limit is justified because
the right-hand side of (8.89) is majorized by
No-1 (8.90)
+
+
IIgk"2 < °°.
L
k=0
Thus (8.89) is valid with g = fM where f E JC and M > 0 are arbitrary. Making M + - and repeating the above argument gives (8.89) with g = f E JC, by Theorem 8.3.
Another proof may be obtained by noting that the left-hand side of (8.89) is a bounded quadratic form on JC, while the right-hand side is
a bounded quadratic function of g = 9'g because of the majorization by (8.90).
Thus (8.17) follows from the boundedness of 'Y and the fact that
(8.89) holds for g in the dense set JC"'.
§9.
NORMAL MODE EXPANSIONS FOR A The normal mode expansions for the acoustic propagator A that are the
main results of this chapter are formulated and proved in this section. The starting point is the representation of the spectral family of A given The main result, Theorem 9.8, shows that the family
by Theorem 8.4.
is a complete orthogonal family of normal modes for A. Theorem 9.9 shows that it provides a spectral representation of A.
These
} and
results are shown to imply that the families
defined in §1, are also complete orthogonal families of normal modes for A and provide alternative spectral representations. The basic representation space for A associated with the family is the direct sum space
No-1 JC = L2 (S2) + L2 (S2) +
I
L2 (Qk)
k=0
introduced in §8.
Theorem 8.3 associates with each f E JC an element f E Jf.
The Parseval relation (8.13) implies that the linear operator
(9.2) defined by
Y'
:
J( -rJC
Normal Mode Expansions for A
3.9.
(9.3)
105
for all f E JC
Yf
is an isometry; i.e.,
(9.4)
IITfIIJC =
IfII3C for all f E JC.
The principal result of this section is Theorem 9.1.
The operator Y` is unitary; i.e.,
(9.5)
'Y*
(9.6)
Y Y`= 1 in R.
4' = 1 in JC, and
Relations (9.5) and (9.6) generalize the completeness and orthogonality
properties, respectively, of the eigenfunction expansions for operators with discrete spectra.
Relation (9.5) is equivalent to (9.4) and thus follows
from Theorem 8.3.
Relation (9.6) is shown below to follow from the unitar-
ity of the operator Y'u associated with Au (Theorem 6.6). The completeness relation (9.5) implies that every f C JC has a normal
mode expansion based on the family relation (9.6) implies that the space JC
The orthogonality is isomorphic to JC and thus
provides a parameterization of the set of all states f E JC of the acoustic field.
These implications of Theorem 9.1 will be developed in a series of
corollaries.
The normal mode expansion will be based on the linear operators
(9.7)
Y`+
(9.8)
Y'k
:
JC -; LZ(Q)
:
JC -> LZ (S2k) .
0 < k < No,
defined for all f E JC by (9.9) (9.10)
Y'kf = fk' 0 < k < No,
where f+, fk are defined as in Theorem 8.3.
(9.11)
Y'f =
It is clear that
(`f+f,T_f,Yaf,T, f,...)
106
3.
SPECTRAL ANALYSIS OF SOUND PROPAGATION
for all f E 3C and, by (9.4), No-i
III+fII2 + II'Y_f1I2 +
(9.12)
I
IIv'kfII2 = IIfrr2.
k=0 In particular, each of the operators 'Y+, Y'k is bounded with norm not exceeding 1.
The normal mode expansion for A, in abstract form, is
given by Corollary 9.2.
The family {Y+,Y-, Yo
satisfies N
(9.13)
1
1 = 'Y+ 'Y+ + Y'* Y'- +
Y'k Y'k
k=0 where 1 is the identity operator in lC and the series in (9.13) converges strongly.
It will be shown that (9.13) is equivalent to the completeness relation (9.5).
The orthogonality relation (9.6) will be shown to be equivalent to
the relations described by Corollary 9.3.
The family
satisfies the relations
(9.14)
'Y+ 'Y* = 1 in L2(),
(9.15)
'Yk q'k = 1 in L2 (2 k), 0 < k < No.
In addition, Y'+ T* _ 'Y- 'Y+ = 0, `Y+ 'Yk = 0, Wk 'Y* = 0 and Y'k T* = 0 for all
k and P. # k such that 0 < k, k < No. Relations (9.14), (9.15) imply that each of the operators Y+1 9' `Yk
(0 < k < No) is partially isometric [11, p. 258].
operators in 3C defined by
(9.16)
0
JC+=P+3C
(9.17) Yk = Pk bC, 0 < k < No. Combining this with Corollaries 9.2 and 9.3 gives
It follows that the
Normal Mode Expansions for A
3.9.
107
Corollary 9.4.
is a complete family of orthogonal
projections in JC; i.e., the range spaces defined by (9.17) are mutually
orthogonal and No-1
P+ + P- +
(9.18)
Pk.
E
k=0
The spaces (9.17) are subspaces of 3C and hence the direct sum space 1
N
"k
3f+ + 3C- +
(9.19)
k=0
may be identified with the set of all
No-i
f = f+ + f- + I
fk
k=0
f+E7C+, fkENk for0
Ny
If+II2 + If-IIZ +
(9.22)
Ifk1I2 <
1 k=0
With this convention, Corollary 9.4 implies Corollary 9.5.
7C has the decomposition
R=x++X-+
(9.23)
No-1
I
k=0
Kk
The definitions of the operators T+, Tk and equations (8.14)-(8.16) imply
Corollary 9.6.
(9.24)
(9.25)
The operators Y+, `Yk have the representations
3 +(x.Y.P.a) f(x,Y)c 2(y)P-1(Y)dxdy,
(Y+f)(p,A) = L2(S2)-lim 1
M-
RM
( (`Yof)(p,A) = L2(0 )-lim J
M-
(Tkf)(p) = L2(S2k)-lim
M-
1 R3_ J
o(x,Y,p,A) f(x,Y)c 2(y)P-1(Y)dxdy,
k(x.Y.P) f(x,Y)c-2(Y)P-1(Y)dxdy,
Ixi < M,
for 1 < k < No where PM = R3 n {(x,y) I
Iyl
< M}.
108
3.
SPECTRAL ANALYSIS OF SOUND PROPAGATION
Of course, the family of sets {RM
M > 0) can be replaced by any
M > 0} of compact sets whose characteristic functions tend to
family {KM 1 almost everywhere in R3 when M -* m. I
Corollary 9.6 implies a similar
representation for the adjoint operators Y',*, W. Corollary 9.7.
The operators T*,
'{
It is formulated as
have the representations
(9.27)
(`Y*g+)(x,Y) = JC-lim
M- f2 M
±(x,Y,P,A) g+(P,A)dpdX,
(9.28)
(Togo) (x,Y) = JC-lim J M
(x,y,p,A) g0 (p,A)dpda,
00
M-(9.29)
(`Ykgk)(x,Y) _ JC- lim f M k(x,y,p) gk(p)dp,
M-
S2k
for 1 < k < No where S2M and 0M (0 < k < NO are families of compact subsets of 0 and 0k whose characteristic functions tend to 1 almost everywhere in C and 0k' respectively, when M By combining Theorem 9.1 and Corollaries 9.2, 9.6 and 9.7 the following explicit formulation of the normal mode expansion is obtained. Theorem 9.8.
Every f E JC has a representation No-l
(9.30)
f(x,y) = f+(x,y) + f_(x,y) +
I
fk(x,y),
k=0
convergent in JC, where f+ E JC+ and fk' 0 < k < N0, are given by
(9.31)
f+(x.Y) = JC-lim 1 M (Y+(x,Y,p,X) f+(p,A)dpdX, M-° C
(9.32)
fo(x,Y) = JC-lim J M V' (x,y,p,X) f0(P,A)dpdA, M 00
(9.33)
fk(x,Y) = 3C-lim J M Uk(x.Y.P) fk(P)dP.
(
M-
for 1 < k < N. and f+, f = (f+,f f E JC such that f+,
Theorem 9.9.
QM
are defined by (8.14)-(8.16). Conversely, if k is any vector in JC then (9.30)-(9.33) define a vector
are related to f by (8.14)-(8.16). k The unitary operator T defines a spectral representation
for A in 3C in the sense that for all f E D(A) one has
(9.34)
(`Y+A f) (P,A) = a(1'+f) (P,A) = A f+(p,a)
Normal Mode Expansions for A
3.9.
109
(9.35)
('P0A f) (p,A) = a(`Yof) (p,A) _ A fo (P.a)
(9.36)
(4'kA f)(P) = Ak(IPI)(''kf)(P) = ak(IPI) fk(P)'
for 1 < k < No. Corollary 9.10.
The complete family of orthogonal projections
{P+,P-,Po,P,, } reduces A; i.e.,
P+A C A P+, PkA C A Pk
(9.37)
for0
Ak = A Pk = Pk A Pk, 0 < k < No , denote the parts of A in x+ and JCk then
No-1
A=A++A_+ I
(9.39)
Ak.
k=0
The Family {O+(x,y,p,q)
I
(p,q) E R3 - N}.
It will now be shown how
the normal mode expansion of Theorem 9.8 can be reformulated in terms of To begin, equation (1.44) for the normalizing
the family
factor c(p,q) will be verified. First, recall that the normalizing factors a+(p,A), ao(p,A) for are related to the factors c+(p,A), co(p,A) in their asymptotic forms, equations (1.18)-(1.20), by
c+ = T+1 a+, co =
(9.40)
see (5.8), (5.14), (5.21).
T01
so;
Combining (9.40) with equations (5.10), (5.16),
(5.23) for T+, To and equations (6.10), (6.14), (6.19), (6.21) defining a+, ao gives
(9.41)
+ (4Tr
1/2
q+(P,X))
provided that the phase factors
ei9±,
l1/2 14Tr
ei0o are defined by
110
SPECTRAL ANALYSIS OF SOUND PROPAGATION
3.
ie+(u,X)
i[0021 [0321
X e A(u),
I
(9.42)
iea w')') e
=
i[
3
21
A E A0(p).
On combining (9.41) and the definition of +(x,y,p,q), equations (1.38) and (1.39), and the asymptotic forms (1.18)-(1.20) for P+, , one obtains the asymptotic forms (1.42), (1.43) for + with the normalizing factor c(p,q) defined by (1.44).
To derive the normal mode expansion for the family that for
from
it is clearly sufficient to restrict attention
to the subspace
1Cf - X+ + 3C- + IC,
(9.43)
and the corresponding orthogonal projection
Pf = P+ + P_ + Pa.
(9.44)
Thus if h
3C and hf = Pfh = h+ + h_ + ho then Theorem 9.8 implies that
+(x,Y,P,A) h+(p,X)dpda + f
hf(x,Y) = J Q
(x,y,P,A) ho(p,x)dpd), Q, Oa
(9.45)
+ J
_(x,Y,P,a) h_(p,A)dpda
SZ
where the integrals converge in X.
Changing the variables of integration in
the three integrals by means of the mappings X+, X0 and X_, respectively,
and using the definition (1.38), (1.39) of + gives
h f(x,y) = J
+(x,Y,p,q) h+(P,X) c(=) (2q) 1/2 dpdq C+
(9.46)
+ J Co 4+(x,Y,p,q) ho (p,A) c(-)(2q)1/2 dpdq. + J C +(x,Y,p,q) h_(p,A) c(-°°)(21gl)1/2 dpdq = JR3 0+(x,Y,p,q) h+(p,q)dpdq
Normal Mode Expansions for A
3.9.
111
where A = X(p,q) is defined by (1.41) and
(9.47)
(2q)1/2 c(°°) h+(p,A(p,q)),
(p,q) E C+.
(2q)1/2 c(m) ho(p.a(p,q)),
(p,q) E Co.
(2lgl)'/2 c(-°°) h_(p,A(p,q)),
(p,q) E C_,
+(p,q) _
(p,q) E N.
0,
It is easy to verify by considering the three cones C+1 Cu and C_ separately that
(9.48)
h+(p.q) = J
3
0+(x.Y.p.q) h(x,y) c-2(Y) P-'(y)dxdy
R
where the integral converges in L2(R3).
Moreover, it can be verified by
direct calculation, using the Parseval relation of Theorem 8.3, that
11hf"X _ Ilh+l1L2(R3)
(9.49)
for all h E K.
These considerations suggest
Theorem 9.11.
(9.50)
exists.
For all h E Jf the limit
+(x,y,p,q) h(x,Y)c 2(Y)P-'(Y)dxdy
+(p.q) = L2(R3)-lim j
M-
3
RM
Moreover, the mapping 0+
Jf - L2(R3) defined by D+h = h} is a
:
partial isometry such that
(9.51)
(D + ID + = 1 and 0+ CD+ = Pf'
and the adjoint mapping hf = 0+ h+ is given by
(9.52)
+(x.Y,p,q) h+(p,q)dpdq.
hf(x,Y) = JC-lim
M-
(R -N)M f'
Finally, 0+ is a spectral mapping for A in the sense that for all h E D(A) one has
(9.53)
(41+A h)(p,q) = A(p,q) (D+h(p,q)
where X(p,q) is defined by (1.41).
112
SPECTRAL ANALYSIS OF SOUND PROPAGATION
3.
Note that Theorem 9.11 is simply a reformulation of Theorem 9.8 and not a new theorem. The Family
The analogue of Theorem
will be formulated as
Corollary 9.12.
(9.54)
(p,q) E R3 - N}.
I
9.11 for the family
For all h E N the limit
h_(p,q) = L2(R3)-lim J
M
exists and the mapping (D_
:
$_(x,y,p,q) h(x,y)c 2(y)P-'(y)dxdy RM
rC-+ L2(R3) defined by 0_h = h_ is a partial
isometry such that
(D- 0 = 1 and (P* (D_ = Pf.
hf(x,y) = JC-lim J
_(x,y,p,q) h_(p,q)dpdq RM
M-
Finally, for all h E D(A), one has
($ A h)(p,q) = A(p,q) @ h(p,q)
(9.57)
These results are direct corollaries of Theorem 9.11.
This follows
from the observations that f(p,q) ' f(-p,q) defines a unitary transformation in L2(R3) while f -+ f defines a unitary transformation in both IC and L2(R3).
This completes the formulation of the results of §9.
The proofs will
now be presented.
The proofs of Theorem 9.1 and Corollaries 9.2 and 9.3 will be based on a lemma concerning bounded linear operators from a Hilbert space into a direct sum space.
To formulate it let X and JCk (k E N) denote Hilbert
spaces, where N is a finite or denumerable set, and define
N=kENXk.
(9.58)
Elements of JC will be written g = {gk} where gk E Nk and
(9.59)
IISkllk <
IIgIl2
kEN
if 11-11k is the norm in Nk. (9.60)
If
3.9.
Normal Mode Expansions for A
113
is a bounded linear operator then
Bf = {(Bf)k} _ {Bkf}
(9.61)
where
(9.62)
Bk
is a bounded linear operator.
Lemma 9.13.
:
With this notation one has
The adjoint operator B* B*g =
(9.63)
JCk
JC
:
JC -+ JC is given by
Bk gk, g = {gk} E JC,
I
keN
where the series converges in X.
Hence
(9.64)
B*B =
Bk Bk,
kEN
where the series converges strongly, and
B B*g =
(9.65)
'{
I
Bk B
gj},
g = (gk}
jEN
where the series converge in "kProof. of B*.
If N is finite then (9.63) follows directly from the definition
If N is denumerably infinite then (9.63) holds for all g C 3C with
finitely many non-zero components gk. 3C by such vectors and B* is bounded. (9.63) follows. (9.61).
But any g E JC can be approximated in The convergence in 3C of the series in
Equation (9.64) follows on applying (9.63) to the vector
(9.65) follows on applying B to the vector (9.63).
Corollary 9.14.
B*B = 1 in 3C if and only if
Bk Bk = 1 in 3C,
(9.66)
kEN
the series converging strongly. Corollary 9.15.
(9.67)
B B* = 1 in 3C if and only if
Bk B = djk for all k,j E N
where 6jk = 1 if j = k and 6 jk = 0 if j # k.
114
SPECTRAL ANALYSIS OF SOUND PROPAGATION
3.
(9.66) is an immediate consequence
Proof of Corollaries 9.14 and 9.15. of (9.64) and B*B = 1.
(9.67) follows from B B* = 1 and (9.65) on taking
g(j) - {6jk gk}, j fixed.
On applying Corollary 9.14 to the direct sum space (9.1) and the operator Y' it is seen that (9.13) of Corollary 9.2 is equivalent to (9.5) of Theorem 9.1.
Similarly, Corollary 9.15 implies that the relations of
Corollary 9.3 are equivalent to (9.6) of Theorem 9.1. Proof of Corollary 9.2.
It was shown that (9.5) follows from Theorem
Thus (9.13) is valid by Corollary 9.14.
8.3.
Proof of Corollary 9.3.
The spectral mapping Y= (Y
for Au is unitary for all p > 0 by Theorem 6.6.
Y
9.15 that the analogue of Corollary 9.3 holds for Y U; see (6.38).
be shown that Corollary 9.3 follows from these relations. the relation To Ya = 1 will be proved.
Y
)
It follows from Corollary It will
For brevity only
The remaining relations can be
proved by the same method. T* = 1 define
For the proof of 'V
(9.68)
Y'o ='YO
1
:
3f - LZ(Q0).
Then
(9.69)
Ya Y* o = o
and it will suffice to prove that 'Yo Pa = 1.
Y* o
The following lemma will be
used.
Lemma 9.16.
(9.70)
For all g E D(120) one has
(Yo g)(P,y) = JA0(JPI) 4o(y,IPI,X) g(P,X)dX = [YJPjo g(P,')](y)
Proof of Lemma 9.16.
Let f e Jf' = F-1 D(R3).
Then f E D(R3) and by
Lemma 8.10 one has (Y'ag,f) _ (g,Yof) = (g,Yof) _ (g,fo) (9.71)
Vo(y, j S2o
g(P,X)
P ,X) f(P,y)c Z(y)P 1(y)dpda.
I
R
Now o(y,jpl,A) is defined for (y,p,A) E R x Q.. all (y,p,X) E R x R3 by
Extend the definition to
3.9.
Normal Mode Expansions for A
115
vO (Y,IPIX) = 0 for y E R, (p,A) E R3 - Qo.
(9.72)
and apply Fubini's theorem to the integral in (9.71).
This gives
f(P,y)c-2(y)P-1
(9.73)
(`fog.f) = JR3 [JR o (y, P ,A)g(P,X)
dXj
(y)dPdy.
which implies (9.70) because D(R3) is dense in 7C and supp Po(y.IPI,-)
C Ao(Ipl), by (9.72). Proof of Corollary 9.3 (completed).
The relation To Y'o = 1 is
equivalent to
(9.74)
(f,g)
('fo 'Yuf+g) =
for all f,g E L2(00).
Moreover, since 'Po is bounded it will suffice to
verify (9.74) for f and g in the dense set D(00).
Now Lemma 9.16 implies
that for all f,g E D(0o) one has
1R3
[TIPIof(P,'))
(y) [Tlplog(P,.))(y)dpdy
(9.75)
=1
(TIplof(p .),T*Plog(P,'))dp. R2
But by (6.38), "Iplo Y'Iplo = 1 for all IpI > 0 and hence the last integral
equals
(9.76)
(f.g)LZ(Sto)
which proves (9.74).
This completes the proof.
Proof of Theorem 9.1. 8.3.
It was shown that (9.5) follows from Theorem
Relation (9.6) follows from Corollary 9.3, by Corollary 9.15. Proof of Corollaries 9.4 and 9.5.
Corollary 9.4 follows immediately
from the definitions (9.16) and Corollaries 9.2 and 9.3.
Corollary 9.5
follows from Corollary 9.4. Proof of Corollary 9.6.
This is just a restatement of Theorem 8.3,
equation (8.12).
Proof of Corollary 9.7.
Equation (8.27) for 'Y+ will be verified.
proofs of the remaining equations are similar.
It is clearly sufficient to
verify (9.27) for functions g+ E L2(S2) with compact supp g+ c SAM. such a function and f E JL'com then
The
If g+ is
116
3.
SPECTRAL ANALYSIS OF SOUND PROPAGATION
(f,Y*g) _ (T f,g+) (9.77)
+(x,Y,P,A) f(x,Y) c-2(Y)P-1 (Y)dxdYJ
10M (IR3
g+(P,A)dpda
='R3 f(x,Y) [TM +(x,Y,P,X)g+(p,A)dpdXI c-2(Y)P-1(Y)dxdy. This relation implies (9.27) for g+ because JCcom is dense in JC.
Proof of Theorem 9.8. Corollaries 9.2 and 9.7.
The representation (9.30), (9.31) follows from The converse follows from the unitarity of 'Y,
Theorem 9.1.
Proof of Theorem 9.9.
Only equation (9.35) will be proved.
equations can be proved by the same method.
The other
To prove (9.35) let f E D(A),
g E D(Q0) and note that (`70A f,g) = (A f,' g) _ (A f,4' g)
(9.78)
Now if h = Y'og then h E JC, by Corollary 9.3 and, by Lemma 9.16,
(9.79)
h(p,y) =
o(y,IPI,A) g(p,X)dX, (p,y) E R3. J
A0(IPI)
A distribution-theoretic calculation of A h gives
A h(p,y) =
(9.80)
JAo(IPI)
o(Y,IpI,X) X g(P,X)dA.
In particular, since A g(p,A) E L2(00), one has A h E IC by Corollary 9.3 and hence h E D(A).
Combining this with (9.78) gives
(T0A f,g) = (A f,h) _ (f,A h) (9.81) = jR3
)o(Y,IPI,X)Ag(p,X)dX c-2(Y)P-l (Y)dpdy
f(p,Y)
JA0(
IP )
Writing k(p,A) = A g(p,A) E D(Q0) and applying Lemma 9.16 again gives
(ToA f,g) =
(f,T k) _ ('Yof,k)
(9.82)
J
fo(p,A) k(p,A)dpda = f
A fo(P,A) g(P,X)dpdA.
This implies (9.35) because the functions g E D(C0) are dense in L2(Qo).
Semi-Infinite and Finite Layers
3.10.
Proof of Corollary 9.10. (9.37).
117
It is only necessary to verify relations
By [11, p. 530] these relations are equivalent to
(9.83)
P+n(p) = n(U)P+, PkR(u) = n(u)Pk
for 0 < k < No and all p E R. Theorem 8.4.
(9.84)
These equations are immediate consequences of
For example (8.17) and the definition of P+ implies that
(f,P+n(u)g) _ (f,R(u)P+g) =
H(U - A) f+(P,A) g+(p,),)dpda J
for all f,g E C, which implies P+11(P) = H(p)P+.
The other relations are
proved similarly. Proof of Theorem 9.11.
Equation (9.50) can be verified by applying the
definition (9.47) to hM and using the convergence statement of Theorem 8.3. Relations (9.51) follow from (9.47) and Corollary 9.3 by direct calculation.
Equation (9.52) can be verified by reversing the steps in the calculation (9.45), (9.46).
Relation (9.53) follows from Theorem 9.9 and equation
(9.47).
Proof of Corollary 9.12.
This was indicated immediately after the
statement of the Corollary.
SEMI-INFINITE AND FINITE LAYERS
§10.
The purpose of this section is to present extensions of the preceding analysis to the cases of semi-infinite and finite layers of stratified fluid.
The methods and results are entirely analogous to those developed
above.
Therefore the presentation emphasizes the modifications required in
these cases.
Proofs are indicated briefly or omitted.
Semi-Infinite Layers.
With a suitable choice of coordinates the
region occupied by the fluid is described by the domain
(10.1)
R+ = {(x,y)
I
y > 0}.
The acoustic field is assumed to satisfy either the Dirichlet or the Neumann boundary condition on the boundary of R+.
Physically, these
conditions correspond to the cases where the boundary plane is free and rigid, respectively.
The functions p(y) and c(y) are assumed to be Lebesgue
measurable and satisfy
(10.2)
0 < Pm < P(y) < PM < -, 0 < cm < c(y) < cm < .
3.
SPECTRAL ANALYSIS OF SOUND PROPAGATION
j0 IP(Y) - P(°°) Idy < .,
Ic(Y) - c(-) Idy < ao,
J
The acoustic propagator A defined by (2.1.12) determines selfadjoint operators A' and Al in
JC+ = L2(R+'c-2(Y)P 1(Y)dxdy)
(10.4)
corresponding to the two boundary conditions.
The domains of A° and A' are
subsets of
(10.5)
L2(A,R+) = LZ(1 ) n {u
I
V
(p -'Vu) E L2(R+)}.
The operator A° corresponding to the Dirichlet condition is defined by
(10.6)
D(A°) = LZ(A,R+) !1 {u
I
u(x,0+) = 0 in L2(R2)}.
Sobolev's embedding theorem for L2(R+) [1] implies that the boundary values u(x,0+) are defined in L2(R2). The Neumann condition will be interpreted in the generalized sense employed in [22]; i.e.,
(10.7)
{V
(p 'Vu)v + P -'Vu
Vv}dxdy = 0
1R +3
for all v E L2(R+).
Thus the operator Al corresponding to the Neumann
condition is defined by
(10.8)
D(A') = LZ(A,R+) h {u
(10.7) holds for all v E L2(R+)}. I
The operators are defined by A]u = Au for all u E D(A3) and one has
A3 = A3* > 0, j = 0, 1.
(10.9)
This is most easily proved by introducing the corresponding sesquilinear forms, as in Chapter 2, and using Kato's representation theorem [11, p. 322]. The spectral analysis of A° and A' may be based on the corresponding reduced propagators Au and Au in
(10.10)
JC(R,}) =
L2(R+,c-2(Y)P-'(Y)dy),
R+ = {y
I
y > 0).
3.10.
Semi-Infinite and Finite Layers
119
They are defined by
(10.11)
D(AU') = L2'(R+) n
E Jf(R+) and (0+) = 01,
(10.12)
D(AU) = LZ(R+) n
E Jf(R+) and (p')(0+) = 01,
E D(AU), j = 0,1,
Auto = Auto for
q
= A - cm u2, j = 0,1.
The results of §4 can be extended to these operators.
Thus
ac(AJ) = ae(A3) ji = Ic2(m)u2,°'),
(10.15)
11
a(AJ) n (__,c2(°°)p2) C aa(AU),
(10.16)
and the eigenvalues in this interval are all simple.
by X
(
) , 1 < k < NJ (P) <
-4
Eigenfunctions of A. 1 < k < NJ(P).
They will be denoted
.
These functions will be denoted by ij (y, u),
They are uniquely defined by the conditions
,u) Ilx(R+) = 1, (10.17)
(10.18)
(AU - ak(4)) k(y,p) = 0 for y E R+, and
iVk(0+,u) = 0, (P-'Ok')(O+,Y) = 0.
The asymptotic behavior of t4k(y,u) for y -+ +m is given by
Uk(y,p) ^'ck(u) exp {-y q'(u,ak(u))1, y
q' (p,)L) = (u2 - A c 2 (°°)) 1/2 > 0. The function tpk(y,u) has precisely k - 1 zeros.
E D(AU),
120
3.
SPECTRAL ANALYSIS OF SOUND PROPAGATION
Generalized Eigenfunctions of Aj.
For A > c2(0o)p12,
family of generalized eigenfunctions
I
has a single
A > c2(m)u2}.
They are
determined up to normalization by the conditions
(10.21)
(AU - A)1p3 (y,p,A) = 0 for y E R+, and
(10.22)
V(0+,P,A) = 0, (P-1
,A) = 0.
Their asymptotic behavior for y - 0O is given by
3 (y,u,A) - c3 (u,A) [e iyq(P,A) + R3 (u, A)
iyq(u,A)
q(p,A) = (A C-2(m) - )J2)1/2 > 0. The normalizing factors c3(p,A) are calculated below. Generalized Eigenfunctions of A3.
(10.25)
These are defined by
$3(x,y,P,A) = (211)-1
,3(y,IPI,A), (p, X) E 0,
(27T)-'
lU
a = {(P,A)
A > c2(0O) IPI2}
& k={p I IPI EOk}, k > 1
(10.28)
and 0k = {U
I
(y,IPI), P E 0k, k > 1
I
N3(11) > k + 1}, as before.
The wave-theoretic interpretations
of the eigenfunctions (10.25), (10.26) may be derived from the asymptotic forms (10.19), (10.23) as in §1.
With these definitions the following
analogue of Theorem 8.4 holds. Theorem 10.1.
The spectral families {113(p)} of A3 satisfy
(f,R3(0f)
H(p - A) J
(10.29)
If3(P,),)I2
dpdX
S2
NJ-1
+
k=1
f2k
H(p - Ak(IPI)) Ifk(P)I2 dp
3.10.
Semi-Infinite and Finite Layers
121
for all f E x+ where Na = sup N3(U) and u>o
M (10.30)
f3(p,X) = L2(Q)-lim j M'
p3(x,Y,p,A) f(x,Y)c z(Y)P-l(Y)dxdy,
j 0
Ixi<M
M (10.31)
fk(P) = Lz(S2k)-lim
$k(x,Y,P) f(x,Y)c 2(Y)P-1(Y)dxdy.
J J j
0
M-+m
IxI<M
Relation to the Infinite Layer Problem.
Theorem 10.1 can be derived
by the method employed for the infinite layer problem in the preceding sections.
However, it can also be deduced directly from Theorem 8.4.
To
this end one extends p(y), c(y) to all y E R as even functions:
(10.32)
P(-y) = P(y), c(-y) = c(Y), y > 0.
Then it follows from (10.2), (10.3) that the extended functions satisfy (2.1.4), (1.1) with
(10.33)
P(-°°) = P(m), c(--) = c(°°).
The corresponding operator in 3C will be denoted by A, as before.
Property
(10.33) implies that
(10.34)
4+(U,a) = 4(U,A), 4+(4,X) = q' (11, X)
where the latter are defined by (10.20), (10.24).
Moreover, the special
solutions 01(y,u,A) of §3 satisfy
$1( Y,U,a) _ y(Y,U,a) (10.35)
and it is easy to verify that the eigenfunction (resp., odd) when k is odd (resp., even).
k(y,u)
of AU is even
It follows that the eigen-
functions of Au can be calculated from those of AU by the rule
$k(y,U) =
Vr2
Vzk(y,u)
y > 0, k = 1,2,---
(10.36) ' (Y,U) =
2k_1(Y,11), y > 0, k = 1,2,...
The factor / is to renormalize k from R to R+.
122
SPECTRAL ANALYSIS OF SOUND PROPAGATION
3.
Concerning the generalized eigenfunctions, note that there is no y0 (y ,p ,a) for AP because c(--) = c(W) and (10.35) implies that
(10.37) It follows that the coefficients R+, T+ in (1.19), (1.20) satisfy
(10.38)
R+(U,A) = R_(U,A), T.(U1,A) = T_(U,A)
and the generalized eigenfunctions of Ai can be calculated from those of Au by the rule
V(Y,U,A) =
Y ? 0,
+(Y,u,a) -
(10.39)
V1(Y,U,A) _ l+(Y,u,A) + V'+(-Y, U,A), Y ? 0.
The factors c3(u,A) and R3(p,A) of (10.23) are given by 1/2
(10.40)
c° (u,A) = c1 (u,A) = i4 P )a))
(10.41)
R° = R+ - T+, R' = R+ + T+.
'
Theorem 10.1 can be deduced from Theorem 8.4 by introducing the
operators (10.42)
Jj
:
JC(R+) - 3C,
j = 0,1,
defined by
u(x,y), (10.43)
(x,y) E R+,
J°u(x,Y) = i
-u(x,-Y), (x,-Y) E R+,
and
u(x,y), (10.44)
(x,y) E R+,
J1u(x,Y) =
u(x,-y), (x,-y) E R.
123
Semi-Infinite and Finite Layers
3.10.
J° and J1 are bounded linear operators and using the fact that A has coefficients satisfying (10.32) one can show that the resolvents of A3 and A are related by
J
R(Aj,C) =
(10.45)
Jj, j = 0,1.
2
From this and Stone's theorem (8.64) it follows that
(10.46)
J 11 (0) JjI j = 0,1.
Hi (P) =
2 Theorem 10.1 follows directly from these relations and Theorem 8.4. Finite Layers.
In this case, with a suitable choice of coordinates
the region occupied by the fluid is described by the domain
Rh = R3 n { (x,y)
(10.47)
where h > 0.
1
0 < y < h}
The case of a fluid layer with a free surface at y = 0 and a
rigid bottom at y = h will be discussed. The acoustic propagator A and boundary conditions determine a selfadjoint operator Ah in
xh = L2(Rh,c-2(Y)P-l(y)dxdy).
(10.48)
To define the domain of Ah let
L2'0(Rh) = L2(Rh) n {u
(10.49)
I
u(x,0+) = 0 in L2(Rh)}.
The Dirichlet condition at y = 0 will be enforced by requiring D(Ah)
2% .
C L1,0(
3
The Neumann condition at y = h will be interpreted in the
generalized sense that
(10.50)
Thus
(10.51)
JR3
{0
(p-10u)v + p 10u
h
D(Ah) = L2i0(Rh) n LZ(A,Rh) n {u
and Ahu = Au for all u E D(Ah).
(10.52)
Vv}dxdy = 0 for all v E LZ'0(Rh).
As before
Ah = Ah* > 0.
(10.50) holds}
124
SPECTRAL ANALYSIS OF SOUND PROPAGATION
3.
The corresponding reduced propagator Au in 3C(Rh) = L2(Rh,c-2(y)p-1(y)dy) (Rh = {y
(10.53)
I
0 < y < h}) is defined by
D(AU) = Lz(Rh) n {
I
(P-',')' E
0},
and Auto = AU,p for all i E: D(AU) .
One has
(10.54)
Ah=Ah*> c2 2 AU mU
U
as before.
In the present case Ah is a regular Sturm-Liouville operator.
Hence
(10.55)
a(AU) = ao(AU)
and if Xh(U), 1 < k < -, denotes the eigenvalues then ak(U) Note that in this case Ok = R+ and Qk = R2.
when k
If th (y,U) denotes the
corresponding eigenfunction and 1Lk(x,y,p) = (27r)-'e'p'x)k(y,Ipl) then the
spectral family {Rh(U)} for Ah satisfies
(f,llh(U)f) _
J
k=1
(h
fk(P) = L2(R2)-lim 1 M-
2
H(U - ak(IpI)) Ifk(P)I2 dp
R
2(Y)P_1(Y)dxdy.
(
Vk(x,Y,P) f(x,Y)c
J
0
xI<M
These results can be proved by the method developed above.
Chapter 4 Transient Sound Field Structure in Stratified Fluids
The purpose of this chapter is to analyze the structure of arbitrary sound fields with finite energy in stratified fluids.
The principal results
of the analysis imply that each such field is a sum of a free component,
which behaves for large times like a diverging spherical wave, and a guided component which is approximately localized in regions ly - y
.I
]
< h
.
where
J
c(y) has minima and propagates outward in horizontal planes like a diverging cylindrical wave.
The methods and results developed below were initiated by the author in 1973 in the special case of the Pekeris profile [23]. were extended to the symmetric Epstein profile [24].
In 1974 the results The general Epstein
profile was treated in 1979 [29] using the spectral analysis of the Epstein profile due to Guillot and Wilcox [7,8]. announced in 1978 [4].
A preliminary version of [29] was
The extension of these results to a large class of
stratified fluids presented below is based on the general normal mode expansions of Chapter 3.
§1.
SUMMARY
Throughout this chapter it is assumed that p(y) and c(y) satisfy the boundedness conditions (2.1.4) and the four conditions
C(±y)-a
Ip(y) - p(±m)l <
for ±y > 0
(1.1)
1c(y) - c(±-) l
C(±y)-a
<
where p(+m), p(-m), c(+W), c(-m), C and a are constants and
a > 2.
(1.2)
125
126
TRANSIENT SOUND FIELD STRUCTURE IN STRATIFIED FLUIDS
4.
Note that (1.1), (1.2) imply condition (3.1.1) of the preceding chapter.
It
will be seen from the analysis that (1.1), (1.2) could be replaced by other order conditions at y = ±-.
It is not known whether the results obtained
below are valid for the entire class of fluids studied in Chapter 3. To begin consider a general sound field with finite energy:
(cos t Al/2) f + (A 1/2 sin t Al/2) g,
(1.3) where f, AI/2 f and g are in
C.
For each fixed time t,
mode decomposition as described by (3.1.58)-(3.1.63).
has a normal Thus
u(t,X) = uf(t,X) + ug(t,X)
(1.4)
where No-1
u (t,X) =
(1.5)
g
k=1
u (t,X) k
and No- 1 is the number of guided normal modes. ug = 0 if No = 1.
Clearly, 1 < No < -f
and
of and ug will be called the free and guided components
of u, respectively.
their behavior for t
The motivation for this terminology is provided by -
- which is described next.
It will be convenient to begin with sound fields that have the complex representation (2.1.19)-(2.1.21); that is
(1.6)
u(t,X) = Re {v(t,X)},
(1.7)
e-itA
1/z
h, h and AI/2 h in Jf.
The spectral property of the normal mode representation, (3.1.57) and (3.1.63), implies that
uf(t,X) = Re {vf(t.X)}, (1.8)
u9(t,X) = Re {vg(t,X)},
where
(1.9)
is the decomposition of v.
v(t,X) = vf(t,X) + vg(t,X)
Moreover, for the same reason,
Summary
4.1.
1.27
vf(t,') = e- itA3/2 hf
(1.10)
and
No-1 v (t,X) _ g
I
k=1
v (t,X) k
where
(1.12)
vk(t,') = e
-itAl/2
hk.
Finally, using (3.1.57), (3.1.60), (3.1.61) and (3.1.63) one gets
(1.13)
$+(X,P) e-itw(p) h+(P)dP
vf(t,X) = R3
and
-itwk(IpI) (1.14)
vk(t,X)
k(X,p) e
hk(p)dp
where
c(°)IPI for P E C+ U C (1.15)
W (P) = Ah/2 (P) =
c(-°)IPI for P E C_, for P E N,
0
and
(1.16)
wk(Ipl) = X1k12 (IPI) for 1 < k < No.
The integrals in (1.13) and (1.14) will, in general, converge only in JC. However, for brevity, the JC-lim notation is usually omitted below.
The integral representations (1.13), (1.14) provide the starting point for calculating the behavior for t + ° of v
f
and vk.
For vf the intuitive
idea is that for large t the wave vf(t,X) will have propagated into regions
where Iyl is large and hence +(X,P) is near its limiting forms for y + ±°, given by (3.1.42), (3.1.43), (3.1.48) and (3.1.49).
On replacing 4+ in
(1.13) by one of its limiting forms one obtains normal mode representations
of waves in homogeneous fluids with parameters p(°), c(°) or p(-), c(-°). Detailed calculations are given in §3 below. here.
Only the results are described
128
TRANSIENT SOUND FIELD STRUCTURE IN STRATIFIED FLUIDS
4.
Equation (1.13) gives two representations of vf, corresponding to the Either can be used for the asymptotic calculation but
choice of 0+ or 4_.
it has been found that the results take their simplest form in the -representation.
To describe them let R+(d) and R3(d) denote the half-
spaces defined by
(1.17)
R+(d) _ {(x,y)
:
±(y - d) > 0}
and let A(4°) and A(--) be the acoustic propagators for the homogeneous fluids with parameters p(°°), c(m) and p(--), c(-m) respectively.
Clearly
A(±-) = c2(±°') A0
(1.18)
where A0 corresponds to the special case p(W) = 1, c(-) = 1; i.e. A0 = -A. Now fix a value of d, arbitrarily, and for each h E X define a wave function vf(t,X) by
(1.19)
where h+ and h
(1.20)
exp (-i t A112 (1) )h+(X) ,
X E R33 (d)
exp (-it A112 (-oo) )h (X) ,
X E R3 (d),
vf(t,X)
are the functions in L2(R3) whose Fourier transforms
h±(P) _ (2) 3/2 jR 3 e
c(am)
h±(X)dX
p E R+(0),
p172 (W) h_(P),
h+(P) = 0,
P E R3 (0)
P E R+(0),
0,
(1.22)
,
h -(P) =
cp1/'2(-OD) h (P) In (1.21), (1.22) the function h
,
P E R3 (0) .
is the 4_-transform of h E JC:
4. 1.
Summary
129
h_(P) = 1R3 _(X,P)
(1.23)
h(X)c-2(y)p-'(y)dX.
With this notation the asymptotic behavior of vf(t,X) may be described by
lim
0.
t- 4-
uf(t,X) = Re (vf(t,X)}
then (1.24) and the inequality IRe zI
<
zi imply that
lim t T+
(1.26)
0.
Equations (1.24) and (1.26) hold for all h E 3C. If also Al/2 h E 3C, so that uf(t,X) has finite energy then one also has lim t-*-
(1.27)
0,
j = 0,1,2,3.
Returning to the definition (1.19)-(1.23) of vf, note that in the complementary half-spaces R+(d) and R3(d) it represents sound fields in homogeneous fluids with parameters p(-), c(W) and p(-m), c(-m) respectively. The latter behave for t - - like diverging spherical waves with speeds c(-) and c(-W), respectively, by the theory of [25].
Hence (1.26), (1.27)
express the property that of behaves for large t like a diverging spherical wave.
Now consider one of the guided components vk(t,X), given by (1.14). Using the definition (3.1.23) of Wk the integral can be written
(1.28)
vk(t,x,y) = 2 I
exp (i(x p- twk(IP ))} k(Y,IP ) hk(P)dp. Pk
For large x and t the exponential factor in the integrand is rapidly oscillating while the remaining factors are slowly varying.
This suggests
that stationary phase estimates can be used to approximate vk for large t. The phase of the exponential is a stationary function of p E 0k C R2 if and only if
(1.29)
Uk(IPI)P/IPI = x/t
130
4.
TRANSIENT SOUND FIELD STRUCTURE IN STRATIFIED FLUIDS
Uk(IPI) = wk(IPI)
is the group speed associated with the dispersion relation w = wk(IPI)
A number of cases may arise, depending on the number of solutions p of (1.29).
Here it will be assumed, for brevity, that Uk(u) is a monotone decreasing function that maps 0k onto (cm,c(°D)).
In this case (1.30) has a unique
solution if IxI/t lies in the range of Uk(IpI); that is,
cm < IxI/t < c(°°),
(1.31)
and no solution otherwise.
The solution is given by
(1.32)
p = Qk(Ixl/t)x/IxI
where Qk = Ukl; that is,
V=Uk(11) c'P =Qk(v).
(1.33)
The stationary phase estimates are formulated in Appendix 2.
They
imply that for IxI/t in the interval (1.31) there is one stationary point (1.32) which makes a contribution
IPI1/2 exp {i(IxIIPI-twk(IPI))} k(Y,IPI)hk(P) (1.34)
v°(t,x,y,p) = k
to the integral in (1.28).
t {ok(IPI)
Uk(IPI)I}1/2
For IxI/t outside the interval (1.31) there is
no point of stationary phase.
Thus the stationary phase approximation to
vk(t,x,y) is
(1.35)
v(t,x,Y) = X(IxI/t) vk(t,x,Y,Qk(IxI/t)x/IxI)
where X is the characteristic function of the interval (cm,c(OD)).
The results in Appendix 2 guarantee that if hk is a smooth function with compact support then vk(t,x,y) is a good approximation to vk(t,x,y) for large t, uniformly for all x e R2 - {0} and y E R.
For general h E Jf
one has only hk E L2(Qk) and the error estimates of Appendix 2 fail. Nevertheless, it is important to recognize that for general h E JC the
4.1.
131
Summary
definition (1.34), (1.35) is still meaningful and one has
(1.36)
E Jf
and
IIvk(t,IIx =
(1.37)
for all t > 0 and 1 < k < N0.
(Pk)
These properties may be verified by direct
calculation.
The central fact concerning vk in the Hilbert space theory states that
for every h E 3C one has lim
(1.38)
vk(t,')IIX = 0.
t It follows as in the case of of that if
(1.39)
uk(t,x,y) = Re {vk(t,x,y)}
then
lim taco
(1.40)
Moreover, if h E D(A112),
uk(t,')IIx = 0.
so that uk is a field with finite energy, then the
derivatives D. uk(t,X) are in JC and asymptotic wave functions uk j for them can be constructed such that
(1.41)
his IID. uk(t,') t+m
for j = 0,1,2,3.
0
The construction of uk j is given in §4 below.
Returning to the definition (1.34), (1.35) note that vk(t,X) is an
inhomogeneous cylindrical wave that propagates radially and dispersively in planes y = const. and is exponentially damped by the factor y-direction.
k(y,Ipl) in the
This behavior and the convergence properties (1.38), (1.41)
motivate the term "guided mode" for vk(t,X). Each sound field u with finite energy has a fixed total energy
(1.42)
see (2.1.9).
E(u,R3,t) = E(u,R3,0) < m;
The same is true of the corresponding modal waves:
132
4.
TRANSIENT SOUND FIELD STRUCTURE IN STRATIFIED FLUIDS
E(uf,R3,t) = E(uf,R3,0), (1.43)
E(uk,R3,t) = E(uk,R',0).
In fact, the Parseval formula (3.1.59) will be shown to imply the energy partition formula No-1 (1.44)
E(uk,R3,0).
E(u,R3,0) = E(uf,R3,0) + k=l
Moreover, the asymptotic estimates (1.27), (1.41) for the modal waves uf, uk yield precise information on the ultimate distribution for t partial energies. proofs.
- of the
The principal results will be formulated here without
The discussion is completed in §5.
The notation
E-(u,K) = lim E(u,K,t)
(1.45)
t-wo
will be used whenever the limit exists.
A first result is the transiency
of all waves with finite energy in stratified fluids:
E-(u,K) = 0 for all bounded K C R3.
(1.46)
Next consider the free component uf(t,X).
(1.27) implies that in each
half-space R+(d) or R3(d) (d is arbitrary) u homogeneous fluid.
f
behaves like a wave in a
Thus if C+ (resp., C-) denotes any cone in R+(d) (resp.,
R3(d)) then (1.27) and the results on asymptotic wave functions of [25] imply that
EW(uf,C ) = c2(±w)
(1.47)
+
IP12 Ih_(P)I2dP.
C
It is noteworthy that the asymptotic distribution in cones of the partial energy E(uf,R3,0) is determined by the i_-transform of the complex initial
state (1.48)
A172
h = A112 f + ig. An alternative form of (1.47) is E-(uf,C±)
= f+ Iw(P)f_(P) + i g_(P)IZdP C-
where w(P) is given by (1.15).
Finally, note that one may take C
in (1.47) and that d is arbitrary.
It follows that if
= R+(d)
Normal Mode Expansions of Transient Sound Fields
4.2.
S = { (x,y)
(1.49)
:
133
d1 < y < d2}
is an arbitrary finite slab parallel to the stratification then
EW(uf,S) = 0;
(1.50)
i.e., the free component of ultimately propagates out of every slab (1.49). Now consider the family of cones defined by
(1.51)
C(e,d) = {(x,y)
where d E R and e > 0.
:
Iy - dI < elxl}
Then if uk is any guided component the estimates
(1.41) imply that
E(uk,C(e,d)) = E(uk,R3,0)
(1.52)
for every d and ever (1.47) for uf.
e > 0.
This is to be contrasted with the result
Finally, if S is the slab defined by (1.49) then (1.41)
implies that, in contrast with (1.50), one has d2
(1.53)
E- (uk,S) = j
k(Y,IPI)c 2(Y)P-1(Y)dY dP
Iak(IPI)hk(P) 2 j
k
d1
Recall that the k(y,lpl) are orthonormal on -Op < y < 00 with respect to c 2(y)p-1(y)dy.
Hence the factor in parenthesis in the integrand of (1.53)
lies between 0 and 1.
Thus, to be sure, for example, that
E-(uk,S) > .9 E(uk,R3,0)
(1.54)
it is only necessary to choose d1, d2 such that 1d2
P2(Y.IPI)c 2(Y)P-1(Y)dy > .9.
(1.55) d1
§2.
NORMAL MODE EXPANSIONS OF TRANSIENT SOUND FIELDS Consider a general sound field with finite energy:
(cos t A112) f + (A-'/2 sin t Al/2) g
(2.1) where f E D(A1/2)
and g E K.
It was shown in Chapter 3, Corollaries 9.4
134
4.
TRANSIENT SOUND FIELD STRUCTURE IN STRATIFIED FLUIDS
and 9.10, that the orthogonal projections
in X defined by
the normal modes form a complete family that reduces A.
Hence the same is
true of the orthogonal projections
Pf = P+ + P- + Po
(2.2)
and
No-1 (2.3)
Pk.
Pg =
k-1
This provides a decomposition
u(t,X) = uf(t,X) + ug(t,X)
(2.4)
into orthogonal partial waves
uf(t,') = Pfu(t,')
u9(t,,) = Pgu(t,').
These components are analyzed in the following two sections.
§3.
TRANSIENT FREE WAVES The normal mode expansions of §2 are used in this section to calculate
the asymptotic behavior for t - W of the free component
Pfu(t,').
The principal result is that in each of the half-spaces R+(d) and R3(d), is asymptotically equal to a wave function for a homogeneous fluid with parameters p(co), c(W) and p(-oo), c(-W), respectively.
It is this
behavior that motivates the term "free component" for uf(t,.). It will be assumed that the total acoustic potential u satisfies Re {v(t,')) where v(t,') = exp f-i t A1I2} h and h E D(A112).
corresponding partial waves = Re {exp (-it A1'2) Pkh}.
The
with k > 1 satisfy uk(t,') This follows from the fact that the normal mode
functions ik(y,p) are real for k > 1, which implies that Pk(h) = Pk (h),
follows by addition that and hence
Re {exp (-i t A1l'2) Pgh} = Re
It
Transient Free Waves
4.3.
135
uf(t,') = Re {vf (3.1)
vf(t,') = exp (-it All?) Pfh = exp (-it A112) hf. The normal mode representations
vf(t,x,Y) = J 3 +(x,y,p,q) exp {-itX1/2 (p,q)} h+(p,q)dpdq R
(3.2)
provide the starting point for calculating the asymptotic behavior of uf(t,x,y) for large t.
Equation (3.2) gives two representations of of corresponding to the
two families + and 0_.
The calculations below are based on the
0_-representation which has been found to yield the simplest form of the asymptotic wave function.
It will be convenient to introduce the charac-
teristic functions X+, X0 and x_ of the cones C+, Co and C_ in (p,q)-space and to decompose h_ as
(3.3)
h_(p,q) = i(p,q) + m(p,q) + n(p,q)
where I = X+ h_, m = Xu h_ and n = X_ h_.
The corresponding decomposition
of of is of = vR+vm+vn
(3.4)
where
V1 = exp (-it (3.5)
4
A112) $*k
vm = exp (-it A1'2) $*m
vn = exp (-i t
Al/2) D*n.
The behavior for t + - of these three functions will be analyzed separately. Behavior of v1.
(3.6)
The partial wave v1 has the representation
_(x,y,p,q) exp (-it w+(p,q)) I(p,q)dpdq
v1(t,x,y) = J C+
where
136
4.
TRANSIENT SOUND FIELD STRUCTURE IN STRATIFIED FLUIDS
w+(P,q) = c(±-)
(3.7)
(Recall X(p,q) = w+(p,q) for ±q > 0.)
"p2
+ q
To discover the behavior of
vi(t,x,y) for (x,y) E R+(d) and t - - it will be convenient to write _(x,y,p,q) in a way that puts in evidence its behavior for y
To
this end recall that by (3.1.38)-(3.1.45), one has
(3.8)
_(x,Y,P,q) =
(21r)-1
c(m)(2q)112
P+(y, -PT, -X)
for (p,q) = X+(P,X) E C+ (and hence A = A(p,q) = c2(°°)(Ipj2 + q2)).
Moreover, by (3.5.5), (3.5.6) and (3.9.41) one can write
(
P (m)
ll 1/2
T+(U,A)
l4Trq (u,X))
(3.9)
_
P(W) 4
1112
T+(Y,u,X) exp {-iY 4_(u,X)}
(u,A)
where
(3.10)
T+(Y,u,A) = T+(u,X) 44 (Y,U,X) exp {i y q_(U,X)} + T+(u,A), y + moo.
Similarly, by (3.5.1) and (3.5.6)-(3.5.11) one can write A (m)
1/2
{4Tr4 (U,1))
[I+(y,p,X) exP {-iyq.F,(u,X)}
(3.11)
+ R+(y,u,A) exp {iy q,(U,X)}] where I+(Y,p,a) _
exp {iy q+(u,A)} + 1
y++m.
(3.12) R+(Y,u,X) = R+(U,A) 1(y,U,A) exp {-i y q. (u,A)} + R+(u,A)
Combining (3.6), (3.8) and (3.11) gives
v(t,x,y)
= c(m)P(m)1/2 (2n)-3/2 J exp {i(x P+Yq - tw+(P,q)} I+(Y, P .a) i(P,q)dpdq C+ (3.13)
4.3.
Transient Free Waves
137
+ c(°°)p(°°)1/2 (2.r)-3/2 (C+ exp {i(x p - y q - tw+(P,q)} R+(Y, p
a) R(p,q)dpdq
(3.13 cont.) It is natural to expect that in R+(d) the partial wave vi(t,x,y) will propagate as t -' m into regions where y is large and hence I+(y,lpl,A) and
Thus the representation (3.13)
R+(Y,Ipl,a) are near their limiting values. suggests the conjecture that
in L2(R+(d)), t i W
(3.14)
where v°2 and vR are defined by
(2n)-3/2
vR(t,x,Y) = c(m)p(°')1/2 (3.15)
exp {i(x p+yq- tw+(p,q)}i(p,q)dpdq
f C+
and
v'(t,x,Y) = c(°°)P(°°)1/2 (27r)-3/2
exp {i(x' p-yq- tw+(p,q)) x
( J
C+
x R+( P ,A) R(p,q)dpdq (3.16) (2n)-3/2
c(m)p(O0)1/2
rJ
exp {i(x' p+yq- tw+(P,q)} x C+
X R+(p , ) R(p,-q)dpdq where -C+ _ {(p,q)
;
(p,-q) E C+}
{(p,q)
:
q < -alpI}.
Note that v°2 and
v1 are waves in a homogeneous medium with density p(m) and sound speed c(oo). More precisely,
exp (- it c (°°) (3.17)
A0'/2) hR
4
vR(t,') = exp (-it c(-) A1/2) h' where A. is the selfadjoint realization in L2(R3) of -A = -(a2/axi + a2/axZ + a2/ay2) and hQ and hR are the functions in L2(R3) whose Fourier transforms are
138
4.
TRANSIENT SOUND FIELD STRUCTURE IN STRATIFIED FLUIDS
(p, q) = c(co)P(°o)1/2 R(p,q) = c(`o)P(co)1/2 X+(P,q) h_(P,q), (3.18)
cR(p,q) = c(°)P(o)1/2 R+( p ,X) R(p,-q)
c(°°)P(m)112 R+( p ,X)
(1 - X+(P,q)) h_(P,-q).
Both functions are in L2(R3) because h_ E L2(R3) and IR+(IPI,X)I < p1/2 (a,)
by the conservation law (3.1.31).
Moreover, supp hR C -C+ and hence the
theory of asymptotic wave functions for d'Alembert's equation [25, Ch.2] 0 in L2(R+(d)) when t -
implies that
Combining this with
(3.14) gives
(3.19)
in L2(R+(d)), t -*
Now consider the behavior of vR(t,x,y) for (x,y) E R3(d), t
oo.
Combining (3.6), (3.9) and (3,10) gives
v1(t,x,y) (3.20) = c(°°) P(°°)1/2
(2Tr)-3/2
(
exp {i(x p+yq_- tw+)} T+(y,IPI,X) i(p,q)dpdq
JC+ where q_ = q_(Ipl,X), w+ = w+(P,q) and X = X(p,q) = w{(P,q).
This
representation suggests that
in L2(R3(d)), t - m
(3.21)
where
vR(t,x,y) (3.22) = c(°°) P(`°)1/2
(2Tr)-3/2
(
exp {i(x p+yq_ - tw+) } T+(IP1,X)
i(p,q)dpdq
JC+ Now the mapping (p,q) -* (p,q') = X'(p,q) = (p,q_(Ipl,w+(p,q))) with domain C+ has range X'(C+) = R+ = R+(0), Jacobian 8(p,q)/8(p,q') = c2(-co)q'/c2(W)q and satisfies w+(p,q) = w_(p,q').
Thus (3.22) implies the representation
Transient Free Waves
4.3.
139
r
V2(t,x,Y) = c(m)P(°°)1/2 (2n)-3/2
exp {i(x P+Yq' - tw_)} x
R+3
(3.23)
x T+(P ,w-(P,q')) R(p,q)(c2(-°°)q' /c2(°°)q)dpdq'
where q = q(Ipl,q') =
a ( p
+ q") + q'
.
Note that
exp (-it c(-W) Ao/2) h2
(3.24)
where hz E L2(R3) has Fourier transform
(3.25)
h2(P,q') = c(oo)P(°°)1/2 T+( P ,w(P,q')) R(p,q)(c2(-°°)q'/c2(°°)q)
Since supp hR C R+ the results of [25, Ch. 2] imply that
0 in
Combining this with (3.21) gives
L2(R3(d)) when t -;
0 in L2(R3(d)), t - -.
(3.26)
Analogous conjectures concerning formulated.
will now be
and
Only the main steps of the calculations will be given since the
method is the same as for Behavior of vm.
(3.27)
vm has the representation
_(x,Y,p,q) exp (-i t w+(P,q)) m(p,q)dpdq,
vm(t,x,y) = J
Co
by (3.5), where
(3.28)
4_(x,Y,P,q) = (27T)_1c(°O)(2q)112
$o(Y, P ,a)
for (p,q) = X0(p,A) E Co (and hence A = A(p,q) = m+(p,q)).
Moreover
(see (3.5.18)-(3.5.24)) 1/2
(3.29) a)
)
T 0 (y,U,A) exp (Y q'(U,A))
where
(3.30)
and
T0(Y,U,A) = T0(U,A) $3(Y,U,A) exp (-Y q'(U,X)) '' T0(U,X), y -* -m
140
4.
TRANSIENT SOUND FIELD STRUCTURE IN STRATIFIED FLUIDS
P (-)
o(Y,U,a)
1112 [Ia(Y,u,a) exp {-i Y q+(u,a)}
(3.31)
+ Ro(Y,u,A) exp {iy q+(p,A)}) where
Io (Y,u,A) _ *2 (Y,u,A) exp {i y q+(u,A) } -' 1
(3.32) Ro(Y,1,A) = Ro(U,A)
(Y,u,A) exp {-1y4+( U,A)} - Ro(P,A)
Combining (3.27), (3.28) and (3.29) gives
vm(t,x,Y) = c(-)p(a')1/2
(270-3/2
tw+)} x
exp
1
Co
(3.33)
x T0(Y,lpl,A) eXp (Y q') m(p,q)dpdq
where w+ = ow+(p,q), A = w+(p,q) and q' = q'(Ipl,A).
Since
T0(Y,lpl,A) exp (Yq'(lpl,A)) -+ 0, y
(3.34)
equation (3.33) suggests that
0 in L2(R3(d)), t -- -.
(3.35)
Similarly, combining (3.27), (3.28) and (3.31) gives
vm(t,x,Y) = c(m)P(_)1/2 (27r)-3/2
exp {i(x p+yq- tw+)} x
(
Co
x I,(Y,Ipl,X) m(p,q)dpdq (3.36)
+ c(-)p(m)1/2 (2Tr)
exp {i(x p - y q - tw+)} x
3/2 ( Co
x R0(Y,lpl,A) m(p,q)dpdq
which suggests that
(3.37)
in L2(R+(d)), t -r
Transient Free Waves
4.3.
141
where v0 and v1 are defined by
(
vm(t,') = exp (-it c(o) A1/2) hm
(3.38)
vm1
(t,') = exp (-it c(o') A01/2) hl
and hm and hm are the functions in L2(R3) whose Fourier transforms are
hm(p,q) = c(o°)P(°°) 172 m(p,q) = c(°°)P(°o) 1/2 X0 (p,q) h_(p,q), (3.39)
hm(p,q) = c(i)P(oo)3/2 Ro( P ,X) m(p,-q) = c(°°)P(o°)'12 Ru(P ,X) (1 - X0(p,q)) h_(p,-q). Note that supp hm C R3 and hence vm(t,') - 0 in L2(R+(d)) when t Combining this with (3.37) gives
in L2(R+(d)), t
(3.40)
Behavior of V. vn has the representation
(3.41)
_(x,Y,p,q) exp (-it w_(P,q)) n(p,q)dpdq,
vn(t,x,Y) = J c
by (3.5), where
(3.42)
_(x,y,P,q) _
(21T)-1
c(-o°)(2Igl)h/2 e'p'x *7Ty-, P ,X)
for (p,q) = X_(p,A) E C_ (and hence X = X(p,q) = w2(p,q)).
Moreover
(see (3.5.5) and (3.5.12)-(3.5.17)),
P (-°°) V_(Y,u,X) _ (43rq_(u,A)J
(3.44)
and
1/2
T_(Y,u,X) exp {iy q+(u,X)}
T_(Y,u,X) = T_(u,X) 41(Y,u,X) exp {-i Y q+(u,X)} - T_(u,X), y
4.
142
(y, 0,
TRANSIENT SOUND FIELD STRUCTURE IN STRATIFIED FLUIDS
P (-°°)
_ (
l 1/2
{I_(y,ii,A) exp {iy q_(u,A)}
(3.45)
+ R-(y,u,X) exp {-i y q_(u,A)}]
where
I_(y,u,X) _ $3(y,u,A) exp {-iy q_(U,A)} i 1 (3.46)
exp {i y q_(u,A)}
R_(y,u,A) - R_(11,A)
R_(u,X)
Combining (3.41), (3.42) and (3.45) gives, after simplification using q_(Ipl,w2(p,q)) _ (q2)1/2 = -q for (p,q) E C_,
vn(t,x,y) = c(-00)P(-00)1/2 (2n)-3/2
exp {i(x p+yq- tw_)} x J
C_
x I_(y, P ,A) n(p,q)dpdq
(3.47) + c(,.°°)P(-°0)1/2
exp {i(x p - yq - tw_)} x
(2n)-3/2 Jf
C
x R(y,Tp ,a) n(p,q)dpdq.
This suggests the asymptotic behavior
(3.48)
in L2(R3(d)), t -> o0
where
= exp (-it c(--)
v0
Al/2) hn
(3.49) v1
n
and h
n
(t,')
exp (-i t c (--) A'/2) h'
and h1 are the functions whose Fourier transforms are n
hn(p,q) = c(-00)P(-0D)1/2 n(p,q) = c(-o)P(-O0)112 X_(p,q)h_(P,q),
(3.50) n(p,q) = c(-0p)P(_oo)1/2 R_( p A) n(p,-q)
= c(-m)P(-00)1/2 R-( p ,X) (1 - X_(p,q)) h_(p,-q).
4.3.
Transient Free Waves
143
Note that supp hn C R+ and hence
0 in L2(R3(d)) when t
Combining this with (3.48) gives
vn(t,-) `
(3.51)
in L2(R3(d)), t
Finally, combining (3.41), (3.42) and (3.43) gives
vn(t,x,y) = c(-oo)P(-°o)1/2
(2Tf)-3/2
exp {i(x p - yq+- tw_)} x
( C
(3.52)
x T_(y, p ,X) n(p,q)dpdq
where w_ = w_(p,q), A = w2(p,q) and q+ = q+(Ipl,X).
(3.53)
This suggests that
in L2(R+(d)), t
where
vn(t,x,y) = c(-°o)P(-m)1/2
exp {i(x p - yq+ - tw_)} x
(21T)-3/21 C
(3.54)
x T_( P ,a) n(p,q)dpdq.
Now the mapping (p,q) i (p,q') = X"(p,q) _ (p,-q+(p,A(p,q))) maps C_ onto X"(C_) _ -C+, has Jacobian 2(p,q)/2(p,q') = c2(o)q'/c2(-oo)q and satisfies w_(p,q) = w+(p,q').
Thus
(3.55)
exp (-it c(am) A'1/2) hn
where h2 has the Fourier transform n (3.56)
hn(p,q')=c(-oo)P(-m)1/2T_( p ,w+(p,q'
Moreover, supp h2 C R3 and hence
n(p,q(p,q'))(c2(°o)q'/c2(-=)q).
0 in L2(R,f(d)), t
Combining this with (3.53) gives
(3.57)
The asymptotic behavior of
0 in L2(R+(d)), t
for t - o, may be obtained from the
three cases analyzed above by superposition, equation (3.4).
Thus
equations (3.19), (3.26), (3.35), (3.40), (3.51) and (3.57) imply
144
4.
TRANSIENT SOUND FIELD STRUCTURE IN STRATIFIED FLUIDS
in L2(R+(d))
t +oo.
(3.58) in L2(R3(d))
On combining this with the definitions of v°o, vm and vn, equations (3.17),
(3.18), (3.38), (3.39), (3.49) and (3.50), one is led to formulate Theorem 3.1.
For every h E Jf let
be defined by 112
exp (-itc(oo) (3.59)
(x,y) E R3(d)
vf(t,x,y) _
exp (-it c(-oo) A1/2) h (x, y) , where h+ and h
(x,y) E R3 (d) .
are the functions in L2(R3) whose Fourier transforms are
given by
c(oo)P(oo)112 h_(p,q),
(p,q) E R+,
h+(p,q) _
(3.60)
0,
(p,q) E R3,
and
0,
(p,q) E R+'
h (p,q) _
(3.61)
c(-oo)P(-oo)1f2 h_(p,q),
(p,q) E R3.
Then
lim
(3.62)
0.
Theorem 3.1 implies corresponding asymptotic estimates for the free component
(3.63)
of the acoustic potential
Re
Indeed, if
is defined by
Re
then Theorem 3.1 and the elementary inequality IRe zi <
Izi imply
145
Transient Free haves
4.3.
Corollary 3.2.
For all h E If one has
lim
(3.64)
0.
tIf the initial state h has derivatives in X then
and
have the same derivatives in 2C and (3.64) can be strengthened to include these derivatives.
In particular, one has
Corollary 3.3.
For all h E LZ(R3) = D(A1/2) one has
0, j = 0,1,2,3.
lim
(3.65)
tEquation (3.65) is equivalent to convergence in energy:
lim E(uf - uf,R3,t) = 0.
(3.66)
tCorollary 3.3 can be proved by applying the method of this section to (j = 0,1,2,3) which are given by integrals of the
the derivatives D.
Detailed proofs for the case of the Pekeris profile
same form as (3.2). were given in [27].
Proof of Theorem 3.1. proof of Theorem 3.1.
The remainder of this section is devoted to the
The decomposition (3.3) is used for the proof.
Moreover, for brevity, only the asymptotic equality (3.53) for proved.
is
The remaining five cases, namely (3.14), (3.21), (3.35), (3.40)
and (3.48) can be proved by the method used for (3.53).
As a first step,
(3.53) will be proved for the special case of n(p,q) E C0(C_), the set of continuous function with compact supports in the open cone C_.
The general
case will then be proved by using the fact that C0(C_) is dense in L2(C-). For functions n(p,q) E C0(C_) the integrals defining vn and vn converge point-wise, as well as in 3C, and one can write
(3.67)
vn(t,x,Y) - v2(t,x,Y) = c(_ )P(-°°) 112
(2F)-3/2
(
2
exp (ix, p)w(y,p,t)dp
R
where r
(3.68)
w(y,p,t) = f°
exp {-i(Y q+ + tm_)}[T-(Y, P ,WZ)- T_( P
w')]n(p,q)dq
Parseval's formula in L2(R2), applied to (3.67), gives
(3.69)
JRZ lvn(t,x,Y) - vn(t,x,Y)I2 dx = c2(_ )P(-°°)(2n)-' JRZlw(Y,p,t)I2 dp.
146
4.
TRANSIENT SOUND FIELD STRUCTURE IN STRATIFIED FLUIDS
On integrating this over y > d one finds
(27r)-1/2 IIw(',t)LZ(R+(d))
(3.70)
The last relation implies that to prove (3.53) it is sufficient to prove that
0 in L2(R+(d)), t ->
(3.71)
To this end it will be convenient to change the variable of integration in (3.68) from q to w = w_(p,q) = c(-°°)
p
Solving this equation
+ q z.
Ipj2)1/2 = -q_(Ipl,wZ)
for q < 0 gives q = -(w2c 2(-OJ) -
with w > c(-m)Ipl.
Hence (3.68) can be written
exp (-it w) W(y,p,w)dw
w(y,p,t) = 1
(3.72)
c
pI
where
W(Y,P,w)
(3.73)
C Z(-°°)exp{-iyq+(IPI,w2)IT_(y, P ,w) -T_( P ,w)]n(p,-q_(PI,w2))w q_(Ipiw2) The assumption that n E Co(C_), together with (3.44), implies that W E Co(R x r) where r = ((p,w)
:
w > c(-°)IpI}.
Moreover, by a standard
partition of unity argument, one may assume without loss of generality that
(3.74)
supp W(y,') C {(P,w)
:
IpI < po and 0 < wo < w < w1}
This in turn implies that
for all y E R where wo > c(-°°)po.
rw
(3.75)
w(y,p,t) -
I
1
exp (-it w) W(y,p,w)dw
JJJU0
and
(3.76)
supp w(y,,,t) C B(po) = {p
for all y E R and t E R.
Thus
:
IpI < po}
Transient Free waves
4.3.
147
IIw(',t) IILZ (R+(d))
Iw(Y,P,t)12 dpdy 1d
1B(p0)
(3.77)
Iw(Y,P,t)Z dpdy Jyo d
1
B(PD)
Iw(Y,p,t)IZ dpdy
+ for any yo > d.
f 1Y D
B(p0)
The proof of (3.71) will be derived from (3.77) and the
following two lemmas. Lemma 3.4.
Let n e CD(C_) and assume that (3.74) holds.
Then for
each d E R, yo > d and pD > 0 one has lim w(y,p,t) = 0,
(3.78)
t-wo
uniformly for all (y,p) E [d,y0] x B(p0). Lemma 3.5.
Under the hypotheses of Lemma 3.4 there is a constant
C = C(n) such that Iw(Y,P,t)I < C yl-a
(3.79)
for all y > 0, p E B(p0) and t e R, where a. > 3/2 is the constant of condition (1.1).
The proof is based on a well-known proof of the
Proof of Lemma 3.4.
Riemann-Lebesgue lemma.
Note that by (3.75) one has
w(Tr/t) exp (-i w t) W(y,p,w+(F/t))dw
w(y,p,t) _ -J
wD-(1T/t) (3.80) w1 2
1
exp (-i t w) [W(y,P,w) - W(y,p,w+(nr/t)) ]dw
WO
WO
1
exp (-it w) W(y,p,w+(Tr/t))dw
2
WD-(n/t) rw
+zI
1
exp (-itw) W(y,p,w+(nr/t))dw.
wl-(7T/t) The limit relation (3.78) is obvious from (3.80) and the continuity of W. The uniformity of the limit follows from (3.80) and the uniform continuity of W on compact subsets of R x r.
148
4.
TRANSIENT SOUND FIELD STRUCTURE IN STRATIFIED FLUIDS
Proof of Lemma 3.5.
Note that by (3.72), (3.73) and (3.44) one has
the estimate
(3.81)
exp {-i y q+(IpI,w2)} - lI M(P,w)dw
Iw(Y,P,t)I <- r1 W°
for all y, p and t where
n(p,-q_(IPIW2))W M(p,W) = c-2(-,) IT_(IPIW2)I
(3.82)
q_(IPIW2) is continuous for IpI
< po, wo < w < W1.
It follows that for any y E R,
p E B(po) and t E R
Iw(Y,P,t)I < Mo sup I$1(Y,IPI,W2) exp {-iy q+(IPI,w2)} - lI
(3.83)
where Mo = Mo(n) = sup M(p,w) and the suprema are taken over all IpI < po and wo < w < w1.
The proof of (3.79) will be based on (3.83), conditions
(2.1.4) and (1.1) on p(y) and c(y) and the proof in Chapter 3 of Theorem
Note that because of the continuity of w it will suffice to prove
3.1.
(3.79) for all y > y1 where y1 = yi(n) is a positive constant. The solution 41(y,u,A) with A >
c2(-")p2 > c2(co)u2 satisfies (see
(3.3.44)ff.)
(3.84)
1(Y,u,A) = exp {i y q+(u,A)} (n1 + n2)
where n = (n1,n2) is characterized on y > y1 as the unique solution of the integral equation (3.3.64) which can be written
(3.85)
n - no + K(u,A)n, no = (1,0).
The kernel K(y,y',u,A) = (Kij (y,y',u,A)) is defined by
U,
(3.86)
and
K15(Y,Y',u,A) =
1
Y1 < Y, < Y.
Transient Free Waves
4.3.
149
0, (3.87)
Y1
Y' < Y,
K2i(Y,Y',u,A) = -exp {-2i(y - Y')9+(v,a) E2.(Y',u,A), Y'
>- Y,
where ((3.3.43))
(3.88)
E(Y,u,A) = B-1()I,A) N(Y,u,A) B(P,A)
From these relations one has ((3.3.64))
(Y,U,A) exp {-i y 4+(U,A)} - 1 = nl + nZ - 1 = -J
E1j(Y',u,X) n.(y')dy' y
(3.89)
JW exp {-21(y- Y')Q+(u,A)} E,i(Y',p,X)nj(Y')dY'
and hence 2
(3.90)
exp {-i y 9+(p,A)} - lI <
J
'k =1
IEjk(Y',u,A)I Ink(Y')I dy'.
Jy
Using (3.3.66) and the continuity of B(p,A) on A > c2(-oo)U2 it can be shown that
(3.91)
IIK(u,A)II < 1/2 for 0 < u < po, Wp < A < tai
provided y1 = yl(n) is large enough.
(3.92)
IInII
for 0 < u < po, Wo < A < Wi.
Thus
1- 11K11
1- lIIK1I
<- 2
Combining this with (3.90) and (3.3.66) gives
exp {-i y 4+(u,A)} - 11
< 2
(°'
C2
1
IEjk(Y',u,a)I dy'
L
y y j,k=l
(3.93)
IP(Y') - p(-) l dy'
< C1 Y
+C2
r'
I c (Y') - c (°°) I dy'
y for the same values of p and A where C1 and C2 depends only on po, wo and W1.
It follows from (3.93) and (1.1) that
150
4.
TRANSIENT SOUND FIELD STRUCTURE IN STRATIFIED FLUIDS
sup I0,(y,Ipl,w') exp {-i y q+(IPI,w')} - 11 < C3 y'-a
(3.94)
for all y > y,(n) where C3 depends only on pa, wo and w, (i.e., n). Combining (3.94) with (3.81) gives (3.75). Proof of Theorem 3.1 (completed).
Lemma 3.5 implies that ya-za
(3.95)
Tyo JB(po)
Iw(y,P,t)IZ dpdy < (itpo Cz/2a - 3)
Thus given any e > 0 there is a ya = yo(e,n) such that
where 3- 2a < 0.
(3.96)
Iw(Y,P,t)I2 dpdy < e I E(Pa
1 ya
for all t E R.
Equation (3.77), together with Lemma 3.4 and the estimate
(3.96), imply that
li sup
(3.97)
e.
t-Ko
Since E > 0 is arbitrary this implies (3.71), as required. The arguments given above, applied to vi, vm and vn, show that the conclusions of Theorem 3.1 hold for all h such that h, E C0(C+ U Ca U C_). Moreover, this set is dense in Lz(R3) and hence (P* Co(C+ U Co U C_) is dense in JCf = Pf,IC by Corollary 9.12 of Chapter 3.
These facts can be used
to extend (3.62) to all h E 7C because the mappings U(t)
Ua (t) =
:
:
JCf i L2(R3) and
Xf ± L,(R3) defined by U(t)hf = exp (-it A112) hf and Uo (t)hf are uniformly bounded for all t E R (see [29, p. 32]).
The
density argument needed to extend (3.62) to all h E 3C has been given in many places; see, for example, [25, Ch. 2] or [27, p. 260].
§4.
TRANSIENT GUIDED WAVES The asymptotic behavior for t -> m of the guided component is a sum in 3C of mutually
is derived in this section.
orthogonal partial waves
Re
1 < k < N,.
The starting point for the analysis is the integral representation
(4.1)
v k(t,x,y) = 1
Yk(x,Y,P) exp (-it wk(IPI)}hk(p)dp
where
(4.2)
hk(p) =
JR3
'k(x,Y,P) h(x,Y) c-'-(y) P-'(y)dxdy
4.4.
Transient Guided Waves
151
and the integrals converge in It and L2(S2k), respectively.
The integral in
(4.1) can be written
(4.3)
exp {i(x
vk(t,x,y) = 2n J
p - tmk(IPI))}
hk(P)dp.
Q k
This is an oscillatory integral that can be estimated by the method of stationary phase when h.k E Co(S2k).
To apply the method define
r = /t2 + 1x 2 (4.4)
t = rC0, xl = r&i, x2 = rE2
&1C2) E S2 C R3
where S2 denotes the unit sphere in R3.
(4.5)
Then (4.3) takes the form
exp fir
vk(t,x,y) = J Q
gk(P,y)dp
k
where
0k(p,
)
_ ElPl + 2P2 - (IPI)
(4.6)
gk(P,Y) _ (2ir)-1 Yk(Y,P) hk(P)
Estimates for large r of the integral in (4.5) are needed that are uniform for
in compact subsets of S2 X R.
Such estimates are provided by a
version of the method of stationary phase due to M. Matsumura [15].
A form
of Matsumura's results applicable to (4.5) is presented in Appendix 2. This result is applied below to estimating vk. The phase function 0k has a point of stationary phase if and only if
Uk(PI)P (4.7)
X t
IPI where
(4.8)
Uk(IPI) = wk(IPI)
is the group speed associated with the dispersion relation w = Wk(Ipl)
It
152
4.
TRANSIENT SOUND FIELD STRUCTURE IN STRATIFIED FLUIDS
will be assumed, for brevity, that Uk(}i) is a monotone decreasing function that maps 0k onto (cm,c(m)).
In this case (4.7) has a unique solution if
IxI/t lies in the range of Uk(IpI); that is
(4.9)
1XI
cm <
and no solution otherwise.
< C(-),
The solution is given by
(4.10)
p = Qk(Ix1/t) x/IxI
where Qk is the inverse function to Uk.
By calculating the Hessian 6k
one can show that
rZ Idet 0j(p,Q I = tZ Uk(IMI) Uk(IPII/IPI
(4.11)
and sgn 6k(p,Q = 0.
In particular, each point of stationary phase is
non-degenerate and makes a contribution
IPI1/2 exp {i(IxIIPI-twk(IPI))} YY,IPI)hk(P) (4.12)
vk(t,x,Y,P) _ I}1/2
t{Uk(IPI) IUk(IPI) to the interval in (4.3), where p is given by (4.10).
For IxI/t outside
the interval (4.9) there is no point of stationary phase.
Thus the
stationary phase approximation to vk(t,x,y) is given by
(4.13)
k(t,x,y) = X(Ixi/t) vk(t,x,Y,Qk(Ixi/t) x/IxI)
where X is the characteristic function of the interval (cm,c(o)) and one has Theorem 4.1.
For all h E 3f such that hk E Co(Ok) there exists a
constant C = Ck(h) such that
(4.14)
Ivk(t,x,Y) - vk(t,x,Y)I < C/t2
for all t > 0, xER2-{0} and yCR. Theorem 4.1 can be proved by application of Theorems A.1 and A.2 of Appendix 2.
The proof that k(yp) has the required p-derivatives is
lengthy but straightforward and will not be given here.
If hk is not a
smooth function then Theorems A.1 and A.2 are not applicable and the
Transient Guided Waves
4.4.
estimate (4.14) may fail.
153
However, the definitions (4.12) and (4.13) are
meaningful for all h E 3C and one has Theorem 4.2.
For all h C JC, all t > 0 and k = 1,2,3, one has
Ex
(4.15)
and
(4.16)
"'(t'-) 113C = IIhkIIL2(0
Moreover, the mapping t -
(4.17)
)
= IIPkhIIx
is continuous from R+ to IC and
0.
I'M t-*
The proofs of these properties are the same as those for the Pekeris profile, given in [27], and are not reproduced here.
(4.18)
On defining
uk(t,x,Y) = Re {vk(t,x,Y)}
one also has Corollary 4.3.
For all h E JC and k =
lim
(4.19)
uk(t,')IIx = 0.
t-
C L2(R3) and asymptotic wave functions for
If h E LZ(R3) then
the first derivatives of uk can be constructed.
Indeed, if hk E CI(Q,,)
then the first derivatives of vk are given by
exp {i(x p-
Dtvk(t,x,Y) = 2n J
(4.20)
exp {i(x' p- t
Djvk(t,x,Y) = 2 J n
(7
k
Dyvk(t,x,y) =
exp {i(x p- twk(IpI))}
I
Qk These integrals have the same form as the integral (4.3) for vk. corresponding asymptotic wave functions are defined by
The
= 1,2),
154
TRANSIENT SOUND FIELD STRUCTURE IN STRATIFIED FLUIDS
4.
vka(t,x,Y,P) = (-iak(IpI)) vk(t,x,Y,P), (4.21)
vk.(t,x,Y,P) = (ip.) vk(t,x,Y,p) (j = 1,2),
vk3(t,x,Y,p) = Dyvk(t,x,Y,p), and
vk.(t,x,Y) = X(IxI/t) vkj(t,x,Y,Qk(IxI/t) x/IxI)
for j = 0,1,2,3. Theorem 4.4.
The analogue of Theorem 4.2 is
For all h E L2(R3), all t > 0 and k = 1,2,3, one has
vkj(t,') E L2(R3), j = 0,1,2,3,
(4.22)
3
(4.23)
11Vk,(t,-)112
+
j-1 Ilvkj(t,')IL2(R3,P-1dxdy) = 2 IA1/2 hkllx,
and
(4.24)
lim lJIDjvk(t,') - vkj(t,')IIL2(R3) = 0, j = 0,1,2,3.
The proof of Theorem 4.4 is the same as for the special case of the Pekeris profile which was treated in detail in [27]. The preceding discussion was restricted to the special case where Uk(U) is monotonic.
If Uk(P) has a finite number of maxima and minima there are
a corresponding number of points of stationary phase and the form of the asymptotic wave function is more complicated but still tractable.
In the
case of the Pekeris profile, treated in [27], there are two points of stationary phase.
Cases that lead to infinitely many stationary points have
not yet been encountered.
§5.
They would require additional analysis.
ASYMPTOTIC ENERGY DISTRIBUTIONS The total energy
(5.1)
E(u,R3,t) =
{IDu(t,X)I2 + C-2(y) ID,u(t,X)121p-'(y)dX
I
R3
of an arbitrary solution wFE is finite and constant.
Moreover, A is the
selfadjoint operator in Jf associated with the sesquilinear form A on JC
4.5.
Asymptotic Energy Distributions
155
defined by D(A) = LZ(R3) C X and
A(u,v) =
(5.2)
f3
vu(X)
Vv(X) P-1(y)dX.
R
It follows from Kato's second representation theorem that D(A1/2) = L2(R3) and for all u E D(A1/2)
(5.3)
one has
IAI/z U112(
= A(u,u) = J
IDu(X) I2P-1 (y)dX. 3
Hence the total energy satisfies
E(u,R3,t) =
(5.4)
Moreover, if h E D(A1/2)
IIA1/2 uII2
+
and e-itA
u(t,X) = Re {v(t,X)},
(5.5)
1/2
h
then a simple calculation gives
(5.6)
E(u,R3,t) = IIA1/2 hllx.
Indeed, Al/2 h = Al/2f + ig where, by assumption, f and g are real-valued.
Now Al/2
is a real operator; i.e., Al/2 h = Al 72 h.
It follows that Al/2 f
and g are the real and imaginary parts of A1/2 h, respectively, and (5.6)
follows immediately since
IA1/2 h(X) 12 = IA1/2 f(X) 12 +
(5.7)
Ig(X) 1 2.
The total energy (5.4) is constant for all t > 0. for the partial waves uf, ug and uk, 1 < k < No.
The same is true
Moreover, as shown in §2,
is a complete family of orthogonal projections in JC that reduces A.
(5.8)
The energy partition theorem follows immediately: NOIIA1/2 hIl
= IIAI/2 hfllx + k=
1 IIAI/2 hkll2
k==1
The partial energies may be calculated from the initial state by (5.9)
w2(P) I'_(p) I2dP
E(uf,R3,t) = IIAI/2 hfllx = J R3
and
156
TRANSIENT SOUND FIELD STRUCTURE IN STRATIFIED FLUIDS
4.
(5.10)
6(uk,R3,t) =
IIA1/2 hkljC
wk(IpI)Ihk(P)I2dp.
= 10
The results on asymptotic energy distributions, formulated above as (1.45)-(1.55), can now be proved.
The results on the free component of
follow immediately from Corollary 3.3 and the results on free waves proved in detail in [25].
The results on the guided components uk follow from
These results were proved in detail in [27) for the case of
Theorem 4.4.
the Pekeris operator.
The proofs are identical for the class of operators
treated here and therefore will not be repeated.
§6.
SEMI-INFINITE AND FINITE LAYERS The preceding analysis is extended in this section to the cases of
semi-infinite and finite layers of stratified fluid.
The extensions are
based on the normal mode expansions for these cases that were derived in Chapter 3, §10.
Only the principal concepts and results are formulated
here since the proofs are entirely analogous to those of the preceding sections.
Semi-Infinite Layers.
As in Chapter 3, §10, the fluid is assumed to
occupy the domain R+ and to satisfy the Dirichlet or Neumann boundary condition.
Here the functions p(y) and c(y) are assumed to be Lebesgue
measurable and satisfy
0
(6.1) and
P(y) - p(°°)I < C Y-0"
(6.2)
Ic(y) - c(-)I < C y-a
for all y > 0 where pm, pM, p(W), cm, cM, c(-), C and a are constants and
a > 3/2.
(6.3)
As in Chapter 3, §10, the acoustic propagators for p(y) and c(y) corresponding to the Dirichlet and Neumann conditions will be denoted by A° and A', respectively.
They are selfadjoint non-negative linear operators in
3C+ = L2(R+,c 2(y)p 1(y)dxdy).
The normal mode functions 3(x,y,p,X) for Aj, as defined in Chapter 3, §10 are parameterized by (p,X) E 0 = f(p,A)
I
A > c2(W)IpI2).
Their
4.6.
Semi-Infinite and Finite Layers
157
asymptotic form for y ± - is given by
(6.4)
c(
43(x,Y,P,q) -
,X) {ei(P'x-yq) + Rjei(p'x+Yq)1
p
where q = q(Ip ,X) = (A c-2(W) - IPI2)1/2 R3 = R3(lpl,N) and lR3(Ipl,A)I = 1.
c(IPI,A) _ (P(°°)/47rq(IPI,a))1/2
As in the preceding sections, it will
be convenient to introduce new parameters:
(p,X) ± (p,q) _ (p,q(lpl,X))
E R+ and normal mode functions
(6.5)
+(x,y,p,q) = (2q)1/2 c(°°) i3(x,Y,P'X)
where
A _ X(p,q) = c2(°°)(IpI2 + q2).
(6.6)
The asymptotic form of
is
{ei(P'x-qY) + RJei(P'x+q))}. +(x.Y,P,q) = c(°°)P-) (27r ) 3/2
(6.7)
y
The second family 3 defined by
(6.8)
3(x,y,p,q) = f+(x,Y,-P,q)
is also needed.
It satisfies
i (x,Y,P,q) -
(6.9)
c(°°)P1/2 (°°) {ei(p'x+qY) + RJ e l(P'x-qY) l y f (27r) 3/2
The expansion theorem of Chapter 3, §10 implies that the limits M
(6.10)
exist.
(6.11)
f+(p,q) = L2(R+)-lim
14
J° JIxI<M
+(x,y,p,q) f(x,y) c-2(Y)P-1(Y)dxdy
Moreover, if
$+
:
3f -> L2 (R+)
is defined by c+f = f+ then (D+ is a partial isometry with range L2(R+) and
158
4.
TRANSIENT SOUND FIELD STRUCTURE IN STRATIFIED FLUIDS
*
Tj
(6.12)
k=1 where `Yk
:
k
Y3
k
if } L2(S2 ) are the partial isometries associated with the guided
wave normal modes k(x,y,p) of Chapter 3, §10. Normal mode expansions for A3 are given by (6.12) with either the + or - sign.
(6.12) implies that the orthogonal projections in 11+ defined by
P3 f = +4'3*+j ='V3*'tj - (6.13)
pk=Tj*Tk, 1
The free component of
a complex acoustic potential
v(t, ) = exp (-i t (A3) 1/2) h, h E X+,
(6.14)
is given by
Pfv(t,') = exp (-it (A3)1'2) P h.
(6.15)
The j-representation of of is (6.16)
vf(t.x.y) =
where w(p,q) =
(x,y,p,q) exp (-it w(p,q)) 0(p,q)dpdq
3 J
R+
Moreover, as in §3 one can write
q2.
(6.17) 3(x,y,p,q) = c(O0)p13 (2Tr)
z(°°)
{ei(p'-+gy) Ij(y,p,q)+el(p-x-gy)Rj(y,p,q)I
l
where
lim 13(y,p,q) = 1, y-
(6.18)
lim R3(y,p,q) = R3(p,A) y-wo
Then, proceeding as in §3, one can prove the following analogue of Theorem 3.1.
4.6.
Semi-Infinite and Finite Layers
Theorem 6.1.
(6.19)
159
For every h E IC+ let
be defined by
vf(t,x,y) = exp (-it c(&)Ao/2) ha (x,y) ,
(x,y) E R+
where ho E L2(R3) is the function whose Fourier transform is c(°O)P112(_) hj(p.4), (6.20)
(p.9) E R"
ho (p+4) _ 0,
(p,q) E R3.
Then
(6.21)
t
lim 11vf(t,
Transient Guided Waves.
form of the guided components of an infinite layer. cases.
)
0.
-
+ For both semi-infinite and finite layers the is precisely the same as for the case
Thus the analysis of §4 applies unchanged to these
Chapter 5 Scattering of Signals by Inhomogeneous Layers
The purpose of this chapter is to study the scattering and distortion of acoustic signals by an inhomogeneous plane-stratified layer separating two homogeneous fluids.
The transmitter or source of the signals is
assumed to lie in one of the homogeneous fluids and the reflected and transmitted signals are calculated. are summarized in §1.
The physical hypotheses and results
The proofs, which are based on the preceding chapters
are developed in the remaining sections.
§1.
SUMMARY
Throughout this chapter it is assumed that p(y) and c(y) satisfy the
boundedness conditions (2.1.4) and are variable only in a finite layer yl < y < y2,
More precisely, it is assumed that
p(y) = p(--) and c(y) = c(-m) for y < y1,
p(y) = p(c) and c(y) = c(m) for y > y2.
These conditions imply, in a trivial way, that p(y) and c(y) satisfy the hypotheses of the preceding chapters. The scattering by the inhomogeneous layer yl < y < y2 of signals whose sources are localized in the half-space y > y2 will be analyzed. described in Chapter 2, this can be modelled by initial values g which satisfy
(1.2)
supp f U supp g U {X
:
xi + x2 + (y- y0)2 < S21.
The sources will lie in the region y > y2 if yo > y2 + d.
161
As f,
162
5.
SCATTERING OF SIGNALS BY INHOMOGENEOUS LAYERS
The Source Radiation Pattern.
The incident signal will be defined to
be the acoustic field that would be generated by the given sources if they were situated in an unlimited homogeneous fluid with parameters p(-), c('). It is characterized by the potential uinc(t,X) that satisfies (2.1.6) and (2.1.8) with p(y) = p(er) and c(y) = c(-) everywhere.
Of particular interest
for applications is the far field form of the incident signal.
It is
described by the asymptotic wave functions [25,28]
(1.3)
uinc,k(t,X)
=
r1 sk(r - c(W)t,0),
k = 0,1,2,3,
where
r = XI = X1 + X2 + y2
(1.4)
and
O=XE r S2. S2 is the unit sphere (= set of all unit vectors) in R3.
It was shown in
Chapter 4 that for suitable functions sk, _
m
(1.6)
ok,
where ok -r 0 in L2(R3) when t
(1.7)
sk(T,O)
k = 0,1,2,3,
In addition, one has [25]
-c-1(C°) Oksu(T,O), k = 1,2,3.
Hence, the far field form of the incident signal is characterized by the single real-valued function so(T,0), defined for T E R, 0 E S2.
Moreover,
for incident fields wFE one has [25] so E L2(R x S2) and
(1.8)
2c-2(_)p-1(_) Ilso11L2(Rxs2) =
3 1
is the total signal energy.
{1Ofl2 + c-2(_)jgj2}P-1(-)dxdy
R
This is verified in §2 below.
The function so(T,0) will be called the source radiation pattern.
It
is uniquely determined by the transmitter or sources through the initial state (f,g).
The exact relationship is given below.
However, it is the
values of so(T,0), rather than f(X) and g(X), that are the primary data of the signal scattering problem studied here.
Indeed so is directly observable
Summary
5.1.
163
through the relations (1.3)-(1.7).
Moreover, it can be shown that a signal
wFE in a homogeneous fluid that is generated in any way by sources confined to a bounded region will have the asymptotic behavior (1.3)-(1.7).
In
applications the design of a pulse mode transmitter with a prescribed radiation pattern so is a primary goal of the design engineer. The Source Momentum Distribution.
(1.9)
9o(w,0) =
converges in L2(R x Sz) (cf.
The Fourier transform of so,
J (22 ) 11
e
iwT
so(T,0)dT,
[25]) and satisfies
so(-w,0) = so(w,0) for all w > 0
(1.10)
if and only if so(T,0) is real-valued.
Hence the transforms of the source
radiation patterns are characterized by their values for positive frequencies w.
For these values it is shown in §2 below that so is related
to the momentum distribution of the sources by
so(w,0) =
where ho = g - i
2w ho(w0)
f E Lz(R3) (for notation see (4.3.17)) and ho is
the usual Fourier transform in L2(R3):
(1.12)
e-iP'X
ho(P) _ (2n)37 J
ho(X)dX,
R 3
with P = (p,q) = (p3,p2_,q) E R3.
with the signal momentum P.
(1.13)
ho(P) is the complex amplitude associated
It is related to the initial state by
ho(P) = S(P) - iw+(P) f(P)
where w+(P) = c(±-)IPI, as in Chapter 4. so is determined by the momentum distribution ho through (1.9)-(1.11). Conversely, if so is known then ho can be recovered by (1.9) and
ho(P) = 2iIPl-1 so(IPI,IPI-1 P).
(1.14)
Moreover,
(1.15)
Iho(P)lzdP = 4 Jo JSz (R3
J
so(w,0)lzdOdw = 2 lsollLz(RxS2)'
164
SCATTERING OF SIGNALS BY INHOMOGENEOUS LAYERS
5.
Thus the correspondence >T so - ho defines a unitary mapping of the Hilbert
space of real radiation patterns wFE onto the Hilbert space of all momentum distributions in L2(R3).
The quantity (1.15) is proportional to the total
energy of the incident signal (cf. (1.8)). The function ho will be called the source momentum distribution.
In
what follows it will be convenient to describe the reflected and transmitted signals by their momentum distributions, rather than their radiation patterns.
The latter can always be recovered from the relations (1.9)-(1.11).
The Structure of the Scattered Signal.
The total acoustic field wFE
u(t,X) generated by the sources in the presence of the plane-stratified scattering layer has a decomposition
(1.16)
ufree + uguided
u
where ufree = P f
u
and uguided =
Pgu.
The structure of uguided was analyzed
thoroughly in Chapter 4 and is not discussed further here.
The component
satisfies, by (4.1.50),
ufree (1.17)
E-(ufree' R2 x (Y1.y2]) = 0.
Hence the free component of the signal is asymptotically negligible in the scattering layer.
The reflected and transmitted signals will be defined for X E R+(y2) and X E R2(y1), respectively, by
(1.18)
(1.19)
urefl(t,X)
ufree (t,X) - uinc(t,X), Y > Y2,
utrans(t,X) = ufree(t,X), Y < y1.
These functions satisfy wave equations for homogeneous fluids with sound speeds c(m) and c(-co), respectively, and have finite energy.
Moreover, the
results of Chapter 4 imply that they have asymptotic wave functions:
(1.20)
Dourefl(t,X) = r-1 arefl(r - c(W)t,0) + orefl
(1.21)
Doutrans(t,X) = r-1 strans(r - c(-)t,0) + otrans
where orefl -' 0 and otrans -* 0 in L2(R+(Y2)) and L2(R3(y1)), respectively,
165
Summary
5.1.
when t
Note that srefl and strans lie in the complementary subspaces
.
of L2(R x S2) defined by L2(R X S+) and L2(R x S2), respectively, where S+ = SZ n R+.
The goal of this chapter is to calculate the radiation patterns srefl and s
trans
and to determine how they vary with the signal radiation pattern
so and the fluid parameters p(y) and c(y).
This will be done by calculating
their momentum distributions
(1.22)
hrefl(P) = 2i PI-'
(1.23)
htrans(P) = 21IPI-'
srefl(IPI,IPI-3
P),
P E R+,
strans(IPI'IPI-1P),
P E R3.
To describe the results it will be convenient to introduce the mappings in momentum space defined by
(1.24)
IIR
R+ -> R3, HR(P,q) = (p,-q),
and
(1.25)
IIT
:
R3 1 -C+, RT(P,q) _ (p,-q+(IPI,X(P,q)))
(See Chapter 3 for the definitions of C+ and q+.) reflection of wave momentum in the plane q = 0.
(1.26)
C(-) IIIT(P) I
RR obviously defines the It is easy to verify that
= c(-°°) IPI for all P E R3.
(1.25) and (1.26) imply that the momenta IIT(P) and P are related by Snell's
law for the refraction of a plane wave passing from a medium with propaga-
tion speed c(°') to one with a speed c(--). With the above notation the principal result of this chapter takes the form
(1.27)
and
re
R+(IPI,A(P)) ho(IIR(P)),
P E C+'
R3(IPI,a(P)) ho(RR(P)),
P E Co,
(P)
166
SCATTERING OF SIGNALS BY INHOMOGENEOUS LAYERS
5.
trans (P)
(1.28)
= C2 (--)P(--) T_(IPI,a(P)) ho(II (P)), P E C
c (oo) p (')
T
,
where R+, Ro and T_ are the reflection and transmission coefficients of the normal mode function $+(x,y,p,q) associated with p(y) and c(y) as in Chapter 3.
Equation (1.27) states that the complex amplitude hrefl(P) associated
with momentum P E C+ (resp., P E CO) is the product of R+ (resp., R0) and the amplitude ho(IIR(P)) associated with the momentum IIR(P) E R3.
Similarly, (1.28) states that the amplitude h
momentum P E
C_
trans
(P) associated with
is the product of c2 (-oo) p (-oo) c-2 (oo)
p-1
(°o) T
and the
amplitude ho(IIT(P)) associated with the momentum HT(P) E -C+'
Relationship to the Scattering Operator.
There is a close relationship
between the scattering relations (1.27), (1.28) and the scattering operator S of the stratified fluid layer characterized by p(y) and c(y).
S is the
unitary operator in L2(R3) defined by
(1.29)
where (D+
+
S
: Xf + L2(R3) are the normal mode mappings of Chapter 3, §9.
It
is shown below that S is determined by the coefficients R+, R0, R_, T+, T_
associated with +(X,P) and the mapping
(1.30)
II
C+ -. C_ = R3
defined by
(1.31)
II(P,q) _ (p,-q_(IPI,X(P,q))
1 1-1 exists and is given by
(1.32)
II 1(P,q) _ (P,q+(IPI,X(P,q)))
With this notation the construction of S is given by
R h(P) + c(-oo)P + (1.33)
S h(P) =
j
2
(W)
T+h(II(P)),
Roh(P), 1/2
R h(P) + c(- )p
_ (°o)) T_h(II 1(P)),
5.2.
Signals in Homogeneous Fluids
where R+ = R+(Ipl,A(P)), etc.
167
Relation (1.33) is derived in §4 below.
The
unitarity of S in L2(R3) imposes certain restrictions on R+, Ro and T+ which are also derived in §4.
In §5 the construction (1.33) is used to
derive an alternative representation of srefl and strans*
§2.
SIGNALS IN HOMOGENEOUS FLUIDS The theory of asymptotic wave functions for sound waves in homogeneous
fluids was developed in [25, Ch. 2].
In that work the constant density and
sound speed were normalized to have the value unity.
Here the theory is
needed for arbitrary constant density p(W) and sound speed c(-), for comparison with inhomogeneous stratified fluids.
The theory may be
obtained as a very simple case of the results of Chapter 4 by the specialization
P(y) - P(m) = const.,
c(y) -} c(°') = const. L
The notation 3C(-) = L2(R3,c
2(_)P-l(m)dxdy) and A(-) _ -c2(W)O2 will be used
for the corresponding Hilbert space and acoustic propagator.
The specialization (2.1) implies that 7C = dC(p,c) - 3C(°) = X(-) f; i.e., there are no guided modes. Moreover (2.2)
(),
q+(p,A) = q_(p,A) - q(11,A) _
C-2(_)
-
112)1/2
and
(2.3)
A(P,q) -' c2(0D)(IP12 + q2) = w2(P,q) for all (p,q) E R3.
The normal mode functions &+(X,P) of Chapter 3 become plane waves
0+' 0+(x,y,P,q) = c(_)pl/2(°°)
(2Tr)-3/2 ei(P'x-qy),
(2.4)
m(x,y,P,q) = c(°O)P1/2(m)
(2,R)-3/2
while the normal mode expansions are essentially the Fourier integral representation in L2(R3).
one has the expansions
Thus for all h e JC(o) (isomorphic with L2(R3))
168
5.
(2.5)
SCATTERING OF SIGNALS BY INHOMOGENEOUS LAYERS
h+(p,q) = c((2
3 e
1
i(p-x+qy)
h(x,y)
c-2(-)P-1(m)dxdy
R
in L2(R3) and
(2.6)
h(x,y) = c( (2
3)
/1
J
3
el(P'x+qY) h+(P.q)dPdq
R
in J((-). Clearly (2.7)
h+(p,q) = c 1(_)p-1/2(°°) h(p, q)
where h is the usual Fourier transform, defined by (1.12), and (2.6) is equivalent to the usual Fourier integral representation.
Parseval's
relation may be written
IlhlI3C(.) = IIh±IIL2(R3).
(2.8)
Asymptotic Wave Functions.
To obtain the asymptotic wave functions
for the incident signal uinc note that it is the solution in J((-) of
(2.9)
D2uinc + A(m)uinc = 0, t E R,
(2.10)
uinc (0) = f and Dpuinc(0) = g.
The solution can be written
(2.11)
uinc
(t A112(m))f + (A 1/2(40) sin t A112(O0))g
If, as will be assumed,
(2.12)
f E D(A1/2(°)). g E D(A 1/2(°°))
then u.inc is a solution wFE and (2.13)
where
uinc (t, X)
Re {vinc(t,X)}
Signals in Homogeneous Fluids
5.2.
169
-itw+(P)
0+(X,P) a
vinc(t,X)
JR
h+(P)dP
3
(2.14) 1(p.x+gy-tw+(P))
c(W)P1/2(m)
(2n
J3
+(p,g)dpdg
e
R and
h+(P) = f+(P) + iw+1(P) g+(P).
(2.15)
On choosing the lower signs in (2.14) and using (2.7) this becomes the Fourier representation
(2.16)
vinc(t,X)
(2
h(P)dP
3
J
R
with
(2.17)
h(P) = f(P) + iw+l(P) g(P).
Note that this is identical with [25, (2.34)] with t replaced by c(-)t. Thus the asymptotic wave functions for vine may be obtained from those of [25, Ch. 2] by the same substitution.
(2.18)
In particular, one has
in X(-) when t
v'
where
(2.19)
vin(t,X)
(2.20)
G(T,0) _ (2,-
= r-1 G(r - c(-)t.O),
1 112
Jo e1Tw
in L2(R x S2) and
(2.21)
IIGIILZ(R), S2)
Derivatives.
= IIhIIL2(R3)
Hypotheses (2.12) imply that vine has first derivatives
in L2(R3) that may be obtained from (2.16) by differentiation under the integral sign.
These integrals also have the form (2.16) and proceeding as
in [25, Ch. 2] gives
170
5.
(2.22)
SCATTERING OF SIGNALS BY INHOMOGENEOUS LAYERS
Dkvinc(t'X) - V00 in
k (t'X) =
r-1
Gk(r - c(m)t,O)
in L2(R3) when t + w where
(2.23)
Go (T,O) _
ho(w0)
(2.24)
(2n
Jo eiTwh0(w0)(-iw)dw,
-i c(=)w h(w0) =
(w0) - i c(°)w f(w0),
and
(2.25)
Gk(T,0) = -c-1(00)Ok Go(T,0), k = 1,2,3.
On taking real parts one obtains the results (1.3)-(1.7) and (1.9)-(1.13) quoted in §1.
Asymptotic Energy Distributions.
The total energy of the incident
signal is
E(u inc,R3,0) = E(u inc,R3,t) (2.26)
{IDuinc(t,X)I2 + c-2(m)IDouinc(t,X)12}p-1(W)dX.
= JR3
It may be expressed in terms of so by combining (1.3)-(1.7) and (2.26) (cf. [25, Ch. 8]).
(2.27)
The result is
E(uinc,R3,0) = 2 c-2(W)p 1N 1IsoI1L2(RxS2)
which verifies (1.8).
Asymptotic energy distributions in cones and other
subsets may be derived by the same method, as in [25, Ch. 8].
§3.
THE REFLECTED AND TRANSMITTED SIGNALS The functions srefl and strans are calculated in this section.
First,
general representations of srefl and strans are derived that are valid under the hypotheses of Chapter 4.
Next it is shown that for finite scattering
layers the asymptotic forms for y + 00 of the normal mode functions is exact
outside the scattering layer.
These results are then combined to verify
the representations (1.27), (1.28) for finite scattering layers.
The
171
The Reflected and Transmitted Signals
5.3.
implications of these results for signal distortion are discussed at the end of the section.
For simplicity it is assumed in the remainder of the report that f E D(A1/2)
D(A-112)
and g E
so that the representation u(t,X) =Re {v(t,X)}
exp (-itA1/2)h is available; of.
[25, Ch. 3].
All of the results
obtained may be extended, by a density argument, to arbitrary fields wFE; i.e., f E D(A1/2)
= LZ(R3) and g E L2(R3), but the details will not be
given here.
The starting point of the calculations is Chapter 4, Corollary 3.3. The hypotheses on f and g imply that
ufree (t,X) = Re {v
(3.1)
free
(t'X)
and for k = 0,1,2,3
(3.2,)
Dkvfree(t,X) ` Dkvfree(t,X) in L2(R3) when t ->
where e-itA1/2 (
(3.3)
h+ in L2 (R+(Y2)).
0
e-itA1/2
and h+ and h
)
h
in L2 (R3(Y1)),
are the functions in L2(R3) whose Fourier transforms are
(Chapter 4, Theorem 3.1)
c(oo)P1/2(oo) h_(P), (3.4)
P E R+,
h+(P) _ 0,
P E R3,
R 3
0,
(3.5)
c(-oo)P1/2(-00) h_(P),
P E R3.
Recall that h = f + iA 1/2 g and hence
(3.6)
h_(P) = f_(P) + ih 1/2(P) 8_(P).
172
SCATTERING OF SIGNALS BY INHOMOGENEOUS LAYERS
5.
The Reflected Signal.
Applying the same formalism to the incident
signal gives
(3.7)
uinc(t,X)
= Re {vinc(t'X)}
where e-itA
(3.S)
vinc(t,-) =
1/z (
)
hinc
and hinc = f + iA 1/z(_) g, whence
(3.9)
hinc (P) = f(P) + iw+1(P) 8(P).
Subtracting (3.7) from (3.1) gives (Definition (1.18))
(3.10)
urefl(t,X)
= Re
{vrefl(t,X)}
where
(3.11)
vrefl(t,-) - vfree(t,-) - vinc(t,-).
It follows from (3.2), (3.3) that
(3.12)
Dkvrefl(t,X) - Dkv0
(t,X) in Lz(R+(yz)), t
where 1/z
(3.13)
v0 free (t,-) - vinc(t,-) =
e-itA
(W) (h+ - hinc).
In particular,
(3.14)
Dovrefl(t,X) -
e-itA 1/2( m )
hrefl in L2(R+(y2)), t -r
where
(3.15)
hrefl = -iA1/z (W) (h+ - hinc) .
Equations (3.14), (3.15) imply the validity of (1.20) with (cf. (2.16)-(2.20))
The
5.3.
Reflected and Transmitted Signals
173
(_lw)
(3.16)
srefl(w,0)
-
2
hrefl(wO )
On taking the Fourier transform of (3.15), using (3.4), (3.6) and (3.9),and recalling that X(P) = w+(P) when P E R+, one finds that the momentum distribution of srefl is
hrefl(P) = (-iw+(P))Ic(-)P11'2(°°) h_(P) - hinc(P)] (3.17)
= c(OD)P112(0°) 8_(P) - g(P) - iW+(P) Ic(°°)PhI2(00) f_(P) - f(P) I, P E R+.
Using (2.7) this can be written (3.18)
hrefl(P) = c(00) P1/2(00) (usc(p) _ iw+(P)fsc(P)I, P E R+
where (3.19)
gsc(P) = S_(P) - gm(P)
and similarly for fsc The Transmitted Signal.
(3.20)
Proceeding similarly gives
utrans(t,X) = ufree (t,X) = Re {v free (t,x)}
where, by (3.2)-(3.6), one has
(3.21)
1/2 Dovfree (t,X) - e itA (-`°) htrans in L2(R3(y1)), t - 0O,
with
(3.22)
htrans
-iA1/2 (-00) h-.
These relations imply the validity of (1.21) with
(3.23)
defined by
s
trans
(w, O)
(-1w) S 2
trans
(wO)
174
5.
(3.24)
SCATTERING OF SIGNALS BY INHOMOGENEOUS LAYERS
htrans(P) = (-iw_(P)) c(-)
P1/2(-a0) h-(P), P E R3.
Using (3.6) with X(P) = w`(P) for P E R' gives the representation
(3.25)
trans(p) = c(_°°) P1/2(-0°)
[
_(P) - iw_(P) f_(P)I, P E R3.
The representations (3.18) and (3.25) are valid for the class of stratified fluids defined in Chapter 4.
They are used below to derive (1.27) and
(1.28).
A Decomposition of the Normal Mode Functions.
It will be convenient
to write the normal mode functions + as
0+(X,P) = 4.
(X,P) + 0+c(X,p),
(3.26)
I
-(X,P) = 0out(X,P) + mac(X,p),
where
(2Tr)-3/2 c(0°)P1/2(°°)
(3.27)
in(X,P) =
P E R+
X E R+,
j
0,
p E R3
PE R+
0,
(3.28)
0in(X,P) =
X E R3
a
(2Tr) -3/2
c(-°°)P1/2(-°°)
P C R3
and
(3.29)
out(x,Y,p,q) =
in(x,Y,-p,q).
Note that bout is given by equations (3.27), (3.28) with q replaced by -q. The behavior of 0+(X,P) for y be described by the equations
which was determined in Chapter 3, may
5.3.
The Reflected and Transmitted Signals
e1P-X,
R+
+c(X,P) = c(P)
(3.30)
175
P E C+
+ a+(X,P)
Ro
e
,
p E C_
,
P E C+
i(P,X-q-y)
T+ d +c(X,P) = c(P)
(3.31)
ezp-x eq!Y,
P E Co
eip'x,
P E C
+ gs_(X,P)
where R+ = R+(IPI,A(P)), etc., q+ = q±(IpI,X(P)), q' = q'(IPI,A(P)), c(P) is defined by (3.1.44), and
(3.32)
Q+(x,y,P,q) - 0 when y
±-.
Similar statements for Esc follow from the relation 4sc(x,y,p,q) _
sc(x,y,-p,q).
The derivation of (1.27) and (1.28) for the case of
finite scattering layers is based on Lemma 3.1.
If p(y) and c(y) are constant outside the interval [yl,y2]
then .+(x,y,p,q) = 0 for y > Y2 and a_(x,y,p,q) = 0 for y < y1. The vanishing of a+(x,y,p,q) for y > y2 when p(y) = p(°°), c(y) = c(00)
for y > y2 follows from the observation that (3.30) with Q+(x,y,p,q) = 0 is a solution of the equation A¢+c(X,P) = 1(P) Osc(X,P) for y > y2 together
with the unique continuation property of the ordinary differential equation for the normal mode function P+(x,y,p,A) _ (2n)-1 e>,p,x P+(y,Ipl,A), etc. (see (3.3.1)).
The same argument applies to Q_ on y < y1.
Verification of (1.27) for Finite Layers. be treated, the other case being similar.
The case where P E C+ will
Assume that p(y) and c(y) are
constant outside [y1,y2] and that supp f C R+(y2), so that p(y) = p(m) and c(y) = c(w) on supp f.
Then
f-(P) = 1R3 _(X,P) f(X) c 2(y)p-1(y)dX (3.33)
= c-2(m)p l(oo)[)R l
3
out (X,P) f(X)dX +
3
Osc(X,P) f(X)dX1111l.
R
Substituting $sc(x,y,p,q) = +c(x,y,-p,q) from (3.30) with P E C+ and
176
SCATTERING OF SIGNALS BY INHOMOGENEOUS LAYERS
5.
a+ = 0 (Lemma 3.1) gives
f- (P) = c-' (°°)P-1 (m) I c(°°)p1/2 (°°)
(2Tr)-3/2
l
e-ip,X
3
J
f (X) dX
R
(3.34)
+ c(=)P1/2(0°)
= c-1(m)P-1/2(00) c-1(00)p-1/2(40)
Moreover, fm(P) =
(21T)-3/2
IR3 R+, e-i(p'x-qy) f(X)dX]
(I (p) + R+ f(p,-q)).
f(P), by (2.7).
Subtracting this equation
from (3.34) gives
fsc(p) = ci(=)p-1/2(°0) R+ f(p,-q) (3.35)
= c1(-)P-1/2(m) R,6(IpI,X(P)) f(RR(P)) for every P E C+.
The same relation holds for gsc.
Substituting these
relations into (3.18) and using the definition (2.24) of ho gives (1.27). Verification of (1.28) for Finite Layers.
If P E C_, p(y) and c(y)
are constant outside [y1,yz] and supp f C R+(y2) one has by the analogous argument, using Lemma 3.1,
f_(P) = c-2(°°)p-1(m)
_(X,P) f(X)dX
J
R3
(3.36)
_
C-2(°°)p-l(CO)C(-°°)p1/2(-00)
= c z(°°)P
(2Tr)-312
- i(p'x-q+y) T_ 1
e R3
1(°°)c(-°°)P1/z(--) T_ f(p.-q+.)
c -2(- ) p 1 (°O) c (-°°) P 112 (-0O) T_ f (11 (P)) The analogous equation holds for g_(P).
Substituting these relations into
(3.25) and noting that for P E C_ one has w+(IIT(p)) = c(O°)jlT(P)I =
= w_(P), by (1.34), gives (1.28). Theorem 3.2.
(-°°)IPl
This completes the proof of
If p(y) and c(y) are constant outside an interval [y1,y2]
and if supp f U supp g C R+(y2) then the reflected and transmitted signal momentum distributions are given by (1.27) and (1.28).
177
The Reflected and Transmitted Signals
5.3.
Signal Distortion.
Combining (1.22) and (1.27) gives iw
srefl(w.0)
2
hrefl(w0)
(3.37)
= - 2 R+(w sin 0,c2(_)w2) h°(IIR(wO))
for 0 E C+ where
0 = (03,02,03) = (cos m sin 6, sin 0 sin 6,cos 6).
(3.38)
Defining OR E R3 by
(3,39)
IIR(w0) = w(01,02,-03) = w OR.
(3.37) can be written
srefl(w,O) = R+(w sin 6,c2(m)w2)g°(w,OR), 0 E C+'
(3.40)
and similarly
(3.41)
= R°(w sin 0,c2(oo)w2) s°(w,OR), 0 E C°.
srefl(w,0)
In the same way, beginning with (1.23) and (1.28) one finds
9trans(w'0)
2
fitrans(w0)
(3.42)
iw c2(-W)p(-W) T (w sin e,c2(-W)w2) 2 c2(m)p(m) for 0 E C_.
a
(IIT(w0))
Now, by (1.26),
(3.43)
IIT(w0) = Y 3w, Y =
Hence (3.42) can be written
(3.44)
s
trans
(w,0) =
T (w sin
where OT E C_ is defined by (3.45)
11 T(w0) = Y-3 W 0 T.
6,c2(-w)w2) 9,(Y- 3 w,0 )
T
178
SCATTERING OF SIGNALS BY INHOMOGENEOUS LAYERS
5.
Equations (3.40), (3.41) and the Fourier inversion formula imply the representations (
srefl(w,0)
i ll = Re IFJ lJ
1/z
eiTW R+(W sin 6,c2(')w2) so(W,OR)d+ 0 E C+1 J o
(3.46) 1/2
(
= Re Oil) l
eiTW Ro(W sin 0,c2(°°)W2) so(W,OR)dw}, 0 E Co.
J
J
o
)))
Similarly, (3.44) and the Fourier inversion formula, followed by a simple change of variable, give
strans(w'0)
=
c (--)P(--) c(°')P(°°)
2ll 1/z
Re {{/
J o
e
iYTw
T(Yw sin 9,c 2(-)w2)90 (w,OT)dwF 11
(3.47)
for 0 E C_.
These results show explicitly how the signal distortion due to
reflection and transmission by a scattering layer is determined by the
reflection and transmission coefficients of the normal mode function +(X,P).
§4.
CONSTRUCTION OF THE SCATTERING OPERATOR The scattering operator S
L2(R3) -+ L2(R3) is defined by
S=0-0+
(4.1)
where cD+
:
:
Jff + L2(R3) are the operators associated in Chapter 3 with the
two normal mode expansions in 'If defined by ( + and 0_.
The unitarity of 0+
and D-, proved in Chapter 3, implies that S is unitary and, in particular
(4.2)
S-1 = S* = + $*.
The defining relations
(4.3)
h+=t+h for all hEJff
imply that the functions h+ E L2(R3) are related by
(4.4)
h_=Sh+, h+ =S*h-.
These relations will be used to calculate S.
5.4.
Construction of the Scattering Operator
179
It was shown in Chapter 4 that, for all h E JCf,
(4.5)
e
-itAl/2 h e-itA1/2
(4.6)
h
where h+ and h
e- itAl/2
(W) h+
e-itA1/2 (-°°) h
w,
in LZ(R+), t
t
in L2 (R3) ,
are the functions in LZ(R3) whose Fourier transforms are
defined by
c(°O)P1/2 (°°) h_(P), p E R+,
h+(P) =
(4.7)
P E R3,
0,
and
P E R+,
0,
(4.8)
c(-0')P1 2(_ ) h_(P), P E R3, respectively. h
Moreover, it is clear from the results of [25] that
fi+
and
are uniquely determined in L2(R+) and L2(R3), respectively, by (4.5) and
(4.6).
The relationship between h_ and h+ will now be determined by
calculating h+ and h
in a second way, using the 0+ representation for
exp (-itA'12) h, and equating the resulting representations to those given by (4.7), (4.8).
Second Calculation of h
Based on 0+.
The starting point is the
representation (4.3.2): (e itA 1/2
h)(X) = J
+(X,P)
e-ita
1/2
(P)
h+(P)dP
R3
(4.9)
= va(t,X) + vb(t,X) + vc(t,X)
where (cf. Chapter 4, §3 for notation)
*
va=e -itA1/2
(D+a
vb=e-itA1/z
(D+b
(4.10)
*
vc=e -itAl/2 +c
180
SCATTERING OF SIGNALS BY INHOMOGENEOUS LAYERS
5.
and
a = X+ h+, b = X0 h+, c = X_ h+.
(4.11)
The asymptotic behavior of the components va, vb and vc will be calculated Only the formal steps of the calculations
by the method of Chapter 4, §3. will be given,
They can be justified by the method of Chapter 4.
Behavior of va,
va has the representation
+(X,P) exp (-itw+(P)) a(P)dP
va(t,X) = 1 C+
4+(x,y,p,q) = (271)-1 c(m)(2q)1/2 eip'x 41 +(Y,IPI,X)
for (p,q) = X+(P,X) E C+ (and hence A = A(P,q) = c2(°O)(IPI2 + q2)).
On
employing the representations (4.3.9) and (4.3.11) for *+, noting that q+(Ipl,),(p,q)) = q for P E C+1 and proceeding as in Chapter 4, §3 one finds
va(t,') in L2(R+), (4.14)
in L2(R3),
for t - m where e-itA1/2(°°)
vl(t,') = a
h1
a
(4.15) e-itA1/2(-`°)
v2(t,') = a
h2
a
are defined by
(4.16)
R+(IPI,X(P)) h+(P)dP
e
va(t,X) = c(°°)P1/2(W) C+
and
va(t,X) = c(_)P'/2(°O)
(27r)-3/2
i(x'p-yq_(IPI,A)-tw+(P))
(
e
J
C+ (4.17)
x T+(IPI,A(P)) h+(P)U.
Construction of the Scattering operator
5.4.
181
To see that va has the form of (4.15) note that the mapping (p,q) - (p,q') = 1T(p,q) defined by (1.31) maps C+ onto C_, has Jacobian a(p,q)/a(p,q') = cz(-oo)q'/cz(W)q and satisfies X(ll(P)) = X(P).
Changing the variables of
integration in (4.17) from P = (p,q) to (p,q') and using (1.32) for H-1 gives 2(t,X)
i(x'p+yq'-tw_(P,q'))
1/z
T+(IPI.X(P,q'))
e
= c((Zp fC
(4.18)
) I q ,i dpdq'
x h+(111(P,q'))
l +
where q+ = q+(IPI,X(P,q')) vb has the representation
Behavior of vb,
¢+(X,P) exp (-itw+(P)) b(P)dP
vb(t,X) = J
(4.19)
Co
where
$+(x,y,P,q) _ (27T)-1 c(-)(2q)"2 elp*x IP,(y, IPI ,X)
(4.20)
for (p,q) E X0(p,a) E Co (and hence X = X(p,q) = w+(p,q)). On employing
the representations (4.3.29) and (4.3.31) for o and proceeding as before one finds
in Lz(R+), (4.21) 0
in L2(R3),
for t-'Wwhere (4.22)
vb(t,') = e
itA 1/z (
)
hb
is defined by
i(X'P-tw+(P))
1/z ,o
c(00(2rr)3z
(4.23)
vb(t,X) =
e
1
)
Ro(IPI,a(P)) fi+(P)dP.
Co
Behavior of vc.
(4.24)
vc has the representation
¢+(X,P) exp (-itw_(P)) c(P)dP
vc(t,X) = 1 C
182
SCATTERING OF SIGNALS BY INHOMOGENEOUS LAYERS
5.
+(x,y,p,q) _
(2Tr)-'
X)
for (p,q) = X_(p,A) E C_ (and hence A = A(p,q) = w2(p,q)). the representations (4.3.43) and (4.3.45) for
On employing
and proceeding as before
one finds
in L2(R+), (4.26)
1
in L2(R3),
for twhere itA1/2(W)
e
c
h'
c
(4.27)
itAl/2 (-O0) h2 v2 c (t, ) = e c are defined by
(4.28)
vc(t,X) -
(2Tr)3 2
i(x.p+yq+ tw_(P))
f
T (pl,a) h+(P)dP
e
C-
and
(4.29)
v2(t,X) = c(_ ) ph/2z
e
f
R(Ipl,A(P)) h+(P)dP.
C
In (4.28), q+ = q+(Ipl,A(p,q)) and A = A(p,q).
To see that v1 has the form
of (4.27), recall that the mapping (p,q) -* (p,q') _ (p,q+(Ipl,A(p,q))) = H-'(p,q) (see (1.32)).
Thus changing to the variables (p,q') in (4.28)
gives
v'(t,X) = c(-c°)p1/2 c (211)
f
C
(4.30)
31 T(Ipl,A(p,q')) h+(H(p,q')) c2( W) q_ ) where q_ = q_(Ipl,A(p,q')).
dpdq'
5.4.
Construction of the Scattering Operator
153
Combining the above results gives
-itAl/2
(4.31)
e
(4.32)
e-itA
when t - W.
h -
in L2(R+),
1/z
0 +
h -
in LZ(R3),
Comparing this with (4.5), (4.6) and using the uniqueness of
h+ in L2(R+) and of h
in L2(R3) gives
h+(P) = c(oo)p'/2(o) R+(IPI,A) h+(P) (4.33) z
h+(11
c(.o)P1/2(_oo) T_(IpI.X)
+
(4.34)
(P) )
cc
q in C +
h+(P) = c(o)P1/2(°°) R0(Ipl,A) h+ (P) in Co
while
h (P) =
c(W)p1/2(m) T+(Ipl,X)
h+(fl 1(P))
c22( ) (o°)
(_q) q+
(4.35)
+ c(-°o)P1/2(-°°) R_(Ipl,X) h+ (P) in C-. Comparing these representations with (4.7), (4.8) gives
R+ h+(P) + c(°°)P c(--)P
(4.36)
h (P) =
(-)
T_ h-(n(P))
q
Ro h(P) R
+
in C+. in Co ,
T+ h+ (]I-1(P)) (-q) in C_, (P) + c(_oo)P122(m) c(-)P (- ) q+
where R+ = R+(IPI,X(P)), etc. Relation (4.36), together with (4.4) provides a representation of S which is not identical with (1.33).
However, (4.36) and the unitarity of
S implies certain relations among the coefficients R+, Ro and T+.
These
may be obtained directly from (4.36) and the integral identity IIh-II = 11h+11-
However, they are more easily obtained from the bracket operation of Lagrange's formula, as in Chapter 3, §5.
There the relations
184
5.
SCATTERING OF SIGNALS BY INHOMOGENEOUS LAYERS
q+
9±-
q
(4.37) P (±W)
P (+°°)
(4.38)
X E A,
IT+12 =
IR+I2 +
P (+-°°)
IR01 = 1, X E A0
were derived in this way.
Similarly, calculating the limits of
for y - ± gives 4+ (4.39)
T
=
where q+ = q+(u,X), T+ = T+(u,X).
T+, X E A,
q
P (-m)
P (°°)
In particular, for P = (p,q) E C+1
q+(Ipl,X(P)) = q and
(4.40)
T_(jPI,A(P)) i_ = P(te)
q_
p(-°0)
T+(PI,X(P))
while for P = (p,q) E C_, q_(Ipl,X(P)) _ -q and
(4.41)
= P(-°°) T_(IPl,X(P)) TF(IPI,X(P)) (-q) P(-) q+
Combining (4.36), (4.40) and (4.41) yields the representation (1.33) for S.
§5.
THE SCATTERING OPERATOR AND SIGNAL STRUCTURE To describe the relationship between S and the scattered signals
srefl and strans it will be convenient to introduce the orthogonal projection Q_ in L2(R3) defined by
Qh=X_h where X_ is the characteristic function of C_ = R3, as before.
Note that
one may rewrite (1.27) as
(5.2)
hrefl(P) = R+(IPI,X(P)) Q_ha(I[R(P)), P E C+
because P E C+ > IIR(P) E C_.
If the unitary operator R
:
defined by
(5.3)
R h(P) = h(11 R(P)) for all P E R3
L2(R3) - L2(R3)
5.5.
The Scattering Operator and Signal Structure
185
is used then (5.2) is equivalent to
(5.4)
hrefl(P) = R+(IPI,A(P))(R Q_ho)(P), P E C+.
Finally, on applying the representation (1.33) of S to the function h = RQ ho and noting that supp RQ_ho C R+ it is seen that
hrefl(P) = SRQ ho(P) for all P E C+.
(5.5)
The same calculation with P E Co gives
(5.6)
hrefl(P) = SRQ_ho(P) for all P E Co.
Now consider (1.28).
It can be rewritten
cz(m)p(oo) T_(IPI,a(P)) Q_ho (IIT (P)), P E C trans (P) = cz(-')P(-0D)
h
(5.7)
Note that definitions (1.24), (1.25) and (1.31) of 11R' HT and TI imply that
IIT = IIR 11
(5.8)
1
.
Substituting in (5.7) and proceeding as before gives z
(5.9)
trans(P)
1/2
ccz(m)PT T_(IPI,A(P)) RQ_ho(n 1(P)), P E C_.
Finally, applying the representation (1.33) of S to h = RQ ho and P E C_ gives, since supp RQ_ho C R+,
(5.10)
(P) = c( -)pl/z( h c(m)p trans
SRQ_ho(P) for all P E C_.
The representations (5.5), (5.6) and (5.10) reveal that the scattered signal depends only on so and S.
Indeed, the occurrence of Q_ho implies
that only the values of so(r,0) with 0 E S2 are needed to find the scattered signals.
Of course, these facts depend on the assumption that
the signal sources lie in R+(y2).
If they lie in R3(yl) then a similar
relation holds with Q_ho replaced by Q+ho = (X+ + Xo)h.
If sources occur
in the scattering layer then the scattered signals cannot be calculated using only S.
Relations (5.5), (5.6) and (5.10) can be given a more symmetrical form by noting that (3.18) and (3.25) can be written
186
SCATTERING OF SIGNALS BY INHOMOGENEOUS LAYERS
5.
p E R+
c(o)P'I2 (o) ksc(p)
hrefl(P) (5.11)
htrans(P) = c(-°°)P112(-0O) isc(P), P E R3 where
k = (-iA1/2) h = g - i A1/2 f.
(5.12)
It follows that, for supp f U supp g C R+(y2), relations (5.5), (5.6) and (5.10) can be combined as
ksc(P) = c-1
(5.13)
(=)P-1/2 (`°)
Energy in the Scattered Field.
SRQ-ho (P) , P E R3 .
The total energy in the scattered
signal field is given by the limit
(5.14)
E'(utrans,R3(yl))
Esc = E'(urefl,R+(Y2)) +
because of (1.17).
ESC can be calculated from the incident signal by means
of (1.20), (1.21) and (5.11), (5.12).
Indeed, proceeding as in [25] one
finds that
EW(urefl,R' (Y2)) = 2c-2(°°)PIlsreflllL2(RXS2) (5.15) Isrefl(a,0)12 d0dw
2c-2(m)P-1(_) 1: -. 12
S+
4c-2(W)P-1(_)
`o
2
1srefl(w,0)12 dOdw
is
+
1
c-2(°°)P-1(°O) jo
J
2
kefl(w®)12 w2 d0dw
S+
C-2(c,)P-1(°°)
I10c1lL2(R+)
and, by a similar calculation,
Ilhrefl11L2(R+)
5.5.
The Scattering Operator and Signal Structure
E'(utrans,R3(y1)) =
187
IlkscIlL2(R3),
Esc = Ilks°IIL (R3).
Combining this with (5.13) and using the unitarity of S and R gives
Esc = c-2
(W)P-1 (°°) IIQ_ho 111-
L, (R3)
(5.18)
c'2(_)P-1(m)
Ih0(P)I2 dP. R3
By using the relation (2.7) this takes the still simpler form
(5.19)
BSc = Ilho_IILz (R3
Thus the total scattered energy is just the portion of the incident signal energy associated with momenta in the half-space R3; i.e., momenta directed from the sources toward the scattering layer.
Appendix 1
The Weyl-Kodaira-Titchmarsh Theory
The general Sturm-Liouville operator may be written
L (y) = W -'(y) {-(p '(y)o')' + q(y)o}.
(A.1)
The basic spectral theory of such operators was established by H. Weyl, K. Kodaira [10] and E. C. Titchmarsh [18].
The purpose of this appendix
is to present a version of the Weyl-Kodaira-Titchmarsh theory that is applicable to the operator Ap of this monograph. It is true that expositions of the Weyl-Kodaira theory are available in [3,5,14,18] and a number of other textbooks and monographs.
However,
in these and most of the book and periodical literature, hypotheses are
made concerning the form of the operator, or the continuity or differentiability of the coefficients, that limit the applicability of the Thus most authors assume that w(y) - 1 and many take p(y) = 1 as
theory.
Moreover, it is usual to assume that the coefficients are smooth
well.
functions or at least continuous.
It is known that if the coefficients
are sufficiently regular then L can be reduced to the Schrodinger form L
= -$" + q(y)o by changes of the independent and dependent variables
(14, p. 2].
However, this technique is not applicable to operators with
singular coefficients.
Here a version of the Weyl-Kodaira-Titchmarsh
theory is presented that is applicable to operators (A.1) with locally integrable coefficients.
The concepts needed for this extension of the
theory are available in the classic book of Coddington and Levinson [3]. The operator (A.1) will be studied on an arbitrary interval I = {y
I
-oo < a < y < b < +W}.
The coefficients will be assumed to have
the properties
189
190
THE WEYL-KODAIRA-TITCHMARSH THEORY
APPENDIX 1.
p(y), q(y), w(y) are defined and real valued for
(A.2)
almost all y E I,
(A.3)
p(y) > 0 and w(y) > 0 for almost all y E I,
(A.4)
ZoC (I), p(y), q(y), w(y) are in L1
where
LRoc(I)
= {f(y)
I
f E L1(K) for every compact K C I).
It is natural
to study L in the Hilbert space JC(I,w) with scalar product
u(y) v(y)w(y)dy.
(u,v) = J
(A.5)
I
In the general theory of singular Sturm-Liouville operators two operators in JC(I,w) are associated with L.
The first is the maximal
operator L1 defined by
D(L1) = JC(I,w) n AC(I) n {u
I
p-lu' E AC(I), Lu E J((I,w)),
(A.6)
L1u = Lu for all u E D(L1).
The second is the minimal operator Lo defined by
D(L0) = D(L1) n {u
I
(Llu,v) = (u,L1v) for all v E D(L1)),
(A. 7)
Lou = Lu for all u E D(Lo).
It can be shown that Lo is densely defined and closed and satisfies
(A.8)
Lo C Lo = L1.
It follows that every selfadjoint realization of L in J((I,w) must satisfy
(A.9)
Lo C L C L1.
If Lo = Lo = L1 then L is said to be essentially selfadjoint.
The classi-
fication of the selfadjoint realizations of L by means of boundary conditions at a and b will now be reviewed here.
For essentially selfadjoint
operators no boundary conditions are needed (Weyl's limit point case).
The
Appendix 1.
191
The Weyl-Kodaira-Titchmarsh Theory
operator A11 of §1 is essentially selfadjoint since its maximal operator
is selfadjoint (cf. (3.2.1), (3.2.2)). The Weyl-Kodaira-Titchmarsh theory provides spectral representations of the selfadjoint realizations of singular Sturm-Liouville operators. Each representation is derived from a basis of solutions of L
1{i
= Aj and a
corresponding 2 x 2 positive matrix measure m(a) = (mjk(X)) [5, p. 1337ff]. The representation spaces are the Lebesgue spaces L2(A,m) associated with m, with norm defined by
(A.10)
J
IE F. (X) Fk(A) m.k(da). A j,k=l
The following version of the Weyl-Kodaira-Titchmarsh theory is adapted from [5, pp. 1351-6]. Theorem (Weyl-Kodaira).
Let L be a selfadjoint realization of L in
3C(I,w) with spectral family {TIL(A)}.
Let A = (X1,A2) C R and let Vj(y,A)
(j = 1,2) be a pair of functions with the properties
(A.11)
lPj(y,A) E C(I x A), j = 1,2,
(A.12)
The pair ij(y,A) (j = 1,2) is a solution basis for
L tp = AiU on I for each A E A. Then there exists a unique 2 x 2 positive matrix measure m = (mjk) on A with the following properties.
For all f E 3C(I,w) there exists the limit
(A.13)
b'
L2(A,m)-lim
f(A) _
(A.14)
a'
a'-a, b'-,-b
The mapping U
:
f(y) (p1(y,A), p2(y,A))w(y)dy
3C(I,w) + L2(A,m) defined by Uf =
is a
partial isometry with initial set IIL(A) 3C(I,w) and final set L2(A,m).
(A.15)
The inverse isomorphism of L2(A,m) onto IIL(A) 3C(I,w) is
given by Jut
2
(U*F)(y) = 3C(I,w)-lim
P1''A1,112J'X2
j(y,a)Fk(X)mjk(da)
U1 j,k=1
192
APPENDIX 1.
THE WEYL-KODAIRA-TITCHMARSH THEORY
For all Borel functions 'f'(A) on R with supp 'Y C A, one has
(A.16)
U D('Y(L)) = L2(A,m) n {f
I
'Y(A)f(X) E L2(A,m)}, and
(U Y'(L)f)(X) = Y(a)f(a). Discussion of the Proof.
The theorem is proved in [5] under the
hypotheses w(y) E 1, p(y), q(y) E C(I) and p(y) > 0.
To prove it under
hypotheses (A.2), (A.3), (A.4) one may first prove it for the special case 0(k-1)(c,A) of the basis 4.(y,A) that satisfies = Sk where a < c < b. The functions 6 (y,A) are entire functions of A and the theorem can be proved
by the classical limit-point, limit-circle method of Weyl as presented in [3].
The general case can then be obtained by a change of basis from
0.(y,A) to i.(y,A). original paper [13].
In fact, this was the procedure used by Kodaira in his The first uniqueness results for m are due to
E. A. Coddington and V. A. Mar6enko (see [5]).
The uniqueness proof given
in [5] can be extended to the case treated here.
As emphasized by Dunford and Schwartz, the utility of the Weyl-Kodaira for different
theorem is due to the possibility of using different bases ip i
portions of the spectrum of L. calculate the measure m.
When a basis has been chosen one need only
A general procedure for doing this, due to
E. C. Titchmarsh [5, p. 1364] is known for cases in which the j.(y,A) have analytic continuations to a neighborhood of A in the complex plane. such continuations are not always available. when the
However,
A procedure that is applicable
.(y,A) have a one-sided continuation into the complex plane is
illustrated in Chapter 3, §6 above.
Appendix 2
Stationary Phase Estimates of Oscillatory Integrals with Parameters
The method of stationary phase provides asymptotic estimates for r -+
of oscillatory integrals of the form
I(r) = JO exp {i r 6(s)} g(s)ds, 0 C Rm
The method was needed in Chapter 4, §4 to estimate the oscillatory integral There the integrand g and phase 6 contain parameters
in equation (4.4.3).
and estimates are needed that are uniform in these parameters, at least on compact sets.
This suggests the study of oscillatory integrals of the form
I(r,E) = JU exp {i r 6(s,E)} g(r,s,E)ds
(1)
where r > 0, s E Rm and E E Rn.
6(s,E) is a real-valued phase function
and it is assumed that there are open sets 0 C Rm and 0' C Rn such that
(2)
D6 6(s,E) E C(0 x 0') for all multi-indices d,
(3)
DS g(r,s,E) E C(R+ x 0 x 0') for all multi-indices
s
where R+ denotes the positive real numbers.
Moreover, it is assumed that
there is a compact set K C 0 such that
(4)
supp g C R+ X K X 0'.
Estimates of I(r,E) for r - - are sought which are uniform in E on compact subsets of 0'.
For large values of r the exponential in (1) is highly
oscillatory except near critical points of the phase function 6(s,E).
193
Two
194
APPENDIX 2.
estimates of
STATIONARY PHASE ESTIMATES
are given here, corresponding to the cases of no
critical points and one critical point, respectively.
The first case is
formulated as Theorem A.1.
Assume that
V. 0(s,Q _ (ae/as1, - ,ae/asm) # 0 for all
(5)
E K x 0'.
Moreover, assume that for each compact set K' C 0', each ro > 0 and each positive integer k there exists a constant M = M(K,K',ro,k) > 0 such that
DS g(r,s,E)I < M for all r > r0, s E K,
(6)
E K' and 16
< k.
Then there exists a constant C = C(K,K',rp,k,g) > 0 such that
II(r, )I
(7)
< C r -k for all r > ro and C E V.
In the second case considered here 0(s,E) has a unique non-degenerate
critical point s = T() for each E E 0'. Theorem A.2. for all
It is formulated as
Assume that there is a function T E C_(0',0) such that,
E 0',
Os 0(s,E) = 0 if and only if s =
(8)
Moreover, assume that the Hessian a"(s,F) =
ask) satisfies
det 0"(T(Q,Q 0 0 for all
(9)
E 0'.
In addition, assume that for each compact set K' C 0' and each ro > 0 there exists a constant M = M(K,K',ro) such that
(10)
IDs g(r,s,E)l < M for all r > ro, s E K, C(=- K' and 151 < m + 5.
Then there exists a constant C = C(K,K',ro,g) > 0 such that if q(r,C) is defined by
+'r sgn 0"(T(E)E))g(r,T( )E)
exp{i r
I(r,C) _ (2ur)"y2 (11)
r2
+ q(r,Q I det e,, (_C
11/2
Appendix 2.
Stationary Phase Estimates
195
then
Iq(r,
(12)
< C
r-ny2-1
for all r > rp and
E V.
In (11), sgn 6"(s,E) is the signature of the real symmetric matrix
e" (s.) The uniform estimates given above are due, in essence, to M. Matsumura [15].
No proofs are offered because the theorems can be proved by following
Matsumura's proofs and recognizing that under the hypotheses formulated above his estimates are uniform for C E K'.
References
[1]
Agmon, S. Lectures on Elliptic Boundary Value Problems. Nostrand, 1965.
[2]
Brekhovskikh, L. M.
[3]
Coddington, E. A. and Levinson, N. McGraw-Hill, 1955.
[4]
Dermenjian, Y., Guillot, J. C. and Wilcox, C. H. Comportement asymptotique des solutions de l'equation des ondes associee a 1'operateur d'Epstein. C. R. Acad. Sci. Paris 287, 731-734 (1978).
[5]
Dunford, N. and Schwartz, J. T. Theory. Interscience, 1963.
[6]
Friedlander, F. G.
[7]
Guillot, J. C. and Wilcox, C. H. Spectral analysis of the Epstein operator. Univ. of Utah Technical Summary Rept. #27, 1975.
[8]
Guillot, J. C. and Wilcox, C. H. Spectral analysis of the Epstein operator. Proc. Roy. Soc. Edinburgh 80A, 85-98 (1978).
[9]
Hartman, P.
Waves in Layered Media.
Sound Pulses.
Van
Academic Press, 1960.
Theory of Differential Equations.
Linear Operators Part II:
Spectral
Cambridge Univ. Press, 1958.
Ordinary Differential Equations.
Wiley, 1964.
[10]
Hormander, L. Verlag, 1963.
[11]
Kato, T. 1966.
[12]
Kneser, A. Untersuchungen uber die reelen Nullstellen der Integrale linearer Differentialgleichungen. Math. Ann. 42, 409-435 (1893).
[13]
Kodaira, K. The eigenvalue problem for ordinary differential equations of the second order and Heisenberg's theory of S-matrices. Amer. J. Math. 71, 921-945 (1949).
[14]
Levitan, B. M. and Sargsjan, I. S. Introduction to Spectral Theory, AMS Translations of Mathematical Monographs V. 39, American Mathematical Society, Providence 1975.
Linear Partial Differential Operators.
Perturbation Theory of Linear Operators.
196
Springer-
Springer-Verlag,
REFERENCES
197
[15]
Matsumura, M. Asymptotic behavior at infinity for Green's functions of first order systems with characteristics of nonuniform multiplicity. Publ. RIMS, Kyoto Univ. 12, 317-377 (1976).
[16]
Naimark, M. A.
[17]
Pekeris, C. L. Theory of propagation of explosive sound in shallow water. Geol. Soc. Am., Memoir 27 (1948).
[18]
Titchmarsh, E. C.
[19]
Tolstoy, I. The theory of waves in stratified fluids including the effects of gravity and rotation. Rev. Mod. Phys. 35, 207-230 (1963).
[20]
Tolstoy, I. and Clay, C. S.
[21]
Weyl, H. Uber gewohnliche Differentialgleichungen mit Singularitaten and die zugehorigen Entwicklungen willkurlicher Funktionen. Math. Ann. 68, 220-269 (1910).
[22]
Wilcox, C. H. Initial boundary value problems for linear hyperbolic Arch. Rational partial differential equations of the second order. Mech. Anal. 10, 361-400 (1962).
[23]
Wilcox, C. H. Transient electromagnetic wave propagation along a dielectric-clad conducting plane, Univ. of Utah Technical Summary Rept. #22, May 1973.
[24]
Wilcox, C. H. Transient acoustic wave propagation in a symmetric Epstein duct, Univ. of Utah Technical Summary Rept. #25, May 1974.
[25]
Wilcox, C. H. Scattering Theory for the d'Alembert Equation in Exterior Domains. Springer Lecture Notes in Mathematics, V. 442. Springer-Verlag, 1975.
[26]
Wilcox, C. H. Spectral analysis of the Pekeris operator. Rational Mech. Anal. 60, 259-300 (1976).
[27]
Wilcox, C. H. Transient electromagnetic wave propagation in a Istituto Nazionale di Alta Matematica, dielectric waveguide. Symposia Mathematics XVIII, 239-277 (1976).
[28]
Wilcox, C. H. Spectral and asymptotic analysis of acoustic wave propagation. Boundary Value Problems for Linear Evolution Partial Reidel, 1977. Differential Equations. H. G. Garnir, Ed.
[29]
Wilcox, C. H. Transient acoustic wave propagation in an Epstein duct. Univ. of Utah Technical Summary Report #36, November 1979.
Linear Differential Equations Part II.
Eigenfunction Expansions, V. I.
Ocean Acoustics.
Ungar, 1967.
OUP, 1946.
McGraw-Hill, 1966.
Arch.
Index
Radiation pattern of source, 162 Reduced acoustic propagator, 22, 32 Reflected signals, 170ff
Acoustic duct, 1 Acoustic potential, 2 Acoustic pressure, 13 Acoustic propagator, 17 Acoustic signals, 161[f Acoustic velocity field, 13 Asymptotic energy distributions, 154ff, 170 Asymptotic wave functions, 131,
Scattering of signals, 161ff. Scattering operator, 166, 178 Semi-infinite layers, 156 Sesquilinear forms, 18, 32 Signal distortion, 177 Signal structure, 161ff, 177 Sobolev embedding theorem,118
168
Conservation laws,
26
space, 17
Solution with finite energy,18 Spectral family,2, 71, 90 representation, 31, 32 Spectrum, 51
Dispersion relations, 85 Distribution theory, 17 Duhamel integral, 16 Energy integral, 15
p oint
,
23
,
51
,
52
continuous, 23, 51, 52 essential, 51, 52 Stationary phase estimates, 151, 193 St urm s osc ill a ti on th eorem, 53 Support (of a function), 14
Finite layers, 156 Fluid parameters, 1 Functional calculus, 2, 15
'
Generalized eigenfunctions, 2, Group speed, 151 Hilbert space,
65
Transient free waves, 134ff Transient guided waves,150ff Transient sound fields 125ff Transmitted signals, 170
2
Kneser's theorem, 56
Wave equation, 1, 13 Wave source densit y, l4 Weyl-Kodaira-Titchmarsh theory, 3, 33, 71, 189
Lagrange formula, 38 Momentum distribution of source, 163 Normal mode functions, 2, 24, 27 expansions, 30, 104 Operational calculus, 15
198
Applied Mathematical Sciences cont. from page ii
36. Bengtsson/Ghil/KallAn: Dynamic Meterology: Data Assimilation Methods. 37. Saperstone: Semidynamical Systems in Infinite Dimensional Spaces. 38. Lichtenberg/Lieberman: Regular and Stochastic Motion. (cloth) 39. Piccinini/Stampacchia/Vidossich: Ordinary Differential Equations in R. 40. Naylor/Sell: Linear Operator Theory in Engineering and Science. (cloth) 41. Sparrow: The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors. 42. Guckenheimer/Holmes: Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields.
43. Ockendon/Tayler: Inviscid Fluid Flows.
44. Pazy:
Semigroups of Linear Operators and Applications to Partial Differential Equations.
45. Glashoff/Gustafson: Linear Optimization and Approximation: An Introduction to the Theoretical Analysis and Numerical Treatment of Semi-Infinite Programs.
46. Wilcox: Scattering Theory for Diffraction Gratings.
47. Hale et al.: An Introduction to Infinite Dimensional Dynamical Systems
-
Geometric
Theory.
48. Murray: Asymptotic Analysis.
49. Ladyzhenskaya:
The Boundary-Value Problems of Mathematical Physics.
50. Wilcox: Sound Propagation in Stratified Fluids. 51.. Golubitsky/Schaeffer: Bifurcation and Groups in Vifurcation Theory, Vol. I. 52. Chipot: Variational Inequalities and Flow in Porous Media. 53. Majda: Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables.
