Advances in Applied Mechanics, Volume 17

Advances in Applied Mechanics Volume 1 7

Editorial Board T. BROOKEBENJAMIN Y. C. FUNG PAULGERMAIN L. HOWARTH WILLIAM PRACER

T. Y. Wu HANSZIECLER

Contributors to Volume 17 P. CHADWICK J. L. ERICKSEN

C. S. Hsu

ROBERTR. LONG T. FRANCIS OGILVIE G. D. SMITH

ADVANCES IN

APPLIED MECHANICS Edited by Chia-Shun Yih DEPARTMENT OF APPLIED MECHANICS A N D ENGINEERING SCIENCE THE UNIVERSITY OF MICHIGAN ANN ARBOR, MICHIGAN

VOLUME 17

1977

ACADEMIC PRESS New York

San Francisco London

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Contents

vii

LISTOF CONTRIBUTORS

Some Aspects of Turbulence in Geophysical Systems Robert R. Long 1. 11. 111. IV. V. VI.

Introduction Introductory Concepts in Turbulence-Homogeneous Fluids Some Basic Effects of Density Variations and Rotation Thermal Convection Turbulence in Stably Stratified Fluids Third-Order Closure Schemes in Turbulence Research References

2 7 21 33 50 12 84

Singular-Perturbation Problems in Ship Hydrodynamics

IT: Francis Ogilvie I. Introduction 11. Slender-Body Theory in Aerodynamics

111. Slender Ships in Unsteady Motion at Zero Speed IV. Slender Ships in Steady Forward Motion V. Slender Ships in Unsteady Forward Motion References

92 95 105 145

169 185

Special Topics in Elastostatics J . L. Ericksen I. 11. 111. IV.

189 192 200 220 24 1

Introduction Basic Equations Semi-Inverse Methods Experiment and Mechanistic Theory References V

Contents

Vi

On Nonlinear Parametric Excitation Problems C . S. Hsu I. Introduction 11. Asymptotic Analysis for Weakly Nonlinear Systems 111. Analysis by Difference Equations

IV. V. VI. VII.

Second Order Difference Systems Global Regions of Asymptotic Stability Impulsive Parametric Excitation An Example: A Hinged Bar Subjected to a Periodic Impact Load References

245 241 251 266 216 283 286 298

Foundations of the Theory of Surface Waves in Anisotropic Elastic Materials P . Chadwick and G. D.Smith

IV. V. VI. VII. VIII.

304 306 310 314 325 332 341 355 359 313 314

AUTHORINDEX SUBJECT INDEX

317 383

I. Introduction 11. Algebraic Preliminaries

111. Elasticity Tensors

The Fundamental Eigenvalue Problem Plane Elastostatics A Uniformly Moving Line Singularity Elastic Surface Waves. Basic Analysis The Uniqueness and Related Properties of Free Surface Waves IX. The Existence of Free Surface Waves X. Supplementary Topics References

List of Contributors

Numbers in parentheses indicate the pages on which the authors’ contributions begin.

P. CHADWICK, School of Mathematics and Physics, University of East Anglia, Norwich, England (303) The Johns Hopkins University, Baltimore, Maryland (189) J. L. ERICKSEN, C. S. Hsu, Department of Mechanical Engineering, University of California, Berkeley, California (245) ROBERTR. LONG, Department of Earth Sciences, The Johns Hopkins University, Baltimore, Maryland (1)

T. FRANCIS OGILVIE,Department of Naval Architecture and Marine Engineering, The University of Michigan, Ann Arbor, Michigan (91) G. D. SMITH, School of Mathematics and Physics, University of East Anglia, Norwich, England (303)

vii

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Some Aspects of Turbulence in Geophysical Systems ROBERT R . LONG Department of Earth Sciences The Johns Hopkins University Baltimore. Maryland

I. Introduction . . . . . . . . . . . . A . The Nature of Turbulent Flows . . B. Dimensional Analysis . . . . . . C. Meaning of Symbol -, . . . . .

. . . . . . . . . . . . . . . . . . . I1. Introductory Concepts in Turbulence-Homogeneous Fluids . . . . A. Reynolds Stresses . . . . . . . . . . . . . . . . . . . B. Turbulent Motion in Pipes and Channels . . . . . . . . . . C. Logarithmic Layer and Drag Coefficient . . . . . . . . . . . D. Eddy Viscosity . . . . . . . . . . . . . . . . . . . . . 111. Some Basic Effects of Density Variations and Rotation . . . . . . A . Governing Equations for Fluids with Density Variations . . . . B. Available Potential Energy . . . . . . . . . . . . . . . . . C. Richardson Numbers . . . . . . . . . . . . . . . . . . D . Thermal Convection . . . . . . . . . . . . . . . . . . E. Double Diffusive Convection . . . . . . . . . . . . . . . F. Molecular and Turbulent Diffusion . . . . . . . . . . . . . G . Neglect of Rotation in the Surface Layer . . . . . . . . . . H . Some Properties of the Ekman Layer in the Atmosphere . . . . IV. Thermal Convection . . . . . . . . . . . . . . . . . . . . A . Similarity Theory of Convection and Comparison with Laboratory . . . . . Observations . . . . . . . . . . . . . . . . . . . . B. Molecular Boundary Layers . . . . . . . . . . . . . . . C . Nusselt Number-Rayleigh Number Relation . . . . . . . . D . Buoyant Convection from an Isolated Source . . . . . . . . E. New Derivation of the Similarity Theory . . . . . . . . . F. Comparison of Theories of Thermal Convection . . . . . . G . Convection with Shear . . . . . . . . . . . . . . . . V. Turbulence in Stably Stratified Fluids . . . . . . . . . . . . A. Experiments without Shear . . . . . . . . . . . . . . . B. Experiments with Shear . . . . . . . . . . . . . . . . C . Comparison of Experiments with and without Shear . . . . . D. Energy Arguments . . . . . . . . . . . . . 1

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2 2

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7 7

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9 16 19

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4

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21 22 25 26 28 28 29 30 . 31

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35 37 40 44 46

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47 48 50

51 54 59 60

2

Robert R. Long E. Implications of Laboratory Experiments and Observations in the Atmosphere and Oceans . . . . . . . . . . . . . . . . . . . . . F. Eddy Viscosity and Eddy Diffusivity . . . . . . . . . . . . . . G. Buoyancy Flux Due to Wake Collapse . . . . . . . . . . . . . VI. Third-Order Closure Schemes in Turbulence Research . . . . . . . . A. Mean Reynolds Stress Model of the Surface Layer of the Atmosphere . B. Ellison’s Derivation of a Critical Flux Richardson Number . . . . . C. Theory of a Mixed Layer of Finite Depth . . . . . . . . . . . . References. . . . . . . . . . . . . . . . . . . . . . . . . .

63 66 70 72 73 80 83 84

I. Introduction This paper discusses turbulence in geophysical systems, particularly atmospheric and oceanographic systems. None of the topics considered is treated exhaustively; rather the emphasis is on the author’s recent research efforts. The reader will need a knowledge of the theory of ideal and viscous fluids, but Section I1 gives a background discussion of turbulence in homogeneous and nonrotating fluids, so that a prior knowledge of turbulence theory is not required. Large-scale, quasihorizontal motions in oceans and atmosphere involve eddies with horizontal dimensions as large as or larger than lo8 cm, and these eddies are sometimes considered to be elements of a turbulent geophysical system. Eddies this large are strongly influenced by the rotation of the earth, however, and we do not consider rotation in this paper except insofar as it affects the properties of the planetary boundary layers in the atmosphere and oceans. This slighting of rotation is a major restriction, but much of the emphasis is on applications of laboratory investigations of turbulent systems in which density variations are of paramount importance but rotation effects are absent. The interpretations of these experimental measurements are often controversial, but they are nevertheless useful in acquiring the rudiments of an understanding of the dynamics of the lower layers of the atmosphere and the upper layers of lakes and oceans.

A. THENATURE OF TURBULENT FLOWS It is not usually difficult to make a decision as to whether a given flow is turbulent or not, but it is difficult to give a precise definition. All turbulent

Turbulence in Geophysical Systems

3

flows, however, have three common characteristics (Tennekes and Lumley, 1972), namely, irregularity, dissipation, and rotationality. A random wave field on an ocean surface is irregular but it is nondissipative and irrotational and therefore nonturbulent. An internal wave field in a continuously stratified fluid may be irregular and rotational but it is not dissipative and is therefore also nonturbulent. Of course, if some of the waves break, turbulence exists locally because dissipation and vorticity appear in the turbulent patches. In saying that wave fields are nondissipative we are neglecting the dissipation arising from the viscous terms in the Navier-Stokes equations. Dissipation is always present in nonturbulent flows but is frequently very weak and for moderate time periods may be neglected. This weak dissipation is characterized by a small coefficient of molecular viscosity v (cm' sec-') which appears in the Navier-Stokes equations and which differs from fluid to fluid. The much larger dissipation of turbulent flows is characteristically independent of v despite the fact that turbulent flows are believed to satisfy the Navier-Stokes equations. In turbulence, therefore, the dissipation is related to the nature of the turbulent flow and not to the nature of the fluid. It is occasionally convenient, for example, to use the concept of an eddy viscosity, K , (cm' sec-I). Although K , has the same dimensions as v, it is usually many orders of magnitude larger and, most importantly, varies from one flow to another rather than one fluid to another. Turbulence is often associated with large values of the Reynolds number Re = UL/v where U is a characteristic velocity and L is a characteristic length. For example, in flow of water in a pipe, U may be the mean velocity along the axis and L may be the diameter. When Re is small (less than 2000 or so), the motion is laminar with velocity parallel to the walls, but above a threshold value of Re the motion is normally turbulent. In engineering problems it is customary to regard the size of the Reynolds number as the deciding criterion but other considerations become important in the flows of interest here in which density variations are important. For example, if a plate is heated, fluid above it will begin to move in a pattern of motion called conuection. Since there is no mean velocity, the Reynolds number is zero; nevertheless the resulting motion may be fully turbulent. Again, in the atmosphere and oceans there is usually a mean velocity and, because L is so large (of order of tens, hundreds, or thousands of meters), Re is normally very large, perhaps as high as 10". Nevertheless portions of these bodies of fluid may be in laminar motion (Businger and Arya, 1974) because the gravitational stability that normally exists may be sufficient to prevent the instabilities usually associated with high Reynolds numbers.

4

Robert R. Long

B. DIMENSIONAL ANALYSIS Dimensional analysis is of the greatest importance in turbulence research to the extent that some of the most fundamental results in theories of turbulence follow from dimensional arguments. The use of dimensional quantities is also essential in interpreting experimental results because it is an economical way to express various dependences. The basic tool of dimensional analysis is the Pi theorem (Bridgman, 1931; Birkhoff, 1950) which states that if in a group of physical quantities, Ql, Qz ... Q,,, the largest number of quantities with independent dimensions is k, a mathematical relationship of the form f(Q1,Qz

... QJ = 0

(1.1)

may be expressed as a function

... %-k)

= 0,

(1.2) where the R’S are independent, nondimensional groups composed of products of the Q s raised to various powers. In the statement of the theorem a collection of quantities Q1,Qz ... are said to have independent dimensions if it is impossible to construct a nondimensional number composed of members of the collection. Results from dimensional reasoning are often limited by the appearance of so many dimensionless groups that little physical understanding can be obtained from a relationship of the form of Eq. (1.2). For example, consider a pendulum composed of a small spherical bob of radius a. The dependence of the period 7 on the gravitational acceleration g, on the length of the string Z, on the amplitude angle a, and on a may be written cp(”1,

722

7(g/l)1’z =f(a, a / / ) ,

(1.3) and in this form there is no simple dependence of 7 on the length l, for example. In many problems, however, one or more of the nondimensional numbers may be very large or very small (in the pendulum problem all and a may both be small) and it may be permissible to neglect a given number as it tends to zero or infinity. In the pendulum problem the concept of a “pointmass” as an idealization of a small body of material is so well established that we have little difficulty in deciding that in the limit as a l l 4 0 the relationship in (1.3) becomes

(1.4) This approximation permits us to increase our understanding of the problem. We may now conclude that the period is proportional to the square root of the length of the pendulum. z(g/l)”z

= h(a).

Turbulence in Geophysical Systems

5

If, as is commonly the case, a is also small, we are tempted to conclude that h(a) in Eq. (1.4) is independent of a as a tends to zero, but without further information we cannot be sure that z remains finite as a -,0. It is conceivable, for example, that h(a) and, therefore, t are proportional to a as a+O. If we solve the problem, using the equations of motion and initial conditions and the approximation a << 1, we find that h(a)3 const, but in fluid mechanics and in turbulence problems especially, it is not usually possible to solve the equations of the problem even though it may often be possible to write down a set of equations, boundary conditions, and initial conditions that we may reasonably expect will uniquely determine the flow problem. In writing down these equations we should incorporate all reasonable approximations, such as 0: << 1 in the pendulum problem, in order to simplify as much as possible. When we do this, we may be able to infer the possibility of neglecting a nondimensional number that is small or large without the necessity of solving the mathematical problem. In the case of the pendulum, for example, if 8(t)is the angle of the pendulum at a given instant of time t, the problem for arbitrary a is determined by the equations

(d28/dt2)+ ( g / l ) sin 8 = 0, 8=a

8=0

att=O,

att=r/4.

(1.6)

Clearly t is a function of a and g/1 since these are the only constant quantities to appear in the complete formulation of the problem. Therefore ~(g//)'/~= h(a) as before. But if we assume a (and therefore 8) is small, sin 8 s 8. Defining t' = t(g/1)'l2, 8' = 8/a, we get the approximate formulation (d28'/dt'2)

8' = 1

at

2' = 0,

+ 8' = 0,

8' = 0

at

t' = t[(g/1)'/2/4].

(1.8)

Now a no longer appears in the statement of the problem and we conclude that z(g/l)1'2 = const. As applied to problems of turbulence, where Re is usually very large, if we could formulate a nondimensionalized set of equations and conditions that determine a given problem and, for Re 3 03, are independent of Re we could conclude that the phenomena in question do not depend on the Reynolds number. In fact we are not sure that this is possible, but in many cases an assumption that Re may be neglected leads to results that are reasonable and agree well with observation (Section 11). Another useful way of approaching the problem of approximation is again illustrated by the pendulum. In the statement of Eqs. (1.5) and (1.6), we may assign arbitrary dimensions to all quantities that enter, demand dimensional homogeneity, and then apply the Pi theorem. Thus, for arbitrary dimensions

Robert R . Long

6

A, B, C . .., we may write

[O] = A,

[t] = D,

[g/d = C ,

[a]= B,

[z]= E,

(1.9)

where the notation [0] = A reads “the dimension of 8 is A.” Requiring dimensional homogeneity in (1.5) and (1.6), we find the O and a must be dimensionless and C = D-’,E = D,i.e., [g/1]= D-2,

[T] = D.

Applying the Pi theorem to the relationship

7

= f ( g / ~a), , we get

7(g/l)”’ = h(a)

as before. On the other hand, if we formulate the problem incorporating the approximation a << 1 (and, therefore, 8 << l), we get sin 8 z 8 and (d’O/dt’) O=a

+ (g/1)O = 0,

att=O,

att=7/4

0=0

The same procedure leads to

[0]= A ,

[a]= A,

[g/q

= 0-’, [t] = D,

[z] = D.

Now there are two dimensions, A and D, and the Pi theorem yields for 7 =f(s/La) ~(g/1)”’ = const as before. Other examples of the latter approach are discussed by Long (1964). We refer to this technique later as generalized dimensional analysis.

-

C. MEANINGOF SYMBOL

-

In most problems there are nondimensional parameters that are large or small, e.g., Re or Re-’. The symbol denotes the order of magnitude of a quantity as the parameter in question goes to zero or infinity. For example, if we say A B and the only small or large parameter is Re, we mean that

-

as Re-co,

A/B-+K

where K is neither zero nor infinity.+Thus K is independent of the particular parameter involved, but if other nondimensional parameters are important, K may be a function of these. Notice that A B implies A / B 1 but this does not imply that K is one, or even close to it, although in cases where we lack any information on the size of K we ordinarily guess that it is about one. An example is the problem of the pendulum. We say 7 (l/g)’” as

-

-

-

This differs from the definition of used by Tennekes and Lumley (1972, p. xii), for example. They require that K lies between f and 5 ! +

Turbulence in Geophysical Systems

7

a + 0 because 7/(Z/g)1’2 -+

K

as a -+ 0.

Exact analysis indicates that K = 2n. 11. Introductory Concepts in Turbulence-Homogeneom Fluids

A. REYNOLDSSTRESSES We begin by reviewing the derivation of the equations of motion of a continuum in terms of stress components zij. We erect a rectangular parallelepiped with sides in the coordinate planes and compute all the forces acting on the element at an instant of time t . We first definet stress components: 7ij ij is the stress force per unit area exerted across a face whose normal is ii on the material pierced by - ii by the material pierced by ii . By the law of action and reaction - 7ij ij is the stress force per unit area exerted by material pierced by - ii on the material pierced by ii . For example, the force per unit area across the face in the x 2 x3 plane on the material in the element is - (zl i , + 71 i, + 2 1 3 i3). The normal component of this force, - 71 is called pressure; 71 is tension. We may calculate all forces on the element in the direction i, and so obtain the equations of motion,

,

+

p(du,/dt) = p x , (d7ij/axi), (2.1) where uj are velocity components and X j are components of the body force. We may show that 7ij = zji (Long, 1961) so that there are 3 equations in 9 unknowns, u j , and the 6 unknown stresses. In addition to the equations in (2.1), we use the continuity equation for an incompressible fluid

auilaxi= 0.

(2.2) This is an excellent approximation for water and a very good one for the atmosphere if we confine attention to phenomena of a small scale, say, hundreds of meters (Monin and Yaglom, 1971, Chapter 4). (See also Section I11 of the present paper.) The problem of indeterminacy of Eqs. (2.1) and (2.2) is overcome by the assumption 7‘.J . =

’

- p & .t J

+ &&.. 13 ’

(2.3)

We use the summation convention that a double subscript means addition over this subscript.

8

Robert R. Long

where p is the pressure, dij is the Kronecker delta, p is the dynamic coefficient of viscosity, and cij is the rate-of-strain tensor,

+ (au,/ax,)].

Eij = 4[(aui/axj)

(2.4) This assumption defines a Newtonianjluid and we are assured by experiments that it is very accurate for laminar flows at least. Our equations are now (2.2) and

+

+

p(duj/dt) = - ( a p / d ~ j ) p X j ( a / d ~ i ) ( 2 p ~ i j ) . (2.5) These are called the Navier-Stokes equations. The viscosity p has a value of 0.00018 gm/cm sec for air at atmospheric pressure and 0.01 1 gm/cm sec for gm/cm3 for air and 1 gm/cm3 for water. If we take for density 1.2 x water, the kinematic coefficient of viscosity, v = p/p, is 0.15 cm2/sec for air and 0.01 1 cm2/sec for water. We ignore the small variations of v or p. The external force X j is usually gravity, - g d j 3 , where the X 3 axis is vertical; g may be taken to be a constant equal to 981 cm/sec2 in engineering and geophysical problems. As we have observed in Section I, fluids are generally capable of two distinct types of motion, well illustrated in the familiar flow of water in a pipe (Reynolds, 1895). Dimensional considerations indicate that the nature of the flow is governed by the Reynolds number Re = uo R/v, where u,, is the mean velocity at the center of the pipe and R is the radius of the pipe. Observations confirm the importance of Re. When Re is small, of the order of lo00 or less, the fluid moves steadily parallel to the walls of the pipe in sheets or layers (laminae) with a parabolic velocity distribution as in Fig. 1.

FIG. 1. Flow in a pipe.

This laminar motion changes to highly irregular turbulent motion at a critical value of Re which varies, however, with the entrance conditions and the level of the disturbances in the entering fluid. In turbulence it is convenient to consider a dependent variable such as ui or p, to be represented as a sum of a mean and a fluctuating part, u.1 = a.I + u!I )

P=B+P’, where the overbar denotes an average and where 4 = 0 by definition.

Turbulence in Geophysical Systems

9

There is one preferred idealized scheme of averaging in turbulence theory and several practical methods. The theoretical average is an ensemble auerage. For example, if we imagine that a turbulence experiment is started up at t = 0 from a state of rest, the ensemble average velocity u at a point (x, y , z) at a later time t, i.e., @(x,y, z, t ) is the velocity at that point averaged over an infinite number of experiments all identical and started from rest, where the measurements at the point (x, y , z) are made exactly t sec after the start. Obviously this is impractical. If the experiment is in a statistically steady state, U = U(x, y , z), and we may average the velocity at the point (x, y , z) over a long period of time. If the experiment is horizontally homogeneous, say, in the x direction, we may average over some portion of the x axis. Either of these is quite satisfactory if performed carefully, although it cannot usually be proved that time and space means converge to ensemble means and an “ergodic hypothesis” (Monin and Yaglom, 1971, p. 216) must be invoked. We need not be concerned with such matters. In the remainder of this section, we consider that the fluid has a constant, uniform density. Substituting into Eq. (2.5), using Eq. (2.2) and averaging, we get

aui - = 0, axi

(2

+ axj

p -++.1 = _ _ ap p x j ) ; : :1

a ( 2 M j - pu:). + axi -

The equations are identical in form to the original equations (2.2) and (2.5) except for the appearance of additional stresses, - pu: uJ called Reynolds stresses. Although the form is similar, notice that we have no information analogous to Eq. (2.3) relating the Reynolds stresses to any other quantities. They are new unknowns and so the problem of turbulence is underdetermined. The situation is the same as that encountered in an effort to deal with Eqs. (2.1) without resort to the stress-rate-of-strain law of Eq. (2.3). The indeterminancy is called the “closure” problem of turbulence. An excellent summary of the various techniques for closing the problem has been given by Reynolds (1976).

B. TURBULENT MOTIONIN PIPES AND CHANNELS Turbulent flows in a pipe or channel or, say, turbulent plane Couette flow between parallel moving planes (Robertson, 1959; Fig. 2), are simple

Robert R . Long

10

/ / / / / / / / / / / / / / - // / / / / 1 UO

x-

UO 4

//-// / / / / / / / / / / / / / / /

FIG.2. Turbulent plane Couette flow.

examples of turbulent motion, and we can gain much insight by considering the theory and observations of these flows. Let us consider plane Couette flow in detail. If we assume a large aspect ratio (width divided by depth of flow), we may neglect variations of any mean quantity in the spanwise direction. We also assume no variation of mean quantities with x or t. The equations become aul

aul

au/

aut

-+((H+u‘)-+v’-+w‘-++’(H at ax ay aZ

=

1 apt ---+

ax

VV2U’ + viiz,, (2.6a)

auf au’ + u’-auf + w’- + (ii + u’)-

aul

at

aZ

ay

ax

1a

= - --

p

p + vV’U‘,

ay

(2.6b)

+ + awl = 0, ax ay aZ

aUI

aUi

-

-

-

where z i z is the derivative of ti. The no-slip boundary conditions are: At z = H/2

At z

=

-H/2

U‘ = U‘ = W‘ = 0

u‘ = U’ = W‘ = 0

ii=uo, =

(2.8)

-uo.

(2-9) In this problem, as in all homogeneous flows, we may absorb gravity in the definition of mod$ed pressure P , P =p

+ pgz + P o ,

(2.10)

where Po is a constant, so that dP/ax, replaces dp/dx, in the equations of motion and gravity g no longer appears. If g is also absent in the boundary and initial conditions, it is apparent that gravity can have no effect on the flow. Using Eqs. (2.7) and averaging Eqs. (2.6a,b,c), we obtain

(a/&)( - ulwl+

vil,)= o

(2.11)

Turbulence in Geophysical Systems

11

a(wlu.)/az = o

(2.12)

(a/az)(P/lp+ 77) =0

(2.13)

-m

The Reynolds stress is especially important. The fluid is in general motion along the x axis with velocity increasing upward and, as a particle from above moves downward (w’ < 0), it will tend to conserve its x momentum. When it arrives below (Fig. 3), therefore, it will have an excess of

FIG.3. An eddy.

momentum compared to its environment so that in thenew position, w’ < 0 and N’ > 0,and there is a negative contribution to u’w’.As a particle rises (w‘ > 0), we would anticipate u’ .c 0 and again UIW) tends to be negative. If we define a correlation coefJicient __

c,, = Urw‘/6,6,,

(2.14)

where 6,and cr, are root mean squares of u’ and w’, e.g., 02 = p, our results imply a nonzero correlation between u’ and w’ because of the mean shear. Since there is no mean shear in the spanwise direction, it is clear that w’u‘ = 0. According to Eq. (2.11),the total stress ~

__

-dW‘

(2.15)

f Va, =T

is a constant in the flow. It is sometimes denoted by u: = T where u* is called the friction velocity. Obviously u)wI is zero at the walls where u’ and w‘ are zero so that T = ~ ( i i , )where ~, is the shear at the wall. As we have indicated, the viscosity coefficients are usually very small in the sense that the Reynolds number UL/v is large, where L is some appropriate length scale. For example, in the atmosphere U E lo3 cm/sec, L 1 lo6 cm, v s 0.15 cm2/sec, so Re 2 10”. Even in a pipe, a speed of 30 cm/sec yields Re z 30,000 or so. Thus, a nonzero value of constancy of the sum T = vU2 - UIW), and equality of z and v q at the wall imply a very large value of a2at the walls compared to ti2in the interior. We have already

m,

12

Robert R. Long

illustrated this in Fig. 2. We expect then to find a thin layer near the wall called the viscous boundary layer where viiz is an important part of the stress, and a large region in the interior where vii, is negligible and z r - u'w' in the layer except just at the wall. This is not surprising because one would expect that although the Reynolds number is large in the interior of the flow as a whole, the ,Reynolds number based on local velocities and lengths should be of order one in the viscous layer, i.e.,+ ii6,/v

- 1,

(2.16)

where 6, is the thickness of the layer. Since the Reynolds number is of the order of the ratio of acceleration to viscous force, this implies that viscous and inertia forces are of the same order, i.e., for example, vuu/6,2

-

a:/6,.

-

(2.17)

This also follows from (2.11) and (2.16) if we assume that ii uuin this layer. It is instructive to form the energy equation, but also important to present for discussion some intermediate steps. Let us multiply Eqs. (2.6)by u', u', w' and then average. Using (2.7), we get

The sum of these equations yields the equation for the kinetic energy of the disturbance

a -

(

'd) : z - l

-(T)=- - w' T'+at

aZ a[

-v-(T)

-U'W'U,-E,

__

(2.21)

where E is a positive quantity,

+ (67)' + (vw')2], is the disturbance kinetic energy, (uf2+ u" + w")/2. E =

v[(p.')'

(2.22)

and There are a number of points to be discussed. One is related to the dissipationfunction E. Turbulence is in general to be considered as composed of random motions on a large number of length scales. Those which are +

From now on we take a coordinate system with the lower plate at rest.

Turbulence in Geophysical Systems

13

found to contain most of the energy have, generally, dimensions that are proportional to the size of the container, e.g., to H in plane Couette flow or to the diameter of the pipe in pipe flow. A quantitative measure of the size of the energy-containing eddies is the integral length scale 1. This may be defined by *

u‘(x, y , z)u’(O, y , Z) dx

l=J, --

(2.23)

0,’

Notice that when turbulence is homogeneous along the x axis, the integrand varies only with the separation so that 1 is independent of x. Equation (2.23) defines a length over which velocities at two points bear some relationship to each other (Fig. 4). The length 1 is proportional to the size of the container,

I

lntegrond

-1-4 I

2

3

4

x , crn

FIG.4. Integral length scale.

although numerically generally considerably smaller. In a pipe of diameter D (Hinze, 1959) the length scale along the axis is about 0.5D while the length scales in the transverse directions are 0.20 - 0.30. Thus the eddies are elongated along the pipe axis. For later purposes we will assume D z 31. There is a decreasing amount of energy in smaller and smaller scales. The __ to originate from the mean-flow energy through the action energy is known of the term - u’w’usin Eq. (2.21).It goes first to the largest scales of motion and this is passed down to smaller and smaller scales by nonlinear interactions, i.e., by virtue of the action of the nonlinear inertia terms. This cascade of energy is opposed by the viscous forces which tend to smooth the velocity field, but these forces, which depend on velocity gradients, are very small for large eddies because v is very small and because gradients associated with large eddies are not large. The viscous forces are negligible unless operating on such small eddies that the viscous forces can be effective. Thus the dissipation term E , involving derivatives, is due to the very small eddies. This brings out one important property. The large eddies in a shearing flow are oriented in preferred ways by the presence of the shear. In the cascade

14

Robert R. Long

process, the smaller eddies lose contact with the mean flow and tend to take on random orientations so that all averages become independent of rotations or reflections of the coordinate system. The small eddies are therefore ’)~ isotropic. Isotropy at small scales is called local isotropy. Thus ~ ( V U = _ _ _ _ _ ~ ( V U= ’ )~~( V W = ’ )~~/ 3in (2.18)-(2.20). We may better appreciate the role of small eddies in the dissipation process by imagining for the moment that there are only two eddies, very different in size so that the velocity may be expressed as u = A sin(x/l)

+ B sin(nx/l),

where n is large. The magnitude of the derivative is au/ax

All

or

Bn/l.

We see that we can have small eddies with very little energy compared to the big eddies, i.e., B << A, but such that the derivative is dominated by the small eddies, i.e., Bn >> A. The smallest scales of motion adjust to the size of the viscosity v. If v decreases, the scale must decrease so as to permit the existence of eddies which can dissipate the energy. Evidently the length, time, and velocity scales q, T, u for these eddies will depend on E and v only. Dimensional analysis yields

,,

= v3/4/E1/4

= v1/2/E1/2,

= v1/4E1/4.

These are called Kolmogorou scales. Notice that the Reynolds number for this scale of motion is precisely one. As we have discussed, the length scale decreases as v decreases. On the basis of the cascade argument, the energy dissipation proceeds at a rate dictated by the action of the large eddies and is in no way related to v . The role of v is simply to dictate the size of the eddies that do the dissipating. Therefore E must be a function of the characteristics of the energy-containing eddies. These have a time scale of l/oUso that the rate of change of kinetic energy should be E o,2/(l/ou) .,”/I. This result is fundamental in turbulence research and asserts that the dissipation time is the time for one rotation of the eddy, so that the system is highly dissipative. The large eddies do not lose much energy to friction. Their dissipation rate is N vo,2/12which is only a fraction,

-

-

of the total dissipation rate. The term -u’w’uz in Eqs. (2.18) and (2.21)an energy source term. In plane Couette flow, we have decided that u‘w’ is negative. Since ii, > 0, ~

Turbulence in Geophysical Systems

15

__

- u'w'u, is positive and represents ageneration of disturbance kinetic energy at the expense of the mean shear. We may see this by considering the effect of thoroughly mixing a patch of fluid of dimensions L (Fig. 5). The energy before mixing is

2

L5a2 6

...

After mixing, the whole fluid is moving at the speed of the center of mass and the mean energy is a2L?/8. The difference, a2L5/24,is available for the kinetic energy of the disturbances. b

-1

/

/

FIG.5. Mixing in shear flow.

An important observation is that the energy-production term directly supplies energy to the longitudinal component of the velocity. The other components must get energy in some other way, namely, through the action of the pressure terms in Eqs. (2.18)-(2.20). We see that these all cancel out and tend to increase, when Eqs. (2.18)-(2.20) are added. Since P du'ldy and P' dw'/dz must be positive while P' du'/dx' is negative. The pressure terms tend to produce isotropy of the motion without, however, ever succeeding. The observed situation is roughly cr,,, z 0.55a,, 6,= 0.70au, so that the component retains the largest proportion of the energy (Zeman and Tennekes, 1975). The term d[w'(T' + P'/p)]/dz in Eq. (2.21) is called an energy transport term or energy-flux divergence. If we integrate from one plate to the other in plane Couette flow, w' is zero at both surfaces so this term vanishes. Thus it can only serve to transport energy from one place to the other. The size of the term is of order Co;/l, where C is a correlation coefficient. Here we have used the fact that the scale of the mean motion H is the same as the eddy scale 1. Since C may well be of order one, this term may be of local importance of the same order as the dissipation term.

16

Robert R. Long

The final term a/az(v d T / a z ) is called the viscous transport term. It is of order vo,2/12,which is small compared to o:/1.

C. LOGARITHMIC LAYERAND DRAGCOEFFICIENT We may now estimate the production term in the energy equation. Since it must balance the dissipation term over the whole flow, its order must also be o:/H in the fluid as a whole. We assume, therefore, that u: ii, o:/H in the main portions of the fluid. Since c,, u * , we have

-

-

Integrating, we get

where uo is the constant of integration and is taken to be the mean velocity at the centerline. Obviously, if we had included Re in x, the result would have been valid over the entire flow right down to the wall. Since we omit Re, we must certainly not expect that the resulting expression will be valid near the wall because of the presence there of the viscous layer. It is reasonable, however, to suppose that Eq. (2.24) is an accurate description of the velocity defect in the main body of the fluid. Obviously x 1 so i i - uo u* as Re -+ 00. The velocity-defect law has been tested by experiment and found to be quite accurate (Laufer, 1954). It is an example in fluid mechanics of an outer solution of the Navier-Stokes equations independent of the viscosity. This independence occurs often and is an example of Reynolds number similarity. Equation (2.24) cannot apply all the way to the wall but if Re + co,it is likely to apply arbitrarily close to the wall. Let us consider the solution very near the wall. Here viscosity is important. If we argue from the energy equation that u’w’u, and E are of the same order and that the integral scale is of order z in this layer, then tiz o,/z u,/z. This implies ii u* and therefore the “law of the wall,”

-

-

- -

-

ii/u* = f ( u * z / v ) .

(2.25)

A fundamental result now follows from reasoning of Izakson (1937) and Millikan (1939) who asserted that there will be an overlap region in which both (2.24) and (2.25) hold simultaneously as Re + 00. We may consider

Turbulence in Geophysical Systems

17

FIG.6. Regions of validity of inner and outer solutions.

Fig. 6 in which we portray, for a given Re = u, H / v , two regions in which the inner and outer solutions are separately valid. In the general case, of course, we must add the argument Re to the two functions. Obviously the outer and inner solutions depend for their validity on the size of Re. As Re goes to infinity (say, H -+ 00) the influence of H (and therefore Re) at a given z should become negligible in the function f so that the region of validity for the inner solution must eventually increase to include the point z in question as Re -+ 00. Thus an overlap region will exist. Since uo/u* is a constant, it must be a function of Re-' = v/Hu, = q so that in the overlap region

f(t)- m(v) = -x(td, = u*z/v,

q = v/Hu, ,

uo/u* = m(q).

(2.26)

This puts severe restrictions on the forms of these functions. Thus, differentiating with respect to t and q and eliminating x, we get

f't = -m'q,

(2.27)

and since t,~and 5 are independent, each side of Eq. (2.27) is equal to a constant, say, 1 / ~ i.e., ,

f = ( 1 / ~ )In

t + A,

m = ( 1 / ~In ) Re

+ E,

(2.28)

where A, E, and K are universal constants. Let us examine the first function. Since u* and v are constants, we have in the overlap region ii/u, = ( 1 / ~In ) z

+ const.

(2.29)

This is the famous log law for turbulent flow over a surface, derived originally by von Karman (1930) and Prandtl (1932). Obviously it cannot hold down to the surface itself and we have seen that it does not hold far from the surface. It has been verified as accurate just above the viscous layer, i.e., for

18

Robert R. Long

v/u, << z << H, in countless experiments (Tennekesand Lumley, 1972, p. 162) and in the lower layers of the atmosphere (Priestley, 1959, p. 20). When z v/u, ,we are in the viscous layer which joins the log layer to the wall. In this layer the velocity profile is linear. The actual observations show a transition from the linear profile to the log profile between values of zu* /v of 5 and 30, i.e., the log layer begins at zu*/v E 30. From (2.28) we get

-

uo/u* = ( 1 / ~ In )

Re + B.

This is called the logarithmicfriction law. Tennekes (1968) has argued that K is a constant only in the sense Re + CQ. His theory leads to an estimate for large Reynolds number of 0.33 instead of the commonly accepted values of 0.40-0.42 in the pipe flow. He claims support from atmospheric measurements of Businger et al. (1971) but more recent measurements by Pruitt et al. (1973) using a very large dragplate lysimeter to measure the stress strongly suggest the same range 0.40-0.42 in the atmosphere too. In geophysics, for example, airflow over the ground, we are concerned with rough surfaces over which v is irrelevant everywhere because no thin viscous boundary layer exists. The drag originates instead from the form drag due to eddies in the lee of the roughness elements. One then argues that some measure of the roughness, say, a roughness length zo ,appears instead of v or, more precisely, instead of v/u, . The analysis is identical and we get @/U*

uo/u* =

=(l/K)

Wzo),

- ( 1 / ~ In(z,/H) )

+ B,

(2.30) (2.31)

where we define zo so that no constant of integration appears in (2.30). In effect then, zo is the height at which the velocity would vanish if the log layer held down to this level (it does not). Experiment in sand-roughened pipes indicates that zo s ( 1/30)h0 where h, is the average height of the roughness. In the atmosphere, the ratio varies somewhat depending on the nature of the roughnesses but is in the range 1/7-1/30 (Plate, 1971, p. 27). The log law may also be derived by dimensional arguments and in a more general case of shearing flow near a surface, say, flow in a pipe. In this case the stress is not a constant but varies across the flow. One argues, however, that in a layer, v/u, << z << H, the only quantities on which, say, u; can depend is u* and z, where u: is the stress at the surface. Then we must have ziiz/u* = const = l / ~ .

(2.32)

Integration leads to the log law. We may apply this to other quantities, for example, cr,/u* = const

E

= const x (u:/z),

(2.33)

19

Turbulence in Geophysical Systems

+ P’Jp) = const. 4

w’( T’

(2.34)

The last reveals that the energy-flux divergence term in the energy equation (2.21) is zero. This is often used to justify neglect of this term in other situations and other regions but this is incautious. Some numbers for the surface layer may be obtained by averaging a number of experiments (Zeman and Tennekes, 1975) CT,

z 2.0u,

CT”

z 1.4u,

ow z

l.lu*.

The u component is largest because the turbulence production is in the x component directly. D. EDDY VISCOSITY

In shearing flows and in many other situations, it is often advantageous to introduce the concept of eddy viscosity K, . We may, for example, define K, by the equation (2.35)

z = K , diiJdz

by analogy with 7 =

(2.36)

v duJdz

for laminar parallel shearing flow. In a turbulent region, z will be almost equal to so that K, u* I, where we assume diildz u* 11. This is in accordance with the analogy since molecular viscosity is of the order of the speed of the molecules times the mean free path and therefore eddy viscosity is of the order of the eddy speed times the size of the eddies. A physical interpretation is sometimes made that K, is of the order of the speed of translation of the eddy times the mixing length, i.e., the distance the eddy moves before it is tom apart or dies out. These, of course, are very closely related because mixing length is obviously of the order of the eddy size. The eddy viscosity in the logarithmic layer is

-m

-

-

(2.37) It may be found by measurement in the rest of the flow in a pipe. There is agreement with (2.37) near the walls but the measured K , / u , R has a maximum of 0.08 about half-way in toward the center of the pipe (Nikuradse, 1932). We see a great difference between eddy viscosity and molecular viscosity in that v is constant but K , varies in space and time. More specifically v is a

Robert R. Long

20

property of the fluid whereas K, is a property of the pow. For engineering purposes K , is often assumed to be constant even when it is obvious that it is not. In some cases this assumption works well, namely, turbulent flow in a jet, wake or mixing layer (layer between two parallel streams of different mean velocity). In a turbulent jet (Fig. 7), for example, the mean motion over most of the core region is well predicted by a constant eddy viscosity model. When rigid surfaces exist, however, turbulent velocities tend to zero and eddy sizes tend to zero near the boundaries so that K, tends to zero at the boundaries and cannot be considered constant.

FIG.7. Turbulent jet.

Notice also the fact that eddy viscosity is almost always many orders of magnitude greater than molecular viscosity. The ratio is K,/v

- ciul/v - Re,

i.e., of the order of the Reynolds number. Even in small laboratory experiments, this will be 100 or larger. there is a possibility that t and dii/dz are of different Since t z --, signs, thereby making K , negative. Observation indicates that this happens only in small regions in shearing flow (Schwartz and Cosart, 1961) although negative viscosity may be more common in more complicated situations. It is interesting that the velocity-defect law (2.24) leads to a Reynolds number Re, 1, where Re, is based on the velocity difference across the interior region, the linear dimension of the region, and the coefficient of turbulent viscosity, i.e.,

-

Re,

- AuHIK, -

u* Hlu,

H

-

1.

Turbulence in Geophysical Systems

21

111. Some Basic Effects of Density Variations and Rotation

There is great interest among geophysicists and engineers in fluid systems with density variations. These are typical of the atmosphere, seas, lakes, and reservoirs, and even when the variations are exceedingly small, they are almost always exceedingly important. In some cases, for example, in the air above the warmed surface of the earth, there may be a layer in which the average density increases with height, i.e., the fluid is unstably stratified (Fig. 8). Then, there is a strong tendency for light parcels of fluid near the ground to rise and heavier parcels to descend in a type of motion called thermal convection. The heating is sufficiently strong to maintain the average density increase with height close to the ground, but at some distance above the ground the mean density increase becomes very weak or may even be replaced by a density decrease with height (Gille, 1967, p. 380; Townsend, 1976, p. 385). In the layer of density increase with height, the motion may

FIG.8. Unstable density field.

have a regular cellular pattern in the form of Benard cells (Benard, 1901). More usually, especially in geophysical situations, the motion is turbulent or irregular, characterized by plumes or thermals. Layers with mean density increase with height are rather rare in the atmosphere and in oceans and lakes. In addition to the thin surface layer above the heated ground, unstable layers occur in water near the water-air interface when the water is losing heat (e.g., by nocturnal cooling or evaporational cooling or when there is overriding cold air). Such layers may also occur near the ocean bottom in regions of formation of bottom water (Turner, 1973, p. 313). Usually, however the mean density decreases (stably stratified) or is uniform with height. If a turbulent fluid has a mean vertical density gradient, turbulence will tend to reduce this density variation. As a simple example of this process, we can imagine a vessel filled with water with a stable temperature stratification-warm near the top becoming colder with depth. If we now

Robert R . Long

22

induce turbulence by stirring mechanically, the density gradient will ultimately be wiped out. If, on the other hand, the fluid is initially unstably stratified-cold above and warm below--convection will set in automatically and the density gradient will again be eliminated. In the first case the warmth near the top is transported downward; in the second case, the warmth near the bottom is transported upward. Thus there is nearly always a flux ofheat from warm regions to colder regions. A. GOVERNING EQUATIONS FOR FLUIDS WITH DENSITY VARIATIONS

Let us now write down the governing equations for the fluid systems considered in these notes. For a liquid’ such as water in which the density variation is linearly related to temperature, for example, a reasonably accurate set of equations is dv/dt = - ( l / p ) V p- gk

dpldt

+ vV’V,

= khV2p,

(3.3) where v is the velocity vector and kh is the coefficient of heat conduction. We have neglected the rotation of the earth although this will be important for larger scale phenomena (Monin and Yaglom, 1971, p. 406;Section 111,G). In geophysical problems, p varies rather little and it is convenient to define a quantity called buoyancy b: b = ( P - PO)S/PO (3.4) where p o is some representative density, often taken to be p o = 1 gm/cm3 in the case of water. Under the assumption that (p - p o ) / p o << 1, we may neglect the difference between po and p everywhere in the equations except in the gravity-force term and obtain 9

dv/dt = - (l/po)VP - bk v*v=o,

+ vV’V,

(3.5)

(3.6)

dbldt = khV2b, (3.7) where P = p + p o gz + const. The approximation we have made is called the Boussinesq approximation (Boussinesq, 1903). Physically, in making this

’

We do not give a separate discussion for the compressible atmosphere. In the lowest few hundred feet, the same equations are valid if we use for the density p in a liquid the potential density in the gas (Yih, 1965, p. 16).The latter is the density ofa parcel of air when its pressure is changed adiabatically to a reference pressure, usually taken to be lo6 dyne/cm2.

Turbulence in Geophysical Systems

23

approximation we have taken into account the tendency for a parcel light or heavier than its environment to rise or fall under the influence of gravity [since we do not assume g(p - po)/po is small compared to dwldt], but we have neglected all other effects of density variation, such as, for example, the small excess of momentum pv - p o v of a parcel because its density is p and not po . In natural circumstames this approximation is excellent in water where, for example, temperature variations cause density differences of order of & of 1%. It is also quite good in the lower atmosphere (Monin and Yaglom, 1971, p. 412). There are some delicate points in a careful derivation of the Boussinesq approximation, especially as applied to compressible fluids. These are considered carefully by Spiegel and Veronis (1960). In some special problems certain phenomena of interest may be lost by making the approximation (Long, 1965), but this does not affect the subject of this paper. Our treatment of the governing equations is very brief and the reader is referred to more careful discussions of the equations by Phillips (1966, pp. 8-19), Monin and Yaglom (1971, pp. 421-425), and Turner (1973, pp. 3-13). In the sea, density variations are caused by both temperature and salinity. We assume a linear relationship, p = po( 1 - a T PS) and we use the following form of the Boussinesq equations:

+

dv/dt = - ( l / p o ) V P

+ (gaT - gPS)k + vV’V,

(3.8a)

d T / d t = khV2T,

(3.8b)

dS/dt = k,V2S,

(3.8~)

v.v=o,

(3.8d)

where T is temperature and S is salinity and where the representative density p are positive known constants, k, is the molecular coefficient of salt diffusivity and is much smaller than kh(kh/ksz 1 0 ) . Notice that even when density variations are due to both heat and salt, if we can neglect molecular processes, we may use the simpler set in Eqs. (3.5)-(3.7) in terms of buoyancy b = g(pS - a T ) after setting the diffusion terms vV2v and k h V 2 b in Eqs. (3.5) and (3.7) equal to zero (but see Section 111,E). In parallel flows of homogeneous fluids, we saw that momentum flux T plays a very important role. When the mean density varies vertically in such flows, a parallel concept is that of buoyancy flux, q = kh b, - w”. The conditions of horizontal homogeneity and statistical steady state then insure that both T and q are constants in space and time. Of course, except near boundaries, q will be very nearly equal to - w” and the flux of heat and salt will be due primarily to the turbulent eddies. po is taken to be the density when T = S = 0; the quantities a and

24

Robert R. Long

The energy equation for this simple flow is the same as for a homogeneous fluid except for a term -w" added to Eq. (2.21). We have

a In addition, if we multiply the perturbation buoyancy equation by b and average, we obtain the equation for buoyancy fluctuations,

where N = kh (Vb')Z is the (positive) dissipation function for buoyancy fluctuations, analogous to E in the energy equation. Obviously, N is also __ to the action of very small eddies. attributable w'b has a sign controlled by the correlation between vertical velocity and perturbation buoyancy in rising and falling parcels of fluid. If the mean density increases with height, as in turbulent thermal convection, rising parcels (w' > 0) are associated with lower density (b'< 0) and sinking parcels with higher density so that q = -w" is positive. Then Eq. (3.9) shows that this effect is a source of kinetic energy helping to drive the motion. The correlation coefficient is of order one. [Deardorff and Willis (1967) measured values around 0.5 and 0.6.1 When the mean stratification is stable, q will have the opposite sign as rising parcels tend to be cool and falling parcels warm. One must be careful in the stable case, however, because of the possibility of wave motions contributing to w' and p'. Thus, if the fluid is at rest and stably stratified, it is capable of internal gravity-wave motion in which the basically level density surfaces move up and down in waves. Obviously, if the waves d o not break, there will be no rupture of these surfaces and therefore, neglecting molecular conduction, no flux of heat or buoyancy despite sizable values of w' and b'. The correlation coefficient will be zero. If the waves break, there will be intermittent turbulence superimposed on the wave motion and q will be negative, although the correlation coefficient may be much less than one. Negative 4 means that the kinetic energy tends to decrease. This is because it requires work to lift heavy parcels up and bring light parcels down. There is a tendency in doing this to increase potential energy at the expense of kinetic energy so that some of the other terms in Eq. (3.9)must be energy-producing if turbulence energy is to increase o r be maintained. The term - w'b'b, in Eq. (3.10) is almost always positive, as we have seen, so that this generates buoyancy fluctuations. The first term on the righthand side of Eq. (3.10) is a transport term, analogous to the energy flux divergence in the energy equation. ~

Turbulence in Geophysical Systems

25

B. AVAILABLE POTENTIAL ENERGY It is useful to define available potential energy per unit mass (Long, 1970) by considering it to be the kinetic energy per unit mass attained by a parcel of buoyancy b = b' + b(z)as it falls from the height z to the height c at which its buoyancy b is equal to the mean buoyancy b(c) at that level. We have b' = 6([)- 5(z)= - b z 5 approximately, where 5 = z - [ and we have assumed that 5 is small compared with the length scale of the vertical variation of mean buoyancy.? Neglecting disturbance pressure, we may then write dwldr = d2{/dt2= - b

= 6,5

(3.11)

because [ is a Lagrangian quantity, so that d[/dr = 0. Integrating, we get ( w 2 / 2 )- (b,t2/2)= const.

(3.12)

Thus available potential energy may be defined as

V' = - b;t2/2

or

V

= b'
(3.13)

We may also identify 4 with potential energy changes. The potential energy of a particle of volume Vo and density p is pgVo z. Let us now define incremental potential energy as pgVo z - p o gVo z so that this potential energy is zero when the particle has the characteristic density p o . If we let V represent the incremental potential energy per unit mass, then, to within the Boussinesq approximation, V = bz. Since b is nearly conservative, putting b = b' + b and assuming no mean vertical velocity, we have ~-

dV/dt

= w'b' =

-4

(3.14)

is the average rate of increase of incremental potential energy per unit mass. We may identify this with available potential energy by differentiating (3.13) and again assuming dbldt = 0. We get dV'/dt = * b d{/dr - $t d6,dt

= iw'b' - *5bzw' = w ' b

(3.15)

so that _ _ _ _ _

dV'/dt = w ' b = - q .

(3.16)

Comparing (3.14) and (3.16), we see that the average rate of increase of incremental potential energy and the average rate of increase of available potential energy are the same. In application, we can conceive of an energy-containing eddy in the form of a whirl with horizontal axis of rotation, with velocity 8,and diameter 1. It 'This may not always. or even usually, be the case but our development here is only suggestive of the definition of Eq. (3.13).

26

Robert R. Long

-

-

will lift parcels from their level of origin a distance 5 1 so that V‘ abI, where b b is the rms buoyancy fluctuation. If the turbulence is not decaying, the kinetic energy must be of this order or larger so that abl/a%I1. In a fully turbulent layer, 1 is of the order of the depth D of the layer.+Thuj we see that if D increases and owis maintained, the layer must become more and more homogeneous. A useful alternative expression for the ratio of available potential energy to kinetic energy is obtained by using b‘ = - 6,5 to eliminate 5 in the expression for V‘. We get R = a;/6,0$.

(3.17)

C. RICHARDSON NUMBERS In most literature on turbulence in stratified fluids (Proudman, 1953, p. 101; Phillips, 1966, p. 201) the role of the energy flux divergence term in Eq. (3.9) is treated rather casually. Typically, the arguments neglect the energy flux divergence and in stable conditions the only energy source term is then - u’w’uz.It is then stated that the sink term q must not exceed the source term if the turbulence is maintained so that ~

14

I /tiiz= Rf < 1,

(3.18)

since E is positive and nonzero. Rf is called thepux Richardson number. In addition, it is possible to define an eddy viscosity K , and eddy conductivity K b by T = K,iiz, q = Kb6, (3.19) so that the condition (3.18) becomes

Ri < K , / K , ,

Ri

=

/6zl/ii:

(3.20)

Ri is called the gradient Richardson number and plays a considerable role in stratified flow theory. We question Eq. (3.18) below but even if the arguments were rigorous, the inequality (3.20) would be of limited usefulness because K , / K b varies strongly with Ri.

’ The large eddies tend to be as large as the dimensions of the region of turbulent motion when there are no density gradients as we saw in Section 11. Here the potential energy may be as large as the kinetic energy but this should not change the order of magnitude of the eddy size. Experiments, for example, those of Moore and Long (1971), have been run in which buoyancy forces are of the same order as the acceleration, i.e., ub u$/l. It was observed that the eddies filled the whole channel in this case.

-

Turbulence in Geophysical Systems

27

The argument leading to (3.18) is quite incorrect in laboratory experiments in which a stratified fluid in a closed vessel is agitated by a vibrator (Section V). There is no mean flow, so cz= z = 0 and Ri and Rf are both infinite, yet turbulence and buoyancy fluxes exist even in the presence of very large density gradients (Turner, 1973). Obviously in this experiment the energy flux-divergence term in the energy equation is the only source of kinetic energy. If shear exists, the flux divergence term tends to be of the same order as the shear term in Eq. (3.9), as we saw in Section 11, and this is the apparent reason for the usefulness of arguments that Rf has an upper limit for the existence of turbulence. A commonly cited value is Rf, = 0.15 but the data in the atmosphere, oceans, and experiments scatter in the range 0.05-0.30 (Kullenberg, 1971; Turner, 1973). As indicated by the energy equation, a source of disturbance energy is in the shear of the flow pattern. This is most simply seen in the case of two fluids with a buoyancy jump and velocity jump across a surface of discontinuity. One finds that small waves on the interface will grow exponentially if the wavelength is small enough, i.e., that such a surface is always unstable (Lamb, 1932). With general velocity and density distributions, the problem becomes more difficult but a famous theorem of Miles (1961) is helpful. It states that all uelocity and density distributions are stable f R i > $everywhere in thepow. Accompanying this is a rule of thumb which says that the situation is usually unstable when Ri < somewhere (Hazel, 1972). Hazel analyzed stability and instability when the velocity difference occurs over a thickness greater (r > 1) or less (r < 1) than the layer thickness for the density variation. If r > 1, the Richardson number falls to zero outside of the region of density variation even when Ri is large in the region of density variation. The theorem of Miles indicates that instability may occur and Hazel has shown that it does occur. The condition r > 1 is typical in an experiment of Moore and Long (1971). It may also be common in oceans and other bodies of water. The instability for Ri < 4 is called Kelvin-Helmholtz instability. A number of experiments have been run to investigate it (Scotti and Corcos, 1969; Thorpe, 1973) and they have shown that this instability can lead to turbulence. Its occurrence in clear air in the atmosphere as detected by ultrasensitive radar (Browning and Watkins, 1970) and in the thermocline (Woods and Wiley, 1972) is well established. In the ocean, at least, the instability can be set off by the increase of shear locally due to the passage of long waves along a density interface. The wave roll-up is called billow turbulence and resembles the drawing in Fig. 9. The whole layer then breaks down into a general region of turbulence with the layer of strong density

a

28

Robert R . Long

FIG.9. Billow turbulence.

variation considerably increased in depth. The Richardson number for the new density interface is just above (Thorpe, 1973).

a

D. THERMAL CONVECTION When the density decreases with height, shear is required to initiate instability, as we have just seen. It seems obvious that a density increase with height should be unstable in all circumstances. This is usually, but not always, the case. A classic example is that of a fluid between two parallel, horizontal plates heated from below and/or cooled from above (Benard, 1901). When the heat is first applied, it is conducted through the fluid by molecular conduction and if the temperature difference is not too great, motion will not occur. The retarding effects are from viscosity v which resists any motion, and conductivity k , which serves to transmit the heat without the necessity of motion. The destabilizing effects are buoyancy difference Ab and the depth H. The situation is controlled therefore by the Rayleigh number Ra = AbH3/vk,. When Ra exceeds a critical value (1000-2000 depending on boundary conditions), the convection begins. The motion is not fully turbulent, but in more or less regular convection patterns, until Ra is quite large. In addition, recent experiments (Garon and Goldstein, 1973; Threlfall, 1975) indicate transitions from one turbulent state to another for Ra below lo6 or so. These are imperfectly understood, but are revealed by the presence of discontinuities in the curve of nondimensional heat flux vs Ra at certain values of Ra below lo6.

E. J~OUBLEDIFFUSIVE CONVECTION A curious type of convection can exist in water stratified by both heat and salt even when the density decreases with height. It depends on the very different molecular diffusivities of heat and salt. Two simple thought experiments bring out the main features of the two kinds of motion that are possible. First, following Stommel et al. (1956), a long, narrow heat-conducting pipe is inserted vertically in the ocean extending from the surface to the

29

Turbulence in Geophysical Systems

depths. If warm salty water overlies cold fresh water and the fluid starts moving upward in the pipe, it would gain heat through the walls of the pipe. If it heats up to the temperature of the surroundings, it becomes lighter than the surroundings at the same level because it is fresher. The pressure at the bottom is the same as the surroundings so that the pressure will decrease in the pipe more slowly with height. The fluid will stand higher in the pipe, and if the top of the pipe is below this height, a perpetual salt fountain will operate. Second, if warm, salty, heavy water lies underneath cooler, fresher, lighter water, consider a parcel of fluid isolated from its surroundings by a thin heat-conducting shell and then displaced upward. It will lose heat but not salt, and the buoyancy force will drive it back down. It will overshoot and the opposite buoyancy force will operate. In this way, an oscillation is produced. If there is no temperature lag, the parcel is negatively buoyant when it is above its starting point and positively buoyant when it is below, so the motion will be damped if any friction exists. If there is a temperature lag, however, the positive buoyancy will exist for more than 50% of the cycle so that the amplitude can increase. As Stern (1960) pointed out, solid boundaries are not essential to the argument. A slower transfer of salt relative to heat is assured by its smaller molecular diffusivity, and motions of both kinds are possible. In the first case, long, narrow convecting cells called salt fingers” have been predicted and observed. In the second, oscillating motion has been verified. “

F. MOLECULAR AND TURBULENT DIFFUSION We have seen something of the contribution of the frictional term in the Navier-Stokes equations. By way of comparison, it is instructive to see how molecular mixing processes proceed. If we consider a container of water at rest, stratified by a variable distribution of salt, the salt will diffuse according to the equation aslat = ~ , v s . (3.21) Without solving exactly, we can see that for a container of dimensions H , the time for diffusion T, is yielded by ASIT, k, A S / H 2 or T, H’lk, . For salt, k, lo-’ cm2/sec,so that for H = 30 cm, the time for the water to become homogeneous is of order 2-3 years! We can see from this example that molecular processes are very slow. Of course, salt has a particularly low coefficient of diffusion but even for heat the diffusiontime in this experiment is of order of a week. This may be compared to turbulent diffusion. In the problem we are considering, the time scale in a turbulent situation will be a

-

-

-

30

Robert R. Long

-

function of gU and 1 or 7; lit,,. If we stir to produce eddies of speed 10 cm/sec and if the eddies fill the box, is of the order of seconds! It is not surprising, therefore that turbulent diffusion of momentum, heat, and additives dominates in almost all problems. We must sometimes be careful in laboratory experiments in which (T, and I may be quite small and in which strong stability locally may cause intermittent rather than fully developed turbulence (Crapper and Linden, 1974). The importance of molecular diffusion is measured by the Peclet number Pe = U H / k , , where U is a representative velocity, and we may argue that phenomena tend to be independent of both Pe and Re when (as is usual) the two numbers are large. On the other hand, the ratio of these is the Prandtl number Pr = Pe/Re = v / k , and there is strong evidence that in thermal convection, at least, Pr is an important parameter even when v and k, are both very small (Rossby, 1969). G. NEGLECT OF ROTATION IN

THE SURFACE LAYER

The horizontal equations of motion, including the earth’s rotation (Holton, 1972, pp. 1-32) are duldt = (1iPo)(~Piax)+fu,

(3.22)

dvldt = - ( l / P O ) ( a P / d Y ) -fu,

(3.23)

where we neglect molecular friction. In Eqs. (3.22) and (3.23),fis the Coriolis parameter, 2R sin 6, where 6 is latitude and R is the angular speed of rotation of the earth. Since we are not concerned with large-scale motions, we may neglect the variation offwith latitude (Holton, 1972, p. 106). Let us consider statistically steady, turbulent flow of a uniform current of air over the earth’s surface. We neglect all variations horizontally and assume w = 0. Then the horizontal equations become

-fa

=

- ( l / p o ) ( a p / a x )+ (a/az)( -m),

f n = - (l/po)(ap//ay)+ (aid.)( -77).

(3.24) (3.25)

There will be a turbulent boundary layer above the surface and above this we will assume nonturbulent flow so that UIW) and become zero there. Then the uniform current is (3.26) (3.27) If we take the x axis along the direction of the wind at the earth’s surface, as in Fig. 10, we have ug = G cos a, ug = - G sin a.

Turbulence in Geophysical Systems

31

FIG. 10. Geostrophic wind.The x axis is along the flow at the surface.

Let us integrate Eq. (3.24)from the ground to height H , . We get -u"(O)

-I-m

( H o )=

I

,H o

[fv

- (l/p,)(d~/dx)] dz.

(3.28)

' 0

Since the pressure gradient force does not vary much with height, we may put it equal t o f G in Eq. (3.28).We get

-u"(O)

+ u"(Ho)


(3.29)

since omittingfc only strengthens the inequality. The stress variation, which is zero if rotation is absent, may be neglected if

[- u " ( O )

+ u"(H,)]/U"(O)

I &,

(3.30)

say. Thus, rotation is negligible if (Monin and Yaglom, 1971)

H o 5 uCv'(O)/lOfG= u i / l O f G

(3.31)

where u, is the friction velocity. In the atmosphere u , / G 2 0.05. Putting f= G = lo3, we get H , 5 25 m. This layer in which rotation is negligible is called the surface layer. A similar computation for oceans and lakes may be inappropriate. Csanady (1972) has argued that in the absence of a solid surface in oceans and lakes, Coriolis effects are not negligible for any range of depths.

H. SOME PROPERTIES OF THE EKMAN LAYERIN THE ATMOSPHERE When we consider conditions above the surface layer in the atmosphere, rotation becomes important. Neglecting effects of compressibility and buoy-

32

Robert R. Long

ancy, and assuming a statistical steady state, we obtain Eqs. (3.24k(3.27)

-f(v-

Ug)=a(-m)/az,

f (a - u,)

__

=

a( - u‘w’)/az.

from (3.32) (3.33)

These equations reveal that the stresses vary with height and this variation causes the wind to deviate from the geostrophic wind. Physically it is evident that the velocity vector is along the lines of constant pressure at z = co,and in the frictional layer the vector is directed at an angle to these lines toward lower pressure. The first investigation of the planetary boundary layer was made by Ekman (1905). If the flow is laminar, dimensional analysis indicates that the depth of the Ekman layer is h

- (~/f)”’.

(3.34)

Let us now consider the turbulence in flow over a rough surface of roughness length z o . Near the ground the velocity components may be expressed as ii/u* = ‘p(z/zo)

v/u* = 0,

(3.35)

where y is the friction velocity at the ground and we take a coordinate system with x axis along the wind direction in the surface layer.fdoes not enter (3.35) since, as we have just seen, rotation is unimportant in the lowest layers. Above the surface layer, we may write

(a - .,)/U*

= Xl(Zf/U* 9

zo f l u * ) ,

6- ug)/u*

= X 2 k f /u*

20

From Eq. (3.33), we obtain (ti - .,)/.*

-

3

u*/f

f h*).

h.

(3.36) (3.37) (3.38)

If we use the analogy between molecular and eddy viscosities, Eq. (3.34) indicates Since 1

-

h h, we get

-

- (K,/f)”2 - ( ~ “ l / f ) l ’ ~ . h - o u l f - u*/f

(3.39)

(3.40)

so that (ii - u,)/u, 1 for all zo flu,. Accordingly, we may write the defect laws for small zo as

(3.41) (3.42)

33

Turbulence in Geophysical Systems

Let us now match the two regions as in Section I1 (Csanady, 1967). Using

5 = z/zo, q = zo f /u, , we have ugh* = m l ( v ) ?

uglu* = 4 q )

(3.43)

and

d5)- ml(q) = x1(5q), The procedure in Section I1 leads to

-m&)

5q’ = - m i q = 1/K

= x&q).

(3.44) (3.45)

= const

(3.46)

mi = - B = const We get (Kazanski and Monin, 1961; Csanady, 1967) ii/u, = (11K) ln(z/zo)

u&,

= G cos alu, = - (11K) In(z, flu,)

- ug/u.+ = G sin alu,

=B

(3.47)

+A

(3.48) (3.49)

As we have seen, the depth of the Ekman layer is h = Cu,/fl where C is a constant. Csanady (1967) and others (Plate, 1971) have accumulated some evidence from the atmosphere for the validity of these results but the data shows enormous scatter, partly, of course, because buoyancy effects have been neglected. The latter has been included by Zilitinkevich (1967,1975)by considering A and B to be functions of the quantity, Po = 4 / 4.L

(3.50)

where q is buoyancy flux. IV. Thermal Convection

When a fluid is heated from below or cooled from above, or both, there is a tendency for the heated or cooled fluid to rise or fall, respectively, because of the resultant density difference between the convection element and the surroundings. The rising or falling element is called a thermal. Common examples are the thermals experienced and used by glider pilots and birds for soaring above the heated ground on a sunny day. In the oceans, cooling of the upper layers of the water by contact with cool air, by radiation of heat, or by evaporational cooling produces colder elements of fluid which fall in exactly the same way as warmed air rises. Much less is known about convection in the oceans than is known about convection in the atmosphere, so we will emphasize atmospheric and laboratory measurements in this section, but convection in lakes and other bodies of water causes the destruction of

Robert R. Long

34

the summer thermocline in the fall and thus has great importance for the flushing and freshening of the deep portions of these bodies of water. As we discussed briefly in Section 111, a fluid can remain at rest in the region between two horizontal surfaces-the lower heated and the upper cooled-provided the Rayleigh number, Ra = AbH3/vkh, is below a certain critical value which defines marginal stability. For simplicity we assume that the fixed buoyancy at the lower plate is zero and the buoyancy at the upper plate is Ab. For two rigid planes, Ra, = 1708 as found by Jeffreys (1926, 1928), who extended an earlier investigation by Rayleigh (1916).Above Ra,, motion first begins in the form of regular cells. Recent investigations (Krishnamurti, 1968, 1970) concentrate on the nonlinear problem, in which case the Prandtl number becomes important for determining the phenomena. The first experiments were made by Benard (1901),but the first laboratory verification of Ra, was by Schmidt and Milverton (1935). The shapes of the cells that form are very sensitive to the geometry of the boundary and even to slight unsteadiness in the boundary conditions. For example, Koschmieder (1967) and Krishnamurti (1968) have shown that two-dimensional roll cells form first for Ra > Ra, but the hexagonal cells, as observed also by Benard, can be induced by gradual increase or decrease with time (a few degrees an hour) of the imposed temperature difference, the sign of the motion depending on the sign of the temperature change. These observations explain some of the discrepancies in past experiments that were not so carefully controlled. Benards hexagonal cells were probably influenced fundamentally by surface tension effects. We are interested here primarily in the higher Rayleigh numbers since large Ra is typical of oceans and atmosphere. In the latter, for example, we may take Ab s 10, v k h 0.1 and a value of H (height of mixed layer) of lo4 yielding Ra One might guess that phenomena must become independent of H as H co,i.e., Ra -+ co,but observations indicate distinct effects for Ra as high as 10'. For example, the nondimensional heat flux has discrete discontinuities at certain values of Ra (Threlfall, 1975),and Garon and Goldstein (1973) believe that they have observed discontinuities in the slope of heat flux curve at values of Ra above lo6, although this is debatable. Results in Section IV,D suggest other important effects of H for Ra lo5 or higher. There has been a very large number of experiments to investigate turbulent convection at values of Ra up to 10" or so (Malkus, 1954a,b; Thomas and Townsend, 1957; Silveston, 1958: Croft, 1958; Townsend, 1959; Globe and Dropkin, 1959; Somerscales and Dropkin, 1966; Deardorff and Willis, 1967; Somerscales and Gazda, 1969; Chu and Goldstein, 1973; Garon and Goldstein, 1973; Threlfall, 1975).Observations have been made of the mean temperature field and a number of other statistical quantities including rms temperatures, rms vertical and horizontal velocities, dissipation functions N

- -+

-

Turbulence in Geophysical Systems

35

and E in Eqs. (3.9)and (3.10), etc. We defer an account of these until we have discussed a theory of convection at high Ra.

AND

A. SIMILARITY THEORY OF CONVECTION COMPARISON WITH LABORATORY OBSERVATIONS

The oldest theory of convection at large Ra is the similarity theory of Prandtl (1932) found independently by Priestley (1954). It is based on an analogy to flow in the turbulent region near a wall in which all mean quantities are assumed to be functions of 7 and z only. Prandtl assumed that in a region 6, << z << H above the thermal boundary layer of thickness 6,, but far from any upper boundary, that all mean quantities, for example, the rms buoyancy ob, are functions of q and z. Then dimensional analysis yields

Kraichman (1962) derived the similarity theory by comparing orders of magnitude in the various equations of the problem. Thus, in the energy __ equation, he assumed that dissipation was E o;/l and that production, w’b’, was of the same order. Since rising parcels are warm and falling parcels cold, he assumed an order-one correlation between w’ and 6‘. He assumed also that o, and ow are of the same order. These assumptions yield

-

If we add the likely requirement that the length scale in the region 6, << z << H is z itself, as in the case of the logarithmic layer in shearing flow,

- -

we derive three of the results in (4.1). The quantity 6,is obtained by the reasonable assumption that ob 6,z. Also enlightening is an equally plausible substitute for the assumption q E, namely, that the buoyancy and inertia terms are of the same order in the vertical equation of motion, i.e., (4.3) Indeed, we find that the similarity theory leads to equal orders of magnitude of all terms in all equations. Although the similarity theory is very old, the evidence is less than convincing that it is correct. One damaging observation in laboratory experiments (Deardorff and Willis, 1967)is that (T, increases with height, roughly as indicated by the similarity theory, but that ouincreases only until the top of the thermal boundary layer or perhaps somewhat higher and then decreases. This has led some investigators, e.g., Bechov and Yaglom (1971), to attempt to modify the similarity theory to allow for two length scales 1,

Robert R. Long

36

and I,. Then from the equation of continuity

-

aulax

- atqay

awlaz

we find 0” 0” ~wlx.l~ and two velocity scales are permitted. This argument is also not convincing because it avoids consideration of the role of pressure forces, which, as we have seen, lead toward equalization of velocity scales. On the whole, laboratory experiments are not favorable to the similarity theory (Townsend, 1976). Comparisons are difficult in part because of problems of accurate measurement and in part because the Rayleigh numbers in experiments are a maximum of 10” or so for reasonably large aspect ratios and this may not be high enough to permit a region 6, << z << H.We will see evidence for this statement in the next section. Many measurements have been made of 5,. The one most quoted is that of Townsend (1959)who fitted a z-’ curve to his data. This is often cited as strong evidence for a z-’ law, but Townsend warned that the data were such that the exponent could lie between - 1.3 and -2.5. This range embraces the similarity theory with an exponent of -$, a theory by Long (1975) discussed below leading to z- 3/2, and a theory by Malkus (1954a) leading to z - ’. In another experiment, Croft (1958) found a z-~’’ behavior for 6, but more recent investigators have asserted that no power law seems to fit the data and this also suggests that Rayleigh numbers are not large enough. There is additional evidence regarding rms velocities given by Malkus (1954b)and DeardorlT and Willis (1967). Over a range of Rayleigh numbers from lo5 to lo9 they find the following N

f ( d 2+ 0’’

+ w”) = const(k,/H)’

Ra,

(4.4) where the double overbar indicates an average over the entire container. A theory of Malkus (1954b)also leads to Eq. (4.4), and this inspired Malkus to make laborious visual velocity measurements with tiny glass fragments as tracers. Deardorff and Willis used a moving hot wire in air. The observations are shown in Fig. 11 and the agreement with the Ra behavior of Eq. (4.4) seems reasonably good. In present terminology, if ou,u w ,cr” are all of the same order, this becomes Q, (HAb)”2, (4.5) where we suppress Prandtl number dependence. This suggests

-

-

uW (ZAb)”’

(4-6) in the region 6, << z << H. The behavior in Eq. (4.5) is different from the similarity theory which yields for the fluid as a whole 6 ,

Iv

q”3H’/3.

(4.7)

37

10

1

I

I

FIG. 11. Experimental observations of average kinetic energy in thermal convection.

If we use the most recent experimental relationship (Garon and Goldstein, 1973; Threlfall, 1975) between q and Ab (see Section IV,C), we have Nu = qH/k,Ab z const Ra””,

(4.8) where the ratio of buoyancy flux to conductive flux is called the Nusselt number. The similarity theory leads to

f(d’4-d2 4-

W”)

Z COnSt(k,/H)’ Rae.",

(4.9) and this is not at all close to the observed behavior as we see in Fig. 11. There are many measurements of ob in the laboratory. If the power law oba Z-” is fitted, n lies between 0.48 (Rossby, 1969) and 0.70 (Somerscales and Gazda, 1969). Townsend (1959) reported n = 0.6. These are not very close to the similarity prediction of n = 0.33. Further comparison may be made with respect to N, the dissipation term for buoyancy fluctuations in Eq. (3.10). The dimensions of N are [ N ] = L2T-’, and similarity theory yields N

-

q5/3z-4/3.

(4.10)

Townsend reported N proportional to z - ~ / ’in his experiments. B. MOLECULAR BOUNDARYLAYERS

Let us now consider the properties of the molecular boundary layers just above a heated plate. In this region, we must consider both conductivity and

Robert R. Long

38

viscosity. If the Prandtl number is small, there will be a thin viscous layer imbedded in a thick conductive layer. If Pr is large, the viscous layer will be thicker. Let us consider the two cases separately: 1. Thick Conductive Layer

Let us first investigate the properties of the imbedded viscous layer. Here inertial and viscous forces balance so that wL6, v. In the perturbation w’6,. In the mean vertical equation buoyancy equation we assume k,):, of motion, we may also assume b or 6, wz/6,. These assumptions, combined with q k , 6/6,, lead to 3/4 - 1/4 p r 7 / 4 6, q - 1/4,,3/4 pr- 114, 6, q3/4v- 114 pr3/4 Obv 4

-

ow,

- (z) -. -

-

-

9

4

1/4 pr1/4

ouu

7

T,

,,1/44 1/4 prl/4

3

-

-

9

q-1/2v1/2 Pr-1/2, (4.11)

where we have assumed o,, ow, and where T, 8,,/ou,, is the time period for the layer. In the thick conductive layer, we may assume that advection and conduction of buoyancy are of the same order, or owcSc/kh 1. In addition we assume that the buoyancy forces and inertial forces are of the same order in the vertical equation - of motion, i.e., (Tb, o;,/S,. Since buoyancy tends to be conserved, obc b, 6, Ab,, where L\b, is the increment of buoyancy over the portion of the layer above the imbedded viscous layer. Finally, assuming q k, db,/S,, we obtain for the buoyancy fluctuations

-

- -

-

-

-- -

obc d b ,

q3/4\,-1/4Pr1l4.

(4.12)

Since this is much larger than 6,,Eq. (4.12) also yields the order of magnitude of 6,. Thus, for the outer boundary layer 3/4 -1/4 pr1/4 6, v3/44- 114 pr- 3/4> 6, q3/4v- 114 pr1/4 Obcv

-

ow,

q1’4v1/4Pr-1/4,

-

ouc

q 1/4v 114 pr-i/4,

-

T,

-

q-

1/2v1/2

p r- 112. (4.13)

Pr’/’ in agreement with the assumption that the viscous Notice that 6,/6, layer is much thinner than the conductive layer. For later purposes we need the equation for the mean buoyancy in the outer boundary layer. We have

6, = q3/4v-1/4Pr1’4f(( Pr3/4,Pr),

( = Z V - ~ ~ ~ (4.14) ~ ’ ~ ~ ,

wheref(5 Pr314, Pr) + g ( l Pr314)as Pr + 0. 2. Thick Viscous Layer

-

We now investigate first the properties of the imbedded conductive layer. Here we assume a balance of advection and conduction, or ~ w c 6 c / k h1,

Turbulence in Geophysical Systems

39

- and viscous forces are of the same order and we assume that buoyancy forces or a b c vow,/6f. Using q k , AbC/6,,abc Ab, b,, we get 114 3/4 pr-l/2 -114 314 ~ ~ 1 1 2 , 6, v-i14 314 pr112, 6,-qv Obc 4 4 n w c - 4 114v 114 pr-i/z, fJuc 4114v 1/4 ~ ~ - 1 1 2 , T, v”24- 112. (4.15)

-

-

9

-

- -

-

-

-

- a,,- a, -

In the thicker viscous layer, we may assume equal orders for the viscous, inertia, and buoyancy terms, i.e., aW,,6,,~v 1 and a,,, vaw,/65. We may where is the increment of buoyalso assume awvnb, q and (Tb, ancy across the portion of the layer above the imbedded conductive layer. We obtain for the buoyancy fluctuation,

-

- -

db,

Ab,

q3I4V- ‘I4.

Since this is much less than b,, we have 6, the outer boundary layer

6,

N

$144-

ow,

-

-

abu

114,

v114q 114,

-

q 314 v -114 ,

a”,

-

- LC--

(4.16)

v-1/4q314Pr”’. Thus, for

6,

v1/44 1/4 7

4 314 Pr1/2, Vri2q-‘I2. (4.17)

v-lI4

T,

Pr-’”, in agreement with the assumption that the conNotice that 6,/6, ductive layer is thinner than the viscous layer. Let us now formulate an equation for 6,. In general

6, = q3I4v- 1’4g(5,Pr).

(4.18)

In the main portion of the viscous boundary layer, we may write

s(5,Pr) =fig)+ pr1’2f2(5) + h(5, Pr).

(4.19)

Our analysis so far shows that we may choose h(5, Pr) 0 as Pr co,so that g(5, Pr) = O(Pr’/’). If we differentiate with respect to 5, we get -+

-+

s’(5, Pr) =fll(t) + pr112f;(t) + h’(5, Pr),

(4.20)

where primes denote total or partial differentiation with respect to 5. The derivative g’(5, Pr) must be of order one because

a6,/az

- SL,-

q314v-

(4.21)

l14.

Now (4.22) and since h + 0 as P r -+ co for all 5, we see that h’(5, Pr) Therefore,f,(() = C is a constant. Thus

6, = q3’4v- ‘/“[fl(;“) + C Pr’”

+ h(5, Pr)],

-+

0 as Pr

-+

00.

(4.23)

40

Robert R. Long

where h(5, Pr) + 0 as Pr + co. It is somewhat more convenient to write this as

6, = q3'"v- ' / " [ f ( 5 ,Pr) + C Pr'/']],

(4.24)

where f becomes independent of Pr as Pr + cc.

C. NUSSELTNUMBER-RAYLEIGH NUMBERRELATION Recent experiments (Goldstein and Chu, 1969; Garon and Goldstein, 1973; Threlfall, 1975) have yielded excellent data on the relationship between the Nusselt number Nu and the Rayleigh number Ra in thermal convection within horizontal layers of fluid heated from below. With the Boussinesq approximation, dimensional considerations require that Nu be a function of Ra and Pr. The experimental results have been somewhat disappointing to theorists. A plausible argument (Howard, 1966) suggests that the buoyancy flux q should become independent of the depth as H gets large (large Ra) so that Nu should be proportional to Ra''3 at large Ra. Instead the data seem to indicate a relationship Nu = const Ra",

(4.25)

where n is unmistakably less than 0.333. Recent estimates of n range from 0.278 to 0.293. We will show below that the observations are nevertheless consistent with the assumption Nu + Ra'I3 as Ra -+co. In the development below, we match expressions for 6 in the interior of the fluid with the expressions for 6 in the upper part of the outer molecular layer near the plate where, according to (4.14) and (4.24),

6 = Cq3/"v-' "A A+O, A+Pr'/',

6 + PI-'/",

6+1,

+ q3/"v- '/"hf(&,

Pr),

Pr3/4

as P r - r 0,

~ + l as

Pr+cc.

E+

(4.26) (4.27)

In Eq. (4.26) the function f becomes independent of Pr as Pr + 0 and Pr + 00. In the interior, we introduce a " buoyancy-defect " law for 6analogous to the velocity-defect law for flow in the interior of a pipe or channel (Section 11). We assume (4.28) where U is some unspecified velocity scale and where we allow a possible

Turbulence in Geophysical Systems

41

variation of the function x with Pr. The basic assumption in (4.28) is the neglect of the nondimensional number (q'/4v1/4)H/v in the function x analogous to the neglect of the Reynolds number in the velocity-defect law in a pipe or channel. We may also define m(q, Pr) = + A b ~ ' / ~ / q ~ q/ ~=, v ~ / ~ / H ~ ' U/q1/4v1/4 /~, = u(q, Pr). (4.29)

Let us now match Eqs. (4.26) and (4.28). We obtain for the region of overlap

41, PrH6f (&,

Pr) + CA - m(q, Pr)] =

-x(&

Pr).

(4.30)

We now differentiate with respect to and with respect to q and use the resulting two equations to eliminate the derivative of x. We get ah$ '5 = quv(6f+ CA) - q(um),,,

(4.31)

where the prime denotes partial differentiation with respect to (&) and subscripts indicate partial derivatives. If we differentiate (4.31 ) with respect to 5. we get

(f'&()I"

= qu,/u = - s,

(4.32)

where s is a function of Pr which we later take to be a constant. Solutions are

f= *A0

+ BO([E)-~,

u

= Pq-',

(4.33)

where A. , Bo ,and /? are functions of Pr. The factor 4 is introduced for later convenience. Equation (4.31) now yields m =fAo6

+ CA - $yoqs,

(4.34)

where yo is another function of Pr. We may now use the first equation in (4.29) to obtain the Nusselt number ~ ~ 1 prw 1 3 NU = (4.35) [ A o 6 + 2CA - y(Nu Ra)-s/4]4/3' where y is a function of Pr. The physical argument that the velocity scale should not increase with increase in viscosity indicates that s 2 4 so that the buoyancy flux becomes independent of H for large Rayleigh numbers. When Pr is large or small, A . is independent of Pr. If, in addition, Ra is large, we get Nu = K , Ra'/3 Pr'l3, Ra large, Pr small, Nu = K, Ra'/3,

Ra large, Pr large,

(4.36)

where K , and K, are constants. These results agree with the theory of

42

Robert R. Long

Kraichnan (1962), using, among other things, the assumption that the buoyancy increments across the conductive layers are of order Ab. Comparison with Experiment

We may try to determine s, A, 6 + 2CA, and y from experiments. The best data, perhaps, are experimental measurements of heat flux in gaseous helium by Threlfall (1975) and in water by Garon and Goldstein (1973). We make two choices for s, namely, s = f and s = 4, corresponding, respectively, to the theory of Long and to the similarity theory. For helium (Pr = 0.8), we obtain 0.04299 Ra'/3 s = - Nu=---1.433(Ra NU)-'/'^]^/^ ' [I 3

I

'\

s=-

'1

Nu=

2

0.04809 Ra'l3 [l - 2.381(Ra NU)-''*]^'^

(4.37)

where we have used two fairly extreme experiments of Threlfall, Nu = 8.060, Nu = 67.183,

Ra = 9.610 x lo5, Ra = 1.804 x lo9.

(4.38)

Threlfall's experiments included measurements at much lower Rayleigh numbers, but transitions, or discontinuities in the curve Nu =j(Ra), are observed below Ra = lo6, and these are not revealed in the present theory. Garon and Goldstein believe that their data reveal abrupt changes in the slope of the curve at Rayleigh numbers above lo6 but this is less certain. An additional comparison may be made using the data of Garon and Goldstein for water (Pr = 6.8). We get s=-

'1

Nu=

3 s=-

'{

Nu=

2

0.04356 Ra'/3 [l - 1.402(Ra

NU)-'/'*]^/^

0.04786 Rail3 [l - 2.544(Ra Nu)- 1/8]4/3

(4.39)

where we have used the lowest and highest Rayleigh numbers, 16.2,

Nu

=

Nu

= 81.3,

Ra = 1.36 x lo', Ra = 3.29 x lo9.

(4.40)

Notice that the coefficients in (4.37) and (4.39) differ by only 1-2% when s = 4 despite the considerable difference in Prandtl numbers. Figure 12 shows the remarkable agreement between the theoretical curve for s = 4and the observations of Garon and Goldstein. We do not show the

Turbulence in Geophysical Systems

43

::' :::I 40 39

3.8

LnNu

3.5

34L

33-

323 1 t

30292ar 16

17

I8

19

20

21

LnRa

FIG. 12. Nusselt number-Rayleigh number relation

comparison with the curve for s = f because the two theoretical curves are almost identical when plotted on these scales. The figure indicates that the curve for s = f is close to a straight line on a log-log plot. The very similar behavior of the data caused both Threlfall and Garon and Goldstein to choose power laws, Nu a Ra0.28 and Nu a Ra0.29, respectively. As illustrated in Fig. 12, the power law gives good agreement but, of course, it has no theoretical basis. In Fig. 12 we have drawn a straight line (power law) in comparison with the theoretical curves for s = f and s = 1 and the data. The theoretical curve for s = f has slight but definite and consistent curvature, and since the data are closer to this curve than to the power law, it seems fair to prefer the present theory corresponding to s = f or s = Threlfall has a plot of his heat flux measurements and there is a good agreement with both the power law and the present theory over the whole range above Ra = lo4 or so, but Threlfall does not supply numerical data and a close comparison is not possible. In Fig. 12 the curve for s = 1 corresponds to the theory of Malkus (1954a).This falls far from the data of Garon and Goldstein, and the Malkus theory seems untenable. We have remarked that the theory of this paper indicates Nu oc Ra'l3 as Ra -+ 00, but there are sensible departures at the Rayleigh numbers of the typical experiment. We may see how large the Rayleigh number must be for Eqs. (4.39) to yield a close approximation to the Ra'/3 law. For s = f, we get Ra > 3 x 10" for an error of less than 1% in the Nusselt number. The corresponding value for s = f is Ra > 1.4 x 10l6.Equations (4.37)give even larger Rayleigh numbers.

t.

Robert R. Long

44

D. BUOYANTCONVECTION FROM

AN

ISOLATEDSOURCE

So far we have concentrated on obtaining an overall picture of thermal convection with no attention to the details of the turbulence. Let us now consider individual thermal elements. They can be conveniently divided into two groups: " plumes and " thermals." The former arises when heat is applied steadily, the latter denotes a buoyant element more or less suddenly released. Let us consider a statistically steady plume and integrate the equation "

ab/at -k v . vb = khV2b

(4.41)

over a circular region of large radius R and thickness Az. We get

\ 5 k , ( g ) r dr do, 2n

wbr dr do = A

'0

R

(4.42)

0

where 6 is the average buoyancy over the area nR2 and A indicates an increment over the distance Az. Since the first term is zero, we have

(4.43) is constant where we have extended the region of integration to infinity because there is no contribution to the integral from any region outside the plume. If we suppose that the plume is caused by applying a buoyancy - Ab to a circular region of the lower surface of radius a, dimensional analysis indicates the following:

6 = F2l3z- 5'3g(r/z,Pr, r/a, F"3a2'3/v), F = Ab3'2a5'2rn(Aba3/vZ, Pr).

(4.44) (4.45)

Let us now consider two separate regions, one near the lower boundary (in the thermal boundary layer) and one far above the boundary. The formulation in (4.44)is valid for all finite a and Ab. As a -+ 0, Ab fixed, the plume disappears. We may obviously obtain a plume of finite properties by letting a + 0, Ab --* co in such a way as to keep F finite. Then we have

6 = F 2 / 3 z -5/3g(r/z,Pr, F 1 / 3 z 2 / 3 / ~ ) .

(4.46)

The last quantity has the form of a Reynolds number and for large z should be negligible. Arguing similarly, we obtain for large F 1 / 3 z 2 / 3 / v ,

6=~

2 ' 3 5/3 ~ -g(r/z,

Pr),

R = zg,(Pr), =~

172(r/z,Pr),

i / 3 113 ~ -

(4.47) (4.48) (4.49)

Turbulence in Geophysical Systems

45

where R is the mean radius of the plume. This dependence on z has been found in experiments by Rouse et al. (1952).It is obvious physically that this formulation is valid for finite a and Ab if z/a >> 1. For v3/'/F1'' << z I O(a), we must include an argument involving a. We write, therefore,

6= ~

2 / 3 513 ~ g(r/z, -

F'/3z2/3/v>> 1.

Pr, r/a),

(4.50)

Let us consider conditions near the plate. If we now assume a large plate compared to the thickness of the molecular boundary layer, the mean buoyancy in the boundary layer will be given by the expressions in Eqs. (4.14) and (4.24)with a term - A b added, with q replaced by F/a2 and with a dependence on q = r/a included. We get

6= F 3 / 4 a - 3 / 2 v - 1 / 4 6 [ f (q,5 ~Pr) , + CA] - Ab,

a >> v3/'/F1/',(4.51)

wherefbecomes independent of Pr as Pr gets large or small and where E, 6, and A are given by A + 0,

E(Pr) + Pr3l4, E(Pr)

-+

1,

A -+ Pr"',

6(Pr) + Pr1/4 6(Pr)

-+

1

as Pr + 0,

as Pr + co.

(4.52)

Since Eqs. (4.50)and (4.51)must agree in the outer portions of the molecular boundary layer,

+ 5'/3(z/a)4'3[6f(5~. q, Pr) + 6CA] = g(r/z, Pr, q), where 5 = z F ' / ~ '/'v~Eq. (4.53)becomes -m-2/3

3/4.

(4.53) Defining Aba3/v2= R , z/a = C, y = ECR3 / 8m 1/4,

+ ~ ' / 8 ~ 1 /[6f(Y, 1 2 q,

Pr) + 6CA] = C-5/3g(v/C, Pr, q). (4.54)

Differentiating with respect to R, we get

(4.55)

= 0.

Differentiating Eq. (4.55)with respect to C, we get Yfyy + $fy= 0.

(4.56)

The solution is

f = ~ ( qPr)y-'/3 ,

+ B(q, Pr),

(4.57)

where A and B are arbitrary functions of q and Pr. Substituting Eq. (4.57)

46

Robert R. Long

into (4.55), we get

(4.58) where we see that B(q, Pr) is a function of Pr only. The solution is [B(Pr) + CA]6R'/8m3'4 - D(Pr)m2/3= 1.

(4.59)

If we now solve Eq. (4.54) for g(q/(, Pr, q), we get for Eq. (4.50) 6 = F2/3a- 5 / 3 ~ ( p+~ ~) 2 1 3 1~ / -3 a - 4 / 3 ~(v, Pr). (4.60) The solution should be valid for arbitrarily large z where 6-0. Thus D(Pr) = 0. If we now combine Eq. (4.45) and (4.59), we get

F/a2 = N(Pr)Ab4/3v'/3,

(4.61)

where N(Pr) cc Pr- u3

as Pr

N(Pr) a Pr-2/3

as Pr

-+

-+

0,

co.

(4.62)

The buoyancy gradient is

6, =f ( ~ ) ( F / a ~ ) ~413' ~ z -

(4.63)

for large or small Pr. Notice that the results are valid for arbitrarily large a but a must be larger than the thickness of the molecular boundary layer.

E. NEWDERIVATION OF THE

SIMILARITY

THEORY

The solution in Section IV,D includes the case a co.Then F/a2 a q and A(r/a, Pr) -+ A(Pr). Thus, for an infinite heated plate and an infinitely deep -+

fluid above

6,= const q2/3z-4/3 q = const Ab4/3~1/3 Prq

= const

Pr large or small, 1/3

Ab4l3v1I3Pr-2/3

as P r -+ 0, as P r -+ co.

(4.64)

The other quantities may be obtained by matching the expressions in Eqs. (4.13) and (4.17). Thus in the interior ow = q

l / 3 ~ ' / (P 3 ~ r

At low Pr in the overlap region q1/3z'/3~(pr)= q1/4y1/4p r-

).

1/4f(zq1/4

(4.65) pr3/4/v3/4)

(4.66)

Turbulence in Geophysical Systems

47

so that H(Pr) = const. At high Pr and H(Pr) is again a constant. The combined results are

Ow

N

q”3z”3

3

O”*q

z

113 113 9

T

(4.68)

q-1/3z2/3

for large or small Pr. This derivation of the similarity theory for infinite Rayleigh number seems quite convincing. Perhaps the most questionable assumption in the present discussion is the neglect of the Reynolds number F1/3z2’3/vin going from Eq. (4.46) to (4.47), but Eqs. (4.47)-(4.49) have been investigated experimentally by Rouse et al. (1952), using a heat source in a large room, and the dependences on z have been confirmed. F. COMPARISON OF THEORIES OF THERMAL CONVECTION

We have referred to three theories that yield the behavior of mean quantities in a region just above the thermal boundary layer. They are the similarity theory, a theory of the author (Long, 1975), and a theory of Malkus (1954a). With respect to the discussion in Section IV,C, the three theories yield

6, K z-‘- ’, where s = 3, $, 1, respectively. The data in Fig. 12 seem to indicate that the Malkus theory is untenable but do not permit us to distinguish between s = and s = f. On the other hand, the matching procedure of Section IV,C seems very successful in comparison with the data for some value of s less than 1 but greater than or equal to On this basis, the similarity theory s = seems preferable because any other choice of s implies a dependence on the Reynolds number r ] - l = ( q ~ ) ” ~ H /in v the expression for U and therefore for the buoyancy defect 6- Ab. In other words, it seems inconsistent to neglect r] on the right-hand side of Eq. (4.28) but retain a dependence on r] on the left-hand side. The discussion in Section IV,E also provides strong support for the similarity theory. An additional remark is suggestive. We saw in Section I1 that the flow in a pipe or channel is such that the Reynolds number Re, based on the velocity defect, the dimension of the region, and the eddy viscosity coefficient is of order one. In thermal convection, using s = i, the buoyancy defect is of order q2’3H-113from (4.28), (4.29), and (4.33) and using K , K, UH q1I3H4l3,the Rayleigh number Ra, based on the buoyancy defect, the

4

4.

-

4

- -

48

Robert R. Long

dimension of the region, and the eddy coefficients of viscosity and diffusion is Ra,

-

1.

4

This result is not obtained for s = and provides an additional indication that the similarity theory is correct. On the other hand, as indicated in Section IV,C, it may be necessary to reach unattainably high values of Ra before one could expect to obtain the behavior indicated by the similarity theory. G. CONVECTION WITH SHEAR

When convection exists in geophysical systems, shear is nearly always present and, as we see in Eq. (3.9), the shear and convection both provide energy sources. A theory of this turbulent situation is described fully by Monin and Yaglom (1971). It assumes that the momentum flux z and the buoyancy flux q are fundamental parameters and neglects molecular viscosity and diffusion. Any mean quantity may then be expressed in terms of 4 , 7 , and z, for example, (4.69) (4.70) (4.71) and where von Kannan’s constant is included to conform with usage. It is remarkable that (4.69)and (4.70)are valid forms for either stable or unstable conditions. There seems little doubt that the Monin-Obukhov theory is a very good one. It collapses the data very well in the lower atmosphere and has been verified in the stable case by Arya and Plate (1969). In the unstable case, it has been argued by Priestley (1954) that large z corresponds to small L or small z so that when z is large z should disappear from the analysis in the expression for buoyancy. We find for (4.70)

6, = const q 2 / 3 z - 4 / 3 .

(4.72)

Other quantities may be obtained similarly, for example, o,, = const q2’3Z-1/3, gu=

const q1/3z1/3,

ow = const q 1 / 3 z 1 / 3 .

(4.73)

This theory has been examined very carefully by comparing with measure-

Turbulence in Geophysical Systems

49

ments on towers in the atmospheric surface layer. Both Dyer (1965) and instead of conformBusinger et al. (1971) find, for example, that 6, K ing with the similarity theory, and the data seem good enough to clearly distinguish between the two behaviors. On the other hand, measurements of ow seem to agree with the similarity theory. The mean velocity distribution certainly depends on T so that it cannot be obtained by the similarity argument. Businger er al. (1971) found ii, a z-’I4

(4.74)

for large z. It is interesting to calculate the behavior of the ratio of turbulent diffusivity to turbulent viscosity. Since T =

K,ii,,

q

=

KbGz,

(4.75)

we find a = K b / K , cc z1I4,i.e., increasing with height. Near the ground, i.e., at small values of z/L+ the energy equation (3.9) reveals that since q is constant the term T U ~takes on increasing importance as the shear increases greatly near the boundary. This implies that the buoyancy becomes a passive quantity, in the sense that the buoyancy term no longer is important in the vertical equation of motion. Then the buoyancy and momentum equations become uncoupled and we see from consideration of generalized dimensional analysis in Section I that buoyancy (or temperature) may be given a dimension independent of the mechanical units of mass, length, and time. Now

6,=f(z,

4, z)

(4.76)

6, = Cq/r’”z.

(4.77)

so that Thus the profile of buoyancy is logarithmic. Of course Zii,/T“2

=

1/K

since there is no effect of buoyancy on the dynamics. These behaviors have been verified by all careful measurements of the first few meters above the ground. In practice, measurements are plotted in terms of (Pm

=

(KZ/U*)iiz 9

(Ph

= (KZu*/q)bz 9

(4.78)

where (P, and (P, are functions of z / L In fact K is chosen to make ( ~ ~ ( =01.) In the measurements of Businger et al. (1971), ( P h ( 0 ) = 0.74 so that the ratio of turbulent diffusivities in neutral conditions, z/L = 0, is

a = K b / K m= ( P m / ( P h = 1.35

(4.79)

approximately. As we have noted, a increases with z/L in the unstable regime.

50

Robert R . Long

The general behavior of (P, and (P, cannot be obtained analytically but we may presume that they can be expanded in a Taylor series about z / L = 0. We get KZ&/U* KZU*b,/q

=

1

+

= 0.74

P,,,Z/L

+PhZ/L.

(4.80)

Upon integration they yield the log-linear laws for the profiles of velocity and buoyancy. These appear to be good approximations for the stable case for all z / L measured but the range of validity in the unstable case is extremely small, certainly less than z / L = 0.3. V. Turbulence in Stably Stratified Fluids

The energy equation (3.9) provides the key to the basic difference between turbulent convection as studied in Section IV and the subject of this section. In the former, the term wlbl= - q provides a source of turbulent energy and in the latter it provides a sink of turbulent energy. It is not surprising that the two kinds of turbulence are very different. Measurements of mean temperatures or salinities in fluid systems normally reveal that the fluid is stably stratified. As we have seen, exceptions are found in the lowest layers of the atmosphere when the air is being heated by the underlying surface and in laboratory experiments with convection, although even in the convection experiments there is often a stable region in the middle of the container away from the heated lower surface and the cooled upper surface (Gille, 1967; Chu and Goldstein, 1973). This phenomenon may be related to the conjecture that the main portion of the liquid core of the earth is stably stratified (W. V. R. Malkus, personal communication, 1975). Recent measurements in the upper layers of the oceans reveal a stable or neutral density distribution even when the surface is losing heat (J. D. Woods, personal communication, 1975)although the failure to find an unstable layer just below the surface may be due to instrumental difficulties. One of the basic features of turbulent, stably stratified fluids is the persistent appearance of layers (sometimes called interfaces), with strong density increase with depth, between layers with little density variation (Stommel and Fedorov, 1967; Woods, 1968; Turner, 1973). Examples in bodies of water are given in these references. The atmosphere also is typified by nearly neutral layers separated by thin layers (inversions) in which potential density decreases rapidly with height. Oceanographers, meteorologists, and engineers have a considerable interest in problems related to the existence, erosion, and motion of interfaces

Turbulence in Geophysical Systems

51

between fluids of different densities and the related problems of momentum, heat, and salt fluxes across these surfaces. An example of the practical importance of these studies is artificial destratification of reservoirs to improve water quality in the hypolimnion (stagnant region below the thermocline). One method involves the pumping of fluid from the hypolimnion through a tube and discharging the water into the epilimnion in a jet downward from the free surface. The heavy discharged water moves down to the interface 'and the mixing in the upper layer and the turbulence generated by the jet erodes the interface causing it to weaken, move downward, and eventually disappear (Brush, 1970). Geophysical implications of mixing across density interfaces are numerous. In the oceans, for example, suddenly increased stress forces exerted by the wind at the water-air surface will increase the turbulence and cause the upper mixed layer to increase in depth at a rate dependent on the stress, the instantaneous depth of the layer, the density jump across the interface, and, perhaps, other effects (Kato and Phillips, 1969; Kraus and Turner, 1967). A. EXPERIMENTS WITHOUT SHEAR

A number of laboratory experiments have been designed to help understand the problem. The first was by Rouse and Dodu (1955) and consisted of a vessel with two layers of liquid of different densities. A grid of metal bars was oscillated in the upper layer with a small stroke a, and observations were made of the entrainment velocity u, or the downward velocity of propagation of the interface. We sometimes idealize by considering the density to be discontinuous. Actually, the interface is a thin layer of thickness 1-2 cm. The thickness is independent of the Richardson number (Moore and Long, 1971; Wolanski, 1972; Crapper and Linden, 1974; Wolanski and Brush, 1975), and may be proportional to the depth of the mixed layer (Long, 1973). A detailed understanding of the turbulence in this experiment has not yet been achieved but Linden (1973) has performed allied experiments and has suggested that the large eddies in the upper mixed layer deflect the interface downward, storing potential energy. When this is released by upward motion, a portion of the heavier fluid is ejected into the homogeneous layer and then carried away by the turbulent eddies, leaving the interface sharp again. The experiment of Rouse and Dodu is typical of those without shear and the only source of energy is the energy flux divergence term of Eq. (3.9).We will see that similar experiments have been constructed with shear (Section

Robert R. Long

52

V,B). This yields an energy supply which is of great importance in oceanic and atmospheric motions. Cromwell (1960) constructed a similar experiment to simulate the pycnocline, but the first reliable data were obtained by Turner (1968). Turner ran two different experiments. One was similar to that described above; the other had stirring in both layers. In the first experiment the lower fluid was agitated and fluid was withdrawn from the stirred layer at a rate adjusted to keep the interface at the same distance from the grid. The entrainment velocity is then defined by Au, = Q, where Q is the volume withdrawn per unit time and A is the cross-sectional area of the tank. In the second experiment both layers are turbulent and, with the same stirring action by the two grids, the interface stays at midpoint. Theoretical considerations of the experiments involve the concept of buoyancy b, defined in Eq. (3.4). In all experiments, the density difference is very small so that the Boussinesq approximation is a very good one. As we see in Eqs. (3.5)-(3.7), since there is only one stratifying element (either salt or temperature), this means that b is the only quantity involving density or gravity entering the analysis. In the one-grid experiment with the upper level mixed, if the interface is allowed to move downward at speed u, (no fluid added or subtracted), the buoyancy flux q satisfies the equation,

aq/az = ab/at

(5.1) obtained by averaging Eq. (3.7) and assuming that all mean quantities are homogeneous horizontally. 6 is the mean buoyancy in the upper layer and z = 0 at the top of the layer. In Eq. (5.1), 6changes little with height in the mixed layer and we may integrate to obtain the flux q at the interface, q = Dd(Ab)/dt, where Ab is the buoyancy jump across the interface and q the layer. We may also write

(5.2) = 0 at the top of

q = [d(DAb)/dt]- u,Ab.

(5.3) Let us apply considerations of mass continuity but to an experiment somewhat more general than this, in which there is initially a basic density variation p ( z ) (Fig. 13). The upper portions are agitated and a mixed layer forms of nearly constant density p1 and depth D. In time dt the interface depth increases to D + dD. Conservation of mass yields (P1

+ dPl)(D + dD) = P P ) dD + P1D.

This may be written d(DAb) = D dD(db/dD),

(5.4)

Turbulence in Geophysical Systems

53 Z

=O

FIG. 13. Erosion of a stratified fluid.

where b(z) is the original buoyancy distribution in the fluid at rest. In the Rouse and Dodu experiment, the lower fluid has a uniform density, so db/dD = 0 and DAb is constant. Therefore, from Eq. (5.3), q = - u, Ab. This result may be used to define an entrainment velocity when both layers are agitated. In this case, if p o is the average of the densities of the two mixed layers, the buoyancy flux at the interface is q = ( 0 / 2 ) d(Ab)/dt and we may define the entrainment velocity to be

ue

= - - 'z(D/Ab)[d(Ab)/dt].

Quantities on the right-hand side are all easily measured. There have been a number of recent experiments similar to those of Rouse and Dodu and of Turner, for example, by Brush (1970). Equipment identical to that of Turner was constructed by Wolanski (1972; see also Wolanski and Brush, 1975), and the one- and two-grid experiments were run with stratification caused by heat, salt, sugar, suspensions of sediments and minute silica spheres. Additional experiments have been run in Turner's apparatus by Linden (1973), Crapper (1973), and Crapper and Linden (1974) using heat and salt. Baines (1975) has studied a similar experiment with entrainment caused by a jet impinging on an interface. An important finding in the experiments by Turner (1968) may be expressed as

u e / o = C[02/(Ab)]", (5.5) where o is the frequency of the oscillating grid and C is independent of o and Ab. A number of lengths are kept constant in the experiment so that the dimensional quantity C may vary with these. Turner found that for larger values of A b / o z , the exponent n = 1 when stratification is caused by temperature differences and n = 3/2 when caused b y differences in salt content.

54

Robert R . Long

Later investigations (Wolanski, 1972) have confirmed these results and, very recently, Crapper and Linden (1974) have shown rather convincingly that the difference in the values of n is due to the influence of the relatively large molecular conductivity k, in the heating experiments (the coefficient k, is much smaller for salt). It appears that whenever the n = 1 law describes the entrainment, the thin layer of strong density variation has an inner layer or core in which molecular diffusion is important. Indeed, earlier unpublished experiments by Claes Rooth (Turner, 1973) support this interpretation. Rooth found a 312 dependence in heating experiments when larger turbulent velocities are generated. Thus it appears well established that the 312 dependence is appropriate for larger Peclet numbers, Pe = ouIlk, or ouIlk,, where o and 1 are the velocity and length units of the turbulence discussed in Section 11. Crapper and Linden suggest a threshold value of Pe z 200 when =a : and 1 = I’ are characteristic of the turbulence near the interface. The dependence on Reynolds number Re has not been established because of the rather small ranges of Re in the experiments, but both Wolanski (1972), who varied Re by a factor of three, and Crapper and Linden (1974) report very weak dependence, if any. We may, therefore, write for large Pe and Re, and strong stability, u,/m = C , w ~ / ( A ~ ) ~ ” ,

(5.6)

where CI is a function of a , D, and the lengths a , , a , ..., characteristic of the grid. It is convenient to introduce a dimensionless quantity K, by the definition

C , = a 4 D - 3 / Z K , ( a / Dala,, , a/a2 ...)

(5.7)

and, therefore, uJu* = K, Ri*-3/2, Ri* = D A b / u i , (5.8) where u* = ma. It is likely that K , is independent of a / D when this ratio is small.

B. EXPERIMENTS WITH SHEAR Several experiments have been constructed to introduce shearing currents in turbulent density-stratified systems in an effort to simulate atmospheric and oceanic phenomena. The first of these of direct relevance to our discus‘The independence of molecular quantities is common in turbulence as we learned in Section 11. In Turner’s original paper (1968). he expressed the belief that the n = 1 law was the fundamental one and that in some manner the very low dihsitivity of salt caused the n = 3/2 law. He has changed his mind on the basis of the evidence we present here (Turner, 1973).

Turbulence in Geophysical Systems

55

sion was that of Kato and Phillips (1969).The apparatus was a large circular annular channel filled with salt water with an initial linear density gradient. A constant stress t = u: was applied by rotating a flat screen at the surface. For larger values of Ri*, they found u,/u* = K ,

Ri*-’,

(5.9)

where Ri* is of the same form as in Eq. (5.8) and Ab is the buoyancy jump from the upper mixed layer to the quiescent region below.+ It is important for later purposes to present an analysis of the relationship between u* and U , where U is the speed of the screen in the experiment of Kato and Phillips. They found that U / u , increased with time (or depth) with u* held fixed. At first glance one might expect this to be an influence of the stable density distribution in the fluid system, but on closer consideration it seems more reasonable to neglect Ab entirely and consider the flow and turbulence in the upper layer as turbulent flow due to the motion of a rough plate at z = 0 with the interface at z = D serving only to reduce the mean velocity to zero at that level. This is supported by a description by Kato and Phillips: “The movement [of a line of hydrogen bubbles] indicated that the mean velocity varied most rapidly near the screen and near the entrainment interface, being almost constant in the central region, where the velocity was typically about half that of the screen.” The mean motion seems very close to that in turbulent plane Couette flow (Robertson, 1959), and a theory for the ratio U / u , may be obtained by use of the technique of Izakson (1937) and Millikan (1939) discussed in Sections I1 and IV. We assume for the mean velocity near the screen (in a coordinate system moving with the screen) tilu* = f ( z / z o ) ,

where zo is the roughness length (Section I1,C) and @(z)is the mean velocity. In the interior we adopt the velocity-defect law (Monin and Yaglom, 1971) (U/2

~-

a) = m l ( i ] .

(5.10)

u*

The defect law should hold near z = zo so that we may match the two expressions for in this region. Writing U / 2 u , = rn,(z,/D), z o / D = q, z/zo = 8, we get m * h ) - f @= ) m1(@). More recent experiments have been run in the Kato and Phillips apparatus using a lower fluid of uniform density (Kantha, 1975). The results d o not yield a simple power law but rather a faster and faster decrease of u, with Ri*. It is possible, however, that molecular viscosity becomes more and more important at the higher values of Ri*.

Robert R . Long

56

The same procedure as in Section I1 yields

u - -2 u*

);(

In

-

+ A,,

K

where K is von Karman's constant and A , is another constant. Kato and Phillips supply data for U ( t )for some of their experiments and, although the paper contains an empirical equation for D(t),it does not agree with the data over the whole time period of the experiment. However, one case permits a comparison and this is shown in Fig. 14 in which the theoretical curve is

U ~

u* 30

I

1

=

(f)

-

+ 5.78.

. . . . ... ..

I

I

1

0

25

In D ( t ) I

.

(5.1 1) '

I...'.

1 .

6

* I

.

O

-

D

-

3

10

-

. O

-

5 -

0

-

1

I

I

I

I

I

I

I

1

I

I

20

40

60

80

100

120

140

160

180

200

220

240

The single constant was chosen to give U = 27.25 cm/sec at t = 200 sec. The solid curve in the figure is a plot of Eq. (5.11) in which D ( t ) is given by curve 11, Fig. 5 of Kato and Phillips (1969). The data points are from Fig. 3 of Kato and Phillips. All experiments have the same buoyancy gradient. The agreement is remarkable and leaves little doubt that U / u , is independent of the Richardson number which varied by a factor of 100 over the course of the experiments. Evidently it is not necessary to invoke sidewall friction as in Pollard et al. (1973) to explain the leveling off of the speed of the driving screen at large times.

Turbulence in Geophysical Systems

57

The work of Pollard, Rhines, and Thompson deserves further discussion. They assume for the Kato and Phillips experiment that the stress exerted by the moving plate is used entirely to accelerate the mixed layer, i.e., d(iiD)/dt = u: or iiD = u i t , where ii is the mean velocity in the upper layer. They also assume, based on several speculative arguments, that the overall Richardson number is equal to one, i.e., DAb/ii2 = 1. Finally from Eq. (5.4) Lib = N2D/2,

where N is the Brunt-Vaisala frequency, N 2 = -db/dz, in the lower, inert layer. A combination of these results leads to D =21/4t1/2~*/N1/2. (5.12) This expression agrees rather well with the data, although it differs from the law chosen by Kato and Phillips. On the other hand, if we assume a relationship of the form u,/u* = const(u:/DAb)”,

Equation (5.12) implies n = 4.Such an entrainment law is not in good agreement with the data over any range. Indeed Niiler (1975)generalized the approach of Pollard, Rhines, and Thompson and obtained a t 1 l 3 law for the initial deepening, again yielding n = 1 as reported by Kato and Phillips. An experiment by Moore and Long (1971) was constructed to permit a steady state. In a large channel shaped like a race track, fluid was injected from nearly horizontal jets at bottom (salt water) and top (fresh water) in opposite directions to obtain a shearing current (Fig. 15). Mean zero vertical velocities were achieved by withdrawing equal volumes of fluid through

-

I

-I

58

Robert R . Long

numerous holes at bottom and top. At larger values of the density difference, two homogeneous layers existed at top and bottom with an interface in the middle. The salt water in the jets comes from a reservoir and the withdrawn fluid at the bottom is pumped back into the reservoir which is kept at a constant level. The jets at the top are of tap water and the slightly salty withdrawn fluid at the top is pumped to waste. Since the fluid returned from the bottom to the reservoir is somewhat less salty than that in the lower jets, salt must be added continually to keep the reservoir at a fixed density. The added salt is transported vertically by the turbulence. The salt flux is known, of course, and this can be used directly to compute the buoyancy flux. The experiment yielded =

K~(Au)~/D

(5.13)

over a range ofDAb/(Au)2of 1-60, where Au is the velocity difference between mean velocities measured near the top and bottom. If we define the entrainment velocity by -u,Ab = q, Eq. (5.13) yields the same result as in Kato and Phillips (Eq. (5.9), if, as seems very likely from the discussion of the Kato and Phillips experiment, Au/u, is independent of the Richardson number, where u: is the constant momentum flux in the tank. Moore and Long also ran unsteady experiments similar to those of Kato and Phillips with a fluid with initial linear buoyancy gradient and subject to the systems of jets and withdrawals at the bottom only. They obtained the result D a t1l3. Since Eq. (5.9) may be written

dDldt

= K,u:

JDAb

and since Eq. (5.4) indicates that Ab cc D in erosion of a linear density gradient, the behavior D a t1I3 is consistent with Eq. (5.9). The velocity observations by Moore and Long apparently relate to those found by Kato and Phillips. Moore and Long report a velocity distribution with strong shears at top and bottom and across the density interface region. This, of course, is exactly the behavior to be expected from plane Couette flow in each layer (see Fig. 17 on p. 68). Finally, in a recent experiment by Wu (1973), the source of energy and shear was a current of air blowing over a vessel containing a two-fluid system. Wu also obtained Eq. (5.9) although his coefficient of proportionality was much smaller. He conjectured that this was because of the very different shear produced in a closed container, but Bo Pedersen (private communication) has suggested that the flow at the interface may have been laminar over much of the length.

Turbulence in Geophysical Systems C.

COMPARISON OF EXPERIMENTS WITH AND

WITHOUT

59

SHEAR

The different dependence on Ri* for the two experiments has been the source of perplexity (Turner, 1973; Linden, 1973) because the mixing processes appear to be very similar. Indeed, Linden has stated that the Kato and Phillips data are also consistent with a -312 behavior, although support for the statement seems lacking. We attempt below to construct a unified understanding of the two results. Turner (1973) has made the valuable suggestion that the erosion of the interface should depend on the properties of the turbulence near the interface, in particular on the rms velocity a: and the integral length scale I' near the interface. Thus, he proposed the form ue/o; = F(Ri,),

Ri,

=

I'Ab/02,

(5.14)

where possible dependence on other quantities is suppressed and it is assumed that Pi: and Re are large. In an attempt to determine the dependence on Ril from his density-interface experiments, in which a: and 1' were not measured, Turner used experimental data by Thompson and Turner (1975) in Turner's apparatus with a homogeneous Jluid and one grid. They measured a, and 1 at many levels, where a, is the rms velocity and 1 is the integral length scale at depth z. They found that 0;was proportional to w and that 1 increased linearly with distance from the grid but was independent of w. Although Thompson and Turner's experiment had no density variation, Turner (1973), Thorpe (1973), and Crapper and Linden (1974) have assumed that the results are directly applicable to the mixing experiments. Thus, at z = D they use al/wa = C 2 ( a / D ,a/a,, a/a2 ...)

I'/D = C,(a/a,, a/a2 ...)

(5.15) (5.16)

so that Eq. (5.8) may be written u,/a: = K, Ril3I2,

K4 = K,(a/D, ala,, aja2 ...).

(5.17)

Neglecting viscosity,' the proportionality of a: and w follows from dimensional analysis but only when theJluid is homogeneous because the presence of an interface introduces a new quantity involving time, namely, Ab. We may also obtain a dependence on Ri, for the shearing experiments. With shear and horizontal homogeneity the averaged x-equation of motion yields

ariaz = aqat,

(5.18)

However, wu is analogous to the speed of the plates Au in plane Couette flow. In the latter case it is known that the Reynolds number is involved in the relation between Au and turbulent velocities. Thus oL/wu may vary with the viscosity coefficient. +

Robert R . Long

60

where ti is the mean horizontal velocity at depth z. In the steady state experiments of Moore and Lon11971), &/az = 0 and therefore t is constant with height. Since t = -u’w‘, and since the correlation coefficient is very likely to be of order one in the mixed layers, it follows that u* = is proportional to a;. The interface introduces the length D and it seems reasonable that the eddies fill the whole depth as they would if the interface were a rigid surface. Thus we use I’ D and obtain

-

u,/a; = K , Ri;‘

(5.19)

for the Moore and Long experiment. In the experiment of Kato and Phillips, we may use Eq. (5.18) to obtain the increment of t over the depth D.It is Ar/r

-

UDlTU:,

(5.20)

where is the time period for a change of depth of order D so that ?I. Dlu, . Therefore, ignoring sidewall effects,

-

At/r = (U/u,)(u,/u,) = Ri*-’.

-

(5.2 1)

This reveals that the stress varies very little over the depth, that u* a;, and that Eq. (5.19) again holds. Thus, two different entrainment velocities, Eqs. (5.17) and (5.19), are indicated in the two cases even when the characteristics of the eroding eddies are the same and this is more perplexing than the difference in the exponent of Ri*. The discussion below questions the applicability of Thompson and Turner’s experiment, in particular Eq. (5.15), to an experiment with a density interface and suggests that Eq. (5.17) is incorrect. D. ENERGY ARGUMENTS A different dependence of a;/u, or a;/oa on Ri* in experiments without shear may be obtained by a plausible argument. When there is shear, we have seen that experiment indicates

q

- oi31D - .:ID.

(5.22)

-

Let us now evaluate q in the mixed layer near the interface. We get q aka; where a; is the rms buoyancy fluctuation near the interface, and we make the plausible assumption that the correlation coefficient is of order one. Thus o;’J&

D

-

1

(5.23)

so that kinetic energy and available potential energy, obD (see the discussion in Section 111) are of the same order in the mixed layer. Although the layer has very little density variation, this result may be obtained by considering

Turbulence in Geophysical Systems

61

first an experiment with very strong turbulence imposed externally. Then a12/abI' will be very large. As we decrease the turbulence in successive experiments, this ratio will decrease, If turbulence continues to exist, the ratio has a lower limit because aL2/abI' <: 1 would imply that T' < V' and a consequent cessation of the turbulence a5 we discussed in Section 111. It seems unlikely that the ratio will get large again as stability increases so that a12/ab1 should approach a constant. The argument is equally valid with or without shear. Using Eq. (5.23) when shear is absent to eliminate ob in the relationship q u, Ab a; a;, the - 312 law leads to

-

-

u,/u*

-

0L3/Du,Ab

or a:/u*

-

-

Ri*-

u:/(DA~)~'~

(5.24)

(5.25)

'I6

instead of Eq. (5.15). The decrease of rms velocity with increase of Richardson number when density variations are present may be caused by the weak density gradient in the mixed layer. In a layer as a whole, the slight density variation still has dynamic importance as indicated by the proportionality of kinetic energy and available potential energy and by the fact that 4,varying linearly in the layer, has a relatively large value near the interface. Such arguments have been advanced earlier by the author (Long, 1972, 1973). The energy argument may be amplified. Rouse and Dodu (1955) and others (Kato and Phillips, 1969; Turner, 1973; Wu, 1973) have suggested that the Ri*-' law implies that the change of potential energy is proportional to the energy supply by the external source. Thus, as we have seen in Section I11 the average rate of increase of potential energy per unit mass is 4, so that the rate of increase of potential energy for the system is proportional to 4D. In the Kato and Phillips experiment, for example, the rate of working of the external force is r U . If these are proportional, q

- rU/D

u:/D

(5.26)

as in Eq. (5.22). The same conclusion cannot be reached for cases without shear and on this basis it may be argued that the Ri*-3'Z law does not conform to any simple energy argument. We may show, however, that the last conclusion is not correctly drawn. The energy equation, Eq. (3.9), for experiments with or without shear is

ar/at= -d[w'( T' + p'/p,)]/az

-

+ ruz + 4 - E,

(5.27)

where ii, = 0 in the experiment without shear. In the shearing experiments, the velocity difference is proportional to r112, and the two energy-source terms, as well as the dissipation E , are of order aL3/D or oL3/l' near the

62

Robert R . Long

-

-

interface. If q u,Ab is also of this order, we obtain u,/a: Ri;' as in Eq. (5.19). When shear is absent, the single source term is the first term on the right-hand side of Eq. (5.27) and is also of order 0L3/l'.The Ri; law again implies equality of all sink and source terms. The correct interpretation of experimental results thus seems to be that the turbulence has a character that causes potential energy to increase at a rate proportional to the rate at which kinetic energy is supplied to the region of the interface and not necessarily proportional to the rate of generation of kinetic energy at the external source. An additional piece of information may be added in relation to experiments without shear. If we assume that the small buoyancy difference across the mixed layer is of order of the rms buoyancy fluctuation (implying an eddy length scale of the order of the depth of the layer), Eq. (5.23) and Eq. (5.25) lead to

'

-

Ab/Ab

-

Ri*-413.

(5.28)

This quantity was measured by Wolanski (1972) and Wolanski and Brush (1975) for the salt experiments (Fig. 16). There is good agreement with Eq. (5.28), especially at higher values of Ri*. 10'

Ai Ab

,o-'

10

FIG.16. Buoyancy difference across mixed layer as function of Richardson number.

When there is shear, a: in Eq. (5.23) is of order u* and ob is of order so that abD/ug 1 and ab/Ab is proportional to Ri*-'. Notice also that in the mixed layer the velocity shear is ii, u* /D,as we see in Eq. (5.10). Thus the

-

-

Turbulence in Geophysical Systems

63

gradient Richardson number Ri = 6z/(Q2is of order one in the mixed layer. Observations in the lower mixed layer in the atmosphere (Businger et al., 1971) indicate that this has a maximum of Ri, z 0.20 and we may speculate that this is the magnitude of Ri in the mixed layer in the experiment. As we have already suggested, Ri 1 means that the slight density gradient in the mixed layer has dynamical significance. We have suggested that

-

u,/o,= K,o:/DAb

(5.29)

is a universal law for entrainment with or without shear, where ouis the rms velocity at or near the interface. Let us apply this to an experiment without shear, as in Turner’s experiment except that the lower quiescent fluid has a linear density gradient. If we also adopt the findings ofBouvard and Dumas (1967) and Thompson and Turner (1975) that ou is proportional to D-3’2, then since Ab = N2D/2, where N 2 is the buoyancy gradient in the lower fluid, we find that D cc t z / l s . This behavior was proposed by Linden (1975) and his experiments with this type of fluid system provided close verification of this time dependence. These experiments, therefore, provide considerable support for the form proposed in Eq. (5.29). E. IMPLICATIONS OF LABORATORY EXPERIMENTS AND OBSERVATIONS IN THE ATMOSPHERE AND OCEANS As we have noted, interfaces or inversions are common in the atmosphere, and properties of the mixed layer near the ground have been measured frequently from meteorological towers. Velocity observations indicate that the mean profile is linear over the upper part of the layer (Businger et al., 1971). This implies that the flux Richardson number Rf = q/zUz becomes a constant if we ignore the small variations of the fluxes in the layer. The constant value of Rf seems to be about 0.20 in the atmosphere (Businger et al., 1971). This compares with the value of0.05 estimated for the oceans by Kullenberg (1971). We may also mention the experiments of Arya and Plate (1969) with air blowing down a wind tunnel over a cooled surface. They found, as in the atmosphere that Rf increases with height from near zero just above the cooled surface. The measurements ended at Rf 2 0.06 and did not extend high enough to establish an asymptotic behavior. Various measurements in the laboratory by Ellison and Turner (1960a,b) indicate values closer to those in the atmosphere, i.e., Rf, 2 0.15. In this connection, we may compute Rf, in Kato and Phillips’ experiment. Since we have decided that the flow closely resembles turbulent plane Couette flow of a homogeneous

64

Robert R . Long

fluid, we may use the data of Robertson (1959) to estimate the shear in the interior region. He measures 17,z 8.2u,/H, where H is the distance between the plates. Taking the average buoyancy flux in the layer as 4 = u,Ab/2 and the relation u, = 2.4u:/DAb found by Kato and Phillips, we get Rf, 2 u,Ab/2u:iiz = 2.4J16.4 z 0.15 (5.30) close to other laboratory results and to estimates in the atmosphere. We may also calculate the constant value of the ratio R = abl/a: in the mixed layer (Eq. 3.17 with abz 6,l). We have (5.31) Rf, = q/zU, = 0.15 Z 0.3a,abD/8.2u:, where we have estimated the correlation coefficient in w" as 0.3 (Arya and Plate, 1969). Using the estimate a,,, = l.lu* (Section II,C) and D = 31 (Section II,B) we obtain R s 1.0. (5.32) Although we compute this in the shearing experiment, the same result is likely to hold in experiments such as Turner's where shear is absent. Thus this critical value of R may be more generally useful than a critical flux Richardson number since Rf -+ co when there is no shear. Typical measurements in the atmosphere show an increase of Ri and Rf with height until both reach asymptotic values near 0.20 and then a sudden transition to laminar flow. This indicates the presence of an inversion commonly found above the mixed layer. In one instance at Tsimlyansk, USSR, the inversion and mixed layer were probed by a thermocouple moved up and down over a depth of 6 meters (Businger and Arya, 1974). The signal reveals an inversion between 2 and 4 meters. Velocities were also measured and Richardson numbers computed. The motion becomes laminar at Ri, r 0.2 1. As found by Businger et al. (1971) both cp, and cp,,, introduced in Eq. (4.78) have the form cp = a + b[ for the whole range of [ = z/L in stable conditions. They give

+ 4.7L (P,, = 0.74 + 4.7[.

cp, = 1

(5.33)

An explanation may be offered for the behavior iiz = const, 6, = const in the upper portion of the mixed layer (Wyngaard, 1973). In the lower portions the relevant length scale is z. As z/L increases the argument contends that this scale should be reduced by the increasing stability until, ultimately, the length should become independent of z. The only length scale is then L. Thus it is argued that iiz and 6, should depend only on T and q and are therefore constants. The explanation is appealing at first glance except that observations indicate a maximum z approximately equal to L so that z/L never

Turbulence in Geophysical Systems

65

becomes large. In addition the success of the assumption that stability may be neglected in computing U / u , in Kato and Phillips’ experiment indicates that the eddies are as in plane Couette flow and are not influenced strongly by the stability. As indicated by Eqs. (4.78) and (5.39, the data show that = KiiIKm = ( P m l q h

(5.34)

is roughly constant with increase of [, but the scatter is considerable. There is some controversy about the value of u in neutral conditions, a, = l/cp,,(O). Other investigators suggest values closer to one (Dyer and Hicks, 1970) in the atmosphere although the higher value of 1.35 is closer to laboratory measurements (Page et a/., 1952). Measurements of u have been made by Ellison and Turner (1960a,b) in a turbulent, stably stratified liquid. They found that u decreased with stability from a neutral value (Ri = 0) somewhat over one to values of perhaps 0.25 at Ri = 0.50. This trend seems reasonable because stability should inhibit exchange of the stratifying element more than exchange of momentum. Momentum can be transmitted by internal gravity waves in a nonturbulent or turbulent stratified fluid, whereas turbulence must exist to transmit any appreciable amounts of heat or salt or other element. It appears fairly definite that turbulence does not exist in the atmosphere when the gradient Richardson number Ri exceeds 0.20 or so. We also came to the conclusion that the Richardson number is of order one (possibly 0.20 or so) in the mixed layer in laboratory experiments. In the Ocean we have the belief of Woods (1968) that Ri does not exceed 1 or so when turbulence exists. This seems to be contradicted by old observations in the Kattegat (Jacobsen, 1913; Proudman, 1953) where diffusivities several orders of magnitude greater than molecular were observed at measured Ri of 4-10. However, in recent years we have come to realize that most bodies of water are typified by thin layers of order of tens of centimeters thick in which velocity and density change rapidly (Kullenberg, 1974), so that one must use observations that are very closely separated in the vertical to obtain the correct local values of ir, and 6,.The Kattegat observations were widely separated vertically. The importance of the separation distance is also revealed by the experiments of Moore and Long (1971). It was possible to obtain overall Richardson numbers based on buoyancy and velocity differences from top to bottom of the channel up to values of 100. Nevertheless the gradient Richardson number Ri nowhere exceeded 2-3 ! The maximum Ri were measured in the stable layer in the middle. Since the transfer of salt through this layer was at least as order of magnitude greater than the molecular transfer, turbulence (perhaps intermittent) certainly existed at values of Ri in excess of one but still of order one.

Robert R. Long

66

T

The Monin-Obukhov length may be examined in the mixed layer. Since , : a q a"a b , we get L T 3 I 2 / q a:/.b . (5.35)

- - 02,

-

Since a b 1 we find that the eddy size is proportional to the MoninObukhov length. On the assumption that the eddies fill the whole layer, we also find L D. Quantitatively, using the same approximations as those above, we have

-

L = u:/K~ E 20.

(5.36)

The atmospheric observations suggest a value slightly larger than D. Kitaigorodskii (1960) found 4 D r 1.2 for the mixed layer in the oceans, and recently Sundaram (1973) has computed L/D E 0.25 for a lake.

F. EDDYVISCOSITY AND EDDYDIFFUSIVITY In almost all applications of turbulence theory (except Lumley, 1972), assumptions are made of the form -

W A= - K ( d A / a z ) ,

(5.37)

where A is some scalar quantity, perhaps velocity, buoyancy, kinetic energy, etc. Among other things these gradient transport models require for their validity that the scale of the eddies be small compared with the scale of the mean property (Corrsin, 1974; Lumley and Khajeh-Nouri, 1974). Since this is generally violated in turbulent systems, doubt is cast immediately on this approach. The almost universal assumption of the gradient transport approach is that K is positive, i.e., that the transport is down the gradient. This is certainly the usual case but exceptions exist. For example, in laboratory experiments with turbulent convection with a lower heated surface and upper cooled surface, the heat transport is always upward but in fact at modest values of the Rayleigh number the temperature actually increases with height over a large interior portion of the fluid (Gille, 1967; Chu and Goldstein, 1973; Townsend, 1976). The same countergradient flux occurs in the upper portions of atmospheric layers under unstable conditions. Countergradient flux for A = w 2 has also been observed in the lower atmosphere (Wyngaard, 1973). Despite these objections, such quantities as eddy viscosity K , and eddy diffusivity Kbdefined in T = K,ii,

(5.38)

(5.39) 4= have proved to be useful concepts. In the simplest case of developing, one-

Turbulence in Geophysical Systems

67

dimensional stratified shearing flow, these definitions lead to the equations (5.40) (5.41)

but this is not immediately helpful unless we can say something about K , and K , . Our hope is to parameterize K , and K,, i.e., to express them in terms of a, 6, z, and t or in some more complicated way to obtain a mathematically determined problem. A common approach in homogeneous flows is to assume K, and K, are constants. This occasionally gives fairly good results, for example, in problems of jets and wakes (Townsend, 1956),but in most cases the approximation is very crude. As we have seen, near a surface K, is not at all constant but is proportional to distance from the surface. Thus the logarithmic layer is lost completely by assuming K, is constant. Constancy of Kb is a very poor assumption in stably stratified fluids. For example, in the Moore and Long experiment, q is a constant but 6, changes enormously from the mixed layer to the stable layer so that K, = q/6z changes enormously, perhaps by as much as three orders of magnitude from the mixed to the stable layer. In homogeneous flows, the eddy viscosity should be a function only of the 6,1 ue H, where H eddy velocity cuand the integral length scale, i.e., K, is the overall dimension of the system, say, the pipe diameter. Locally, in the u* z, of course. Since we have decided that the turlogarithmic layer, K, bulence in a mixed layer in stratified flow is similar to that in a homogeneous fluid, this should be a valid estimate for the mixed layer. If we use Robertson's data that aZ 8.2ue/H, we obtain for the mixed layer

-

-

-.

=

K , = U: D 1 8 . 2 ~z ~0 . 1 2D. ~~

(5.42)

Atmospheric evidence suggests that K , / K , is a constant near one in the stable regime so that this is also an estimate for K b . On the other hand, laboratory investigations of Ellison and Turner (1960a,b) suggest the possibility that Kb is somewhat less than K , , perhaps Kb Z 0 . 0 6 D. ~~ The eddy coefficient K , in the stable layer may be computed for the Kato and Phillips experiment and the Moore and Long experiment. In the former the buoyancy flux q falls from a maximum just above the stable layer to zero at the bottom of the stable layer. Thus q = fu, Ab = K b Ablh,

(5.43)

where h is the thickness of the layer. From the Moore and Long experiment we estimate h = 016, approximately. Using u, = 2.4ui/DAb, we get

K , = .20ueD Ri*-'.

(5.44)

Robert R . Long

68

- -

In the Moore and Long experiment, q ( A u ) ~ / Dand , K b AuD Ri*-',

(5.45)

so that the behavior is similar since Au/u* is independent of the overall Richardson number. The turbulence in the stable layer is presumably intermittent. A computation of K , in the stable layer is more speculative because velocities were not measured at all accurately in either experiment. As we have pointed out, there is no good evidence that any turbulence can exist at large gradient Richardson numbers in shearing flow so that we may assume in the Moore and Long experiment that the gradient Richardson number is a constant of order one as Ri* varies. Also it appeared that the layer over which the strong shear existed had a thickness of order L1 different from h. Thus Ri 1 L:Ab/h(Au)2. (5.46)

Since h

-

-/D -

D, this leads to L1 K,

-

Ri*-li2. Then

zL, /Au

-

-

u, D Ri*-l1*.

(5.47)

This indicates that K b / K , Ri*-'/' so that the disturbances are less capable of transferring heat than momentum. Again this is reasonable because gravity waves can transmit momentum. The assumed constancy of the gradient Richardson number has support from observations in the Moore and Long experiment (Long, 1973). The velocity distribution resembles that in

FIG. 17. Velocity distribution in the Moore and Long experiment.

Fig. 17, similar to two plane-Couette velocity profiles. The shear in the middle increased, in fact, as Ri* increased so that Ri was roughly constant with a value of 2-3 perhaps. The earliest theory for K , / K , was given by Ellison (1957).' As discussed by Ellison and Turner (1960a,b), the theory may be put in the form Kb/K,,, = a,( 1 - Rf/Rf,)/(1 - Rf)2,

(5.48)

The Ellison theory has been reviewed by Yamada (1975) and is discussed at greater length in Section VI. +

Turbulence in Geophysical Systems

69

where a, is associated with the value of a in Eq. (5.34) in neutral conditions, where Rf is the flux Richardson number Rf = q/.rii, = Ri(K,/K,), and Rf, is a complicated function of time scales and rms velocities. All shearing experiments indicate that Rf is bounded as Ri gets large; indeed, if ti, is regarded as the important energy-supply term in the energy equation, i.e., if we neglect the energy flux divergence, Rf must always be less than 1. Therefore, for large Ri (assuming turbulence exists), Eq. (5.48) shows that Rf -,Rf, so that a critical value of Rf exists above which turbulence presumably dies out. The data suggest Rf, = 0.05-0.30. At larger values of Ri, perhaps Ri > 0.1,the quantity 1 - Rf/Rf, is small so that we may write, approximately, 1 - Rf/Rf, = E,

K b / K , Z u , E / ( ~- Rf,)2

= [u,/(l - Rf,)2][1

- Ri Kb/Rf, K,]

(5.49)

or

+ [u,/(l

- Rf,)2] Ri/Rf,}.

(5.50) Since the coefficient of Ri in (5.50) is 15-20, for Ri >> 0.1,we would expect the simpler equation Kb/K, = const/Ri. This was indeed found to be true by Kullenberg (1971). A number of investigators, beginning with Rossby and Montgomery (1935) have chosen for stable conditions the relationship K b / K , = [u,/(l - Rf,)2]/(l

Kb = KbO(1

+

(T

Ri)-',

(5.51)

where (T is a constant and KbO is the value of Kb in neutral conditions. This can be derived from Eq. (5.50) if we ignore the (weaker) variation of K, with Ri. For example, Sundaram and Rehm (1973) have applied Eq. (5.51) to a one-dimensional model of a lake. Their numerical time-dependent integrations yield a thermocline. Equation (5.51) has great limitations, however, in that Ri is small in deep water but Kb is certainly small there since it is protected by the thermocline from disturbances of the mixed layer. This was overcome by arbitrarily choosing a low value in deep levels. Sundaram and Rehm form Ri as Ri = 6 , z 2 / ~ ,0 where T~ is the (given) stress at the surface. The diffusion equation now determines the problem, (5.52)

and no additional dynamics is employed. Although Eq. (5.51) is often used in practical applications it is exceedingly crude when compared to recent results from third-order turbulence closure models (Section VI).

70

Robert R . Long

Kullenberg (1971) has proposed a relationship for K , by assuming a constant flux Richardson number Rf = Rf,. Then (5.53)

If the flow is steady, the kinematic stress in the water t equals the surface kinematic stress 70 = t,pa/pw , where pa and pw are the densities of water and air, respectively, and 7, is the kinematic stress exerted by the air. Kullenberg also assumed a constant drag coefficient in the expression 7, = cd U:. His result is

(5.54) The data on the dispersion of die patterns (assuming the same coefficient of diffusion for heat and dye) fit well. Indeed, the theory assumes mainly that q is proportional to ztl, and this is verified well by experiment. He takes Rf, = 0.05, (close to unpublished estimates by Stigebrandt, but rather low compared to our estimates for Kato and Phillips and other estimates) and A limitation as far as parameterization is concerned is the cd = 1.2 assumption that the stress everywhere in the water is equal to the surface stress. According to Eq. (5.18) this would not permit time integration, for example. Also, of course, the assumption that Rf = Rf, is not permissible for Ri < 0.1. G . BUOYANCY FLUX DUETO WAKECOLLAPSE

Long (1970) has attempted to apply the concept of wake collapse to the exchange of buoyancy (heat or salt) in a density-stratified medium. We suppose that a turbulent region (or patch) of linear dimensions I,, volume Vo, is produced by the breaking of internal gravity waves. We assume the region is initially undisturbed with density given by p ( z ) .Waves break and thoroughly mix the patch. It will then tend to flatten out at a level at which the environmental density equals the density of the patch. If the new level is different from that of the original center of mass of the material, heat will be transported vertically by this process. If z, and z , are the heights of the centroid and center of mass, the average value of density for the mixed patch is

or

p p = p ( z c )+ pzz(Z,)Aj 1; + . . .,

(5.56)

Turbulence in Geophysical Systems

71

where A, is of order one. Expanding about z = z , , we get

p p = p(z,)

+ p,(z,)Lc + pzz(zm)$L:+ ... + pzr(Z,)A3 1; + ...,

(5.57)

where L, is the height of the centroid above the center of mass. The patch flattens out at height & above the center of mass given by p p = P ( Z , + 4)= p(zm) + Pz(zm)& + . . or ’ 9

~p

+ [Pzz(Zm)A,& ~ z ( z r n ) l .

~c

(5.58)

The distance L, is easily computed. Taking z = 0 at the centroid

where M is the mass of the patch, or (5.59)

or

L,

=

+

(El pz(0)l,” . ..)/(I?, p(0)l;

+ . ..),

(5.60)

where B , , B , ... are of order one. Thus, since we take I, much less than the vertical scale of the motion,

Lc

- PZ(O)IPz/P(O).

(5.61)

The ratio of Lc to A, pZz$ / p , is of order

i r , 2 ( ~ ) / ~ ( O ) P Z , ( O ) (Aiv1,2/1,2PoAP W P O where A p is density difference across the patch. Thus Iv

Lp

Iv

3

- bZL1;/6,

(5.62) (5.63)

and is proportional to the curvature of the buoyancy profile. Notice that the patch rises when the curvature is negative and falls when the curvature is positive. In either case, heat is transported downward. The flux of buoyancy is 4

-

Kb&

(5.64)

7

where K , is the coefficient of eddy diffusion and is given by the product of the mixing length 4 ,the vertical velocity of the patch wp 16, l l i 2 & , and the ratio y of the volume of the turbulent patches to the whole volume, i.e., the intermittency factor (Hinze, 1959). Thus

-

4

- rezI;/ I 6p2.

(5.65)

This formula plays a considerable role in a theory of turbulence in stratified

72

Robert R. Long

fluids by the author (Long, 1970). For example, we may use the theory to indicate that continuous density gradients tend to be unstable. Thus, in

FIG. 18. Instability of continuous density gradients.

Fig. 18, suppose a profile (solid curve) develops an inflection point. Then the buoyancy flux distribution is as shown so that the mean equation ab/at =

aq/az

leads to the development of a profile of the form of the dashed line. Thus, a discontinuity tends to form and the process continues until a sharp interface develops and some other process, perhaps Kelvin-Helmholtz instability, arises to transfer the heat. VI. Third-Order Closure Schemes in Turbulence Research As discussed in Section 11, the central problem in turbulence is to devise of obtaining a closed set of equations for the mean quantities, for some way-~ example uior u: u> . The problem may be circumvented by direct integration of the Navier-Stokes equations on a high-speed computer-also called numerical simulations of turbulent j o w s . This has been done for threedimensional homogeneous, isotropic turbulence by Orszag and Patterson (1972) and for turbulent shear flow by Orszag and Pao (1974). To save computation, the assumption is often made that the turbulence is twodimensional (Herring et al., 1974). The numerical simulation technique holds promise, at least for computers with the higher efficiencies likely in the future. The maximum Reynolds number that can be simulated is far below those of the atmosphere, for example, although present results tend to indicate that phenomena of large scale differ little with an increase of Reynolds number above the “moderate” range (Orszag and Pao, 1974).

Turbulence in Geophysical Systems

73

Another technique is numerical simulation with subgrid scale modeling (Deardorff, 1974). Here the grid scale is made as small as possible, bearing in mind the economics of the computations, and the motions on scales below the grid scale are modeled in a convenient way, recognizing that the choice of the modeling assumptions is not particularly important since the precise nature of the dissipation does not bear strongly on the nature of the largescale motions. Again the technique is expensive, prohibitively so perhaps for complicated geophysical phenomena (Lumley and Khajeh-Nouri, 1974). Finally, a number of authors have used the equations for second moments, modeling the unknown third-order terms (Daly and Harlow, 1970; Daly, 1972; Donaldson, 1972a,b; Donaldson et al., 1972; Jones and Launder, 1972; Ng and Spalding, 1972; Mellor, 1973; Naot et al., 1973; Launder, 1975; Launder et al., 1975; Reynolds, 1976). This is a difficult task and the modeling is often done on an ad hoc basis with little or no justification for the assumptions made. The technique permits a large number of constants to enter the analysis. Some authors argue that these constants are universal and need only be calculated once, but the arguments are not convincing and the “constants ” appear likely, in fact, to be unknown functions of the Richardson number, the Rossby number, etc. Nevertheless, some of the calculations are in reasonable agreement with observation both in nature and in the laboratory, and we will present a discussion of one of these investigations in some detail. A. MEANREYNOLDS STRESSMODELOF THE SURFACE LAYER OF THE ATMOSPHERE We cannot present an adequate discussion of the many second-order closure models. Instead we will consider one approach which represents perhaps the simplest possible one for a case in which density variations are fundamentally important (Mellor, 1973). This investigation relates to the surface layer of the atmosphere, i.e., the lowest layer in which rotation may be neglected and the fluxes of buoyancy and momentum may be considered constant. The mean equations are very simple:

a - -+ vuz), az a 0 = - (- b’w’ + k, bz), az

0 = - (- u’w’

a -(w‘w’)= - ap - - b. dZ aZ -

Robert R. Long

74

The equations for the fluctuating components of velocity and buoyancy are aui

-

ax

ati

ay

aZ

auf aP‘ + (u, + ii-au‘ ax + u’- ay + w’-az + w’tr, = - ax + VV2U‘, aul

-

at

awl + av’ - + - = 0, -

(6.5)

+ (u, + a)-auf + u’-ati + w’-au’ = - + VV%‘, (6.6) ax ay aZ ay awl , aw’ a __ aP‘ + v’+ w’- - -(w’w’) = - b + VVW‘, + (u + a)at ax ay az aZ aZ a01

apt

-

-

at

awl

awl

-

-

(6.7)

aw -+ (u’+~)-+v’-+ at ax ay ati

ati

ab w’-+

w’b;=k,V2b’.

aZ

(6.81

We now take the various moments of Eqs. (6.5)-(6.8) and average, assuming a steady state and that all mean quantities are independent of x and y. We obtain

”I

I-

aut

W’U‘U’- v (u’u’) = 2p’ aZ aZ ax - ~u’w’u,- ~ v ( V U ’ ) ~ , a -

-

’1aZa IaZ

-

-

a-l

r

I

aZ

1

aZa (w’w’)

~

~

= 2p’

a aZ

w’u’v‘ - v - (u’u’)

”1

-

I

aZ

-

(ax

auf +-

a

aZ

I-

aui

~

a aZ

aY)

- 2vVu‘

= - - (u’p’)+ p‘

. VV‘,

(6.11) (6.12)

(E+ E) -

-

~~

(w’v’)

a-

= - - (u‘p’) + p‘ aZ

- 2VVV‘ -

(6.10)

-

- u’b’ - w’w‘u, - 2vVu’ . VW‘,

”]

-

I

aul

= p’

(6.9)

a aZ - 2 aZ (w’p’)

awl ~

- 2V(VW’)Z - 2w”,

a -w’u’w‘ - v

-

aY

aZ

aZa a aZ w’u’w’ - y aZ (u’w’)

-

avl __ = 2p’ - - 2V(VU’)Z,

w‘v‘u‘ - v - (v’v’)

w‘w’w’ - v

~

~

-

(6.13)

(TZ) -

+-

. VW’ - u”,

(6.14)

~

w‘u‘b’ - v b - - kh U’

aZ

=

+ p ’ a- b - (V + kh)Vu’ . V b ,

ax

(6.15)

Turbulence in Geophysical Systems

75

(6.16)

I

a -w’w‘b‘- vb’

-

a2

awl

--

aZ

khw‘ __

aab = - - (p’b’)+ p ’ - ( V + kh)Vw’ . Vb‘, az az =

- 2khVb ‘ V b .

(6.17) (6.18)

These are difficult and complicated equations and one may take the viewpoint that little has been accomplished in going from (6.1)-(6.8) to (6.9)(6.18).Indeed it is certain that information is lost in the process. One may, on the other hand, adopt the philosophy that the crudity of the approximations which must now be made in modeling such terms as 2p’ du‘/ax, for example, may be a less serious matter than similarly crude approximations for the nature of the Reynolds stress terms in the mean equations by employing, say, the eddy viscosity concept. This is a matter of faith and it must be realized that the successes scored in this approach may reflect the fact that many adjustable constants enter the analysis below and that these may permit agreement of theory and observation. With these reservations we may, however, present the arguments of Mellor (1973). One of the very important terms appearing in Eqs. (6.9)-(6.14)has the form au; au; p ’ - + - . ax)

(axj

(6.19)

Treatment of this pressure-strain correlation dates back to a suggestion by __ that this term may be related to the sought-for Reynolds Rotta (1951) stresses ui u; and derivatives of the mean velocity by the tensor relation (6.20)

A further assumption is that the coefficients are isotopic fourth-order tensors, i.e., Cijkm = c 1 d i j d k m C28ikdjm + c 3 d j k d i m (6.21) with a similar expression for Cijkm.If we contract (6.20), i.e., let i = j and sum, we get, from continuity,

+

(3c1 -k so that C , = - ( C 2

_ _ _ _ _ _ U ; -k U;U; U;U;) = 0

c, + c , ) ( U ;

+ C3)/3.

+

(6.22)

Robert R . Long

76

We obtain = Cl[bijd2 -

an.

aa.

3Ur.: ]+ C2-axj + C’3 2 J axi’

(6.23)

where c‘ is the magnitude of the turbulent velocity. Notice, however, that Eq. (6.20) is a symmetric tensor so that aii.

0 = (c;- C 3 ) >

axj

so that C ;

= C 3 . We

+ (c;- c;)-afij axi

(6.24)

write the results as

au: aUj p‘ 2+- -(axj a x )

aij 3

u!u’.--c

,j+

KC” (aai -+ aaj) , axj

axi

(6.25)

where K is a constant, and Il is an undetermined length. As we have seen in Section 11 and as is obvious in Eq. (6.25), the contribution of this term to the energy equation is zero so that the pressure serves to redistribute energy from the UIz component (which is contributed to directly by the shear production term) to the other two components. It serves therefore to work toward a return to isotropy. In Eq. (6.9),for example, the first term on the right-hand side of (6.25) is

- (c’/3/,)($u.2

-

+p- $p),

(6.26)

and since will generally exceed the other two components (at least this is true in the neutral case), this is negative and the tendency is for to decrease. In (6. lo), the corresponding contribution is (6.27)

z).

This will tend to be positive and cause to increase (at the expense of These are quite reasonable behaviors. __ one In the second term on the right-hand side of (6.25), there is only contribution, namely, in Eq. (6.13). If we combine it with the term - w’w‘uz, we get _(Kc’2 - w’2)uz.

(6.28)

We cannot really judge whether this has a reasonable form. In the case of a homogeneous fluid, K is apparently quite small compared to one and this contribution is not very important. Then the effect of the quantity in (6.28) is to cause u)wI to decrease, i.e., to cause T to increase. In Eq. (6.12), Eq. (6.25) yields - (cr/31,)(ko’) and this is zero because there is no reason for any correlation between u’ and u’. Thus, there is no contribu-

Turbulence in Geophysical Systems

77

tion toward an increase o f u ” which is certainly correct. Indeed, we may reasonably presume that all terms in Eq. (6.12)are zero. The same comments apply to (6.14). The pressure terms in (6.15)-(6.17) may be handled similarly. We assume ~

p’(db’/dxJ = ciju;b‘

(6.29)

and assuming that C i j is isotropic, we get C i j = const d i j . The result may be written p’(db’/dXi)= - (Cf/312)u/bl,

(6.30)

where 1, is another length. The dissipation terms in (6.9)-(6.14) may be approached by recognizing that the contribution to these is negligible except in very small eddies. The hypothesis of Kolmogoroff (1941) was that in the energy cascade from large to small eddies, the natural tendency to isotropy would cause these small eddies to be accurately isotropic. Then ~~~

~ ( V U ’ )= ’ v ( V O ’ )= ~ ~ ( V W ’ )= ’ 43

= d3/3A1,

(6.31 )

where A, is another length. Since we expect no correlations, the dissipation terms in (6.12)-(6.14) are zero. The dissipation terms in (6.15)-(6.17) involve correlations between the velocity gradients and buoyancy gradients in the various directions and, since these eddies are isotropic, these correlations must be zero. In (6.18), however, the dissipation term is not zero. It will depend on the properties of the large eddies. With use of the notation of Ellison (1957), energy dissipation may be taken to be E

cc ct2/2T2,

(6.32)

where 2T2 is a dissipation time taken to be 2T2 cc A, /c‘. In (6.18) we may write k , , m cc b’2/2T1

or

k , ( V b ) 2 = c’b’2/A2 ,

2T,

= A2/c’,

(6.33)

where A 2 is another length. The other terms in the moment equations to be modeled are troublesome. Mellor assumed, for example, (6.34) ~

where A, is a length, and then assumed that ufu; are all constants. Altogether

Robert R. Long

78

Mellor ignores all terms in (6.9)-(6.18) involving a/az. This is hardly defensible, but is certainly the simplest situation and we will assume this also. The modeling is now complete, and we may write our results as follows: (c'/31,)(u" - +ct2)+ 2u)wIuz + + C ' ~ / A=~0,

(6.35)

+ 4cf3/A1= 0, (c'/3l1)(w" - 4~")+ $C"/A~ + 2 w " = 0,

(6.36) (6.37)

UIV' = 0,

(6.38)

(c'/311)(v" - ic") __

(C'/311)u"

+ (w"- K C 1 2 ) U Z + u" = 0,

(6.39)

~

wlu' = 0,

+

(6.40)

- _ _

(C'/312)u'b' w'bu,

+ u'w'b,

= 0,

(6.41)

-

v'b = 0,

(6.42)

-~

+ w'w'bz + (c'/3l2)w'b' = 0, 2(d/A,)b'b' + 2w"b, = 0.

b b

(6.43) (6.44)

We now make the following definitions: __

- w'b' = q, C = C'/U,,

C = z/L,

__

- u'w' = u2 * (Pm =

,

(u'2,

KZa,/U,,

P = (U:/q')b'Z,

_

_

d 2 , w'2) = ,'.(:it (Ph

=

,'v

(KZU*/q)6=,

ub = m/q.

w')

L = -U:/qK (6.45)

Mellor assumed that all the lengths were proportional to z, as is true for example in the neutral case (logarithmic layer). This may well be true in the unstable case as well but there is strong reason from the discussion in Section V to believe that the length scale reaches a maximum in the middle of the mixed layer in the stable case, and then decreases with further height, perhaps to zero again at the upper inversion. We may put aside this difficulty temporarily by defining

I , = KA, ZS

A, = K B zS, ~

(6.46)

l2 = KA,zS A2 = KB'zS, where S is an arbitrary function of z. We obtain

(6.47)

+ 3c3/B, + (c/3A,)[u2 - ~ c ' J= 0, ++c3/B1 + ( C / ~ A , ) [ V~ 421 = 0, 2CS + $c3/B, + (c/3A1)[w2- ic'] = 0, -C/3A1 + S(P,,,(W~ - KC') - [Sub = 0,

-29,s

(6.48) (6.49) (6.50) (6.51)

79

Turbulence in Geophysical Systems

If we add Eqs. (6.48)-(6.50), we obtain c3 = B,(rp, - 5)s.

(6.55)

Equation (6.54) may be written -

(6.56)

b2 = B 2 q h S / c .

Eliminating w2 in (6.53) by using (6.50) and then using (6.55) and (6.56), we get

(6.57) Eliminating obtain

ub in (6.51) by using (6.52) and using (6.50) to eliminate w 2 ,we

where y = 3 - 2 A , / B l . Equations (6.57) and (6.58) may be used to compute rp, and (Ph as functions of 5 once choices have been made for the behavior of S and values of the constants. Mellor chooses S = 1, thus assuming length scales proportional to z. He assumes values of the constants appropriate to laboratory measurements in the neutral case:

B , = 15.0,

B 2 = 8.0,

A,

= 0.78,

A,

= 0.79,

K

= 0.056.

(6.59) The agreement of rpm and rph with the atmospheric data of Businger el al. (1971) is quite good (Mellor, 1973, Figs. 1 and 2 ) . 1. Similarity Theory for the Unstable Case

The assumption that the lengths are proportional to z, i.e., S = 1, is probably valid in the unstable case since the convection can probably only act to increase the length scale and it is difficult to imagine it increasing with height faster than z. We have three possibilities as 1 [ + co : (a) qmdecreases or increases slower than i. (b) qm+ const <. (c) rp, increases faster than [. We find that (c) is impossible if we examine the asymptotic behavior of

<

80

Robert R. Long

(6.55), (6.57), and (6.58). In (a) we find the results,

This, of course, is entirely in accordance with the similarity theory for the unstable case. Using (6.59), we obtain

1 p3,

c + 2.47 [

(Ph

0.159 I 11-

(Pm -+

1/3,

0.230 I [ 1-

‘I3.

(6.61)

2. Asymptotic Theory for the Stable Case There is another possible solution of (6.57) and (6.58) for S = 1, I[ 1 + CG, corresponding to the possibility mentioned under (b) in the above section. It is

+ (6Al + W B l Y? c3 1(6Al + B2)/Yt q h / l ( B l / 3 A Z ) [ 1 + ( 6 A l + B 2 ) / y B l ] [ ( y - K)(6Al + BZ)/yB1 - ( 6 4 + 3A2)/B11. (6.62) v,/i

+

1

-+

With Mellor’s values for the constants, we have

4.681,

(P,,

c + 3.8111’3,

q h+

5.071.

(6.63)

This result is obviously appropriate to the stable case if we suppose that S = 1. The solution indicates turbulence for arbitrarily large 1, contrary to our findings in Section V that [ has a maximum value of order one a t the top of the mixed layer. We regard the assumption of S = 1 inappropriate although adopted in the paper by Mellor.

B. ELLISON’S DERIVATION OF A CRITICAL FLUX RICHARDSONNUMBER As we mentioned in Section V, Ellison (1957)gave an interesting derivation of a critical flux Richardson number for the stable case by using three equations which may be identijied as identical to three equations of Mellor. They are Eq. (6.44),

the sum of Eqs. (6.35)-(6.37), i.e., the energy equation -7ii,

-q

+ ( ~ “ / 2 T z =) 0,

T2 = A1/2c’

(6.65)

Turbulence in Geophysical Systems

81

and Eq. (6.43)

wlZbz + b'2 - (q/2T3) = 0,

T3 = 312/2c',

(6.66)

where we have introduced certain characteristic times used by Ellison. We now introduce the flux Richardson number Rf = - q/Tiiz .

(6.67)

Then Eq. (6.65) may be written q = C" Rf/2T2(Rf - 1).

(6.68)

Equations (6.64) and (6.66) yield q/2 T3 - wlZbz - 2 TI qb, = 0.

(6.69)

If we now introduce the eddy coefficients of diffusion and viscosity, K, and Krn 9

4 = Kbb,,

?

= K,&,

(6.70)

then R f = - (&,/K,,,)(bZ/D:),

6,

=

- (q2/T2)/(Kb/K,) Rf.

(6.71)

Equation (6.69) becomes (6.72)

If we use (6.68) to eliminate q in (6.72), we obtain 1 -+ 2T3

c12fl

2T2(Kb/K,)(Rf - I)?'

+ 4T:(1

2T1 d4 Rf = 0 (6.73) - Rf)2T2(Kb/&,)

or

We may write this as (6.75)

where (6.76)

Since a, > 0, if Rf increases faster than Rf,, K, tends toward zero. It becomes

Robert R. Long

82

zero at Rf = Rf, and then, obviously turbulence no longer exists. We may then regard Rf, as the upper limit of Rf. In computing Rf, ,Ellison assumed TI = T, and took the value c f 2 / pz 5.5 from neutral data. This gives Rf, z 0.15. We have already discussed that Rf seems to have an upper limit in the laboratory and in nature although there is considerable controversy about the value of Rf,. We may subject Ellison’s result to further scrutiny by using the rest of Mellor’s equations. Since Pmli = 1IRf

(6.77)

manipulation of the solution in (6.55)-(6.58) yields i S = Rf(1 - Rf)”’

(1

Rf),/‘/( -Rf2

1 - _Rf)3’4( _ 1 - -Rf)l’“ , (6.78) Rf3 Rf l

where

(6.79)

and where we have used 3B:’3A1(y - K ) = 1.

(6.80)

This follows from (6.58) if we notice that S = 1 and qm= 1 at = 0 since we have the logarithmic layer for small [ with length scale proportional to z. Notice that Rf, < Rf, so that as R f - + Rf,, the length scale increases to infinity. Mellor (1973) and Yamada (1975) have adopted Rf, as the critical flux Richardson number although one can really only say that this is an upper bound for Rf. Mellor’s data yield Rf, = 0.213. We may relate these results to that of Ellison by finding w2/c2 from Eqs. (6.48)-(6.54). We get w’/c’ = y - [6A, Rf/B1(1 - Rf)]

(6.81)

so that from Eq. (6.76),

Rf,

=

B,y(l - Rf) - 6A1 Rf (B2 + B1 y)(l - Rf) - 6A1 Rf

(6.82)

Ellison evaluated Rf, at Rf = 0, i.e., in neutral conditions, so that he obtained, in effect, Rf4 = Bl Yl(B2 + Bl Y).

(6.83)

Turbulence in Geophysical Systems

83

Since Rf, > Rf, , Eq. (6.83) is an overestimate of the critical flux Richardson number. Clearly from the viewpoint of Mellor’s theory, Rf, in (6.82) should be evaluated when Rf = Rf, . We then find Rf, + Rf, so that the two theories are not in contradiction. Arya (1972) was the first to emphasize that c2/wz in Ellison’s theory of Rf, should be considered a function of the stability. C. THEORY OF A MIXEDLAYER OF FINITE DEPTH

As we have seen, the flux Richardson number Rf, is reached when the turbulence length scale becomes infinitely large. This presumably can only occur when z is infinity, so that if Rf, is taken to be the critical flux Richardson number, the theory of Mellor does not predict a mixed layer of finite depth as observed. We may examine this by considering the concept of available potential energy introduced in Section 111. We neglected the disturbance pressure gradient in the vertical equation of motion and obtained

w”

- t26, = const.

Using b‘ z - 6,5, we may eliminate wJ2 -

c2 to obtain

(b”/6,) = const = w; ,

where w o is the vertical velocity when the parcel is at the level at which its density is equal to that of the environment. This suggests the importance of the ratio

R = oz/ai Ib,l.

(6.84)

If R exceeds 1, there will be a tendency for the eddies to die out and turbulence to cease. Using Mellor’s theory, we find

+

R = B2 Rf/[yB, - (YB, 6x41) Rf],

(6.85)

and it is remarkable that R precisely equals 1 when Rf = Rf, . On the other hand, our argument neglects the disturbance pressure gradient. This will have the tendency to cause energy to flow from the component of turbulence kinetic energy to the component. This associates eddy motion with values of R less than one. We conclude that Rf < Rf, and therefore that the layer has a finite thickness. When the mixed layer has a finite depth, it may be reasonable to use a simple expression for S which yields a behavior of the length scale similar to that in turbulent plane Couette flow. The simplest choice is

s=

- (C/CD)?

(6.86)

where C D = D/Land D is the depth of the layer. Elimination of q,, between

Robert R. Long

84

(6.57)and (6.58)yields

where

(

Rf)3'4( 4 I , ) ; : 1- _ 1Rf3

a = Rf(1 - Rf)'I2 1 - Rf2

~

. (6.88)

Clearly the maximum flux Richardson number is attained in the middle of the layer at CD = 4a.The value of C D is uncertain but the data of Businger et al. (1971) and Wyngaard and Cote (1971) reveal very few observations above of approximately 1. If we take CD = 1, then a = 4 and the corresponding maximum Rf is 0.135. We may tentatively take this to be the critical flux Richardson number. Finally, with some manipulation, the correlation coefficient __

cwb= w'b'/a,ob

(6.89)

is given by

(6.90) The value corresponding to Rf = 0.135 is Cwbz 0.40.This is perhaps reasonable although somewhat greater than measured values by Arya and Plate (1969).

Launder (1975) has developed a theory with a set of assumptions quite different from those of Mellor. Nevertheless the results are identical except for different values of the constants A,, A,, B1 ...,and the appearance of several new constants. His value for Rf, is 50% larger than that of Mellor. ACKNOWLEDGMENTS The support of the Office of Naval Research under Contract N00014-75-C-0805 and the National Science Foundation under Grant DES 74-23500-A01 is gratefully acknowledged. REFERENCES ARYA,S. P. S . (1972). The condition for the maintenance of turbulence in stratified flows.Q. J. R. Meteor. SOC.98,264-273. ARYA,S . P. S., and PLATE, E. J. (1969). Modeling of the stably stratified atmospheric boundary layer. J. Atmos. Sci. 26, 656-665. BAINES,W. D. (1975). Entrainment by a plume or jet at a density interface. J. Fluid Mech. 68,

309-320.

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BECHOV,R., and YAGLOM,A. M. (1971). Comments on the theory of similarity as applied to turbulence in an unstably stratified fluid. Izo. Atmos. Ocean. Phys. 7, 829 (Engl. ed.). B~NARD, H. (1901). Les tourbillons cellularies dans une nappe liquide transportant de la chaleur par convections en regime permanent. Ann. Chim. Phys. [7] 23, 62-144. G. (1950). “ Hydrodynamics.” Princeton Univ. Press, Princeton, New Jersey. BIRKHOFF, BOUSSINESQ,J. (1903). “Theorie analytique de la chaleur,” Vol. 2, p. 9. Gauthier-Villars, Paris. BOUVARD,M., and DUMAS,H.(1967). Application de la methode de fil chaud a la mesure de la turbulence dans I’eau. Houille Blanche 22, 257 and 723. BRIDGMAN, P. W. (1931). “Dimensional Analysis.” Yale Univ. Press, New Haven, Connecticut. BRUSH,L. M., Jr. (1970). “Artificial Mixing of Stratified Fluids Formed by Salt and Heat in a Laboratory Reservoir,” R e . Proj. B-024. New Jersey Water Resources Research Institute. K. A., and WATKINS,C. D. (1970). Observations of clear air turbulence by high BROWNING, power radar. Nature (London) 227,26&263. BUSINGER, J. A,, and ARYA,S. P. S. (1974). Height of the mixed layer in a stably stratified planetary boundary layer. Ado. Geophys. 18A, 73-92 1. BUSINGER, J. A., WYNGAARD, J. C., IZUMI,Y., and BRADLEY, E. F. (1971). Flux profile relationships in the atmospheric surface layer. J. Atmos. Sci. 28, 181-189. R. J. (1973).Turbulent convection in a horizontal layer of water. J. CHU,T. Y., and GOLDSTEIN, Fluid Mech. 60,141-159. CORRSIN, S. (1974). Limitations of gradient transport models in random walks and in turbulence. Ado. Geophys. 18A, 25-60. CRAPPER, P. F. (1973). An experimental study of mixing across density interfaces. Ph.D. Dissertation, University of Cambridge. P. F., and LINDEN,P. F. (1974). The structure of turbulent density interfaces. J. Fluid CRAPPER, Mech. 65, 45-63. C R O ~J.; F. (1958). The convective regime and temperature distributions above a horizontal heated surface. Q. J. R . Meteor. SOC.84, 418. CROMWELL, T. (1960). Pycnoclines created by mixing in an aquarium tank. J. Mar. Res. 18, 73-82. CSANADY, G. T. (1967). On the resistance law of a turbulent Ekman layer. J. Atmos. Sci. 24, 467-471. CSANADY, G . T. (1972). Frictional currents in the mixed layer at the sea surface. J. Phys. Oceanogr. 2, 498-508. DALY, B. J. (1972). “A Numerical Study of Turbulence Transitions in Convective Flow,” Lab. Rep. LA-DC-72-81. University of California, Los Alamos. DALY,B. J., and HARLOW,F. H.(1970). Transport equations in turbulence. Phys. Fluids 13, 2634-2649. DEARDORFF, J. W. (1974). The use of sub-grid transport equations in a three-dimensional model of atmospheric turbulence. Trans. ASME, Ser. I Pap. No. 73-FE21. DEARDORFF, J. W., and WILLIS,G. F. (1967). Investigation of turbulent thermal convection between horizontal plates. J . Fluid Mech. 28, 675. DONALDSON, C. DuP. (1972a). Calculation of turbulent shear flows for atmospheric and vortex motions. A I A A J. 10, 4-12. DONALDSON,C. DuP. (1972b). “Construction of a Dynamical Model of the Production of Atmospheric Turbulence and the Dispersal of Atmospheric Pollutants.” Am. Meteorol. Soc.Workshop Micrometeorol., Boston, Massachusetts. DONALDSON, C. DuP., SULLIVAN, R. D., and ROSENBAUM, H. (1972). A theoretical study of the generation of atmospheric clear air turbulence. A I A A J 10, 162. DYER, A. J. (1965). The flux-gradient relations near the ground in unstable conditions. Q. J. R. Meteor. SOC.91. 151-157.

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DYER,A. J., and HICKS, D. B. (1970). Flux-gradient relationships in the constant flux layer. Q.J. R. Meteor. SOC. %, 715-721. EKMAN, V. W. (1905). On the influence of the earths rotation on ocean currents. Ark. Mat., Astron. Fys. 2, No. 11. ELLISON, T. H. (1957). Turbulent transport of heat and momentum from an infinite rough plane. J. Fluid Mech. 2,456466. ELLISON, T. H., and TURNER, J. S. (1960a). Mixing of dense fluid in turbulent pipe flow. J. Fluid Mech. 6, 423-448. ELLISON, T. H., and TURNER, J. S. (1960b). Mixing of dense fluid in turbulent pipe flow. Part 2. Dependence of transfer coefficients on local stability. J. Fluid Mech. 8, 529-544. GARON,A. M., and GOLDSTEIN, R. J. (1973). Velocity and heat transfer measurements in thermal convection. Phys. Fluids 16, 1818-1825. GILLE,J. (1967). Interferometric measurement of temperature gradient reversal in a layer of convecting air. J. Fluid Mech. 30, 371-384. GLOBE,S., and DROPKIN, D. (1959). Natural convection heat transfers in liquids confined by two horizontal plates and heated from below. J. Heat Transfer 81, 24. GOLDSTEIN, R. J., and CHU,T. Y. (1969). Thermal convection in a horizontal layer of air. Prog. Heat Mass Transfer 2, 55. HAZEL,P. (1972). Numerical studies of the stability of inviscid stratified shear flows. J. Fluid Mech. 51, 39-61. HERRING, J. R., ORSZAG, S. A., KRAICHNAN, R. H., and Fox, D. C. (1974). Decay of twodimensional homogeneous turbulence. J. Fluid Mech. 66, 417-444. HINZE,J. 0. (1959). “Turbulence.” McGraw-Hill, New York. HOLTON, J. R. (1972). “An Introduction to Dynamic Meteorology.” Academic Press, New York. HOWARD,L. N. (1966). Convection at high Rayleigh number. Appl. Mech. Proc. Int. Congr., Ilth, 1964, pp. 1109-1115. IZAKSON, A. A. (1937). On the formula for velocity distributions near walls. Tech. Phys. USSR 4, 27-37. JACOBSEN, J. P. (1913). Beitrag zur Hydrographie der danischen Gewasser. Medd. Komm.Harunders. Ser. Hyd. 2. JEFFREYS, H. (1926). The stability of a layer of fluid heated from below. Philos. Mag. [7] 2, 833-844. JEFFREYS, H. (1928). Some cases of instability in fluid motion. Proc. R. Soc., London, Ser. A 118, 195-208. JONES,W. P., and LAUNDER, B. E. (1972). The prediction of laminarization with a two equation model of turbulence. Int. J. Heat Mass Transfer 5, 301-314. KANTHA, L. H. (1975). “Turbulent Entrainment at the Density Interface of a Two-Layer Stably Stratified Fluid System,” Tech. Rep. Department of Earth & Planetary Sciences, Johns Hopkins University, Baltimore, Maryland. 0. M. (1969). On the penetration of a turbulent layer into a stratified KATO,H., and PHILLIPS, fluid. J. Fluid Mech. 37, 643-655. KAZANSKI, A. B., and MONIN,A. S. (1961). On the dynamical interaction between the atmosphere and the earth’s surface. Izo. Akad. Nauk SSSR, Ser. Geofz. No. 5, pp. 786-788. KITAIGORODSKII, S. A. (1960). On the computation of the thickness of the wind-mixing layer in the ocean. Bull. Acad. Sci. USSR, Geophys. Ser. 3,248-287. KOLMOGOROFF, A. N. (1941). The local structure of turbulence in incompressible viscous fluid for very large Reynolds number. Dokl. Akad. Nauk SSSR 30, 301-305. KOSCHMIEDER, F. L. (1967). On convection under an air surface. J . Fluid Mech. 30, 9-15. KRAICHNAN, R. H. (1962). Turbulent thermal convection at arbitrary Prandtl numbers. Phys. Fluids 5, 1374-1389.

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KRAUS,E. B., and TURNER, J. S. (1967). A one-dimensional model of the seasonal thermocline. 11. The general theory and its consequences. Tellus 19, 98-106. KRISHNAMURTI, R. (1968). Finite amplitude convection with changing mean temperature. J . Fluid Mech. 33, 44-463. KRISHNAMURTI, R. (1970). On the transition to turbulent convection. J . Fluid Mech. 42, 295-320. KULLENBERG, G . (1971). Vertical diffusion in shallow waters. Tellus 23, 129-135. KULLENBERG, G. (1974). Investigation of small-scale vertical mixing in relation to temperature structure in stably stratified waters. Adc. Geophys. M A , 339-352. LAMB,Sir H. (1932). “ Hydrodynamics,” 6th ed., p. 373. Cambridge Univ. Press, London and New York. LAUFER, J. (1954). The structure of turbulence in fully developed pipe flow. Natl. Aduis. Comm. Aeronaut., Rep. 1174. LAUNDER, B. E. (1975). On the effects of a gravitational field on the turbulent transport ofheat and momentum. J . Fluid Mech. 67, 569-581. LAUNDER, B. E., REECE,G. J., and RODI,W. (1975). Progress in the development of a Reynoldsstress turbulence closure. J. Fluid Mech. 68, 537-566. LINDEN,P. F. (1973). The interaction of a vortex ring with a sharp density interface: A model for turbulent entrainment. J. Fluid Mech. 60, 467-480. LINDEN,P. F. (1975). The deepening of a mixed layer in a stratified fluid. J. Fluid Mech. 71, 385-405. LONG,R. R. (1961). “ Mechanics of Solids and Fluids.” Prentice-Hall, Englewood Cliffs, New Jersey. LONG,R. R. (1964). “Engineering Science Mechanics.” Prentice-Hall, Englewood Cliffs, New Jersey. LONG,R. R. (1965). On the Boussinesq approximation and its role in the theory of internal waves. Tellus 17, 46-52. LONG,R. R. (1970). A theory of turbulence in stratified fluids. J . Fluid Mech. 42, 349-365. LONG,R. R. (1972). Some aspects of turbulence in stratified fluids. Appl. Mech. Rev. 1297-1301. LONG, R. R. (1973). Some properties of horizontally homogeneous, statistically steady turbulence in a stratified fluid. Boundary-Layer Meteor. 5, 139-157. LONG,R. R. (1975). The influence of shear on mixing across density interfaces. J. Fluid Mech. 70, 305-320. LUMLEY, J. L. (1972). A model for computation of stratified turbulent flows. Proc. Int. Symp. Stratified Flows, 1972, Nocosibirsk, USSR, paper 14, 1-9. LUMLEY, J. L., and KHAJEH-NOURI, B. (1974). Computational modeling of turbulent transport. Adu. Geophys. MA, 169-192. W. V. R. (1954a). Discrete transitions in turbulent convection. Proc. R . SOC.London, MALKUS, Ser. A 225, 185. MALKUS, W. V. R. (1954b). The heat transport and spectrum of thermal turbulence. Proc. R . SOC.London, Ser. A 225, 196. MELLOR,G. L. (1973). Analytical prediction of the properties of stratified planetary surface layers. J. Atmos. Sci. 30, 106-1069. MILES,J. W. (1961). On the stability of heterogeneous shear flows. J. Fluid Mech. 10,496-508. MILLIKAN, C. B. (1939). A critical discussion of turbulent flow in channels and circular tubes. Appl. Mech., Proc. Int. Congr., 5th, 1937 pp. 386392. MONIN,A. S., and YAGLOM,A. M. (1971). “Statistical Fluid Mechanics: Mechanics of Turbulence,” Vol. 1. MIT Press,Cambridge, Massachusetts. MOORE,M. J., and LONG,R. R. (1971). An experimental investigation of turbulent stratified shear flow. J. Fluid Mech. 49, 635-655.

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NAOT,D., SHAVIT,A., and WOLFSHTEIN, M. (1973). Two-point correlation model and the redistribution of Reynolds stress. Phys. Fluids 16, 738. NG, K. H., and SPALDING, D. B. (1972). Turbulence model for boundary layers near walls. Phys. Fluids 15, 2Ck30. NIILER,P. P. (1975). Deepening of the wind-mixed layer. J. Mar. Res. 33,405-422. NIKURADSE, J. (1932). Gesetzmassigkeiten der turbulent Stromung in glatten Rohren. Forschungsarb. Ingenieurwes. No. 356. ORSZAG,S. A,, and PAO,Y. H. (1974). Numerical computation of turbulent shear flows. Adu. Geophys. H A , 225-236. ORSZAG,S. A., and PAITERSON,G. S., Jr. (1972). Numerical simulation of turbulence in statistical models and turbulence. Lecr. Notes Phys. 12, 127-147. PAGE,F., SCHLINGER, W. G., BREAUX, D. K., and SAGE,B. H. (1952). Point values of eddy conductivity ,and viscosity in uniform flow between parallel plates. Ind. Eng. Chem. 44, 424-430. PHILLIPS, 0. M. (1966). “The Dynamics of the Upper Ocean.” Cambridge Univ. Press, London and New York. PLATE,E. J. (1971). “Aerodynamic Characteristics of Atmospheric Boundary Layers.” U.S. At. Energy Comm., Washington, D.C. R. (1973). The deepening of the wind-mixed POLLARD, R. T., RHINES, P. B., and THOMPSON, layer. Geophys. Fluid Dyn. 4, 381-404. PRANDTL, L. (1932). Meteorologische Andwendungen der Stromungslehre. Beitr. Phys. Atmos. 19, 188-202. PRIESTLEY, C . H. B. (1954). Convection from a large horizontal surface. Aust. J. Phys. 7 , 176. PRIESTLEY, C. H. B. (1959). “Turbulent Transfer in the Lower Atmosphere.” Univ. of Chicago Press, Chicago, Illinois. PROUDMAN, J. (1953). “ Dynamical Oceanography.” Methuen, London. PRUITT, W. O., MORGAN,D. L., and LAURENCE, F. J. (1973). Momentum and mass transfers in the surface boundary layer. Q. J. R. Meteorol. SOC.99, 37Ck386. RAYLEIGH, Lord. (1916). On convection currents in a horizontal layer of fluid where the higher temperature is on the underside. Phil. Mag. [6] 32, 529-546. REYNOLDS, 0. (1895). On the dynamical theory of incompressible viscous fluids and the determination of the criterion. Phil. Trans. R. SOC.London, Ser. A 186, 123. REYNOLDS, W. C. (1976). Computation of turbulent flows. Annu. Rev. Fluid Mech. 8, 183-208. ROBERTSON, J. M. (1959). On turbulent plane-Couette flow. Proc. Annu. Con$ Fluid Mech., 6th pp. 169-182. ROSSBY,C. G., and MONTGOMERY, R. B. (1935).The layer of frictional influence. in wind and ocean currents. Pap. Phys. Oceanogr. 3, 1-101. ROSBY, H. T. (1969). A study of Benard convection with and without rotation. J . Fluid Mech. 36, 309. ROTTA,J. C. (1951). Statistische Theorie Nichthomogener Turbulenz. Z. Phys. 129, 547-572; 131, 51-77. ROUSE, H., and DODU,J. (1955). Turbulent diffusion across a density discontinuity. Houille Blanche 10, 405410. ROUSE,H., YIH, C. S., and HUMPHREYS, H. W. (1952). Gravitational convection from a boundary source. Tellus 4, 201-210. S. W. (1935). On the instability of a fluid when heated from SCHMIDT,R. J., and MILVERTON, below. Proc. R. Soc.London, Ser. A 152, 586-594. W. H., and COSART, W. A. (1961). The two-dimensional wall-jet. J. Fluid Mech. 10, SCHWARZ, 48 1-495. SCOTTI,R. S., and CORCOS,G. M. (1969). Measurements on the growth of small disturbances in

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a stratified shear layer. Radio Sci. 4, 1309-1313. SILVFSTON,P. L. (1958). Warmedurchgang in Waagrechten Flussigkeitschichten. Forschungsarb. lngenieurwes. 24, 29. SOMERSCALES, E. F. C., and DROPKIN, D. (1966). Experimental investigation of the temperature distribution in a horizontal layer of fluid heated from below. Heat Mass Transfer 9, 1-1 189. SOMERSCALES, E. F. C., and GAZDA, I. W. (1969). Thermal convection in high Prandtl number liquids at high Rayleigh numbers. Heat Mass Transfer 12, 1491. SPIECEL,E. A., and VERONIS,G. (1960). On the Boussinesq approximation for a compressible fluid. Astrophys. J. 131, 44-447. STERN,M. (1960). The salt fountain and thermohaline convection. Tellus 12, 172-175. STOMMEL, H., and FEDEROV, K. N. (1967). Small-scale structure in temperature and salinity near Timor and Mindinao. Tellus 19, 306-325. STOMMEL,H., ARONS,A. B., and BLANCHARD, D. (1956). An oceanographical curiosity: The perpetual salt fountain. Deep-sea Res. 3, 152-153. SUNDARAM, T. R. (1973). A theoretical model for the seasonal thermal cycle of deep temperate lakes. Proc. Conf. Great Lakes Res. 16, 1009-1025. SUNDARAM, T. R., and REHM,R. C. (1973). The seasonal thermal structure of deep temperate lakes. Tellus 25, 157-167. TENNEKES, H. (1968). Outline of a second order theory of turbulent pipe flow. A I A A J. 6, 1735-1155. TENNEKES, H., and LUMLEY,J. L. (1972). “A First Course in Turbulence.” MIT Press, Cambridge, Massachusetts. THOMAS, D. B., and TOWNSEND, A. A. (1957). Turbulent convection over a heated horizontal surface. J. Fluid Mech. 2, 473. THOMPSON,S. M.,and TURNER,J. S. (1975). Mixing across an interface generated by an oscillating grid. J. Fluid Mech. 67, 349-368. THORPE,S. A. (1973). Turbulence in stably stratified fluids: A review of laboratory experiments. Boundary-Layer Meteorol. 5, 95-1 19. THRELFALL, D. C. (1975). Free convection in low temperature gaseous helium. J. Fluid Mech. 67, 17-28. TOWNSEND, A. A. (1956). “The Structure of Turbulent Shear Flow,” 1st ed. Cambridge Univ. Press, London and New York. TOWNSEND, A. A. (1959). Temperature fluctuations over a heated horizontal surface. J. Fluid Mech. 5, 209. TOWNSEND, A. A. (1976). “The Structure of Turbulent Shear Flow,” 2nd ed. Cambridge Univ. Press, London and New York. TURNER, J. S. (1968). The influence of molecular diffusivity on turbulent entrainment across a density interface. J. Fluid Mech. 33, 639-656. TURNER, J. S. (1973). “ Buoyancy Effects in Fluids.” Cambridge Univ. Press, London and New York (see Chapter 9). VON KARMAN,T. (1930). Mechanische Ahnlichkeit und Turbulenz. Nachr. Ges. Wiss. Goettingen, Math.-Phys. K l . pp. 58-76. WOLANSKI, E. (1972). Turbulent entrainment across stable density-stratified liquids and suspensions. Ph.D. Dissertation, Johns Hopkins University, Baltimore, Maryland. WOLANSKI, E., and BRUSH,L. M., Jr. (1975). Turbulent entrainment across stable density step structures. Tellus 27, 259-268. WOODS,J. D. (1968). Wave-induced shear instability in the summer thermocline. J : Fluid Mech. 32,791-800. WOODS,J. D., and WILEY,R. L. (1972). Billow turbulence and ocean microstructure. Deep-sea Res. 19, 87-121.

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Singular-Perturbation Problems in Ship Hydrodynamicst T. FRANCIS OGILVIE Department of Naval Architecture and Marine Engineering The University of Michigan Ann Arbor. Michigan In Memory: Reinier Timman. 191 7-1975

I. Introduction . . . . . . . . . . . . I1. Slender-Body Theory in Aerodynamics . . 111. Slender Ships in Unsteady A. Problem Formulation B. Radiation Patterns . C. Forced Oscillations . D . DifTraction Problems

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. . . . . . . . . . . . . . . . IV. Slender Ships in Steady Forward Motion . . . . A. Introduction . . . . . . . . . . . . . B. Ordinary Slender-Ship Theory . . . . . . C. Failure of Ordinary Slender-Ship Theory . . D. The Bow-Flow Problem . . . . . . . . . E . The Low-Speed Problem . . . . . . . . . V. Slender Ships in Unsteady Forward Motion . . A . Introduction and Formulation . . . . . . B. Solution of the Heave/Pitch Forced-Oscillation C. High-speed Slender-Ship Theory . . . . . References . . . . . . . . . . . . . . . . .

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'This article was written while the author was Visiting Professor in the Department of Naval Architecture. Osaka University. Japan. and in the Mathematics Department. ManChester University. England . These visits were supported partially by grants from the Japan Society for the Promotion of Science and the North Atlantic Treaty Organization. respectively. 91

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I. Introhction Michell(l898) laid the foundation for modern ship hydrodynamics in his extraordinary paper on the wave resistance of a thin ship. Havelock (1923) discovered and extended Michell's work, and for 40years thereafter progress in ship hydrodynamics was almost synonymous with development of thinship theory. Havelock pointed out in his first paper on this subject that one may question the validity of thin-ship theory in practical applications since real ships are hardly thin. Nevertheless, study of the thin ship prospered, and the theory provided some guidance in the reduction of wave resistance, even if it did not give accurate quantitative predictions. The early and continuing success of thin-ship theory was due to two factors: (i) An explicit solution can be found for the case of a ship in steady straight-ahead motion. (ii) The first approximation is essentially a uniform approximation. Khaskind (1946) and Peters and Stoker (1957) also obtained an explicit solution for a heaving and pitching thin ship with forward motion. In all such problems, the ship is replaced by a centerplane distribution of sources, steady or pulsating, as appropriate. The body boundary condition is transferred to the centerplane, thus providing an expression for the normal velocity component on each side of the centerplane, from which an explicit expression for source density can be obtained. If the potential for each source in the distribution satisfies the free-surface condition and a radiation condition, the potential for the superposition of sources satisfies approximately all of the conditions of the problem. Then it is a straightforward matter to find the pressure and to integrate it appropriately over the hull to obtain the force on the ship. Alternatively, the momentum theorem or an energy-flux theorem can be used to obtain similar results from the solution at great distance from the ship. [The latter can even be used to obtain some of the desired quantities in the unsteady-motion problem, as shown by Newman (1959).] If the ship is not symmetrical, either because of its actual shape or because it is yawed, this procedure does not work. A distribution of transverse dipoles must also be placed on the centerplane in the corresponding mathematical problem, and an integral equation must be solved to determine the density of the dipoles. This is a two-dimensional (2-D) singular integral equation with a very complicated kernel, and only the advent of the modern high-speed digital computer made its solution possible. This was first done by Daoud (1973). Similarly, if one tries to satisfy the boundary condition exactly on the true hull surface instead of on the centerplane, a complicated integral equation

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must be solved, and there is still some controversy about the proper way to do this. Nevertheless, the thin-ship model (perhaps with modifications) still provides the best strictly analytical information available about the wave resistance of a ship, largely because an explicit solution is obtainable. Corresponding results for ship-motion problems have not been so useful, at least partly because a better mathematical model has evglved. (This other model will be discussed in this article, but we may note that it has not been successful in the steady-motion problem.) The thin-ship solution is the first term in a perturbation expansion of the corresponding exact problem. The small parameter may be considered as the ratio beam/length. As this parameter, say, B, approaches zero, the ship reduces to just its centerplane projection, and the hull boundary condition must be satisfied on the two sides of this surface. If, on the other hand, one formulates a slender-ship problem, in which both beam and draft approach zero with the small parameter, say, E, the hull shrinks down to a line, and the limit problem is not a well-posed problem in potential theory. Since the limit problem cannot even be posed properly, let alone solved, one cannot obtain a first approximation to the solution in this way. Thus, in spite of the fact that a ship appears to be more “slender than “thin,” the mathematician was forced until recently to consider it only as thin. The slender-ship problem is a singular-perturbation problem of a kind that aerodynamicists have learned to solve in the last several decades (see, for example, Ward, 1955). Thus it was to be expected that similar approaches should have been made to ship problems in recent years. However, when the aerodynamic techniques were applied in a straightforward way to ship problems, the results were generally disappointing, being sometimes trivial, sometimes nonsensical. Only in the last 10 years have these techniques been satisfactorilyadapted to the special conditions of ship problems so as to lead to reasonable and sometimes useful results. The treatment of these singular-perturbation problems is the subject of this article. The concept of the slender ship-and all that that implies for ship hydrodynamics-should not be considered as competition or a replacement for the thin-ship concept. Rather, it should be considered as a complement to thin-ship theory, offering some new understanding of the physical phenomena of ship hydrodynamics and perhaps eventually leading to some alternative methods of calculating quantities of practical interest. There is a growing body of investigators in ship hydrodynamics working on slender-ship theory. This article is not intended primarily for them, but rather for two other groups: (1) those who are still working largely with the older, more established concepts, who wish to learn quickly the principles, techniques, and possibilities of slender-ship theory, and (2) people who are ”

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T. Francis Ogilvie

familiar with singular-perturbation techniques in other areas, especially with slender-body theory of aerodynamics, who are interested in finding out what is being done with these techniques in ship hydrodynamics. The section on slender-body theory in aerodynamics is provided as a brief introduction for the first group only. There are some singular-perturbation problems of ship hydrodynamics that are neglected, largely because they did not fit in with the general theme as we have organized it. In this category, we mention especially the work by Rispin (1967) and Wu (1967) in which they show how to solve the 2-D planing problem by matching a locally nonlinear description (without gravity in the first approximation) to a far-field gravity-wave solution. In a similar way, Shen and Ogilvie (1972) developed a theory for a high-aspectratio planing surface in which the local picture is a 2-D nonlinear flow, just like the near-field solution of Rispin and of Wu, while the far-field solution is a lifting-line approximation. There has also been considerable interest in recent years in high-frequency problems and in low-speed, steady-motion problems, in both of which the waves characteristically have a very small wavelength, and so the wave motion is confined to a thin layer near the free surface. These problems are barely mentioned in Sections III,C,3 and III,D,5. From a scientific point of view, they are very interesting as singular-perturbation problems, since they are truly boundary-layer problems. However, some choices had to be made, and the theory for such problems is not discussed in any detail. Also, as implied in the mention of “potential theory,” we assume that the fluid is ideal. Thus, problems of a viscous fluid are not considered at all. Of course, classical boundary-layer theory is one of the best-known of all singular-perturbation problems in fluid mechanics, and an understanding of viscous boundary layers is of critical importance in ship hydrodynamics. The boundary layers on ships are, however, always turbulent, and so the mathematical theory of boundary layers is less useful in ship hydrodynamics than in some other branches of fluid mechanics. To have included such problems in this article would have required the introduction of a tremendous and largely independent subject, probably with marginal benefit, and so there will be no further discussion of nonideal fluids. In general, we have tried to emphasize the physical understanding of problems and, where possible, to indicate the lines of thought that have led to the various formulations and solutions. The research worker first gets an idea that something may possibly be true, then tries to find out whether it really is true, and, if he convinces himself, he then tries to work out a proof (or performs experiments) to convince others. It is unfortunate that only the last of these stages is usually reported, and we have taken the conscious risk of trying to report something from the first two stages as well.

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In this respect, much of the work discussed here (especially our own work) is based on the method of matched asymptotic expansions. We prefer to think about thesi problems along much the same lines as Van Dyke (1975). Now, Van Dyke’s book is perhaps unsatisfactory to mathematicians, but we believe that he provides an extremely valuable way of thinking about physical problems, a way that often suggests practical solutions. Thus his approach is a useful part of the first stage of scientific investigation as described above. By itself, of course, it proves nothing, and a lot of nonsense has been “derived” with the help of the method of matched asymptotic expansions. In providing references, we have tried to give those that might be of interest for further reading, rather than trying to establish priority of publication or presenting a comprehensive bibliography of the subject. To those who thereby feel neglected, we can only apologize and ask their understanding; we did not want to produce an annotated bibliography, which this article would surely have become had we tried to include all pertinent references. In matters of notation, we consistently follow the usage of Abramowitz and Stegun (1964). 11. Slender-Body Theory in Aerodynamics

The basic concept of the slender body in aerodynamics originated with Munk (1924). He was concerned with the flow around an airship. He noted that such a body has one long axis, parallel to which the fluid velocity component is generally much smaller than the components in the crossplanes. If the reference frame is fixed to the fluid at infinity, one may imagine that the forward motion of the slender body causes the fluid to part-to move sideways-without being greatly disturbed in the longitudinal direction. There is approximately a 2-D flow in each crossplane. Thus the 3-D problem is greatly simplified in being replaced by a set of 2-D problems, and furthermore the powerful methods for solving 2-D flows become available. Of course, such a viewpoint is only valid rather close to the body. Very far away, the flow field caused by any moving nonlifting body appears as if it might have been created by a moving dipole, and such a flow field is in direct contrast to the notion of a 2-D flow pattern. Thus, it must be recognized from the outset that the slender-body description of the flow field is not valid uniformly far away from the body. Furthermore, there generally are stagnation points at the nose and tail of the body, at which points the fluid velocity is approximately equal to the

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T . Francis Ogilvie

velocity of the body itself, i.e., the longitudinal fluid velocity component dominates the transverse components. At such points, the slender-body concept is completely wrong. Notwithstanding these two difficulties, it has been found that predictions of slender-body theory are fairly good in many important aerodynamic problems, and so the theory has been studied and further developed over a period of several decades. Starting in the 1940s especially, when many aerodynamic vehicles became effectively slender, basic understanding of slender-body theory improved, the accuracy of its predictions increased, and the scope of problems to which it could be applied widened greatly. The final stage in the development of aerodynamic slender-body formalism occurred in the 196Os, when the method of matched asymptotic expansions was applied to such problems. A good textbook treatment has been provided by Ashley and Landahl (1965). This method has not received universal acceptance by workers in the field, but its practical usefulness cannot be denied, and it will be used here. In particular, it provides some simplified viewpoints on slender-body problems and it also provides a formal procedure for relating those viewpoints mathematically. One may also comment that it gives the illusion of proving various conclusions, and so it can be dangerous when used improperly. In slender-body theory, we infer the existence of a small parameter E, which is a measure of the “slenderness” of the body under study. This parameter may be simply the ratio of a typical transverse dimension, say diameter, to the length, or it may be the maximum slope with respect to the longitudinal axis of planes tangent to the body surface. For developing the general theory, its precise definition does not matter. What does matter is that the entire theory is developed in an asymptotic sense as E + 0. That is, any conclusions will be more nearly valid as E becomes smaller and smaller. Whether such conclusions are valid for any finite value of E will always be a matter for further investigation in any particular problem. For convenience in discussing slender-body problems of all kinds, we shall generally consider that ,!J the body length, is O( 1) as E + 0, i.e., L is fixed and its value is not affected by the limit operation. Then, a transverse dimension can be described as being O(E)as E -+ 0. If any ambiguity is likely to arise from such usage, we can revert to the more usual statement, B/L = O(E), where B is the transverse dimension of interest. The basic concept of slender-body theory can be stated mathematically as follows: Iff(x, y, z) is any flow variable, then, in some region near the body,

where the x axis is chosen as the longitudinal axis of the body. This state-

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97

ment says simply that derivatives in the transverse directions are very large compared to derivatives in the longitudinal direction. The region in which these statements are assumed to be valid has transverse dimensions that are O(E).These statements are assumptions, and so they cannot be proven. However, they can be made to appear plausible, as is suggested in the following two typical situations: (1) In Fig. 1 there is depicted a section of a slender body which is moving at speed U in the longitudinal direction through fluid at rest. Let v = ( u l , u,, u 3 ) be the fluid velocity caused by the motion of this body. At

FIG. 1. A slender body in steady longitudinal motion.

points A and B, the u2 component has almost the same magnitude but opposite signs. Thus Au, ,defined as u2 IA - u2 IB, has a value comparable to twice u 2 J k The distance D between the two points is O(E),and so A u , / B = O(V,/E).It follows that the derivative of u2 along a path connecting points A and B must be O(U,/E), and thus that derivatives in the transverse planes are large. On the other hand, there is no such reason to expect that d v , / d x should be large. (2) In Fig. 2 there is depicted a cross section of a slender body moving with speed V in the transverse direction. The velocity component u2 at point A must equal V, but at point B v2 is very nearly equal to - V . Again, the is O(E),and so the same argument leads to the conclusion that distance

au,/ay = o ( ~ , / & ) .

Such arguments are not so clear in the case of some flow variables, but nevertheless the relationships indicated in (2.1) can generally be justified,

T. Francis Ogilvie

98

ty

FIG.2. A slender body moving in the transverse direction.

especially if one is willing to overlook the difference in asymptotic behavior between, say, O(E")and O(E"log E).' In order to use the assumption stated in (2.1), let us consider a slender body moving with velocity U = ( U l , U,, U 3 )(possibly a function of time). We assume that the fluid motion can be described in terms of a velocity potential, 4(x, y, z, t), which satisfies the Laplace equation in three dimensions,

The orders of magnitude of the terms in (2.2) have been noted. Although we have not yet determined the dependence of 4 on E, we can assert that the first term in (2.2) is of higher order than the other two terms, and so it can be neglected in the determination of the first approximation for 4. The potential also satisfies a kinematic body boundary condition, &$/an = n U, (2.3) or (2.3') '14.x + n Z 4 y + n 3 $ 2 = n l U l + n Z U , + n 3 u3

-

--

O(#JE)

7

v

O(4lE)

Y

O(E)

Of course, the asymptotic behavior of the two is not the same. Nevertheless, I shall treat them as if they were. This never seems to lead to trouble, and there are two very practical reasons for doing so: (i) A function such as log AE can always be broken into two terms, log A + log E, and, if A = O( 1) as E + 0,these two terms are O( 1) and O(log E ) , respectively. However, the value of A can be changed by making a new definition for E , and so it is clear that the two terms cannot be treated independently, even if they are formally of different orders of magnitude with respect to E . (ii) We derive asymptotic formulas with the intention of applying them for finite values of E, although the formulas are only valid asymptotically as E + 0. If we had to restrict the range of E so that the difference between log E and 1 were large, we would probably never obtain a useful formula.

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99

(2.4)

is a unit vector normal to the body surface, directed into the body. For a slender body, it is important to note that n, = O(E)

and

n , , n, = O(1)

as ~ 4 0 .

(2.4)

This fact has been used already in noting the orders of magnitude of the terms in (2.3’).Also, it has been assumed in (2.3’)that U , = O(1) and that U , , U , = O(E),which are useful choices (if arbitrary). These are the largest possible orders of magnitude for the U:s which still allow “small disturbances” to result. Finally, (2.1) has been used in estimating the left-hand side of (2.3’). It is evident that we can drop n , 4x from (2.3’),at least for the first approximation, and we then find that

4 = O(E2),

(2.5)

which is a typical result of slender-body theory. One further approximation, consistent with the above, is usually made. The left-hand side of (2.3) is a directional derivative of 4 taken along the normal vector n. On a slender body, this vector is almost perpendicular to the longitudinal axis. In finding the first approximation for 4, we can replace &$/an by a$/diV, where N is a unit vector, lying in a crossplane, perpendicular to the contour of the body in that crossplane. The relative error incurred in this replacement is O(E’). Thus the first approximation for 4 (strictly, the first term in an asymptotic expansion) satisfies the following conditions:

4yy+ 422 = 0 ; ab/aN = N 2 4 , ,+ N34z= n , U , + n2 U , + n3 U 3 .

(2.6) (2.7)

This is strictly a set of 2-D problems, as was anticipated. The solutions can be found analytically or numerically by standard methods. Apparently the solution at each crossplane is independent of the solution at all other crossplanes, except insofar as they are related by the body geometry, which enters into (2.7). But in fact this is not true, for the simple 2-D solutions in the crossplanes are not unique: At any x, the solution of (2.6) which satisfies (2.7) is arbitrary to the extent that any constant may be added without affecting the validity of the solution. This constant may be different at various crossplanes, and so we must add an arbitrary function of x to any solution determined from (2.6) and (2.7). We must somehow determine this

100

T. Francis Ogilvie

additive function of x, and it turns out to represent a further interaction among the various crossplanes. The additive function of x cannot be determined from the near-field analysis that we have been considering. We must construct a far-field description of the flow in order to determine the interaction among the crossplanes. Before doing this, we should note how the approximate solution z')'''. Since 4 of the above near-field problem behaves for large r = (y' must be harmonic everywhere outside the body contour, and its gradient must vanish at infinity, 4 can be represented as the real part of a complex potential, which can in turn be represented by a Laurent series supplemented by a logarithm term. The potential 4 itself then takes the form

+

4(x, y, z;

t ) = [a(x; t)/2n] log r

+c

+ Ao(x; t )

A,,(x;t ) cos n6 + B,,(x; t ) sin no 9

r"

n= 1

(2.8)

where 0 = arctan y/x. Since 4 = O(E') and (in the near field) r = O(E),we may conclude that ~7= O(E')

and

A,, B,, = O(P+').

(2.8')+

At very large r, the first two terms in (2.8) dominate the others, and so

4(x, y, z;

t)

- [~(x;t)/2nJ

log r

+ Ao(x; t )

as r + co.

(2.8")

Thus, very far away (but still in the near field), the flow caused by the body appears to have been caused by a 2-D source at the origin. We note that the strength of the source can be computed from (2.7),(2.8),and the equation of continuity, with the result that O(X; t ) = - U,[dS(x)/dx], (2.9) where S(x) is the cross-sectional area of the body. From the far-field point of view, as E --+ 0 the body shrinks down to a line, the segment of the x axis between x = 0 and x = L. It is this fact that makes slender-body theory a singular-perturbation problem even in the first approximation, for it is not a well-posed problem in 3-D potential theory to prescribe boundary conditions along a line. If we require that the potential function be harmonic everywhere other than on this line and that it be bounded at infinity as well, then 4 must be singular on the line segment. A convenient way to obtain a general form of the far-field solution is to restate the problem in terms of Fourier transforms with respect to x; the resulting 2-D problems can be solved in a general way and then transformed again to

' We are ignoring differences that are O(log E ) .

See the footnote on p. 98.

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Ship Hydrodynamics

give solutions of the form

\

W 1 " dkeikX Kn(I k I r)[a:(k; t ) cos no 4(x, y , z, t) = 2n.-, n=O

1

+ b:(k; t ) sin no], (2.10)

where the functions a:@; t) and b,*(k;t) are unknown functions that can themselves be interpreted as Fourier transforms.' Near the x axis, which includes the singular line, r is small, and the terms in (2.10) can be approximated by using the small-r series expressions for the Bessel functions 1 " dkeikxKo(I k I r)ag(k; t ) 271 .-"

\

-1

27t

5"

-m

-

1 28

5-" dkeikxag(k; m

- - log

r

t)

dkeikxag( k ; t ) log ~. Clkl 2 '

(2.1la)*

1 " dkeikxKn(I k I r)c,*(k; t) 'OS no 27t sin

[

-.-m

2"-'(n - I)!cos

t),

N

n > 0.

(2.11b)

In (2.1 la) we see that the first term on the right-hand side behaves like log r in a crossplane, and so it appears to represent a 2-D source of strength 6(x; t) = -2na,(x; t), where ao(x; t ) =

\"

27t.-,

(2.12a)

dkeikxa$(k;t ) .

The full 3-D expression on the left-hand side of (2.1la) can also be rewritten to show that it represents a distribution of sources in three dimensions

The second term on the right-hand side of (2.11a) is simply a function of x; using (2.12a), we can rewrite this function as follows

=--j1

"

d ( & ' ( ( ; t ) log 2 Ix - 4; I sgn(x - < ) = f ( x ; t ) , 47t -"

'

(2.13)

All special functions are denoted as by Abramowitz and Stegun (1964). Here, K. is the modified Bessel function of the second kind. log C = 7 is the Euler constant.

102

T. Francis Ogilvie

where the prime denotes differentiation with respect to 5. The left- and right-hand sides of (2.1 lb) can also be interpreted in terms of 3-D and 2-D distributions of multipoles. It is important to recognize that there is no simple function of just x left on the right-hand side of (2.11b). It is thus seen that the general far-field solution can be expressed in terms of distributions of sources, dipoles, etc., along the x axis. The strengths of these singularities are of course not known. From (2.12b), we might infer that the distribution extends along the entire x axis, but in fact it must be limited essentially to the line segment that represents the remnant of the body in the limit E + 0; we can simply set 6(x; t ) = 0 outside of this segment. A similar step follows also for the multipole distributions. Near the line of sources, the potential corresponding to just the sources takes the following approximate form (1/2n)6(x; t ) log r + f ( x ; t);

(2.14)

this is the right-hand side of (2.1 la) after we use (2.12a) and (2.13). A comparison with (2.8”) indicates that this is precisely the kind of behavior of the far-field potential that we would have liked to find. It follows directly that 6(x; t) = o(x; t),

the value of which is known from (2.9). The expression in (2.8”) represents the near-field potential in its limiting behavior for large r ; the expression in (2.14) comes from one term of the far-field potential, giving the limiting behavior of that term for small r. This is the kind of “matching” that forms the basis for the method of matched asymptotic expansions. It has been a well-known procedure ever since Prandtl first developed classical boundarylayer theory, although much of the formalism has been developed only recently. This matching does not appear to be perfect yet, for we have considered just one term in the series in (2.10). The subsequent terms are more and more singular near the x axis, as (2.11b) shows; it appears that the terms with n = 1 should dominate the n = 0 term if r is small enough, and the n = 2 terms should dominate the n = 1 terms, etc. If this were true, the matching would be destroyed. Physically it does not make sense either; if we start very close to the body, where the flow appears as if it might have been generated by sources, dipoles, quadrupoles, and so on, and then we move farther away, the sourcelike behavior decreases most slowly. Then, if we start very far away and approach the body, the sourcelike behavior should appear first. We conclude that in the far field the lowest-order term in the solution expansion should be sourcelike; other terms are of higher order with respect to E, that is, a: and b: are o(a$)as E .+0 for n > 0. Then the first term in the

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outer expansion is $(x, y , z, t)

-

-

1 -

2n

“

dkeik“K,( I k I r)a,*(k;t )

(2.10)

t) log r + f ( x ; t)

(2.14‘)

-“

1 --(x; 2n

as r + O .

Now the one-term outer expansion is completely known, since 6, and thus a,, is known. Also, the one-term inner expansion is known, including the additive function of x, for A, in (2.8”) must be the same asf(x; t), given by

(2.13). In the aerodynamic problem, one can proceed to obtain higher-order solutions, although “end effects” are likely to cast doubt on the validity of such extended expansions. In any case, in the free-Surface problems that are the focus of this article, it is difficult enough to obtain a first approximation, and so we are not likely to be concerned with anything beyond. Only a few comments on the interpretation of the above results need yet to be made. The additive function in the near-field 2-D solution, A , orf, has a simple interpretation. At any x it represents an axial flow that is needed to correct the 2-D result represented by the log r term in the approximate solution. To see this, one can start with the right-hand side of (2.12b),which is genuinely three-dimensional in nature. Integrate it once by parts with respect to 9 ; the result is 1 3(x; t) log r 2n -

1

--

4n

5

“

-m

d98’(9; t ) log{ Ix - 5 I

+ [(x - 9)2 + r2I1’*}sgn(x - 9 ) .

As r -P 0, the second term here approaches a well-defined limit, which in.fact is equal tof(x; t ) . If the problem really were two-dimensional, so that 6 were a constant, this extra term would vanish. But the problem is not twodimensional, and f ( x ; r ) represents the value on the axis of the potential which corrects for the nonzero longitudinal rate of change of 6. The 2-D solutions in the various crossplanes are related in two ways: (i) through the body geometry, which enters into the boundary condition on the body at every cross section, and (ii) through the additive “constant at each section,f(x; t ) . The latter depends only on the sourcelike behavior of the flow in the cross-sectional planes, and so it vanishes if there is no sourcelike behavior. There are two cases in which this happens. ”

(1) The body has no thickness. Although this cannot happen in reality, we often assume that a body has no thickness if we are interested primarily in

104

T. Francis Ogilvie

the effect of the body as a lifting surface. In this case, in the near-field solution as given in (2.8), the source term vanishes, and the lowest-order term in the solution expansion is a dipole term. In the far field, we must then start the solution with a line distribution of dipoles, and, as seen from (2.11b), there is no function of x alone when this singular potential is estimated near the line of singularities. So, A, in (2.8) vanishes. Slender-body theory for the lifting body of zero thickness reduces to the simplest kind of a strip theory, with no interaction among the cross-sectional flows. Even if we consider that a lifting body does have thickness, the interactions among sections depends only on the thickness effects. (2) There is noforward motion. In the body boundary condition (2.7), let U , = 0. Again, the source strength is zero. This is indicated by (2.9), but it could also have been deduced from the fact that the 2-D problems represent a translating body, which has no sourcelike behavior. As in the case of the lifting body of zero thickness, all source terms vanish and the interaction term A, (or f) vanishes with them. The slender-body theory for vertical or lateral oscillations of a body at zero speed is thus a primitive strip theory. This reduced capability of slender-body theory to represent interactions among cross sections leads to a special difficulty in the case of slender lifting surfaces: It is generally impossible to satisfy a Kutta condition. A cross section forward of the trailing edge is unaffected (in the first approximation) by what happens at the trailing edge, and so in the theory there can be no adjustment of the pressure and velocity fields ahead of the trailing edge to ensure a smooth flow from the edge. This difficulty has been rectified in recent years by the use of an acceleration potential. See especially Newman and Wu (1973), who treat the problem of unsteady motion of a slender lifting body with thickness by such a method. Also, Rogallo (1969) has shown the nature of the failure of liftingsurface theory near the trailing edge of a slender wing. More generally, errors at the body ends cause the worst problems in aerodynamic slender-body theory. The basic assumptions are violated in such regions, even without difficulties in satisfying a Kutta condition. Similarly in applications to ship hydrodynamics, the theory is most limited in describing the fluid motion near the ends. The nature of the end effects is quite different in the two fields, however, largely because of the presence of the free surface in ship problems, and little benefit would be gained from a discussion here of problems that are specifically aerodynamic. As a final comment about aerodynamic slender-body theory, we should mention how the pressure computation is affected by slenderness. In order to keep the results concise, let us suppose that, if there is a steady forward motion, we introduce instead an opposite uniform stream at infinity. The

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105

pressure is computed from the Bernoulli equation,

PIP= - u4.x - 4 r - 3-6x2 + 4; + 4:). P

q E 2 )

-_c

o(E4)

q E 2 )

The orders of magnitude are noted below the terms, the estimates being based on (2.5). We have assumed that 4r = O(4). Thus, in the first-order approximation for the pressure, we have

PIP= - u4.x - 4 r - 34; + 4:)

= O(E’)-

(2.15)

The appearance here of two of the quadratic terms is a characteristic of slender-body theory. Even if the velocity-potential problem itself can be treated as a linear problem, a consistent computation of the pressure generally requires that the two quadratic terms be included. A partial exception will be found later when we develop the solution of some motion problems in terms of two small parameters, in which case we shall neglect quantities that are quadratic in terms of the second parameter. However, the argument will still stand with respect to quantities quadratic in terms of the slenderness parameter E . 111. Slender Ships in Unsteady Motion at Zero Speed

A. PROBLEM FORMULATION Now we consider a slender body whose longitudinal axis is approximately parallel to the undisturbed free surface. Since we have it in mind to apply the results to ship problems, we assume that the body is partially submerged, that is, it intersects the free surface. The body may oscillate vertically or horizontally, it may be fixed and exposed to incident waves, or it may be free to respond to incident waves. We shall not consider the case of longitudinal oscillation of the body (surge, in nautical terms), largely because it is illsuited for the application of slender-body theory; we do not believe that anyone has yet shown whether it is valid to simplify this case through the use of the slenderness property, although such an analysis has been performed formally. It will be assumed throughout this section that the ship has no forward speed. In the notation of Eq. (2.3’) or (2.7), we have U , = 0. Because of the presence of the free surface, the resulting theory is not necessarily a primitive strip theory, as was always the case in the corresponding aerodynamic problem. Since we seldom go beyond linear theory in ship-motion problems, it is

106

T . Francis Ogilvie

convenient at many points to assume that the unsteady motion is sinusoidal. If we are concerned with just the linear problem, other motions can be studied as superpositions of sinusoidal motions, and there is no loss of generality in restricting ourselves to sinusoidal motions. It is still assumed that the body geometry allows for the definition of a slenderness parameter E with implications essentially the same as in the infinite-fluid problem. Again we treat the fluid-motion problem as a problem in potential theory, and so the governing partial differential equation is the Laplace equation in three dimensions. The kinematic condition on the body is the same as before. Thus we start out requiring that (2.2) and (2.3) be satisfied, but with U , = 0 in the latter. Also, the arguments that were presented with respect to Figs. 1 and 2 are still valid, and so we still accept (2.1) as a reasonable hypothesis in the near field, although we shall have to introduce some modifications presently. The new aspect is the presence of the free surface. The boundary conditions to be satisfied there are well known [see, for example, Stoker (1957) or Wehausen and Laitone (196O)l. On the free surface, z = [(x, y, f ) , the pressure is constant, and we can set it equal to zero in the Bernoulli equation

+ +

on z = [(x, y, t). (3.1) p / p = -4, - g i - &(4: 4; 4:) = 0 There is a kinematic condition to be satisfied on the free surface

4 L X+ 4JY- A

+ C, = 0

on z

= i(x,

y, t).

(3.2) In addition, we must satisfy a radiation condition; we postpone specifying this condition until we have a linear problem to work with. We want to simplify the free-surface boundary conditions in a way consistent with the concepts of slender-body theory, as described in Section 11. In the near field, we initially apply (2.1) as before. We must also consider what effect differentiation with respect to time has on orders of magnitude; this is necessary if we are to interpret the time derivatives in (3.1) and (3.2) correctly. There is a range of possible choices, but only three of these lead to ~); = significant results: (i) a4/at = O(4); (ii) a4/at = O ( ~ / E ’ / (iii) O ( ~ / E It ) . will be necessary to provide an acceptable interpretation for presuming that time differentiation is related (in an order-of-magnitude sense) to the slenderness parameter E, and we shall do this presently. It will also be necessary to satisfy (3.1) and (3.2) in the far field. Corresponding to the three possibilities just mentioned for the order-of-magnitude effect of the operator a/&, we shall find that some modifications of the infinite-fluid ideas are needed in the far field. Before examining these cases separately, it is worthwhile to anticipate some simple results and to use them to provide some physical insight into the meaning of the several choices. That is the purpose of the next section.

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B. RADIATION PAITERNS

In the far field, it turns out that the most reasonable free-surface condition is

& + g4= = 0

on z = 0.

(3.3)

This will be discussed further in Section II1,C. For now, we note only that it is easily obtained from (3.1) and (3.2) by retainingjust the linear terms and eliminating [ between the two equations. Equation (3.3) is the classical free-surface condition for linearized problems involving unsteady motion of a fluid with a free surface. We shall show later that (3.3) should be used in the far field regardless of what assumption is made about the effect of the operator alat. Consider the case of a ship heaving (translating in the vertical direction). If the ship is sufficiently slender, the disturbance far away appears as if it might have been created by a distribution of time-dependent sources on the x axis. This is somewhat analogous to the result for a slender body in an infinite fluid, as expressed in (2.10), but there are two important differences to be noted: (i) In the present case, the simple sources in (2.10) must be replaced by free-surface sources, that is, the corresponding velocity potentials must satisfy (3.3). (ii) In the infinite-fluid problem, vertical oscillation of the body is represented in the far field by a distribution of vertically oriented dipoles on the singular line. In the free-surface problem, the fluid fills only a half-space, and we cannot really distinguish between the effects of timedependent sources at z = 0 and time-dependent vertical dipoles at z = 0. It seems to be somewhat simpler to think in terms of sources, and we do so. Let the source density be given as Re[6(x)ei"'],

0 < x < L.

(3.4)

Then the potential is Re{4(x, y , z)ei"'},

(3-5)

where @(x,y, z) is given by [see Wehausen and Laitone (1960), Eq. (13.17")]:

r ' \

Here, R = [(x - <)z + yz]1/2 and v = wz/g. The integral is to be interpreted as a contour integral indented above the pole at k = v. Far away, where Ro = (xz + yz)"2 is very large, the above potential can be approximated as

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T . Francis Ogilvie

follows iv

(3.7a)

iv

= -evz x a n H:2) ( vR , ) cos

2

no

n=O

IOLd 5 w J , ( V 5 ) ,

(3.7b)

where an=

{t;

n = 0, n > 0.

The second form is obtained by applying an addition theorem for Bessel functions, as given in Eq. (9.1.79) by Abramowitz and Stegun (1964). The Hankel functions in (3.7b) can be further approximated by their large-r asymptotic representations.

FIG.3. Angular distribution of amplitude of radiated waves from a line of pulsating sources.

We do not know 6(x) in general; it must be determined somehow from the near-field solution. But we obtain useful insight by considering a special case: 8(x) = oo,a constant. This is close to what one might find for a boxlike ship heaving at moderate frequency. For this case, in Fig. 3 the radial distance to the curve at any angle represents the relative magnitude of 4 for the wave radiating at that same angle. This quantity, computed from (3.7b), is proportional to the amplitude of the outgoing wave, and so we can tell from

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109

Fig. 3 whether the source distribution causes waves to go out symmetrically in all directions or whether the waves are focused in certain directions. Four cases are shown. For V L= 2 (that is, l / L = IL, where l = 2n/v), the waves propagate outward with essentially the same amplitude in all directions. The source distribution might just as well have been circular, rather than rectilinear. At the other extreme, for V L= 20 ( l / L= rc/lO), the waves go out almost exclusively in the broadside direction ; practically no wave energy escapes in the endwise directions nor even over a wide range of oblique angles. Two in-between cases are also shown. The two extreme cases shown are typical results in radiation problems of acoustics, electromagnetic wave theory, and so on. If the radiator is small in comparison with the wavelength of the radiation, it is not possible to focus the outgoing waves; the distribution of wave energy with direction is essentially uniform. On the other hand, if the radiator is large compared with wavelength, sharp focusing is possible. What is of interest in this example is what is meant by " large " and " small " in the comparison of wavelength and radiator size. Apparently, a l / L ratio of about 3 is very large, and a l / L ratio of about 1/3 is very small. The above results are true only very far away, in an asymptotic sense. It is difficult to make precise statements about the distribution of wave amplitude in a region at finite distance from the oscillating ship. However, the nearby behavior in the short-wave case is suggested qualitatively in Fig. 4. Alongside the body, there are waves propagating in the directions perpendicular to the axis of the body. The shorter the waves, the farther out this

--- .

1-1

r---I \ --I

I--+I-\-

1

I

I--\

I \ ----

FIG.4. Short waves generated by a line of pulsating sources.

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T. Francis Ogilvie

behavior extends. Analytically, the description corresponding to Fig. 4 can be derived from (3.7a) by using the large4 asymptotic formula for the Hankel function and then applying the method of stationary phase. It turns out that there is no point of stationary phase unless 0 I x I L, but within this range the potential function in (3.7a) is given approximately as follows 4 ( x , y, z ) e ' ~ ' iG(x)evzei(W'-vI~I). (3.8) This result says that there are waves propagating outward in the f y directions, that these waves move as if they were strictly 2-D waves, and that the amplitude at any point ( x , y) of the free surface depends only on G(x), the source density at the same x. Of course, the abrupt change in character of the solution along the lines x = 0, L is fictitious,a result of using the method of stationary phase. If frequency is considered to become higher and higher (wavelength becomes shorter and shorter), the actual wave motion would come closer and closer to this discontinuous pattern. But, for finite frequency and finite wavelength, there must be ever-widening regions spreading out from both ends of the ship in which this idealization is invalid. This is suggested in Fig. 4 by the broken lines emanating from the body ends; in the regions bounded by these lines, the sharp distinction becomes fuzzy between the two regions defined by the stationary-phase procedure. A similar qualification must be made to the result in (3.8) that relates the wave amplitude at any (x, y) exclusively to the value of 6 ( x )for the same x . This relationship becomes more nearly true closer and closer to the singular line. In this discussion of radiation patterns from a heaving ship, the important relationship between wavelength and frequency has been used or inferred several times, that is, 2n/1= v = o 2 / g . (3-9) This is, of course, the dispersion relationship for free-surface gravity waves on deep water. It provides the means to rationalize possible order-ofmagnitude relationships between the operator a/at and the slenderness parameter E. [See the discussion following Eqs. (3.1) and (3.2).] For example, if we consider sinusoidal motion at radian frequency o,there is a wavelength 1 = 27cg/oz associated with the resulting fluid motion, and we may expect the relationship between this 1and the dimensions of the ship to provide a characteristic parameter for describing the motion. This is clear from Fig. 3. In this way, the initially strange idea of relating the effect of time differentiation to the slenderness parameter becomes quite natural: "Time " implies a frequency, which implies a wavelength, which can be compared with body dimensions from which the slenderness parameter is defined. Furthermore, suppose that we assume that o = O ( E - ' ' ~ )so, that a/& =

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111

O(E-lI2), which is one of the possibilities mentioned in the discussion of (3.1) and (3.2). A simple plane wave at this frequency might be described in terms

+

of a potential of the form exp(vz io t - ivy). Differentiation of this potential with respect to y or z is equivalent in magnitude to multiplying by v = O ( E - ’ ) .This is an order-of-magnitude effect not encompassed in (2.1), and so we shall have to complement (2.1) with a formalism that adequately accounts for this effect. We shall treat this situation when we come to the case of high-frequency oscillations.

C. FORCED OSCILLATIONS 1. Low-Frequency Oscillations We define low frequency to mean that w = O( 1) as E + 0, or I = O( 1) [see (3.911. We shall also refer to this as the “long-wave’’ problem. In practice, we are likely to think of wavelength in this case as being more or less comparable to ship length, and this is a useful way of thinking. But it is not really correct and it may occasionally be misleading. The statement I = O( 1)” implies that I / B -P 00 as E + 0, where B is ship beam. The ratio I / L may have any value, in principle, just so it remains fixed in the limit process. It could be 1/10 or even 1/100. However, the theory will not be valid unless I / B is very large, and so a small value of I / L may require that B/L be really infinitesimal. Such a theory would not be useful because no one is interested in a ship with such dimensions. So the meaning suggested when we call this a low-frequency or long-wave case is important for practical purposes, although it is not entirely correct. Formally, we most often introduce the assumption o = 0 (1 ) when we differentiate with respect to time, in which case we assume that a/& = O( 1). Also, v and L are O(1). Near Field. There is assumed to be a velocity potential, 4(x, y, z, t) = Re[4(x, y, z)exp(iot)], to which we can apply (2.1). Then the first approximation for $(x, y. z) satisfies the 2-D Laplace equation in the near field, as in (2.6). The kinematic boundary condition on the body is as stated in (2.7) for the infinite-fluid case. It is convenient now to redefine U j “

”

“

U , = 0;

U j = Re(io<jeimt), j

= 2,

3.

(3.10)

on the body.

(3.1 1)

Then the body boundary condition can be rewritten: ac$faN

= io (n ztz

+ n3t3)

We linearize the boundary conditions, and so we apply (3.11) on the body surface at its undisturbed position. (This is not valid in the forward-speed problem, even in the linearized case.) We assume that U j = O ( E ) ;this is

112

T. Francis Ogilvie

adequate for the present case, although we shall need a more precise statement later for the high-frequency case. This assumption concerning U j implies that

4 = O(EZ),

(3.12)

just as in the infinite-fluid problems [see (2.5)]. Now we simplify the free-surface boundary conditions, (3.1) and (3.2). First consider just the linear terms. If we keep both linear terms in (3.1), we must expect that

r = O(4) = O(EZ),

(3.13)

which then implies in (3.2) that

4* = 0,

(3.14)

r,

) . the other hand, we since 4z= O(E)[the effect of (2.1)] and = O ( E ~On might choose to retain both linear terms in (3.2), which would imply that Then, in (3.1), to

r = O(4J = O(E).

(3.15)

r = 0.

(3.16)

4, would be of higher order than r, and so (3.1)would reduce

But this is a trivial result; it means only that we do not yet have the lowestorder term in the expansion for [. Therefore we reject (3.15); it appears that (3.13) is the more reasonable choice. Of course, (3.14) follows directly from (3.13), and this means that the free surface has been replaced by a rigid wall in the problem that defines the first-order approximation of 4. It might seem that we have thereby discarded the free surface completely, but this is not true. It must be recognized that the rigid-wall condition, (3.14), does not mean literally that there is no vertical fluid motion at z = 0; it means only that the vertical component of fluid motion is much smaller than the horizontal component. From the linear terms in (3.1),t we can calculate the corresponding deflection of the free surface [(x, Y,t ) = Re[ - (l/g)4,] = Re[ - (Wg)4(x,

Y,o)ei"'].

(3.17)

From this formula, we can calculate [,,and we observe that it is O(E').The kinematic free-surface condition, (3.2), then requires that +z = + .... The right-hand side is O ( E ~ but ) the left-hand side is O(E).If we want to calculate

r,

+ The term - (1/2)4; in (3.1) is O(E'),the same as @,, and so it should in principle be retained here. However, we generally assume that we may linearize oscillation problems with respect to the amplitude of oscillation, in which case the quadratic term is of higher order in that sense.

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113

the left-hand side to an accuracy which is only O(E),we can set the righthand side equal to zero. And this is what (3.14) means. The near-field problem represented by (3.1l), (3.14), and (3.17) was intensively investigated in the mid-1960s by Newman (1964), Joosen (1964), Newman and Tuck (1964), and Maruo (1967a). This was the basis for the early slender-body theory of ship motions, which gave reasonable predictions at zero speed (but only at zero speed). The solution of this near-field problem is no more difficult than in the infinite-fluid case. Since the free surface is replaced by a rigid wall in the 4 problem, we can extend the definition of 4 analytically into the upper halfspace as an even function with respect to z. The extended problem can be interpreted as an infinite-fluid problem, and so it is not necessary here to discuss methods of solution further. The solution of the problem has somewhat different properties according to whether the body oscillates vertically or horizontally. The two cases are depicted in Fig. 5. If the body oscillates vertically, the image of the body

1'

1'

(a)

(b)

FIG. 5. Motion of the body image in the low-frequencyproblem. (a) Vertical oscillation, (b) horizontal oscillation.

moves oppositely, so that the two act together somewhat as a pulsating body. Far away from the body, the potential can be expanded as in (2.8), and the logarithm term is the most important, that is, $(x, .y, z)eiw'

-

{ [ o ( x ) / 2 a ]log r

+ Ao(x)}eiw',

(3.18)

as r = (y' + z2)1/2+ co. [Cf. (2.8").] From conservation ofmass, one can show easily that the source strength is given by D(X) =

-2ioB(x)<, ,

(3.18')

where B(x) is the beam at the waterline. It may be noted that the kinetic

114

T. Francis Ogilvie

energy of the fluid motion per unit length of the body is infinite, a fact that leads to a prediction of infinite added mass per unit length of the body. If the body oscillates horizontally, the image moves in the same direction, so that the total effect is the same as for an oscillating double body in an infinite fluid. In this case then, there is no sourcelike behavior, and the leading term in (2.8) for this problem is the horizontal dipole term, the A , term, that is, $(x, y,

-

~ ) e ’ ~[A,(x) ‘ + A , ( x ) cos O/r]eiut

as r - ,

00.

(3.19)

Whatever method of solution is used, allterms in the expansion (2.8) can be determined except A,(x). There is no information available in the near field alone that can be used to determine this quantity, since the near-field boundary-value problem is a Neumann problem, the solution of which is always nonunique to the extent of an arbitrary additive constant. The additive “constant” is not trivial, since it is really a function of x, and so it contributes to the evaluation of fluid pressure. It does not vanish, as in the infinite-fluid case of an oscillating body; this is a consequence of the presence of the free surface. We must solve the far-field problem in order to determine the additive term. Far Field. The velocity potential must satisfy the 3-D Laplace equation and the free-surface conditions, (3.1) and (3.2). There is no body boundary condition, only a condition that the far-field solution be consistent with the near-field solution. The far-field solution must satisfy a radiation condition, that is, it must represent outgoing waves far away from the body. Since we are here assuming that o = O( 1) as E -,0,differentiation with respect to time and space coordinates does not alter orders of magnitude, that is,

apt, a p x , spy, a p z = O(1)

as

E -,0.

(3.20)

The potential in the far field will presumably be o( I), and so the quadratic terms in (3.1) and (3.2) are all of higher order of magnitude than the linear terms. It is also consistent in the first approximation to satisfy the conditions on z = 0. Then it is a simple step to show that the free-surface conditions can be combined into the form given by (3.3),which, for sinusoidal time dependence, becomes

v4 - 4z= 0

on z = O ,

with v = 0 2 / g .

(3.21)

The reasoning that follows is parallel to the infinite-fluid case: In the limit as E -+ 0, the body shrinks down to a line, and so the far-field solution can be represented in terms of a distribution of singularities on that line. In the presence of the free surface, the simple singularity potentials must be modified so that they satisfy (3.21),and they must represent outgoing waves.

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115

Otherwise the distributions are arbitrary. The densities of the sources, dipoles, etc., cannot be determined solely from the far-field problem. Furthermore, in the lowest-order approximation in the far field, there will be only one kind of singularity required and possible-whatever kind matches the leading-order term in the near-field solution as r -+ co. The reasoning still follows that for the infinite-fluid problem. In the case of vertical body oscillation, the near-field solution at large r behaves as in (3.18), and so the matching far-field solution represents a distribution of sources. In the case of horizontal body oscillation, the near-field solution behaves as in (3.19), and so the matching far-field solution represents a line of transverse dipoles. The general problem that was discussed in connection with radiation patterns is precisely the far-field problem for the case of a slender body undergoing vertical sinusoidal oscillations, and so the solution expressed in (3.6) can be used here directly. The estimates given in (3.7a) and (3.7b) are not useful now in setting up the matching to the near field, since they are valid only far away; however, they do show that the solution represents outgoing waves, since ei"'Hi2'(vR,)

- (C/R~12)ei("'-"Ro) as

Ro + 00,

where C is a complex constant. In order to match (3.6)with (3.18), we must approximate the expression in (3.6) for small r. The procedure, in brief, is as follows: Rewrite (3.6) as

l L 4 ( ~Y,, Z) = - - d t h ( 5 ) 2n 0

J"

J"

m

dkpJo(kR)

0

(3.22) r\

The first term can be expressed

(3.23)

as r + 0, that is, it represents the potential for a line distribution of sources of density 26(x) in an infinite fluid; the small-r approximation for this potential was already given in (2.11b(2.13), and we have taken it over directly here except for the change by a factor of 2. The second term in (3.22) can be divided into two parts, a principal-value integral and a contribution from the contour indentation; as r + 0, they can be expressed in terms of standard

116

T. Francis Ogilvie

functions

where Ho is a Struve function. Thus the small-r approximation for $(x, y, z) is (3.25) where

(3.25') After being multiplied by exp(iot), this must match with (3.18), and so we have &(x)= - ioB(x)S,;

AOb) = f ( x ) .

(3.26a) (3.26b)

Just as in the infinite-fluid problem, the near field has provided the information that was lacking in the far field, namely, the value of 8(x), and the far field has provided the information that was lacking in the near field, namely, the additive function A o ( x ) .The latter, in particular, is essential information, for this " constant " in the solution of the 2-D crossplane problems contributes to the pressure field acting on the body. This extra pressure does not depend on y and z, but it may nevertheless cause a net force in the transverse directions, since the pressure is integrated over just the wetted portion of the hull to give net force. In any case, the term A o ( x )in the near-field solution is the only manifestation of interactions among the cross sections at various values of x; it represents a longitudinal wave at a particular x caused by the sources (in the far-field description) at all other values of x. If Ao(x) were equal to zero, the above slender-ship theory would be just a primitive kind of strip theory. In the case of horizontal body oscillations, the resulting theory is just a primitive strip theory, with no interactions among cross sections. The situation is analogous to the infinite-fluid case. The additive function of x cannot be determined in the near field alone; all that can be asserted from the near-field problem is that, for large r, the solution behaves like a constant or like cos O/r [see (3.1911. This fact eliminates the possibility of there being a

Ship Hydrodynamics

117

line of sources in the lowest-order far-field solution; the first term must represent a line distribution of horizontal dipoles of unknown density. The potential for such a distribution can be written down immediately, for example, by differentiating (3.22) with respect to y . This potential can be approximated for small r by a single-term expression that represents a 2-D dipole, that is, it matches the A , term of (3.19), and there is nothing to match with A,. Therefore we must set A, = 0 in the near-field solution for the case of horizontal body oscillations, and there are no interaction effects among the various cross sections. Nonunqormity of the Near-Field Solution. Slender-body theory is by its nature a study in singular-perturbation problems, marked by strong mathematical nonuniformities in the solutions. The far-field solution is nonuniform near the singular line. The near-field solution is nonuniform far away, for it gives no solution (or a trivial solution) outside of the domain 0 < x < L The low-frequency ship-oscillation case presents an even stronger nonuniformity in the near-field solution: It admits of no wavelike motion [except for the add-on longitudinal wave component included in A,(x)]. Physically, one expects a heaving ship to create outgoing waves and these waves ought to be evident in the regions off the sides of the ship. In fact, this is what Fig. 3 showed on the basis of a simple mathematical model. co However, the near-field solution displays no such behavior even as y -+ in the near field. The interpretation of this fact is that the wavelength is so very, very long that the behavior identified as “near-field behavior” occurs entirely in a region which is small compared with a wavelength. Of course, we have already described this as a long-wave theory,” and we implied that ship beam was small compared with the wavelength. But much more is implied: the whole n e a r j e l d is small compared with the wavelength. This seems to be a rather severe constraint on the possible utility of the theory. It also implies that there is no damping (to lowest order) associated with the waves radiated out to the sides. Another aspect of possibly singular behavior may be noted, this being a local singularity. In the case of vertical oscillation of the body, the fluid velocity at the level z = 0 on the side of the body can have only a vertical component. [We are assuming that the body is wall-sided at the waterline.] However, the rigid-wall condition at the free surface says that there can be only a horizontal component of velocity at that same point. Accordingly, the solution must indicate a stagnation point at the juncture of the body surface and the undisturbed free surface. Such a flow condition at that point seems highly unlikely, however. Our general interpretation of the rigid-wall condition seems to be valid enough: The vertical velocity component is very small compared with the horizontal component. However, at the body side the “

118

T. Francis Ogilvie

latter is precisely zero in magnitude, and so this comparison is meaningless there. Our assumptions have led us into setting a poorly posed problem in this respect, but we do not know of any careful analysis of this point. 2. High-Frequency Oscillations; Strip Theory High frequency” will be defined by the order-of-magnitude statement o = O ( E - ’ / ’ ) .The physical interpretation of this should be made in terms of the corresponding wavelength, A = 2ng/oZ, which is now O(E). Since B = O ( E )as well, we presume that the ratio A/B is O( 1) as E -,0. Thus we are assuming that the frequency becomes higher and higher as the body becomes more slender, and it does so in just such a way that A/B can remain fixed. This may seem to be highly artificial, but in fact it leads to an extremely useful theory. As a heuristic explanation, we may say that it is important under certain circumstances to relate wavelength primarily to the ship beam (or possibly draft), rather than to ship length, as was done in the foregoing low-frequency case. The resulting theory corresponds generally to what is usually called strip theory” in naval architecture. The ordinary derivations of this strip theory are quite different from what we shall present here, although it has long been recognized that “ high frequency is generally implied. In the current literature on ship hydrodynamics, we sometimes encounter an objection being made to the use of a high-frequency theory, usually on the basis of the argument that the wavelength is not much smaller than ship length in the most important problems of wave loads and ship motions. To some extent, such an objection misses the practical significance of asymptotic solutions: We develop an asymptotic expansion in terms of a small parameter which is supposed to be approaching zero, and then we apply the theory to cases in which the small parameter has quite a finite value. We can seldom tell from the theory alone whether the finite value of the parameter is really small enough for the approximation to be useful; we never seem to know how small is “small.” In one problem, the small parameter might have to be and in another problem it might be unity. We must have some other basis for judging the validity of the approximation, perhaps experiments, a higher-order calculation, or a special case that can be solved exactly. Experiments, in particular, have demonstrated that the assumption of high frequency in the ship-motion problem gives more accurate answers than the assumption of low frequency. Near Field. The assumption that o = O(E-’/’) arises naturally from a consideration of the near-field problem. If we combine the two free-surface conditions, (3.1) and (3.2), and retain only the linear terms, we obtain (3.3), that is, & + g+= = O on z =O. (3.27) “

”

“

“

”

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119

In the near field, 4z= O(&-I). If the motion is sinusoidal at radian frequency w, then &r = O(&02). There are obviously only three possibilities = O ( E ’ ~ ’ ) (iii) ~]; to consider: (i) o = o(c-l12); (ii) w = O ( E - ’ / ’ ) [and o- = o ( E ’ / ~ )The . fir& case led to the low-frequency problem of the last section, with the rigid-wall free-surface condition. The second case is the present one, and we see that both terms in the free-surface condition must be retained; they are strictly of the same order of magnitude. In the third case, the frequency is so high that only the first term in the free-surface condition is retained, that is, $,?= 0, which is true for all time, and so 4 = 0 on z = 0. It should be observed that only case (ii) above allows directly for the occurrence of gravity waves in the near field, and so it is in many ways the most interesting case. This is the problem that we now consider. The argument above is based on consideration of just the linear terms in the free-surface conditions, (3.1) and (3.2). In problems of small oscillations of a body in the free surface, we generally expect that it will be legitimate to linearize the conditions, and so we expect that the conclusions above will stand when the nonlinear terms are considered too. This formal linearization will be carried out in an explicit way in Section V,where we encounter more difficulties if we simply try to linearize by inspection. For the present, we d o not need that degree of formalism. However, the consequences of linearization should be noted: We lose all harmonic responses and the occasionally important steady (mean) force. The body boundary condition is exactly as in the low-frequency problem, Eq. (3.11). We can even include pitch and yaw motionsf in the formulation by defining t5as the amplitude of pitch (rotation about they axis) and t6as the amplitude of yaw (rotation about the z axis). These angular displacements are assumed to be small enough to be treated as components of a rotation vector. A generalized form of nj is also required: For j = 5, 6, we define

’

nJ. = e1.- 3 ’ (r n), (3.28)’ where ej is the unit vector parallel to the xi axis, r is the position vector of a point on the body surface, and n = (nl, n2. n3j is the usual unit normal + Note that, without this alternative statement for case (ii), we should have to accept the situation that case (ii) encompasses case (i), whereas we want them to be mutually exclusive. The reader is reminded that the statement y = O(x) as x + 0 means that I y/x I remains bounded in the limit. The statement y = o(x) means that 1 y/x I + 0. If y and x satisfy the latter statement, then they also satisfy the former. A similar extension is, of course, possible for roll and surge motions, but the resulting theory is of doubtful validity. The question about surge was mentioned at the beginning of Section III,A. With respect to roll motion, the difficultyprobably lies more in the assumption of an ideal fluid. In practical situations, large amplitude may also invalidate the theory. ‘See also Eqs. (5.5a) and (5.5b).

120

T. Francis Ogilvie

vector on the body surface (always taken outward from the fluid). We now rewrite the body boundary condition, (3.11), in the compact form a@/alv= i o

1nitj,

(3.29)

i

wherej is summed over 2, 3, 5, 6. As usual in applications of slender-body theory, we expect to be able to simplify the 3-D Laplace equation to a 2-D Laplace equation for use in the near field. In fact, we can do this, but the step is not so straightforward as in the previous problems. The near-field potential will satisfy Eq. (3.27) on the free surface, which, for the case of sinusoidal oscillations, reduces to the familiar condition v$ - $z = 0

on z = 0,

with

v

= w2/g.

(3.30)

The solution of the problem will represent, at least in part, a wave motion with the characteristic wavelength I = 24v. If that solution is differentiated in the direction perpendicular to the wave crests, the differentiation has an order-of-magnitude effect like multiplying by v, which is O(E-'). This effect must be considered along with the basic assumption of slender-body theory, (2.1). In particular, the statement in (2.1) that a f d x = O(f)is not true unless the wave crests are parallel to the x axis. Fortunately, the wave crests are approximately parallel to the x axis in the case of high-frequency oscillations of a slender ship. This was already suggested by Fig. 4. One might also be led to this conclusion by applying a gravity-wave analogue of Huygens' principle (although no such analogue exists in a strict sense). We shall assume then that the waves generated in the nearjeld by the oscillating ship do move out with their crests approximately parallel to the ship axis, and so the assumption about d/ax in (2.1) is retained. This assumption should be a very good one over most of the length of the ship, but we must recognize that it cannot really be true near the ship ends, and so some error is thereby introduced. The higher the frequency, the better is this assumption. The short-wave nature of the solution does not introduce any complication with respect to the assumption in (2.1) about derivatives in the transverse directions. We continue to assume that a / a y and d / d z have order-of-magnitude effects like multiplying by E - ' ; it does not matter whether this is a result of slenderness or of the short-wave character of the fluid motion. So finally we are free to use the 2-D Laplace equation in the near field. The solution must satisfy the body boundary condition (3.29) and the freesurface condition (3.30). We expect the solution to represent outgoing waves at large I y 1, but a more basic approach is to require that the solution should match properly with a far-field outgoing-wave solution. This 2-D problem has been studied extensively over a period of 30 years,

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starting with Ursell's (1949a) famous paper on oscillations of a semicircular cylinder. In another paper (not so well known) in that same year, Ursell (1949b) showed how to generalize the circular-cylinder results to other body shapes; his method was later used systematically by Tasai (1959)and Porter (1960). Frank (1967) solved these problems by an integral-equation method. Both methods are now in general use in many parts of the world for application to the ship-motion problem, and there is no need to describe them here. Some basic properties are worth noting however: (1) The solution of such a 2-D problem is probably unique, although this has not been proven with complete generality. Uniqueness implies that there is no additive arbitrary constant comparable to the A, in (2.8) or (3.18),and so there can be no interaction among the cross sections in this lowest-order approximation. A primitive " strip theory " always results. (2) At large I y 1, the solution represents outgoing waves. It can be expressed in the following forms Vertical body oscillations 4(x, y , z)eiw'

Horizontal body oscillations $(x, y , z)eiw'

-

-

(ig/o)a(x)ev'ei("'-v~~~),

( i g / o ) b ( x )sgn yev'ei("'-"IYI',

(3.31a) (3.3lb)

where a ( x ) and b ( x ) are the (complex) wave amplitudes in the respective cases. The waves represented by (3.31a) could have been produced by any kind of laterally symmetric oscillatory singularity at the origin, that is, by a pulsating source, vertical dipole, etc. It is generally most convenient to assume that the singularity is a source, in which case the complete near-field solution outside of a circle circumscribing the body can be represented in terms of that source plus an infinite set of "wave-free singularities," as discussed by Ursell(1949a). The extra terms not included above in (3.31a)all decrease algebraically with increasing I y I. Similarly, it is convenient to identify the waves expressed in (3.3lb) with a pulsating antisymmetric singularity, and the lowest-order possibility is a horizontal dipole. (3) The force per unit length on the oscillating body can be calculated directly from this near-field solution. Far Field. If our goal is simply to compute the force on the body to a first approximation, the only role of a far-field solution is to confirm that (3.31a) and (3.31b) are correct asymptotic relationships. Of course, they cannot really be correct if taken quite literally: Far off to the sides of the ship, the wave amplitude must decrease in proportion to the inverse square root of the distance, and furthermore the above relationships say nothing whatever about the wave motion outside the strip 0 < x < L We must interpret (3.31a) and (3.31b) as being valid estimates at relatively large values of I y 1, but not in any absolute sense as I y I + co. The meaning is quite analogous

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122

to that implied in classical boundary-layer theory, in which one speaks of the behavior of the fluid as y co inside the boundary layer, where y is the coordinate measured in the direction locally normal to the body surface. Of course, one means that y 4 co on some relative scale appropriate to the normal dimension of the boundary layer, and even then one cannot literally let the limit process occur. However, just as Prandtl used such arguments so effectively in the development of boundary-layer theory, so one can use them here for obtaining both a physical understanding of the problem and a first approximation to the solution. If one requires better approximations, the procedure must be developed more formally. The formalism is, of course, the method of matched asymptotic expansions. In the far field for this problem, we must modify the formal procedures that were used previously. The necessity for doing so arises when we consider the free-surface condition in the far field. This condition, in linearized form, has been given several times, e.g., Eq. (3.30). Should we retain both terms in this condition? In the far field, we usually assume that differentiation has no order-of-magnitude effect, that is, the assumptions in (2.1) do not apply. In (3.30), the result of this normal procedure is to require us to neglect term, since it is O(+),whereas the first term, v+, is O ( + E - ' ) .Thus we the are led to the free-surface condition = 0 on z = 0. But this is patently unacceptable, for it does not allow for the existence of a wave motion, and yet we know that very far away there is only a wave motion. Clearly, the formalism has led us to a wrong boundary condition. The error is in assuming that 4z= O(c$). If the fluid motion far away is dominated by a wave motion characterized by the wave number v, differentiation of the solution with respect to x or y will have the order-ofmagnitude effect of multiplication by v. Since the solution satisfies the Laplace equation, we must conclude that differentiation with respect to z has a similar effect. Perhaps this is seen most clearly just by examining the potential for a plane wave, which might take the form exp(vz - ivx i o t ) , for example. Differentiation with respect to a space variable has an effect comparable to multiplication by v, which is O(E-'). Thus one must conclude that we should retain both terms in (3.30) when applying it in the far field. We note in passing that it would not have been wrong to keep both terms in (3.30) even without the benefit of the above argument. In an asymptotic analysis, we may retain selected terms of higher order a t any time that it is convenient; keeping such terms does not necessarily make our analysis any more accurate, but it is perfectly proper mathematically to do so if we wish. Accepting (3.30) now as the free-surface condition in the far field, we are back to the problem discussed in Section III,B, at least for the case of symmetrical problems, that is, vertical body motions. As E + 0, the body shrinks down to a line, and the far-field solution can be represented in terms --f

+*

+

+

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of a line distribution of sources on the x axis. The potential is given in (3.5) and (3.6). Far away from the singular line, it represents outgoing waves in a genuine 3-D sense. Near the singular line, it represents outgoing 2-D waves, as given in (3.8). In fact, a comparison of (3.8) and (3.31a) shows directly that

6(x) = ga(x)/w.

(3.32)

The far-field solution has served its purpose: It matches with the near-field solution and shows that the outgoing 2-D wave solution in (3.3la) is indeed correct in the near field. No further information is obtained from the far field in this case. The potential in (3.5) and (3.6) has been used for the far-field solution of both the low-frequency and the high-frequency cases; the statements of the respective far-field boundary-value problems were identical. However, the estimates of this solution near the singular line are notably different: Compare (3.25) and (3.25’) with (3.8). In the high-frequency case, it was possible to use the method of stationary phase, and the resulting estimate (3.8) is valid in a region that is still many wavelengths away from the singular line (even though the lateral distance is small compared with L). In the lowfrequency case, a direct estimate of the solution at small lateral distance was possible, and that estimate contained no suggestion of an outgoing 2 - D wave. In both cases, we obtained an explicit value for the 3-D source density on the singular line. In the low-frequency case, it is directly related to the body geometry, as given in (3.26a). In the high-frequency case, it depends on the net wave-making effectiveness of the 2 - D section, as expressed in (3.32). The amplitude of the generated 2-D wave is not simply related to body geometry, because the body’s transverse dimensions are comparable to a wavelength. Completely analogous results can be obtained for the case of horizontal body motions, with the source distribution replaced by transverse dipoles. The potential for the dipoles can be obtained from (3.6) by a differentiation with respect to y. Thus the lowest-order problem can be completely solved, and it is not much more complicated than in the low-frequency case. The only real extra complication is that rhe near-field solution must satisfy the free-surface condition (3.30). This must generally be done numerically, but most institutions concerned with problems of ship motions have an adequate computer program for doing it.

3. Very High Frequency Oscillations In the discussion following Eq. (3.27), it was pointed out that there are three possibilities to be considered with respect to the orders of magnitude of the two terms in (3.27). In the third of these, the frequency o is so high that

124

T. Francis Ogilvie

= O ( E ” ~ )as E + 0, or, what is the same thing, that w2e -+ co.The natural conclusion to draw is that the free-surface condition should be taken as 4 = 0. As noted previously, such a boundary condition does not allow for the existence of gravity waves-not, at least, in the first approximation. However, no matter how high the frequency of oscillation of the body, we really do expect that gravity wavest will occur and that they will be characterized by the wave number v = w2/g. In this case, following arguments already given, we must expect that derivatives with respect to space coordinates may have order-of-magnitude effectscomparable to multiplication by v, and so the term g4= in (3.27) cannot be neglected. The question then arises whether any valid results can be obtained by solving problems in which the condition 4 = 0 is imposed on z = 0. It is important to resolve this question for a practical reason: In studies of ship vibration, the added mass of the water is generally computed from the solution of the boundary-value problem with 4 = 0 on z = 0. The argument is made that the frequency is so high that the radiation of waves (and thus damping) is negligible, so that the simpler boundary condition is acceptable. Ursell (1953) showed the relationship between these two points of view by solving a particular problem, vertical oscillation of a half-submerged circular cylinder at very high frequency. In an earlier paper, Ursell (1949a) had solved this problem for arbitrary frequency, but his solution in that case was ill-suited for drawing any conclusions as to what happens at high frequency. So he developed a new method, an iterative method which gives a series solution appropriate for the high-frequency case. The first term is precisely the solution of the simplified boundary-value problem, in which 4 = 0 on the undisturbed free surface. The subsequent terms represent a wave motion that has the characteristics expected. In particular, the waves are very short and the entire wave motion is confined to a very thin layer just under the free surface. The latter was to be expected, of course, since even these very short waves have the usual exponential decay with depth that is characteristic of gravity waves. Ursell showed that the added-mass coefficient for the circular cylinder at high frequency is given by

w-

1 - (4/3aN)

+ O(l/N),

where N = 02 a / g, with a equal to the radius of the cylinder. (“Added-mass coefficient” is the ratio of the added mass to the mass displaced by the

’

Of course, surface tension must be considered at some point as frequency becomes higher and higher. But the meaning of “high frequency” and of “short waves” are related here to the size of the body generating the waves, and we are thinking in terms of ship problems for the most part, so that even “very short waves” are much longer than surface-tension or capillary waves. In model experiments, on the other hand, one might have to consider seriously the effects of surface tension.

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125

cylinder.) The first term obviously comes from the solution of the problem with 4 = 0 on z = 0, since that problem can be solved in terms of an analytic continuation into the upper half-space, with the image of the half-cylinder moving together with the actual body. Ursell also obtained the “waveamplitude ratio,” that is, the amplitude of the radiated waves divided by the amplitude of body motion, as 4/N. There does not seem to be much point in trying to relate these facts to the slender-body problems that have concerned us up to this point. In this veryhigh-frequency problem, we are considering the case in which A = 2ng/o2 is very much smaller than any dimension of the oscillating body, and so it is irrelevant whether the body is slender. However, it should be pointed out explicitly that this is a kind of singularperturbation problem of special scientific interest. Since the wave motion is limited to a thin layer at the free surface, we have a problem analogous to the boundary-layer problem for viscous fluids. There is an “outer region” in which there are no effects of the wave motion, but this region is separated by that very thin layer from the surface on which the boundary condition must really be applied. Ursell (1953) considered only the case of vertical oscillation of the body, and the conclusions above apply just to that case. The case of horizontal oscillation is somewhat more difficult, and we do not believe that it has been treated satisfactorily. The difficulty can be seen in a study of the limit problem. If the body moves horizontally and the potential satisfies the condition 4 = 0 on z = 0, there is a weak singularity at the juncture of the body and the undisturbed free surface: The horizontal velocity component is finite but discontinuous across the surface z = 0, and the vertical component thus has a logarithmic singularity. It appears that some of Ursell’s arguments then cannot be applied.

D. DIFFRACTION PROBLEMS 1. Introduction

We now consider problems concerning a ship fixed in incident waves. Since we are linearizing all of our time-dependent problems, there is no loss of generality in so dividing the problems between this section and the previous section. The force (or force distribution) on the ship is the sum of the forces (or force distributions) associated with (i) the fixed ship in incident waves and (ii) the motions of the ship itself. It will appear that the formulation of the appropriate boundary-value problems is very similar to that of the last section. The body boundary condition has to be modified, and the diffraction potential (which we seek)

126

T. Francis Ogilvie

has to be combined with the potential for the incident waves to give the complete description of the fluid motion. But formally there seems to be little difference. However, the physics of the problems is quite different. In particular, the flow variables may vary rapidly along the length of the body, a phenomenon that did not occur in the forced-oscillation problem, with the result that the near-field 2-D problem does not always involve the 2-D Laplace equation. There are also complicated diffraction phenomena around the ends of the ship. These differences will be emphasized here, although the general outline of this section follows that of the last section. Before considering the resulting diffraction problems in detail, we should note the possibility of avoiding having to solve the diffraction problem at all. If the forced-oscillation problem has been solved (for any mode of motion), and if only the total force (for that mode) is required, one can use a reciprocity formula to obtain that force without solving the diffraction problem at all. A general discussion of such reciprocity formulas is given by Ogilvie (1973). They were first applied to the problem of computing wave loads on a ship by Khaskind (1957). Important extensions were developed by Hanaoka (1959) and Newman (1962,1965). Reciprocity formulas will not be discussed further here, although they are extremely important to the shiphydrodynamics practitioner. The diffraction problems remain important nevertheless, not only for their scientific interest but also because they must be solved for some practical problems, for example, the prediction of the relative displacement and velocity between the ship and the water surface. We assume that the typical incident wave can be described in terms of the velocity potential

where h is the wave amplitude, v = 0 2 / g , and /? is the heading angle of the waves with respect to the positive x axis. As before, we assume that the ship is oriented parallel to the x axis, generally extending from x = 0 to x = L. In (3.33), we should use only the real part of the complex quantity, of course, but by convention we shall not indicate this explicitly hereafter unless it is essential for some reason. Because everything is linearized, the restriction to sinusoidal waves represents no loss of generality. Figure 6 shows the geometry in plan view. The ship will be considered as a slender body, in the same sense as previously. It should be noted that there are now two fundamental preferred directions in the x-y plane: (i) the x axis is the longitudinal axis of the slender body (as before); (ii) the direction with respect to the x axis is the direction of propagation of the incident

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FIG.6 . Plan view of a ship in incident waves.

waves. The introduction of the latter complicates the problem and increases the number of possible cases that must be considered. We first examine the possibility of obtaining a solution without using the method of matched asymptotic expansions. Let the diffraction potential be given by $(x, y, z ) exp(iot). The function 4 satisfies, as usual, the Laplace equation in three dimensions. On the basis of the various discussions of Section III,C, we suppose that the appropriate linear free-surface condition is

v4 - 4z= O

on

z=O.

(3.34)

[The potential for the incident waves, (3.33), obviously satisfies this condition.] The condition on the body surface is

At great distance, 4 should represent outgoing waves. The solution can be written formally through the use of a Green function

128

T . Francis Ogilvie

where S is the wetted surface of the body, and

1

1

r

r’

-----2

jo kk -dkv 2(’+‘)Jo(kR), a,

~

(3.36b)

_-r4

r = [(x - t)’ + (y - q)’

+ ( z - C)’I1’’, + (Z +

+ ( y - q)’ R = [(x - t)’ + (y - q)’I1/’. r‘

= [(x -

5)’

C)’]liZI

The Green function here is essentially the same one used in (3.6) (in which case we had q = c = 0); if G is multiplied by exp(iot), it represents the potential at (x, y , z) of a source of strength -47~ exp(iot) located at (l,q, c). In general, in an expression such as (3.36), the domain of integration should be a closed surface. However, the contributions from the free surface and a closing surface at infinity vanish because 4 and G satisfy the same free-surface and radiation conditions, respectively. In (3.36), everything is known except for 4, and so (3.36) becomes an integral equation to be solved for 4 if the point (x, y, z ) lies on S. Such an integral equation can be solved numerically; this has been done by Faltinsen and Michelsen (1974). However, such solutions are expensive and very demanding in terms of computer capability. Moreover, they d o not provide insight into the mechanisms governing the fluid motion, although they are invaluable in providing detailed numerical predictions in individual cases. Therefore we try to simplify the general problem. It was probably Vossers (1960) who first started with a result like (3.36) and then simplified it by taking advantage of the slenderness of the body. The following analysis is, however, due to Newman (1964). [A similar approach to the much more difficult problem of a ship advancing through waves was worked out by Maruo (1967a).] If the point (x, y, z ) is in the near field, that is, (y’ z2)l/’= 0(E Newman showed that 4 could be written in either of two simplified ways, the first being

+

)?

is a potential function satisfying the body boundary condition, where (i) 4RW

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129

(3.37’)

(iii) C(x) is a girthwise contour around the wetted hull. The concept of slenderness need not even be used in defining 4RW, although it has been used in obtaining the additive function F(x). The significance of (3.37) is that it represents the solution in terms of the double-body (rigid-wall) solution plus a distinctive free-surface contribution. The latter, F(x), should be compared with the expression in (3.24). Presumably such a decomposition can be found in the case of a nonslender body oscillating in the free surface or acted on by incident waves, although F(x) will then be more complicated, not just a function of x. Newman’s second simplified expression for 4 in the near field is “

”

4(x~ Y,

-

42D

+f(.)?

(3.38)

where (i) 4zDis a 2 - D potential function in the y-z plane (x may be considered simply as a parameter), and (ii)

In order for 4zDto be unique, we require that it should behave like c log r + o( 1 ) as r = (y’ + z’)”’ + co,where c is some constant. With these definitions, &,, is the potential in the cross-sectional planes for the most primitive kind of strip theory, containing no interactions among sections at all. The additive term f ( x ) contains all of the interactions, part of which depend on the presence of the free surface [those included in F ( x ) ] and part of which would exist in an infinite fluid. If we identify the line integral in (3.37’) and (3.38’) with -o(x) of Section III,C,l, that is, (3.39)

we see that the abovef(x) is identical with thef(x) given in (3.25’) for the low-frequency oscillation problem. In that earlier problem, we started by assuming that there is a far-field representation of 4 in terms of a line distribution of sources, and then we obtained (3.25) and (3.25’) as an approximation to that representation, valid near the singular line. Newman had to make no such initial assumption ; he just used the Green-function representation for the potential, which he then evaluated near the body. Newman’s

130

T. Francis Ogiluie

approach can be used for the forced-oscillation problem as well, and he did just that; only the body boundary condition needs to be changed. It is a very convincing way of handling these problems. If the only problem of interest here were the low-frequency (long-wave) case, we would recommend an approach like Newman's throughout. He introduces the slenderness approximation into a formal solution that is already known, such as (3.36),rather than into the boundary-value problem, and clearly his procedure is the safer one. Unfortunately, no one appears to have done this for the high-frequency case. Newman (1964) specifically pointed out that his results were based on the assumption that v = O( l), and no conclusions can be drawn from his results if v = O ( E -I ) . Vossers (1960)did briefly consider the short-wave case. but without significant results. Ursell (1962) performed a very careful analysis of a problem quite similar to Newman's, but his results are similarly restricted to the low-frequency range. Maruo (1967a),although he extended Newman's analysis to the forward-speed case, still considered only the case of low frequency. There seems to be nothing better available for the high-frequency case than the method of matched asymptotic expansions. As will be seen presently, this requires some surprising assumptions about various orders of magnitude, and it is difficult to justify these assumptions except to say that they apparently work-to the extent that we can check them.

2. Long Waves " Long waves " and " low-frequency waves " here have precisely the same significance as in Section III,C,l, although now the length and frequency are set by the incoming waves rather than by a forced motion of the ship. The diffraction velocity potential must satisfy the 3-D Laplace equation, the free-surface condition (3.34),the body boundary condition (3.35), and a radiation condition. Since the incident waves are long compared with the transverse dimensions of the body, we assume that the basic assumption in (2.1) applies without any complications; the only large gradients in the problem are those caused by the slenderness of the body. Thus the 3-D Laplace equation reduces approximately to the 2-D Laplace equation, just as in Section III,C,l. When the incident-wave potential (3.33) is substituted into the body boundary condition (3.35) and the slenderness of the body is noted, we obtain the approximate form of the condition

&plan

= oh(in, sin

- n3)e-"" cos o[ 1 + O(E)].

(3.40)

The method of solution may follow that of Section III,C,l or Newman's method. In either case, one must solve the near-field problem for arbitrary body shape by numerical methods. However, the near-field solution can still be expanded in a series like (2.8), and the source strength, a(x; t) =

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131

o(x) exp(iwt), is given by o(x) = ohB(x)e-""

'OS

(3.41)

D.

Of course, this comes from the term in (3.40) containing n 3 , which can be interpreted in terms of an effective instantaneous change in immersion of the cross section. The term in (3.40) containing n 2 is antisymmetric with respect to the plane y = 0, and so its effect is like that of the body oscillating horizon tally. There is an interaction among the cross-sectional flows as a result of the source distribution o(x) given by (3.41). As one might expect from the study of the forced-oscillation problem, the sections interact with respect to the symmetric solution of the 2-D problems, but not with respect to the antisymmetric solutions. These conclusions can come from the Newman-type analysis or from a separately constructed far-field solution that is required to match with the near-field solution. The practical conclusions from this approach were very disappointing when they were first recognized in the early 1960s. A discussion will be found at the end of Newman (1964). The most important deficiency does not appear until force is computed and the equations of motion set up. Then it appears that the hydrodynamic force is entirely of higher order with respect to E than the buoyancy force, which implies that the rigid-body response of the ship to waves is determined primarily by effects that are essentially hydrostatic in nature. Even the inertia of the body itself is of higher order, so that there is no resonance phenomenon in the first-order theory. 3. Short Waves from Oblique Angles Short waves " has the same physical significance as high-frequency waves" (see Section III,C,2). In particular, 1= 27cg/oz = O(E).Now, of course, 1is fixed when the incident waves are given, and we are saying that 1 becomes smaller and smaller as the ship becomes more slender. There is no physical basis whatever for supposing that such a relationship exists. The justification for this approach lies entirely in the possibility that the resulting theory may give us better answersforfinite values of& than a theory in which such an unnatural relationship is not supposed. Practical experience of the past 20 years indicates that this possibility is indeed realized. For the case of waves from a general oblique direction, as in Fig. 6, we can define three distinct wavelengths: I'

"

(i) 1,the actual wavelength, = 2zg/02; (ii) 1, = 1/cos p, the apparent wavelength along the x axis; (iii) 1, = 1/sin p, the apparent wavelength along the y axis. We shall assume in this section that all three of these characteristic lengths are O(E),that is, that does not equal 0, i-z/2, or K. As it turns out, the

132

T . Francis Ogilvie

analysis remains valid in the beam-sea case, p = +n/2. It is not valid if the waves are incident end-on, and the following section will be devoted to that problem. In an asymptotic sense, B can have a value arbitrarily close to 0 (or n) and the theory of the present section remains valid. But in practice, for finite value of E , we must anticipate that there is some range of p on either side of these special limit values in which the theory cannot be used with confidence. Unfortunately, we do not know how large that range is, nor does any kind of solution exist, even in a restricted range of p, in which B can be allowed to approach one of those limits. That is a separate singularperturbation problem. From the expression representing the incident waves, (3.33), it is evident that all three components of the gradient of 4o are “large,” that is, 4ox= - iv cos p 4o +oy

= -iv sin

4oz= v40

1

p 4o

I

= O(40&-1).

(3.42)

Thus one can hardly expect any component of the gradient of 4, the diffraction potential, not to be large in a similar sense. It appears that we cannot take advantage of (2.1) and thus that we cannot reduce the boundary-value problem in the near field from three to two dimensions. However, the ship is still considered to be slender, and so we might guess that it affects the incident waves in a “gradual” way in the x direction. This idea is expressed by writing the diffraction potential in the following form Re[4(x, y, z)ei(’’“-’’x O)I? (3.43) where we now assume that (2.1) can be applied to the function 4(x, y, z). This means that the diffraction wave is nearly periodic along the x axis, but its amplitude and phase vary “slowly.” In the transverse planes, 4 varies rapidly, both because of body slenderness and because of the shortness of the incident waves. (The latter reason will carry over into the far field too.) When (3.43) is substituted into the 3-D Laplace equation, we obtain o = 4xx- 2iv cos p 4 x - v 2 C O S 4~ +~ 4yy+ + z z .

--

O(4)

---

Y

O W 2, (3.44) In solving for the first term in the expansion of 4, we use only the lowest-

order terms

O(4E -

4,)).+ 4zz- (v cos p)24= 0.

(3.447 Thus the appropriate partial differential equation is a Helmholtz equation. One can confirm easily that the linearized free-surface condition is still given by (3.34). We must modify the body boundary condition (3.35) somewhat, because 4 in (3.35) must be replaced by 4 exp( - ivx cos p). It is found

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133

easily that the condition is &$/an

- a4/aN -

oh(in, sin /? - n3)e"(Z-iysin8).

(3.45)

If this is multiplied by exp(-ivx cos /?), it corresponds to the boundary condition (3.40) for the long-wave case. The only difference is that now it is not permissible to expand the exponential factor in (3.45) in a series to be truncated after one term. It is now necessary to find a solution of the Helmholtz equation (3.44'), satisfying the body boundary condition (3.45) and the usual free-surface condition (3.34). We might suppose that the solution will represent outgoing waves, in the sense that ($(x, y, Z)ei(or - yx c- 8 ) ~* (x)evzei(wr- vx CB 8 - vlyl sin 8 ) (3.46)

-

as 1 y I 00, where c , or c - is chosen according to whether y -+ 00. [One should compare this with (3.31a) and (3.31b), representing the outgoing waves in the case of high-frequency body oscillations.] Rather than just assuming outgoing waves at this stage, we might have insisted that the solution eventually match satisfactorily with a far-field solution ;the result is the same. Ursell (1968) investigated the solutions of such problems, and he showed explicitly that, outside of a circle circumscribing the body in a cross section, the solution can be written as the sum of a sourcelike term, which includes outgoing waves, and an infinite sum of wave-free terms, representing higherorder singularities at the origin. Bolton and Ursell (1973) computed the force per unit length on an infinitely long circular cylinder in oblique waves, but they did this by solving the corresponding forced-oscillation problem? and using the Khaskind formula, which was mentioned near the beginning of Section III,D,l; thus they did not actually solve the diffraction problem, although their method of solution in a series could have been adapted to the purpose. Troesch (1976) has solved the above 2-D problem for rather arbitrary conditions and body shapes; he uses an integral-equation technique, and it appears that this is the only technique available for solving problems with arbitrary body shapes. In the forced-oscillation problem of Section III,C,2, one has the choice of solving an integral equation or of using conformal mapping and complex function theory. The latter is not possible when the Helmholtz equation is to be solved. The Bolton/Ursell method can be applied confidently only if the body is really a circular cylinder. -+

In the forced-oscillation problem for vertical motion, the velocity of a point on the bod) is given by

iotaei(or-

8)

II ~ 0 6

which may be compared with the expression in (3.10).

T. Francis Ogilvie

134

As in the discussion of the high-frequency forced-oscillation problem, we can only say that the solution of the problem before us can be obtained numerically and that it has all of the required properties. Everything works well even in the beam-sea problem, for which B = kn/2. However, as Ursell (1968) showed, the head-sea problem (for which fl = 0) cannot be treated as the limit of a sequence of problems with /3 + 0. The limit problem must be considered by itself, and a certain aspect of the full 3-D problem must be retained. The far-field solution of this problem is not required if we want just to compute, say, force on the body. The far-field solution only confirms that we have used the appropriate radiation condition in the near-field problem. Nevertheless, it is worthwhile to set up at least part of the far-field solution. As usual, the far-field solution is fully three dimensional, that is, it is a solution of the 3-D Laplace equation. As E + 0, the ship shrinks down to a line, just as before, and so the potential function is harmonic in the closure of the lower half-space except on the x axis. The singularities on the x axis will be a mixture of symmetrical and antisymmetrical singularities, and so we can work with sources and transverse dipoles. The sources provide the more interesting results, for the antisymmetric part is well-behaved even in the limit p -+ 0. So consider just the source part of the singularity distribution. We have already argued that the diffraction wave near the body is almost periodic along the x axis, and so we should expect the source density in the far-field solution also to be almost periodic in the same way. Therefore we assume that the source density can be expressed as Re[a(x) exp(iot - ivx cos B)], which should be compared with (3.4). The function ).(a varies “slowly,” that is, a’(x) = O(a). We continue to represent the potential in the far field as in (3.5), letting the x dependence come out as it will. Then 4 can be written as in (3.6),but with &(x) replaced by ) . ( a exp( - ivx cos /I) At.distances many wavelengths away from the singular line, we can use a stationary-phase approximation on +,just as we did previously to obtain (3.8). Now, because of the modified source density, the result is as follows $(x, y, z) (i/sin B)a(x - ( yI cot ~ ) e v z e - i Y ( X C 0 6 ~ + ~ y ~ (3.47) sin~)

-

9

provided that o<x-

lyl cot/3
(3.47‘)

Figure 7 depicts this result: To the leading order of magnitude, there is (diffraction) wave motion only in the region indicated by wave-crest lines. At any point within this region, the wave amplitude depends on the source density at only one point of the x axis, namely, the projection back along a wave ray at the angle B or -/3 (according to whether y 3 0). Compare with

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135

FIG. 7. Skewed short waves generated by a line of pulsating sources.

the discussion following (3.8). What shows clearly in (3.47) but not in the . /l is very small, sketch is that the amplitude is proportional also to l/sin /IIf the wave motion is confined to a very thin region in which its amplitude is very large. The description above, represented in Fig. 7, becomes more nearly valid as wavelength becomes closer to zero. Aside from the difficulties that we must now anticipate when p -+ 0, we can have a somewhat better concept of how the diffraction waves must appear in the far field. Close to the body, there are reflected and transmitted waves almost as if the body had infinite length and uniform cross section. These waves are seen for some distance away from the ship, but more so toward the right-hand side in the figure. They gradually disperse in all directions, so that at very great distance there are waves propagating outward in all directions, but with more energy moving in the directions suggested in Fig. 7. Just as Fig. 7 can be compared with Fig. 4, we could certainly draw a radiation pattern for the present case that would look like a skewed version of Fig. 3. If we distribute transverse dipoles with density Re[p(x) exp(iwt - ivx cos

p)]

along the x axis between x = 0 and x = L we can express the far-field potential readily by differentiating (3.6) with respect to - y and substituting this dipole distribution for n. Then, if we estimate the potential in a region

136

T. Francis Ogilvie

near the singular line, we find that $(x, y , z)

-

-vsgn y p ( x - ( y ( cot p)e’ze-i’(xcasP+IYlsin~) , (3.48)

under the same condition, (3.47’), imposed on the validity of (3.47). The amplitude of the disturbance does not grow as p decreases toward zero, as was the case with the distribution of sources, although the picture otherwise is very much the same as indicated in Fig. 7. (Of course, the disturbance is now antisymmetric with respect to y . ) One may expect that the magnitude of p will decrease as p --* 0, as suggested by the body boundary condition, (3.45), in which the antisymmetric term is proportional to sin 8; thus we have good reason to expect the antisymmetric part of the solution to be well-behaved in the same limit that causes much trouble with the symmetric part. At present, none of the more-or-less standard methods for computing ship motions and wave loads on ships involves the solution of the Helmholtz equation for the 2-D cross-sectional flows. Only Salvesen et al. (1970) avoid this situation in a way that is fundamentally correct: They use the Khaskind formula to eliminate the need for finding the diffraction solution. In practice, the various methods in use give predictions that are quite similar, even quantitatively, but this must be considered as a matter of luck. 4. Short Waves from Directly Ahead

Two facets of the problem in the preceding section suggested that there will be trouble in the case of waves from directly ahead, that is, for p = 0: (i) The near-field problem has no physically acceptable solution in this limit. This was shown by Ursell (1968). (ii) The far-field solution is ill-behaved near the singular line as p + 0. It is convenient to begin by discussing the far-field problem this time, for its properties suggest how to modify the near-field solution. Once again, we assume that the lowest-order term in the far-field expansion can be expressed in terms of a distribution of sources on the x axis, the density of the sources being given by

Re[a(x) exp(iot - ivx)]. The potential in the far field is then given by (3.5) and (3.6), with &(x) there ) - ivx). Clearly, this potential cannot be evaluated near replaced by ~ ( xexp( the singular line by the same method as previously, since the result is undefined for p = 0, as (3.47) shows. Faltinsen (1971) treated this problem by restating the potential in a Fourier-transform manner. m dleily+“k’ +l’)’i’ dkeikxo*(k v ) lim 4(x,y, z) = 4n2 - m p+o - (kZ P)’’2 - (o- ip)’/g ’ (3.49) 00

j

+

1

+

137

Ship Hydrodynamics where o*(k)is the Fourier transform of o ( x ) o*(k)=

dxo(x)e-'"". --m

The quantity p is a fictitious viscosity coefficient in the manner of Rayleigh. This form of the potential has been used by several investigators, e.g., Newman and Tuck (1964) and Ogilvie and Tuck (1969); it is completely equivalent to the previous expression. It satisfies the 3-D Laplace equation and the usual free-surface condition, (3.34),and it represents outgoing waves far away. In order to define the Fourier transform completely, we set o(x) = 0 outside the segment 0 < x < L of the x axis. Near the singular line, Faltinsen obtains the following estimate of the above potential' &(x, y , z)

-

evz-ivx

d50(5)

O(ae-

-

f

v

1

+ v I y Io(x) + io(x)/2

(3.50)

Ob)

112)

The orders of magnitude shown are valid for y = O(E),that is, in the near field. It is this expression that should match with the near-field expansion. An important point to notice about the result in (3.50)is that the leadingorder term is a function of x only, except for the factor exp(vz - ivx). When a one-term near-field solution is found, it must match this term. The near-field problem is formulated just as in the preceding section. The potential is expressed in the form

(3.51)

Re[$(x, y, z)e'(O'- v x ) ] ,

which should be compared with (3.43).It is assumed that = 0(4), so that the assumption (2.1) applies. This potential 4 must satisfy the Helmholtz equation (3.447, with B = 0, and it must satisfy the body boundary condition (3.45). Ursell (1968) studied the general problem so set, at least in the region outside a circle circumscribing the body in a cross section. He showed that the solution in such a region can be expressed as the sum of three kinds of terms, 4)

+(x, Y , Z) = A(x)evz+ BO(x)S(y, Z )

+ C Bn(x)Fn(y, z),

(3.52)

1

'

The analysis required to obtain this estimate is long and tedious. The reader is referred to Faltinsen (1971) for full details.

T . Francis Ogilvie

138 where

(i) evz[after being recombined with the t and x dependence, as in (3.51)] represents a regular wave propagating along the x axis, (ii) S ( y , z) represents a sourcelike term, and (iii) F,(y, z) represents a wave-free term that is more singular than the source term at the origin, but which decreases rapidly to zero as lateral distance increases.

[Of course, there is no x dependence in Ursell's analysis except for the sinusoidal variation of all quantities, as in (3.51).] The sourcelike term has a strange behavior: As 1 y 1 -+ co,it behaves as v 1 y 1 exp(vz), that is, its amplitude increases linearly with distance, without limit. In (3.50), we observe that the second term has just this kind of behavior, but that term is not the lowest-order in (3.50), and so the matching cannot be performed directly. In fact, the first term in the near-field solution cannot contain a sourcelike term at all: In the solution represented by (3.52), the strange behavior of the source term dominates far away, and so the better-behaved terms are not available for matching at the same time. However, without the source term, it is doubtful that the body boundary condition can be satisfied. The resolution of this difficulty is to assume that the one-term near-field solution consists solely of the first kind of function listed above, that is, the solution represents a regular wave propagating along the body, and all of the coefficients in (3.52) except A ( x ) are identically zero. This eliminates the source term, but then, in order for the body boundary condition to be satisfied, one must set this traveling-wave term precisely equal to the negative of the incident wave

4(x, y, z)

-

- (gh/w)eYZ.

(3.53)

This matches the far-field solution if it corresponds to the first term of (3.50), that is, (3.54) Since ~ ( xhas ) not been specified previously, we consider (3.54) as an integral equation to determine it. In fact, it is an Abel equation, for which a solution is known G(X

) = (2/.rrv~)"~ (gh/~~)e'"/~.

(3.55)

Thus we have the first approximation: In the near field, it is given by (3.53), and, in the far field, we must substitute the source density as given in (3.55) back into the expression (3.49) or its equivalent. This result is not really useful, for we have reached the conclusion that

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139

there simply are no waves in the near field in this approximation. It will obviously be necessary to obtain another term in the near-field expansion. However, the interpretation of even this result is not trivial. It implies that the ship creates disturbances starting right at the bow in such a way that they cancel the incoming waves locally. These disturbances propagate outward, of course, but mostly they propagate along the body, in the same direction as the incident waves; this preferred direction appears because the disturbances are created with a very special phase relationship, controlled by the incoming wave that has to be canceled. To the order of magnitude that we are considering here, there is no effect in the bow direction from the disturbance emanating from any particular cross section ; this appears explicitly in the upper limit of the integral in (3.50). Physically, it means that such disturbances moving toward the bow cancel each other through interference. Before considering the second approximation, we should note that the solution obtained so far really violates an assumption on which it was based, namely, that the derivative of o(x) is a “small” quantity. In fact, o(x) is weakly singular at x = 0, and its derivative is much worse. Faltinsen argues that a different method of solution is needed in a very small region near the bow and that such a method would give a o(x) that does behave properly there. The fact that o(x) becomes infinite at x = 0 suggests that o(x) is really of lower order than supposed here, at least in some small neighborhood of x = 0. Otherwise, in regions not too near the bow, Faltinsen’s solution is probably valid. We shall return to this subject presently, after we have seen how Faltinsen’s solution develops further. The orders of magnitude in (3.50) suggests that the second-order terms in the near-field expansion are an order of magnitude & I i Zsmaller than the first-order solution. Then it may be expected that the second approximation will satisfy the 2 - D Helmholtz equation (3.44’)and the free-surface condition (3.34) satisfied by the first-order term. On the body, there will be a homogeneous Neumann condition, since the normal velocity component of the incident waves has been canceled by the normal velocity component of the first-order diffraction waves. Thus all of the conditions in the near field are homogeneous conditions, except possibly the condition at infinity. This kind of problem is perhaps not encountered very often, but it should not be surprising here: The diffraction waves have canceled the incident waves in the nearfield but certainly not in the far field, and so the only disturbance that can appear next in the near field must come directly from the far field, which is the same as saying from infinity in the near field. This is still the same kind of problem that Ursell (1968) studied, the solution of which can be expressed as in (3.52). It may be noted that the source term and the wave-free terms together constitute a complete set of “

”

T . Francis Ogilvie

140

functions in terms of which an arbitrary boundary condition on a circle circumscribing the body can be satisfied. We can set A ( x ) equal to an arbitrary value, say, A ( x ) = 1, and find a set of coefficients for the other terms such that (3.52)still satisfies an arbitrary boundary condition. (While this is possible strictly only for a circular boundary, there is another form of the solution in terms of a Green function, and that form allows a similar result to be obtained for any boundary shape of interest to us.) Suppose that we do find a solution [that is, we find B j ( x ) , j = 0, 1,2, such that A ( x ) = 1 and the normal derivative of the potential vanishes on the body boundary. Then the solution for any other value of A ( x )is simply A ( x ) times the solution just found. Now let the two-term near-field solution be expressed as follows 4 ( x , y, z ) = - ( g h / u ) e V Z+ A ( x ) evs + Bo(x)S(y,z ) +

[

00

C B,(x)F,(y, z) 1

1

.

(3.56)

The quantity in brackets is understood to satisfy the homogeneous conditions of the near-field problem. For large l y l , the summation vanishes rapidly, and the potential has the following asymptotic behavior 4 ( x 3y,

2)

-

[ A ( x )- ( g h / ~ ) ] e "-"[A(x)B&)]nv I Y I evz,

(3.57)

the second term coming from Ursell's result that S(y, z )

N

-nvIyIevz

as

( y l + co.

(3.57')

Expression (3.57) can now be compared with (3.50), and we see that the troublesome terms match if we set 7LA(X)BO(X)= - o ( x ) .

Since B o ( x ) was already determined, and o ( x ) was found as part of the first-order solution,+we now know A ( x ) and thus the two-term near-field solution, as given in (3.56). Since (3.56)represents just the diffraction potential in the near field, one + The reader may note that the matching between (3.57) and (3.50) is still not perfect. It appears that the last term in (3.50) should match the term A ( x ) exp vz in (3.57), but it does not. The reason is that (3.50) is not entirely adequate for determining fully the second-order solution. At the present stage, we should start with a two-term far-field expansion and evaluate it in the same way that we did previously to obtain (3.50). All that is needed for this purpose is to replace ).(a by u,(x) u2(x), where u 2 = o(ul). The final matching then determines u 2 . However, this quantity is not needed in the near field unless we contemplate obtaining yet a higher-order solution. See Faltinsen (1971) for details.

+

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141

must add to it the potential of the incident wave to have the description of the complete fluid motion. As already noted, this process will simply cancel the first term in (3.56), and so the quantity A ( x ) (with the terms that it multiplies) is all-important for finding the interesting physical variables, for example, pressure on the ship hull. Also, it is necessary to solve the near-field boundary-value problem involving the Helmholtz equation. Until Troesch (1976) did this recently for arbitrary geometry, it had been accomplished for circular cylinders only. Faltinsen performed some calculations based on his analysis, and he was able to compare his results with some unpublished experimental data for a spheroid in head waves, the experiments having been performed at the University of California (Berkeley) by Dr. C. M. Lee. As expected, of course, the agreement was poor near the bow, but over most of the length of the body Faltinsen’s theory predicted fairly well the cumulative effects of the diffraction waves. This good agreement was realized when 1/L = 0.5, but it deteriorated rapidly with longer incident waves. Considering that the theory is based on an assumption of “short waves,” in the sense that 1/L = O(E),one may consider it remarkable that the theory gave good results for such a large value of 1/L The agreement was better on the bottom of the body than near the waterplane, a fact which may have been due to the experimental difficulties of measuring pressure close to the free surface. For a body of uniform cross-sectional shape and size, two features of Faltinsen’s solution are worth noting explicitly: (i) The amplitude of waves and of pressure fluctuations decreases with x - ‘ I 2 . (ii) The phase of the waves is advanced n/4 with respect to the incident waves. These phenomena have been confirmed independently by Ursell (1975). He formulated more idealized problems, but he then obtained his results by rigorous asymptotic methods, rather than by using matched asymptotic expansions. In one of his problems, he postulated the existence of an infinite horizontal circular cylinder, half-submerged, on which the normal velocity component is

a4/ar = - e i v x h ( x ) ( ~ / a r ) e v z , where on the forward part, - m < x I -1, on the aft part, I I x < 00, 1 and h ( x ) is an infinitely differentiable increasing function in the part -I 5 x I 1. The boundary condition on the aft part is just the condition that would be imposed on a semi-infinite hull of circular cross section which is located in head waves. Since Ursell’s intention was to investigate the wave motion far behind the bow of a ship, the nature of the disturbance near the h(x) =

i

0

T. Francis Ogilvie

142

bow itself was of little interest, and so the condition to be satisfied in - 1 < x < 1 could be left vague.+ At any fixed y, Ursell found that the solution of his problem had just the limiting behavior for x + cx) that Faltinsen discovered. He found one more result too: The region in which these phenomena occur grows in the positive x direction as x ’ / ~Ursell . describes the situation by saying that the incident waves are refracted away from the cylinder. There is a “shadow region near the cylinder (or ship), in which the wave motion becomes steadily smaller in the direction of increasing x. It may also be noted that the amplitude of the wave motion in this shadow region increases linearly with increasing 1 y I, at least in a region near the ship; this is the meaning of the last term in (3.57), for example. The most important defect in Faltinsen’s theory is the singularity at the bow, and of course Ursell’s analysis does not shed any light on that difficulty. Maruo and Sasaki (1974) developed a modified version of Faltinsen’s theory in order to remove this defect. They effectively treat the firstand second-order problems simultaneously in a kind of “ composite problem.” Thus, suppose that the first-order near-field solution is just C(x)e”’-and leave it unspecified otherwise. Then a two-term near-field solution can be expressed ”

4(x,

+

i

y, z) = C(x)evZ A(x) e”’

30

+ Bo(x)S(y,2) + 1 B,(x)F,(y, 1

I

z) . (3.58)

Except for the first term, this is identical to (3.56). For large I y 1, this potential shows the following behavior 4(x, y, z )

- [ C ( x )+ A(x)]e”’

I I

- A(x)Bo(x)av y evr.

(3.58’)

[Cf, (3.57).] This should match (3.50), which then gives us the following relationships

- aA(x)B,(x) = a(x);

’

(3.59a)

In the same paper, Ursell considered another problem in which the nonhomogeneous body boundary condition was restricted to a finite length of the cylinder. The resulting wave motion diminished rapidly enough along the aft part of the cylinder that it was quite negligible compared with what he found in the problem above. Thus the “ vagueness ” mentioned causes a negligible error in the results.

Ship Hydrodynamics

143

Substituting these relationships back into (3.58), we obtain for the near-field solution

Except for a factor o ff in the integral term (which we cannot explain), this is the same as Maruo and Sasaki's potential, their Eq. (17). To show this, one must set a(.) = -ni?(x), this new source density being the same as their. The source density a(.) and the coefficients B,(x) are still to be determined. Let us do the latter first, in the following way: Solve a near-field problem such that, on the body surface, m

( ~ nlev')

+ ~ , ( x ) ~ (z y) +, 1 B , ( x ) ~ , ( y ,z ) 1

J

= 0.

Note that this is exactly the same procedure used by Faltinsen for his second-order problem, as discussed just before (3.56). If we multiply this quantity by A ( x ) = -a(x)/nB,(x), we obtain - - a1( x ) - dS - - a ( x1)

n

dn

n

jy!L----B ( x ) d F , 1 a(.) B o ( x ) an

d

(e"). nBO(x)dn

(3.61)

Now we write the body boundary condition for the actual problem at hand (d/dn)[$

We obtain

+ (gh/co)eVZ]= 0.

4 from (3.60) and then

use (3.61), obtaining

(3.62)

Since the normal derivative of the exponential function cannot generally be zero, the factor in braces ahead of it must be zero. This is a Volterra integral equation for a ( x ) , which must be solved numerically. Except for a factor in the integral term, this is Maruo's equation, and he did solve it. Now all quantities are known, and the physical quantities of interest can be computed from (3.60). Maruo and Sasaki performed experiments with a model made entirely of circular cross sections and compared the experimental data with their computations. They generally found better agreement than Faltinsen did. In particular, their predictions are well-behaved and quite reasonable near the

4

144

T. Francis Ogilvie

bow, and their theory seems to remain valid at considerably larger wavelengths than Faltinsen’s. In addition, they even found good agreement for the pressure on the body near the free surface. Both theories are in principle short-wave theories. When Faltinsen’s theory gave incorrect predictions for AIL > 0.5, it was believed that this was simply a manifestation of the restrictiveness of the initial assumption. The results of Maruo and Sasaki suggest that the error at the bow in Faltinsen’s theory has a more pervasive effect, especially for larger wavelength, and that the theory may not be greatly restricted by the assumption of short waves. 5. Extremely Short Waves If A = o(E),we have a class of diffraction problems that correspond to the very-high-frequency forced-oscillation problems of Section III,C,3. There are several interesting scientific aspects of such problems, but the practical importance of such considerations in ship hydrodynamics is rather questionable. Therefore we shall only mention the kinds of phenomena that occur when wavelength is small compared even with the transverse dimensions of the body. Sinusoidal incident waves can still be described by the velocity potential given in (3.33), even if the wavelength is very small. As usual with such waves, the amplitude of all flow variables decreases exponentially with depth. Thus, when such waves encounter a partially submerged body, only the shape of the body near the free surface has an effect on how the waves are diffracted. If the sides of the body are perpendicular to the undisturbed water surface, one can represent the diffraction potential in the form $(x, y, z) . exp(iot) = &x, y)exp(vz + iot),and the new potential satisfies the equation

4xx+ 6yy+ v 2 4 = 0, instead of the Laplace equation. This is, of course, the wave equation, and so the problem reduces to a 2-D diffraction problem of a kind studied in several branches of physics. Since the waves are assumed to be very short in comparison with body dimensions in the waterplane, a first approximation can be obtained in the manner of geometrical optics analysis. Diffraction effects at body edges and the propagation of waves into shadow regions can all be studied by the methods of physical optics (or of acoustics). This is a large subject, which it would not be appropriate to discuss here. What is of more interest in the gravity-wave problem is to consider the diffraction effects of a body which is not strictly wall-sided at the undisturbed free surface. If the tangent to the body surface is not vertical, little progress can be made on the mathematical analysis of the problem: There is then generally a singularity in the solution at the juncture of the body surface and the undisturbed free surface, and the validity of linearized theory is questionable. (Nonlinear theory seems to be out of the question at present

Ship Hydrodynamics

145

for such problems!) And so we must restrict ourselves to problems in which the tangent is vertical but the curvature is nonzero at the waterplane. The prototype problem of this type was studied by Ursell (1961). He considered the reflection and transmission of incident 2-D waves by an infinite horizontal cylinder. Not surprisingly, he found that the amount of wave transmission depends essentially on the curvature of the body at the waterplane. The analysis required to show this is very complex, and it will not be discussed here. The physical phenomena involved in the corresponding 3-D problems must be greatly more complicated, and so even less is known about 3-D problems. Although little can be said about the diffraction of very short water waves, a motivation has arisen in recent years for studying this subject more intensively. There is some indication in problems of steady forward motion that the ship-generated waves can be considered as very short waves that are diffracted by the ship itself. Such problems are yet more difficult than the simple diffraction problems posed here, and so it makes sense to study the simpler problems first, in the hope that the results will suggest better how to analyze the more difficult forward-motion problems. In both kinds of problem, the wave motion is confined to a thin surface layer near the level of the undisturbed free surface, and the geometry of the ship right at the waterplane is undoubtedly of primary importance in determining how the waves are diffracted, although there is a superposed steady flow in the forwardmotion case. These are still problems for the future.

IV. Slender Ships in Steady Forward Motion A. INTRODUCTION

Of all of the problems in ship hydrodynamics to which one might try to apply the slender-body theoxy of aerodynamics, the problem of steady forward motion of a ship seems to be the most obvious. And yet attempts to do this have resulted generally in failure. In the analysis of loads on a ship in a seaway and in the prediction of ship motions, the standard methods in use around the world are all based to some extent on the concept of the slenderness of a ship, and these methods are quite successful in providing predictions for engineering purposes. Yet, in the prediction of the flow around a ship in steady forward motion, little usable information has come from slender-ship ideas. Still, a ship is a “slender body,” and one ought to be able to realize thereby some advantages in terms of simplifying the intractable exact problem for a ship in steady motion. As we shall try to show, there is still hope of

146

T. Francis Ogilvie

doing this, and some of the negative conclusions that have been voiced about slender-ship theory are based on an incorrect application of the concepts of slender-body theory. + The basic assumption of slender-ship theory is the same as in the slenderbody theory of aerodynamics, as stated in (2.1). However, those relationships must be used with care and they must be modified or augmented as the occasion arises. There are no simple rules for knowing how or when to do this. We must carefully study the physics of every problem that we consider and check whether the assumptions are likely to be valid. . The first attempt to apply slender-body theory to this problem was made by Cummins (1956). Unfortunately, he did not obtain any firm results that could be checked experimentally, and his work was forgotten for many years. During the early 1960s, several investigators undertook to develop a slender-ship theory, notably Vossers (1960), Maruo (1962), and Tuck (1963). Their approach was notably different from that of Cummins, and they (and their followers) all reached an impasse when they found that their theory led to unreasonable conclusions. Only since 1970 has an understanding developed of the reasons for such results, and one can say with some justification now that their approach had to fail, because some critical aspects of the physical problem were not adequately treated. At this writing, we believe that we know what was wrong and perhaps even how to correct it, but the results of this new insight have not yet been tested either. In spite of the tentative nature of recent results, we shall concentrate on them here, for that is the current status of slender-ship theory for steady forward motion. The formulation of the problem itself is easy. We assume that there is a uniform stream moving in the positive x direction. The velocity potential will be written consistently in the form

u x + $(x, y , z), where $ satisfies the Laplace equation in three dimensions, the body boundary condition

&plan

’

= -Un,

(4.1)

It must also be recognized that the computation of a quantity such as wave resistance places greater demands on the theory than the computation of, say, heave and pitch force. Resistance depends on the difference between quite large forces on the forebody and the afterbody, and these separate forces may be strongly affected by errors in the theory associated with the ends. Vertical and transverse forces, on the other hand, result from the integration of the pressure distribution over the entire length of the ship, a large part of the total generally coming from the midbody, which really is “slender.” So one should expect better results in the ship-motion problem. Similar good fortune is found in the computation of sinkage and trim of a ship in steady motion, as shown by Tuck (1966, 1967).

147

Ship Hydrodynamics

on the hull surface, a radiation condition, and two free-surface conditions, (i) the dynamic condition

+ +; + $:}

p / p = - ~4~ - g i - ${&

=0

on

z

= i(x,

Y ) , (4.2)

and (ii) the kinematic condition (I/

+ &)L + & i,- 4%= 0

on

z = i(x, Y ) .

(4.3)

It will be assumed that the ship surface is given a priori, perhaps in the form S(X,

y , z ) = 0.

(4-4)

First we shall briefly review the theory of the early 196Os, following the approach of Tuck (1963). Under certain rather restrictive conditions, we find that this approach gives fairly good predictions of some aspects of the flow around a ship, particularly along the midsection of the hull if forward speed is fairly high. A section will be given over to a discussion of the failure of this theory, with special emphasis on understanding the failure so that a better theory can be constructed. Then some recent attempts to develop a theory valid near the bow will be described. The approach here will be remarkably similar to that of Cummins (1956), already mentioned. The essential added element is the thorough consideration of the divergingwave system. Finally, a section is presented on the problem of ships at very low speed. In all of these cases, it is important to recognize that what we call “low speed or “high speed depends on the fluid mechanics of the problem. These are, of course, relative terms, and they have no essential relationship to what a naval architect might mean by the same terms. Furthermore, one must face the situation that, from a fluid-mechanics point of view, the flow around part of a ship may be “low-speed” and the flow around another part may be “high-speed.” The meaning of this statement will be discussed presently. ”

”

B. ORDINARY SLENDER-SHIP THEORY Let us assume that (2.1) can be applied in a formal way to the problem formulated above. The body boundary condition (4.1) becomes

a4/aN = -un,. The right-hand side is O(E),since n, suppose that

=

O(E).Therefore, in view of (2.1), we

4 = O(E2),

(4.5)

T . Francis Ogilvie

148

just as in the aerodynamic case [see (2.5)]. The Laplace equation reduces approximately to a 2-D Laplace equation. The free-surface conditions become

U$x

+ g[ + ++: = 0 $== O

on z

= 0,

on z = O .

(4.6) (4.7)

The reasoning is so similar to that involving (3.13) to (3.16) that it will not be presented here. Equation (4.7) comes from the kinematic condition, and it says that we should consider the free surface as a rigid wall when we seek the first-order approximation. As in the case of low-frequency oscillations, we must interpret this condition to mean that the vertical component of fluid velocity at the free surface is small compared with the horizontal component. When the potential problem has been solved, we can then find the corresponding free-surface elevation [ from (4.6); note that (4.8) The pressure in the fluid can be computed from (2.15) if we augment the right-hand side by the hydrostatic pressure term -92. This near-field problem can be solved by classical means, although numerical procedures must be developed for handling arbitrary body shapes. Outside a circumscribing circle around the body cross section, the solution can be represented as in (2.8). From the body boundary condition and the continuity condition, we can determine readily that [ = O(E2).

.(x) = 2US’(X),

(4.9)

where S(x) is the area of the submerged cross section at x. Thus the near-field potential has the form +(x, y , z ) = [ US(x)/x] log r

+ Ao(x) + .. .,

(4.10)

where the omitted terms vanish as r + co.As in the forced-oscillation problem, Ao(x) cannot be determined from near-field conditions, since the boundary-value problem is a Neumann problem, to which an arbitrary constant (at each x) can always be added. In the far field, the ship shrinks down to a line, and the fluid disturbance appears as if it might have been caused by a line distribution of singularities, which we can take to be sources. The source potential must satisfy the free-surface condition $xx

+

=0

on z = 0,

with

K =g/U2.

(4.11)

This condition can be obtained from (4.2) and (4.3) by eliminating i and

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149

retaining just the linear terms. The logic for keeping precisely these terms is quite analogous to that used for obtaining (3.21) in the case of low-frequency ship oscillations. The source potential must also satisfy a radiation condition [see Wehausen and Laitone (1960) for a thorough discussion]. In the slender-body problem, it is convenient, as in Section 11, to use a Fourier-transform representation of the potential for the line of sources. One such form, given by Tuck (1963), is as follows +(x, y , z ) = -

1 2 2 ~

J’ “

-m

dkeik”6*(k)Ko(I k I r )

(4.12) The first term here is twice the expression in (2.1la), with the substitution for a,* there made from (2.12a). Thus the approximations in (2.11a) and (2.13.) can be introduced. In the second term of (4.12), we can set y = z = 0 to obtain an evaluation of that term near (and on) the x axis. This term can be reduced to an expression in terms of standard functions; the reader should see Tuck (1963) for details. One obtains from (4.12)

(4.13) The first part of the integral in (4.13)can be recognized as the same quantity that was calledf(x) in (2.13) (except for a factor of 2). The rest of the integral in (4.13) represents interactions among the cross sections resulting from the presence of the free surface. Once again, the potential in the far field has been evaluated near the singular line, and it reduced to a sum of two parts: (i) a part that represents the flow caused by a 2-D source at the origin, and (ii) a part representing the interactions among the cross-sectional flows. Matching of (4.13) with the near-field expression in (4.10) immediately yields the value of 6(x) 6(x) = US’(x).

(4.14)

Then the interaction terms in (4.13) can be evaluated, and this gives Ao(x),so that the one-term near-field expansion is completely known.

T. Francis Ogilvie

150

It is now evident that the procedure for solving this problem is very similar to that in Section I1 for the aerodynamic problem. The only significant difference is the form of the far-field solution (4.12)and its evaluation near the singular line. The meaning of the solution is quite different. In particular, we recall that ship-generated waves are located predominantly downstream of the cause of the disturbance, and this fact ought to be evident in the expression for interactions among the cross sections, (4.13). In fact, the Bessel function in the integrand of (4.13) has a logarithmic behavior for small values of its argument, and it will be found that this part of the Bessel function cancels the logarithm term upstream and doubles it downstream. The total amount of upstream effect in (4.13) is very small indeed, as expected. The difference by a factor of 2 between the source strengths in the infinitefluid and the free-surface problems is simply the result of our defining S(x) as the cross-sectional area of the submerged part of the body in the latter problem. In the rigid-wall problem of the near field, this area is effectively doubled with respect to its hydrodynamic effects. One of the physical quantities of most interest in such problems is the wave resistance. In principle, this can be computed by integrating the pressure over the body or by using the momentum theorem to obtain an expression involving the solution at infinity only. In the latter case, one can use the far-field representation in terms of a line distribution of sources to obtain an especially simple result, namely, that the resistance is

p

-K

IK

a

k 2 dk (k2- K ~

)

I&*(k) (’, ~ / ~

(4.15b)

where 6 * ( k ) is the Fourier transform of 8(x). In (4.15a), it is necessary to define &(x) for all x by setting it equal to zero outside of the line 0 < x < L. From (4.15b), it is evident that the wave resistance as predicted by slendership theory is always positive. This has been the subject of some controversy, largely because incorrect manipulations have sometimes been made on the integral in (4.15a). The form of the free-surface disturbance can be found in the near field by substituting the potential into (4.6). The term 4x involves the A, term of (4.10), which is the same as the integral term in (4.13). Thus the water elevation includes a part that is associated with Froude number. The quadratic term in (4.6) yields no such effects, of course.

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151

C. FAILURE OF ORDINARY SLENDER-SHIP THEORY It has long been known that the quantitative predictions from the above theory are poor. This has generally been attributed to the fact that the source distribution in the far-field solution is located right at the level of the undisturbed free surface. In contrast, for example, in thin-ship theory [see, for example, Stoker (1957) or Wehausen and Laitone (1960)], there is a distribution of sources over the centerplane with density given by 6 ( x , Z ) = 2Ub,(x, z),

(4.16)

where b(x, z ) is the hull offset. Here the total source strength at any cross section is US’(x),in agreement with the slender-ship result, (4.14). However, the wave-making effectiveness of the sources in thin-ship theory is attenuated strongly by their depth, whereas there is no such attenuation in slender-ship theory. There must obviously be a significant difference in the respective conclusions of the two theories. However, this is an oversimplification of the situation. There is an even greater effect that can arise from the peculiarities of the flow caused by a line distribution of sources which is insufficiently smooth at the ends. The failure of slender-ship theory can be catastrophic in this case, and it gives some insight to consider such a case in some detail. It will be seen that “end effects” lead to a completely wrong answer. Such a catastrophic case arises if we try to apply slender-ship theory to a hull which is wedge-shaped at, say, the bow. In a region near the bow, let the offsets be given as follows b ( x , Z) =

ux

+ ...,

- T < z < 0, x > 0, otherwise,

where the omitted terms represent a smooth downstream continuation of the hull shape. The cross-sectional area and the slender-ship source density are, respectively,

+ 6 ( ~=)~ U T U H ( X+)..., S ( X )= ~ T u x H ( x ) ...,

(4.17a) (4.17b)

where H ( x ) is the Heaviside step function, equal to 0 for x < 0 and 1 for x > 0. The first formula for wave resistance, (4.15a), requires the value of 6 ’ ( x ) ,which is 6 ’ ( x ) = 2 U T a 6 ( x ) + ..., where 6(x) is the Dirac function. It is easy to see that (4.15a) gives an undefined (infinite) answer for wave resistance, especially since the Bessel

T. Francis Ogilvie

152

function is logarithmically infinite for 5 = x . It does not matter what the rest of the ship is like; the predicted wave resistance is always infinite because of the wedge-shaped bow. As will be seen presently, this anomaly can be attributed to the lack of a finite depth for the source distribution, but it is clearly not the kind of error that one would expect from such a shortcoming. In fact, the difficulty is more closely related to the degree of singularity of the flow caused by a point disturbance located on the free surface. It is well known, for example, that a pressure point applied to the free surface of a stream causes an infinite wave resistance; physically, this occurs because the point disturbance causes large-amplitude very short waves on its track, and these waves contain an infinite amount of energy. Apparently the same thing happens if the disturbance takes the form of a line distribution of sources with a discontinuity in source density. We can devise a line distribution of sources on the free surface that gives the same wave resistance as the thin-ship centerplane distribution of sources. Let this distribution be given by C ( x ) . It will be defined in terms of its Fourier transform, as follows m

~ * ( k=)

j

0

m

j

dxe-ik"C(x)= (K/n)1/2

dxe-ik"

dz&(x, Z ) P / K , - T(x)

-m

-03

(4.18) where 6 ( x , z ) is the usual thin-ship source distribution. [Note that we cannot set X ( x ) equal to the inner integral in the last expression, since it is a function of k.] If we substitute this expression for C * ( k ) into (4.15b)in place of &*(k), and if we then change variables by letting k = K sec 8, we obtain for the wave resistance d8 sec38 (P'

R=--

+ Q2),

(4.19)

where

P(8) + iQ(8) =

5 1&(x,z)eKZ

secZ

eeiKx

' dx dz,

(4.19')

CP

the last integral being taken over the centerplane of the ship. Equation (4.19) will be recognized as a standard form of Michell's integral, the wave resistance of a thin ship. Thus the line distribution of sources, (4.18), generates the same wave resistance as the centerplane distribution usually used in thinship theory. Since thin-ship theory does not lead to disastrous end effects (although it has other defects), we can learn from this artificial distribution of sources on

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153

a line something about how and why slender-ship theory comes to such grief. In order to understand the nature of the artificial line distribution, we consider some special cases. Point Source. Suppose that q x , z) = rJoS(x)6(z- zo). Then C*(k)= ~

~

d

‘

~

~

~

~

~

,

which implies that

1

W

C(x) =?! ! 2a

dkeikxek’zdr

-m

I I )1/2exx2’4z0

= ao(‘l-/4n zo

< 0). The total strength of C,that is, the integral of C over all x, is the same as the strength of the original source c0,but it is spread out over the longitudinal axis. In principle, it is spread out over an infinite length, but actually it is fairly well concentrated, especially if the source is shallow. In no sense does this line distribution produce local effects comparable to those of the original point source. In fact, the potential corresponding to the line distribution is analytic in the lower half-space, even at the location of the “equivalent” point source. What is important for our purposes is that a free-surface line distribution of sources can produce the same waves as the submerged point source, but the line distribution extends fore and aft of the point source. This spreading out shows, incidentally, that the usual picture of the Kelvin wave system bounded by a wedge-shaped region of half-angle sin- 4 1 19”28’is not so clear as is usually thought to be the case; the apex of the wedge cannot be defined precisely. It may be noted that the above line distribution is analytic. Vertical Line of Sources. Now consider the case 8(x, z) =

(zo

-T
Then X * ( k ) = (icao/k2)(1 - e - k 2 T /),N and Z(x) = go{- f I~x I [l - e r f ( ~ x ~ / 4 T ) ’+/ ~( ] ~ T / n ) ~ 1./ ~ e - ~ ~ ~ ~ A longitudinal distribution of sources given by this Z(x) produces the same wave system as the vertical line of sources of uniform density no per unit

T. Francis Ogiluie

154

height. It may be noted that C(x) is continuous, but its first derivative is not. It is convenient to define a function which is proportional to the above W ) ~ ( x =) - (K I x I / 2 ~ )1[ - erf(~x~/4T)”’] + (4.20) It may be noted that ( T / K ) ~ ” M ( xis) dimensionless and depends on the ) ’ ~ is, ~, single variable, X = ( K X ~ / ~ Tthat (T/K)”’M(x)

=

-X(1 - erf X)

+ (l/n)”ze-x2

(4.20)

Strutlike Body. Let a symmetrical strut be defined in terms of its offsets as follows y = &b(x),

O<X
-T
The corresponding thin-ship source distribution is given by

0 < x < L,

&(x,Z) = 2Ub’(x),

-T

< z < 0.

From the previous result (for a vertical line of sources), we can immediately write down the corresponding longitudinal source distribution C(X)= 2 UT

[

L

d
(4.21)

‘0

This reduces to the ordinary slender-body-theory result if M(x) is replaced by 6(x), that is, C(x) = 2UTb’(x). The representation in (4.21) shows that the ordinary slender-body-theory source distribution is smeared out. Since (4.21) has the form of a convolution integral, we can readily rewrite it in terms of Fourier transforms

1

1 -m ~ ( x=) dkeikxM*(k)C,*(k), 2n.-,

(4.22)

M * ( k ) = ( K / ~ ’ H ) (I e-kzT/K),

(4.22‘)

where the transform of M ( x ) as given in (4.20),and C,*(k)is the transform of the source density function in ordinary slender-body theory. For a strut with the shape of a double wedge of half-angle a, that is, b(x, Z) =

-T
ordinary slender-body theory gives

+

Co(x) = 2aUT[H(x) - 2H(x - L/2) H(x - L)], where H(x) is again the Heaviside step function. We can also find C(x)

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Ship Hydrodynamics

explicitly C(x) = 2 a U i 7 7 ( x ) - 2 t ( x - L/2)

+ 7(x - L)],

where ~ ( x= ) sgn x [ -

x 2- (X/n’”)e- + +( 1 + 2 X 2 ) erf XI, x2

with X = I X I = ( I C X ~ / ~ Tas) ”before. ~ , Figure 8 shows C(x)/2ccUT and C0(x)/2aUTfor the case LIT = 9, F _= U/(gL)’” = 0.2. According to the ordinary slender-ship theory, the difference between X ( x ) and &(x) should cause a higher-order effect than CO(x). It is always recognized, of course, that the far-field expansion is not valid in the near field, especially not in any uniform sense, but the far-field expansion ought

ordinary slender-body theory (%)

X P

-I

--

FIG.8. A line source distribution for a ship in steady motion, modified for end effects

not to be sensitive to a smoothing over of the source density function in a very short region. [The smoothing takes place for the most part in a longitudinal distance which is O ( E ’ ’ ~ )Nevertheless, .] the wave resistance, which can be computed from the far-field solution, depends critically on the precise amount of rounding off of the steps in the C , ( x ) curve. Furthermore, in the example presented in Fig. 8, the amount of smoothing would hardly be negligible, although the body is quite slender in any conventional sense, and the ratio A/,!+ which is supposed to be 0(1),has the value 4. It is clear that end effects are much more critical in the slender-ship problem than in slender-body problems of aerodynamics. There is another failure of ordinary slender-ship theory, and this failure is shared with thin-ship theory. Waves are generated at each cross section of the body, and these waves must undergo diffraction by the rest of the body.

156

T. Francis Ogilvie

However, such effects are considered formally to be of higher order, and so they are disregarded in the lowest-order theory. It is easiest to see how this happens by considering thin-ship theory. Sources are distributed on the centerplane according to a rule that involves only the forward speed and the body geometry, as prescribed in (4.16).The flow caused by such a source at all other points on the centerplane is entirely tangential to the centerplane, and so it cannot affect the manner in which the body boundary condition is satisfied, since this condition is transferred to the centerplane too. Essentially the same situation prevails in slender-ship theory, in which the interference among cross sections appears in the form of a function of x alone; this function can be interpreted as a local modification of the incident free stream, but its magnitude is small in comparison, and so the local solution is not modified to account for it. [See, for example, the integral term in (4.13), which implies the existence of a nonzero velocity component normal to the body.] It is possible to modify slender-ship theory to take account of this effect, and this possibility will be discussed briefly in a subsequent section. It is not possible to modify thin-ship theory in a similar way; the only alternative there leads to the necessity of solving a numerical problem of incredible complexity.

D. THEBOW-FLOW PROBLEM 1. Problem Formulation

The exact free-surface conditions for the steady-forward-motion problem were given in (4.2) and (4.3). If we retain just the linear terms in those equations and eliminate c, we obtain & x + ~ ~ , = O

on z=O,

(4.23)

which was already given, (4.1l), as the free-surface condition in the far field of ordinary slender-ship theory. In the near field of ordinary slender-ship theory, we retained just the second term, on the grounds that the first term was of higher order than the second in a systematic perturbation expansion. We now inquire under what conditions it might be legitimate to retain both terms in (4.23) even in the near field and what the consequences would be. Formally, we can assume that K = g / U 2 = O(E),and then the two terms in (4.23) are of the same order of magnitude, provided that we still apply (2.1). This suggests that (4.23) should be used just as it stands if the speed is high, in the sense that U = O(E-I/’). This was recognized many years ago, but it was not followed up, largely on the grounds that “high speed for a ship might mean a Froude number U/(gL)”’ of about 0.3, corresponding to the ”

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157

value K Lrr 10, which is hardly small. Such an argument is not convincing, however, because high speed” may very well not mean the same thing for the naval architect and in the fluid mechanics problem. Regardless of how that controversy is resolved, it should be observed that the flow around the bow of a ship may have the characteristics of a highspeed flow even if this is not true for the overall flow around the ship. One may refer to the interpretation of Froude number given in many elementary textbooks on fluid mechanics: It is a measure of the relative importance of inertial forces to gravity forces, higher values of the Froude number Gorresponding to higher levels of inertial forces. As a ship passes through the water, the particles of water are at rest until they are rather suddenly accelerated by the oncoming ship bow, and gravity begins to be effective (as a restoring force) only after the particles have been displaced considerably from their rest positions. Therefore, near the bow, one might suppose that gravity forces have relatively small effect on the water, at least locally. It will appear, in fact, that we can usefully define a Froude number based on distance from the bow, U/(gx)’”, in somewhat the same way that one frequently defines a “running Reynolds number” in the mechanics of viscous fluids, and this suggests that the Froude number is effectively very high near the bow. There is yet one more way in which to understand (4.23).We may discard (2.1) and assume instead that “

df

ay’

af = O ( f & - l ) , -

az

af = O(fi-’/’) ax

as

E +

0,

(4.24)

wherefis any flow variable in the near field. This kind of assumption cannot be justified so readily as (2.1), but it is nevertheless a plausible possibility to consider: One needs only to observe the bow wave created by a fine ship to recognize that the amplitude and the longitudinal slope of the bow wave are much greater than the amplitude and slope of the waves along most of the length of the ship. In (4.24), we might have assumed that aflax = O ( ~ E - ~ ) , with 0 < /3 < 1. However, the choice made in (4.24) is the only one that leads to a plausible problem statement. (A choice p = 1 leads to a fully 3-D problem. For blunt bows, this may be the only acceptable assumption, to the extent that linear theory applies in that case at all. We shall not pursue such a possibility further here.) The approach that we shall describe here is based on the assumption (4.24). However, it is completely consistent with the concept of the bow flow as a high-Froude-number problem. It will be assumed that

U = O( 1)

[i.e., U/(gL)’”

= 0(I)]

as

E +

0.

In order to apply (4.24) formally, one may stretch the x coordinate in the

158

T. Francis Ogilvie

downstream direction from the bow X = XE1l2,

and then assume that a/dX = O( 1). We shall not use such a formalism, but it may always be implied. Furthermore, the solution to be obtained will be valid in principle only in a region near the bow, over a length which is O(E"'), and the solution will have to match satisfactorily an acceptable solution in other regions. Let us now consider the consequences of using the condition (4.23) in the near field around the bow. The problem is easily formulated: Because of condition (4.24), the 3-D Laplace equation once again reduces to a 2-D Laplace equation,

4yy+ 4 2 2 = 0,

(4.25) in the fluid region, and the body boundary condition is just as before, a4ialv = - Un,. (4.26) The free-surface condition was given by (4.23). Thus we have a set of 2-D problems to be solved in the y-z crossplanes, but the solutions in the different planes are related through the appearance of the x derivative in the free-surface condition (4.23). For the purpose of understanding this conceptually, it is convenient for a moment to make a substitution t = x / U , with t interpreted as a time variable. Then (4.23) becomes the usual free-surface condition for timedependent problems, as given already in (3.27), for example. We can think of a 2-D body of changing cross-sectional shape and size, on which the normal component of fluid velocity @/aN is prescribed [as by (4.26)]. Then we have the appropriate partial differential equation for a time-dependent 2-D problem, (4.25), and appropriate free-surface and body conditions, and the entire problem can be interpreted easily. We still need conditions equivalent to initial conditions in the timedependent problem. For this purpose, we assume that there is no disturbance at all ahead of the ship, so that

4EO

for x < 0.

(4.27)

We also assume that the solution represents outgoing waves as 1 y I + a. Both of these conditions should be justified through a matching to a far-field solution. Condition (4.27), of course, really means only that the disturbance ahead of the ship is of higher order than the solution that we shall obtain.

2. A Solution and Its Properties This problem is a generalization of the classical Cauchy-Poisson problem (see, e.g., Lamb, 1932),and the solution can be constructed with this in mind.

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Ship Hydrodynamics

The velocity potential for an impulsive source of unit strength at (<; q, i) can be expressed

cos ky sin[(lck)"'(x - <)I,

71

+

(4.28)

+ +

where r = [(y - q)' (z - 1)2]112, r' = [(y - q)' ( z i)2]112, and 6(x) and H ( x ) are the Dirac delta and Heaviside step functions, respectively. This quantity satisfies the 2-D Laplace equation except at (x; y, z ) = (<; q, i), and it satisfies the free-surface condition for all x (and y ) . We can multiply (4.28) by a source density function and integrate over a cross section at x = 5, thus representing the potential caused by a distribution of impulsive sources on that section contour, and then we can further integrate in the 5 direction to represent the effect of source distributions at all cross sections, as follows

where C(x) is the contour of the section at x. This expression can be used with an arbitrary distribution of sources over the body surface. The effects of the sources at any x are felt only at cross sections downstream of x. The source density must, of course, be found so that the body boundary condition is satisfied. In order to simplify the result, let us reflect the contour C(x), calling the image C(x), and require that a(x; y , -z) = -a(x; y , z). When (4.29) is substituted into the body condition (4.26),we find that

=

2un1

-

2f(x; y , z ) ,

(4.30)

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T . Francis Ogilvie

where f(x;y,z)=-

K

I

a m

II . o

x

dkkh’

___ (Kk)”’

lo d( sin(Kk)1’2(x

dlB(4; q, [)eke [ - n 2 sin k(y - q )

.(.a<)

-

+

n3

t)

cos k(y - 03.

(4.30)

The integral in (4.30) is a principal-value integral. The functionf(x; y, z) represents the normal velocity component on C(x) that has been caused by the distribution of sources upstream of x. Thus, at any section x, there must be a local distribution of sources over C(x) to accomplish two purposes: (i) generate the geometrically determined normal velocity component [from (4.2611,and (ii) offset the normal velocity component caused by the presence of other cross sections upstream. Over any cross section, (4.30)is a Fredholm equation of the second kind, and so it can be solved by appropriate numerical methods. In fact, the wavelike nature of the problem appears only on the right-hand side of (4.30);the left-hand side represents a distribution of sources over the body and its image, and the density on the image is just opposite to that on the body itself. As far as the solution of the integral equation is concerned, we might as well be solving a boundary-value problem with the potential equal to zero on the undisturbed water surface. The last term in (4.30),written out in detail in (4.30), contains the unknown function B inside the triple integral. However, this term does not depend on the source density at ( = x, but only at ( < x, because of the sine factor in the t integrand. We can solve (4.30)step by step, starting at x = 0, and the quantity f ( x ; y, z ) will then be completely known at each x; it belongs on the right-hand side above. In fact, (4.30) is a Volterra integral equation with respect to the x variable. The analysis above is fairly close to what was done by Cummins (1956). The first actual solution following this approach was obtained by Ogilvie (1972) for the case of a fine wedgelike bow. One additional approximation was made, namely, that the wedge is thin enough so that the body boundary condition can be applied on the centerplane and so that the sources can be located there too. (This is an extra assumption not implicit in the general formulation just presented.) Thenf(x; y, z ) = 0 on the centerplane, and the integral term on the left-hand side of (4.30) also vanishes. So an explicit solution for (T is immediately available; it is the same as in thin-ship theory, although the corresponding potential is different. The wave shape along the wedge can be computed from the following analytical result ((x, 0) = (2a/n)(T/K)”2Z(X, O),

(4.31)

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161

where (4.31‘)

X = x(K/T)”’, (4.31”) and a and Tare, respectively, the half-angle and draft of the wedge. Experiments were performed to check this simple result, and it was found that fair agreement was realized, thus encouraging the further investigation of this approach. It should be noted from (4.31) that all lengths may be nondimensionalized by the quantity ( T/K)~/’,which is proportional to the geometric mean of the draft and the characteristic wavelength. We can rewrite (4.31”)

X = (KX)”’ . (x/T)”’, the first factor of which can be recognized as l / F x , with F , the Froude number based on length x from the bow. It was predicted earlier that this quantity would appear as a natural parameter of the problem. It is of The above analysis is valid only in the region in which x = (I(&’/’) interest to examine the solution as x + 03, since it should match with the solution in the region in which x = O(1). For the special case of the simplified wedge problem, Ogilvie (1972) showed that

- -4UaT/mcx2

4z

[-4aT/nux

as x -+

03,

as x-03.

Since a and Tare O ( E and ) x = O(E”’),we observe that 4, = O ( E)in the bow near field. If now we consider that x = 0(1), so that we are looking at the near field of the ordinary slender-body theory, we find that 4z= O(E’).This is precisely what (4.7), the rigid-wall condition, implied; the vertical component of velocity at z = 0 is of higher order than the lateral component. The second relationship above, giving the asymptotic behavior of as x 03, agrees even more explicitly with the matching result from ordinary slender-body theory. We see ‘that [ = O(c3/’)in the bow near field, but this becomes O(E’) if we let x be O(1). If the wedge problem is solved as in ordinary slender-ship theory, one obtains precisely the above form for ( in the limit as x ---r 0, that is, [ goes to infinity as 4aT/mcx. In that case, [ is O(E’) in the near field (of ordinary slender-body theory) and the tendency toward infinity as x approaches 0 really means that [ is of lower order than E’ near the bow. The solution of the bow-flow problem agrees with this completely. The matching is entirely satisfactory. This fact gives us some confidence that the formulation of the bow-near-field problem is correct, even though no matching conditions were established a priori.

-

T. Francis Ogilvie

162

The wedge-bow problem discussed above is precisely the same problem discussed at length in Section IV,C, in which the complete failure of the ordinary slender-ship theory occurred. Considering that the ordinary theory leads to a prediction of infinite wave height at x = 0, we should not be surprised that it also leads to a prediction of infinite wave resistance. The modified theory for the bow region predicts finite wave height throughout the bow region (although the slope is infinite at x = 0). From the modified solution in the bow-flow problem, one may expect to be able to find a modification of the far-field source density, in the manner shown in Fig. 8. In fact, with the extra assumption made by Ogilvie in the solution of this problem, the modification is exactly the same as in Fig. 8. For the more general problem, in which the body boundary condition is satisfied exactly on the body surface, no one has yet performed the required matching between the far-field solution and the bow-near-field solution, and so the proper far-field source distribution cannot yet be obtained in this way. The physical interpretation of the above problem and its solution deserves further comment. First of all, consider what is nor included. Since the solution above satisfies a 2-D Laplace equation, there can be no representation of transverse waves. In the ordinary slender-ship theory, such waves did appear, but only in the additive function of x that had to be determined from the far-field solution [the second term of (4.13)].There is no possibility in the bow-near-field problem of adding such an extra term, and the transverse waves cannot be represented as a solution of the homogeneous problem in the bow region. So they are missing, and the situation cannot be corrected within the above framework. This means that the above form of solution can be valid only in a region near the bow which is short compared with a wavelength, I = 2ng/U2.Behind this region, the solution of ordinary slender-ship theory must be used (possibly with modifications to be discussed presently in Section IV,E). Maruo (1967b) developed a theory for high-speed planing craft in which this shortcoming is accentuated; his first approximation is equivalent to solving (4.30) with f= 0. Tuck (1973) and Baba (1974) have produced flat-ship theories which are essentially equivalent to the bow-near-field theory described above, but with the additional assumption that the body boundary condition can be applied on the underside of the plane z = 0. Tuck in particular showed that Maruo’s first approximation is useful only at such high Froude number that it is not likely to be valid even in planing problems. Even the results of Tuck and Baba do not include any consideration of the transverse wave part of the Kelvin wave system. What is included in the bow-near-field solution is the diverging-wave system, and this is done to an extent not achieved in any standard method of analyzing ship wave systems. Near a ship bow, the diverging-wave system has rather large gradients in the transverse direction, since the waves are “

”

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short and their crests are nearly parallel to the track of the disturbance. This suggests very strongly that it is not legitimate to transfer the body boundary condition from the exact hull surface to the centerplane, as is standard practice, for example, in thin-ship theory. This transfer was also done in the wedge example above by Ogilvie, but that was only for convenience in obtaining an immediate approximate solution for comparison with experiments (really to determine whether the method was worth further investigation). It is not an essential step in the above analysis. Therefore we have a theory here with a formal justification for satisfying the body boundary condition on the exact location of the body surface, but the theory is still a linear theory. The fact that the theory does predict diverging waves can be shown readily. Consider just a point source in three dimensions. The corresponding velocity potential is given in many references, for example, Wehausen and Laitone (1960). One can estimate the waves generated by a point source of unit strength at (0, 0, [) by applying the method of stationary phase. Two wave components are obtained, the diverging waves and the transverse waves. When these are evaluated close to the track, we obtain, respectively, 4(x, y, z ; 0, 0, [)

N

- ( ~ / 7 1 y ) ~ ’ ~ e -sin[(~x’/4y) ~ ~ ~ ~ ~ +- 5n/4] ~ ’ ~ ~ ~ -

+

( 2 ~ / n x ) ~ ’ ~ e - ~sin(lcx 1 ~ + 5 1 71/4). (4.32)

If the source is located on the free surface ([ = 0), the approximation for the diverging-wave system is actually valid near the entire track of the singularity, even close to the singularity itself, as shown by Ursell(1960). The same is not true for the second term, representing the transverse waves, but we are not concerned here with that term. If, on the other hand, we consider again the impulsive source represented by (4.28), we can also obtain an estimate by the method of stationary phase, and it is precisely equal to the first term of the above result obtained for the 3-D source. This is a rather unusual situation, for it means that the fluid motion close behind a translating 3-D source can be described fairly well by the solution of 2-D problems in the cross sections behind the point source; such a statement would be pure nonsense in the infinite-fluid counterpart. If we now construct source distributions that vary smoothly in the longitudinal direction, our conclusion ought to be even more accurate, and so the solution of the bow-near-field problem must represent the diverging waves of the actual 3-D hull. The above argument can actually be made somewhat more precise. The 3-D source potential can be written as a Fourier transform with respect to x and then divided into odd and even parts. With an error which is n o larger than the even part, one can evaluate the 3-D potential near the track and “

”

164

T. Francis Ogilvie

obtain (4.28), which contains considerably more information about the wave motion than the stationary-phase approximations. 3. Extensions The work by Tuck (1973) and Baba (1974) for the Rat-ship problem has already been mentioned. Daoud (1975) has developed a practical procedure for solving the full problem as set above, with the body boundary condition satisfied on the exact body surface. He has been able to compute the solution of the wedge-bow problem for which Ogilvie obtained an approximation under extra assumptions, and he finds that the waves are shifted somewhat forward and outward, as one might expect. It may be noted too that Daoud’s method of solution has a somewhat more rigorous foundation than that discussed above, in the sense that he did not assume ab initio that a source distribution can be used for constructing a solution, as in (4.29). Instead, he started with an exact statement of Green’s theorem for the fluid region, and then he systematically derived a form of solution corresponding to (4.29). In his integral equation [corresponding to (4.30) above], the unknown quantity is the potential itself on the body surface. Hirata (1972) solved the problem of a yawed flat plate (of zero thickness) in steady forward motion, using essentially the approach described above. More recently, Chapman (1976) solved the same problem by quite a different method. His approach can be extended even to nonlinear problems, and it is worth noting how he approaches the problem. Suppose that the 2-D problems have been solved at all sections to and including a particular value of x. Then the gradient of the potential in the transverse planes and the shape of the free surface are known too. In the boundary conditions, (4.2) and (4.3), solve for & and (, (4.33a) u4x= - [ S i + H4; + 433 y) UYX = - 4 y Y y ) lz=<(x, y ) (4.33b) 9

’

The term -*& has been neglected in the first and the term &tx in the second, since these are of much higher order than the retained terms if the body is slender. However, it is not necessary to linearize. With the values of 4 (on z = Y) and l‘ known at x , the equations above can be used to extrapolate to a section Ax farther downstream, and so the free-surface condition for the problem at x + Ax is a simple Dirichlet condition. On the body, there is a Neumann condition. Chapman solves this 2-D problem by applying a finite difference method to the Laplace equation, although he could just as well have solved an integral equation based on Green’s theorem and a simple log r Green function. Considerable care is needed to ensure that the solution behaves properly far off to the sides of the body, but Chapman succeeded in developing a fairly simple method for that too. There is also a

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difficulty about the stability of this step-by-step method of solution, but Chapman has considered this adequately and shown how to ensure stability by using a system of staggered intervals for extrapolating the potential and the wave shape. Chapman first solved the linear equivalent of the above problem. Then he formulated a second-order sequel and solved that. Finally, he solved the fully nonlinear problem. Predictions of force and moment on the plate did not vary greatly between the linear and the fully nonlinear solutions, although the predicted wave shapes were quite different. Agreement with experimental data was fairly good, but the effects of separation and vortexshedding from the long edge of the plate (the keel) are not negligible,’and they are not included in the theory. The second-order theory was, in some ways, less satisfactory than the linear theory. This might have been anticipated, since the analytical solution of the linear problem is discontinuous at the leading edge, and so a second-order analytical solution would probably be singular there. Chapman did not formulate his boundary-value problem by a systematic perturbation analysis, but this had been done already by Ogilvie (1967).He showed that the assumption of high speed, in the sense that U = O ( E - ~ / ~ ) , leads to precisely the problem that Chapman solved. In the discussion in Section IV,D,l, it was pointed out that “high speed” could provide the logical basis for retaining the term 4xxin the linearized free-surface condition (4.23). But it also leads to the retention of certain nonlinear terms, as in (4.33a) and (4.33b). A method of solution similar to Chapman’s was proposed by Ogilvie but never carried out. In order to get his step-by-step method of solution started, Chapman assumed that there is no disturbance ahead of the plate, just as Ogilvie (1972) did. This seems to be a somewhat stronger assumption in the flatplate problem than in the symmetrical-wedge problem. If a plate of zero thickness moves with an angle of attack in an infinite ideal fluid, the velocity around the leading edge is infinite. At the nose of a wedge in an infinite fluid, there is a stagnation point (or line). For both of the corresponding bodies in the presence of the free surface, we assume no disturbance whatever ahead of the nose, and this is clearly a more drastic misrepresentation in the plate problem. However, the presence of the free surface greatly attenuates upstream effects of any kind, and the quality of Chapman’s results suggests that this attenuation is sufficient to justify their neglect. Finally, it should be noted that Chapman’s approach avoids one of the most troublesome aspects of using a Green function satisfying a free-surface condition. The function given in (4.28) is dreadfully singular near the x axis if tj = i= 0; it oscillates wildly, and this oscillation can be removed only by submerging the source. Since the source distribution in (4.29) extends right to the plane z = 0, such a possibility is not available to us. But the potential

166

T. Francis Ogiluie

for the complete fluid motion has no such wild oscillations; they are canceled out through the interference of the wave motions caused by the continuous distribution of sources. Since these oscillations do not really exist then,. it would appear desirable to develop methods of solution that avoid them completely. Chapman’s method does this.

E. THELOW-SPEED PROBLEM At the end of Section IV,C, we pointed out that ordinary slender-ship theory at the lowest order does not include any effects of the diffraction by the ship of its own waves, This characteristic is shared with thin-ship theory. It is a natural consequence of two factors: (i) The assumption K = g / U 2 = 0(1)means that the transverse waves have length comparable to ship length, or-what is more to the point-length much greater than ship beam. These waves have an associated velocity field that violates the body boundary condition, but the error is formally of higher order than the 2-D motion in the crossplanes. (ii) The diverging waves, which always have small wavelength near the x axis, are never really considered in ordinary slendership theory, and so their diffraction by the ship is also overlooked. Section IV,D was devoted to the latter factor. Daoud’s (1975) and Chapman’s (1976) analyses both include the diffraction of the diverging waves by the body itself, and they do this in a plausible way. In this section, we shall discuss briefly how the first factor above may be reconsidered. It should be noted that this is not just an academic problem. Ever since Professor Takao Inui at Tokyo University initiated an energetic program more than 15 years ago to measure complete ship-generated wave systems, it has been recognized that the waves alongside a ship decrease in amplitude toward the stern more rapidly than any theory had predicted. This phenomenon is stronger the lower the ship speed. But it is still apparently observable even if there are only about two wavelengths along the ship, which means that the Froude number is almost 0.3. A naval architect would consider this Froude number to indicate a fairly high speed. Nevertheless, as will be seen, the formal derivation of a theory to account for this phenomenon requires that we assume low speed, in the sense that U = O ( E ’ ’ ~Alternatively, ). we may assume that ship length is very great, i.e., L = O(E-’), as proposed by Newman (1976). Adachi (1973) conducted experiments with an extremely long model, to measure accurately the rate of decrease of wave amplitude along the parallel middle body. His model had an entrance length of 1.75 m and a parallelmiddle-body length of 20 m, and so he was able to observe many wavelengths along the side. His data show that the amplitude decreases , x measured downstream from the bow. approximately as x - ~ . ~with According to conventional theory, waves are generated in a small region

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near the bow (in this case, along the first 1.75 m) and these waves should eventually decrease as along the parallel middle body. The difference between the measured and predicted rates of decrease is very great, and it is mostly the result of diffraction of the bow-generated waves by the rest of the body, an effect which has not been included in the theory. Ursell (1975) formulated and solved an idealized problem which essentially corresponds to Adachi’s experiments. He considered an infinitely long cylinder, half submerged, past which a stream is flowing. On some finite length of the cylinder a normal velocity component is imposed, generating shiplike waves, and a condition of zero normal velocity component is imposed aft of this section. Under these conditions, Ursell showed that the wave amplitude along the cylinder eventually decreases as x- 3’2, which agrees much better with Adachi’s result than the ordinary theory does, although the agreement is not entirely satisfactory. This paper by Ursell is the same one in which he studied the diffraction of incident head waves, as discussed in Section lII,D,4; perhaps the concept of wave diffraction is somewhat clearer in that case, but the basic interpretation is the same in both cases. Ursell’s asymptotic analysis and Adachi’s experiments are valuable for showing the basic nature of the diffraction by the ship of its own waves, but a theory is needed for application to ordinary ships. The first step in this direction was made by Reed (1975). He assumed that K = O ( E -’), so that the wavelength of the transverse waves is O(E),which we interpret as meaning that wavelength is comparable to ship beam. His far-field expansion starts the same as in the previous slender-ship theories, representing a line distribution of sources on the axis of the ship. When this expression is evaluated near the x axis, he obtains the same result as Tuck (1963, 1964), given in (4.13), except that there is some difference in order of magnitude: The integrals containing the Struve and Bessel functions are of higher order than the other terms in (4.13). The lowest-order terms are exactly the same as the corresponding result for a slender body in an infinite fluid (except for a factor of 2, which results from the definition of 6 in terms of the submerged cross-sectional area). This indicates that the first term in the near-field expansion should be just the solution of the problem of the body and its image in an infinite fluid (the double-body or zero-Froude-number problem). The terms containing the Struve and Bessel functions in (4.13) are highly oscillatory with respect to x, a fact that makes it plausible that they should be of higher order than in the moderate-speed case. The highly oscillatory nature of these two terms also suggests how to proceed with the next terms in the expansions. Reed assumes that the second-order source distribution in the far field should be represented in the form C(x) exp ilcx with X ( x ) “slowly varying.” This is comparable to what Faltinsen (1971) did in the head-sea diffraction problem, and the evaluation of this source potential

168

T. Francis Ogilvie

near the singular line is quite similar to Faltinsen’s result (3.50).In the near field, Reed assumes that the potential should be expressed 4(x, y, z ) exp I’Kx, and, as in Faltinsen’s problem, he obtains a 2-D Helmholtz equation to be solved. The body boundary condition is a homogeneous condition, since the normal component of the incident stream was already canceled by the double-body flow, the first-order solution. The solution then proceeds very much as in Faltinsen’s problem. There is one step that is particularly interesting: The highly oscillatory source density must be determined in such a way that it cancels the leftover Struve- and Bessel-function terms that came from the first-order far-field solution. We can make a rather precise analogy between this fact and what Faltinsen did. These leftover far-field terms can be interpreted as waves that are incident on the hull, and, as in the Faltinsen-type analysis, they are nonexistent (that is, canceled out) in the corresponding near-field solution. Reed performed some experiments to confirm his analysis, and he found that he had a great improvement over ordinary slender-ship theory in his predictions of wave amplitude along the side of the model. There is still room for further improvement, especially since Reed’s mathematical model exhibits the same bow singularity as Faltinsen’s; obviously, Reed’s assumptions are not valid there, but it is not clear how to modify his formulation of the problem. Baba and Takekuma (1975) proposed to formulate the steady-motion problem as a very-low-speed problem, in such a way that IC = o(E). They point to an experimental justification for doing this in the case of full-form ships, which generally exhibit a breaking-wave phenomenon just ahead of the bow: In flow-visualization studies by Taneda (1974), colored dye was injected into the incident stream at various depths ahead of the bow. The major wave disturbance, especially around the bow, clearly occurred in a thin layer of water near the free surface; at depths much less than the draft, the flow was quite similar to the flow in the corresponding double-body problem. Such a picture can really only be understood in terms of a very-lowspeed problem, for only in that case is the wave motion restricted to a thin surface layer (just as in the cases of very-high-frequency oscillations and extremely short waves, discussed in Sections III,C,3 and III,D,5). If speed can be considered as extremely small in some practical problems, then there is not much need to restrict ourselves to slender bodies-a fact that makes this approach especially attractive. Linearization can be justified on the basis of the small speed rather than on the basis of slenderness. In this case, the lowest-order solution corresponds to the problem of the nonslender body and its image in an infinite fluid, and succeeding terms then introduce the wave effects. Such a problem was first studied in the case of a 2-D submerged body by Ogilvie (1968), and Dagan later considered several similar problems (see, for example, Dagan and Tulin, 1972).

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Inui and Kajitani (1976) report experimental observations on ship waves that further suggest the above very-low-speed approach. They found that, if ship waves are considered to propagate on the nonuniform flow corresponding to the flow around the double hull in an infinite fluid, there are amplitude and phase shifts from ordinary linear theory that greatly improve agreement with experiment data. This effect is so great that one is reluctant to relegate it to the category of “ higher-order effects,” but the only way to avoid this is to assume, as above, that the entire problem should be considered as a very-low-speed problem. This subject of low-speed and very-low-speed problems is obviously so new in its development that no strong conclusions can be drawn. There may be a basic conflict with the concept of the bow flow as a high-Froudenumber problem, as discussed in Section IV,D,l. In any case, we have here some new approaches to some very difficult problems, and they are highly recommended for further study. V. Slender Ships in Unsteady Forward Motion

A. INTRODUCTION A N D FORMULATION We now consider the problem of a ship that is acted on by incident waves and/or forced to oscillate while it also moves forward. This problem is much more difficult than those of the preceding two sections, and the state of our knowledge about it is correspondingly poorer. But notwithstanding the fundamental deficiencies in our understanding of this problem, we have remarkably good techniques for predicting the response of a ship in a seaway. See, for example, Salvesen et al. (1970). This paradoxical situation must be viewed as the result of extraordinary good luck on the part of the ship hydrodynamicists who have been concerned with the prediction of ship motions and of wave loads on ships. It should also warn us not to accept any predictions of our theory that are not similar to what has been thoroughly checked experimentally. The analysis of Salvesen, Tuck, and Faltinsen will be taken as a standard of good engineering practice in this field, although several other investigators have developed analyses that are quite similar, for example, W i n g (1969), Tasai and Takagi (1969), and Borodai and Netsvetayev (1969). As we briefly sketch out how the problem should be formulated relatively rigorously, we shall try to emphasize the points at which such practical methods compromise to avoid unsolved and unresolved aspects of the rational mechanics approach. This does not imply a criticism of existing prediction techniques, for some of the investigators named above have been very ingenious in handling difficulties that still cannot be treated rigorously, and

170

T. Francis Ogilvie

they have developed useful prediction techniques. But there are shortcomings and they should be understood before one tries to apply similar analyses to problems that may emphasize the shortcomings, as happens, for example, in the treatment of “end effects” and in the prediction of waveexcited vibrations. Let the ship be located in a stream of speed U in the positive x direction. Two coordinate systems will be defined: (1) O x y z is fixed to the mean positibn of the body, with the z axis upward (as usual), (ii) O‘x‘y‘z‘ is fixed to the ship.

The systems coincide when the ship is at rest in the stream. Let r and r’ denote the position vectors of a point of the body in the two coordinate systems, respectively. The displacement of any point on the body at any instant is then given by

a(r, t ) = r - r’ E g ( t )

+ Q ( t ) x r,

(5.1)

where

= (tl?t 2 Q(l) = ( t 4 5 5 S(t)

9

7

7

t3)ei0’,

(5.1‘)

t6)ei0‘.

(5.1“)

In ship terminology, the coefficients represent complex amplitudes of translations and angular displacements as follows

tl:

surge,

t4: roll,

t2:

sway,

t5: pitch,

t 3 : heave, t 6 : yaw.

It will be assumed that all of these amplitudes are small; the angular amplitudes, in particular, should be small enough to allow them to be considered as the components of a vector R. Later it will be convenient to assume that

5j

=O(~E),

(5.2)

where E is the usual slenderness parameter and 6 is an independent “motionamplitude parameter,” assumed to be small. This convention implies that the amplitude of motion is small compared with the ship cross-section dimensions, even as E + O . In fact, we shall linearize all conditions with respect to 6, which justifies the consideration of just sinusoidal motions. The radian frequency o is the actual frequency of oscillation of the ship. If there are incident waves, it is the frequency of encounter, often denoted by we in the ship-motion literature. The hull surface is defined by the equation SO(X, y, z ) = 0

(5.3a)

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for the ship in its mean (rest) position and by the equation (5.3b)

S(x, y , z, t ) = S,(X', y', z') = 0

in its instantaneous position. The body boundary condition must be satisfied on the exact location of the hull, as given in (5.3b). If the complete velocity potential is expressed as Ux

+ 4(x, y , z, t ) ,

the boundary condition can be written as follows: 0 = DS/Dt = ( U

+

+ 4 y S y+ &Sz + S ,

&$x

on

S = 0.

(5.4)

It is more convenient generally to have a condition to impose on the mean location of the body, S,(x, y , z ) = 0, and one can derive such a condition by expanding the derivatives of the potential in (5.4) in Taylor series about the mean hull position. The result is that

- + U V x (a x v)} - U n ,

@/an = n { a

on So(x, y, z ) = 0, ( 5 . 4 )

where Uv is the fluid velocity in the steady-motion case (the problem of Section IV). The above condition was derived in this form by Timman and Newman (1962), and the derivation will not be repeated here. The meaning of the several terms should be noted, however: (1) The term n * a is equivalent to the boundary condition given in (3.1 l), except that it has been generalized to six degrees of freedom. (2) The next term results from transferring the body boundary condition from the instantaneous position of the hull to its mean position. (3) The final term is the same condition that was used previously in the steady-motion problem. In order to include all six degrees of freedom in our formulation,+ we shall introduce a compact notation, as follows: Generalize the definition of nj

+ n2 j + n3 k = n,

(5.5a)

+ n5 j + n6 k = r x n.

(5.5b)

n, i n4 i

Define another set of six quantities mj

+ m2 j + m3 k = -(n - V)v = m, m4 i + m5 j + m6 k = -(n V)(r x v) = r x m + v x n. m, i

*

(5.6a) (5.6b)

Let the potential be written

+ 4(x, y , z, t ) = U x + Ux(x, y , z ) + $(x, y , z)eiW'+ 0(6'), (5.7) [ U x + U x ( x , y , z ) ] is the solution of the steady-motion problem.

Ux

where +

It is questionable whether the results are valid for j = 1 (surge) and j = 4 (roll).

T. Francis Ogilvie

172

[Thus v

+ x).] Then +(x, y, z ) satisfies the body boundary condition a+/& = C (ionj + U m j ) l j on s,(x, y, z) = 0, (5.8)

= V(x

6

j= 1

with an error which is O(h2). In all present-day strip theories except that by Ogilvie and Tuck (l969), the condition (5.8) is further simplified. The quantities mj all involve the gradient of the steady-motion flow field, as is obvious from (5.6a) and (5.6b). The most interesting of these quantities can be expressed simply if the body is slender m2

+ n3xyz)? = - ( n 2 x y z + n3 x z z ) ,

=

-(n2Xyy

m5

=

-(n3

= (n2

+ xm3),

+ xm2)*

(5.9) These are all accurate to within a multiplicative factor of the form [1 + O(c2)]. The usual practice is to ignore all terms involving derivatives of x. This is valid only if the difference between v on the exact and mean body surface locations is negligible; however, according to the basic assumption of slender-body theory, derivatives in the transverse planes are large, and the difference should not be negligible. Of course, the motivation for ignoring these terms is to avoid having to introduce the steady-motion potential into the boundary-value problems for the unsteady motion. But this actually causes no difficulties whatever, thanks to an extension of Stokes’ theorem proven by Tuck, and so there is no need to introduce this extra simplification. We shall return to Tuck’s theorem presently. The orders of magnitude of the terms in (5.8) should be mentioned. Both n j and mj are 0(1) for j = 2, 3, 5, 6. Since ti= O(&),as stated in (5.2), it appears that the right-hand side of (5.8) is O ( ~ Eand ) , so we may suppose that = O(de2), since the operator a/an = O(E-I). However, this conclusion is based on the implied assumption that o = O(1)as E -+ 0. From the results of Section Ill, we are aware that other assumptions relative to frequency may lead to more useful conclusions. This is indeed the case here, and we shall generally assume that w = O ( E - ’ ” ) . In addition, we assume that U = 0(1) as E + O.+ Under these conditions, the right-hand side of (5.8) contains terms of two different orders of magnitude, O ( ~ E ”and ~ ) O ( ~ E Thus ). the that m3

m6

+

+

’

Faltinsen (1971) assumed that w = O ( E - ’ ’ ~ -and ~ ) U = O ( E ” ~ - “ with ), 0 < a 5 4. If wo is the radian frequency of incident waves moving in the direction fl with respect to the positive x axis, the frequency being measured in a reference frame moving with the stream, these assumptions give w o = O ( E - ’ ’ ~so ) , that the wavelength of the incident waves is O ( E )(provided < n/2). In view of the discussion in Section III,D,4, this is seen to be a reasonable that choice, and it led to an elegantly simple result in Faltinsen’sanalysis. However, it implies that the value of the frequency of encounter is controlled more by the Coppler shift than by the actual wave frequency. Thus, w = coo + vo LI cos fl.., vo CJ cos fi, where vo = co: /g.

173

Ship Hydrodynamics

satisfies (5.8) will consist of two terms Y,

$(x9

-

Y , 4 + $z(x, Y? z )

4 -.,$ I k

(5.10)

-"-

o(w2)

-

O(dE2)

In the lowest-order solution, we have a$l / a n = i o n j t j ,which is identical to the boundary condition in the zero-speed problem. Thus the lowest-order solution is a primitive strip theory containing no effects of forward speed. (This conclusion will be supported also when we consider the free-surface conditions and the partial differential equation.) There are the two usual conditions to be satisfied on the free surface. Let the shape of the free surface be expressed as follows {(x, Y , t) = U V ( X ,Y )

1

+ e ( x , y)eio' + 0(6'),

(5.11)

where Uq(x, y) gives the wave shape in the steady-motion case. The two boundary conditions to be satisfied on this surface are

0= g i

+ 4t + U 4 x + i(4; + 4; + 4;)

0 = (U + 4 x K x

+ 4 y i y - 42 + r,

on z = ((x, y , t), (5.12) on z = ((x, y , t). (5.13)

We assume that the potential and its derivatives in these expressions can be expanded with respect to the plane z = 0, and we arrange all terms according to their orders of magnitude with respect to both E and 6. From (4.5) and (4.8) [after the change of notation implied by (5.7) and (5.1l)], we have

From (5.10), we have

x = O(E2),

(5.14a)

V = O(E2).

(5.14b)

+=

(5.14~)

o(w2),

and a comparison of the time-dependent terms in (5.12)+suggests that 8 =O(~E). (5.14d) When all of this information is used in (5.12) and (5.13), the results are as follows 0 = go

+ iw$ + U ( $ , + x y $ y ) + -..,

L__

W E )

0(aE 312

+ xy$hy - $xZz) + ..., -+-io8 - -( i o U / g ) ( $ xv

0 = -$=

I

0(6&'/2)

' In particular, the terms g[

(5.15)

v

O@&)

+ 4, include the combination gf3 + iwJl = O(6e).

(5.16)

T . Francis Ogilvie

174

both to be satisfied on z = 0. If we are to find a two-term solution for $, both terms must be used in the lower-order terms of these conditions, but we can and 8 8 , in the higher-order terms. We can eliminate 8 from let $ these two conditions, obtaining (5.17) 0 = $ z - v$ + (ioU/g)(2$, + 2xy$y - $x,J + ’.. .

-

-

--

v

O(d&1’2) O@) The lower-order terms in (5.17) are identical to the terms in the zero-speed high-frequency problem of Section III,C,2 [see (3.30)], and so, just as with the body boundary condition, the lowest-order condition on the free surface contains no forward-speed effects. If we had not introduced the slenderness approximations, but instead simply linearized the two full free-surface conditions, we should have obtained the following condition on $

0 = $z - v$

+ (2ioU/g)$, + (U2/g)$xx

on z = 0.

(5.18)

This is the condition used in earlier work on the heave and pitch motions of thin ships; see, e.g., Stoker (1957).The first two terms here appear also as the first two terms in (5.17). The third term here appears in (5.17) too, but it is of higher order than the first two. The final term in (5.18) would be O ( ~ E ~in’ ’ ) the high-frequency slender-body scheme, and so it is missing altogether from (5.17). This shows clearly the way in which the assumption of high frequency deemphasizes the forward-speed terms and so it leads to a zero-speed strip theory for a first approximation. There are two more terms in (5.17) which do not appear at all in (5.18). They represent an interaction between the unsteady oscillatory flow and the perturbation of the incident stream in the steady-motion problem. The effect is quite similar to those terms in the body boundary condition (5.8) involving the mj quantities. The slender-body formalism requires that these terms be present, and, as found by Ogilvie and Tuck (1969), the problem represented by the second-order terms of (5.17) becomes even more difficult if they are omitted. Actually, the standard ship-motion analyses of the present day are all based on the neglect of all of the second-order terms of (5.17), even the strictly linear term involving t+hx. The justification for this is solely that it gives predictions that check fairly well with experimental data. As one might expect, the predictions are poorer at high speed than at low speed, and in Section V,C we shall discuss an alternative recently developed by Chapman (1975) for obtaining good high-speed predictions. In the case of the forced-oscillation problem, the fundamental slenderbody assumption (2.1) can be applied to the 3-D Laplace equation, and the result is that the fluid motion is governed by the 2-D Laplace equation in the near field; this is true for at least the first two terms in the expansion. If

Ship Hydrodynamics

175

incident waves are present, this conclusion should be modified. The incident waves may be assumed to vary with x as exp(-iv,x cos /?), where vo = m i /g, with mo the radian frequency of the incident waves in a reference frame fixed to the stream at infinity. This is not a “slowly varying” dependence on x, and the 3-D Laplace equation reduces to a 2-D Helmholtz equation $yy

+ $zz

- (vo

cos

PI2$

= 0.

[See (3.44) and (3.44‘).]This implies that the diffraction waves have approximately the same kind of x dependence as the incident waves. Apparently, no one uses this equation in the prediction of wave loads on a ship; either the diffraction problem is avoided completely through the use of the Khaskind relation or on the basis of some kind of relative-motion hypothesis,” or the Laplace equation is used without formal justification. This situation has been discussed in some detail by Newman (1970) and Ogilvie (1974). The first term in the near-field expansion must represent simple outgoing 2-D waves far away, just as in the zero-speed problem. But the radiation condition on the second-order term is not so simple, and one must take great care to ensure that the two-term solution properly matches the corresponding far-field expansion. This is discussed briefly in the following section. In the far field, the potential should satisfy the 3-D Laplace equation and the free-surface condition (5.18). The reasoning is similar to that in Section III,C,2 for the far field. Formally, one might expect to set $ = 0 on z = 0 in the far field, but the far-field solution would then not represent any wave motion, and we expect on physical grounds that the far-field flow will be dominantly a wave motion. In the forced-oscillation problem, the far-field solution represents a line of sources and dipoles on the x axis, with densities given by ~ ( xexp(iwt) ) and p ( x ) exp(iwt), where ~ ( x and ) p ( x ) are “slowly varying” functions. In incident-wave problems, one expects the source and dipole densities to vary primarily in a wavelike way, corresponding to the wavelike nature of the incident waves, and so we may suppose that the densities may be expressed ~ ( x exp(iwt ) - vo x cos 8) and p ( x ) exp(imt - vo x cos P). Again, we suppose that +) and &) are slowly varying. In order to solve these problems, one must find general two-term solutions of the near-field problems. Then these must be matched to the far-field solutions, representing the line distributions of singularities. The matching yields the functions o(x) and p(x), and it ensures that the near-field solutions satisfy proper radiation conditions. To accomplish such a matching is not a trivial task, and the results are not obvious beforehand, as they were in the zero-speed problem. The general procedure will only be sketched in the following section, and it will be discussed only in the context of the problem of forced heave and pitch motion. The analysis for forced sway and yaw is “

T. Francis Ogilvie

176

similar. The problem of a ship in incident waves will not be considered further, largely because it still involves many unresolved questions. Indeed, even the forced-oscillation solution has some unsatisfactory aspects, as will be evident. €3. SOLUTION OF THE HEAVE/PITCH FORCED-OSCILLATION PROBLEM

The unsteady-motion potential can be written in the form $(x, y, z)eiu'

- [$l(x,

y, z)

+ $2(x,

y, z)]e'"',

where $ 1 and $2 both satisfy the 2-D Laplace equation in the near field. In addition, $ satisfies the body and free-surface conditions = iw(n3c3

+ nSts)

$lz-v$l

and

tj2

=O

on So(x, y, z ) = 0, on

(5.19a)

z=O,

(5.20a)

satisfies the following conditions

a$2/an *&- v$2 =

=

+

U ( m 3 t 3 m,t,)

on s,(x, y,

- (iwU/g)(2hx + 2XY$lY- $1

Xzz)

2)

= 0,

on

z

(5.19b) = 0.

(5.20b)

problem is identical to that discussed in Sec[See (5.8) and (5.17).] The tion III,C,2, and so it requires no further consideration here. In general, the boundary-value problem must be solved numerically, either through a conformal mapping technique (Tasai, 1959; Porter, 1960) or by the solution of an integral equation (Frank, 1967). For large I y 1, the solution represents outgoing waves, as expressed in (3.31a). ,tz problem could be solved in a similar way except for the nonhoThe h mogeneous free-surface condition (5.20b). As it turns out, we shall be able to avoid completely the necessity to find $2 at all, but the nature of this term must be understood if this approach is to be used (or even evaluated), for its properties will be implied in the process of removing it from the final formulas for force on the ship. Since the only new difficulty comes from (5.20b), let us ignore the presence of the body for the moment. When we have found some kind of a potential that satisfies the nonhomogeneous free-surface condition, we can always add to it another potential that satisfies the homogeneous free-surface condition and that, together with the first, satisfies the body boundary condition. The right-hand side of (5.20b) can be interpreted as if it came from an applied pressure field on the free surface, say,

P(x, y)eiu' = P U ( ~ @+ ,2~x y @ l y- $

1 ~ ~ ~ ) e ~ ~ (5.21) ~ .

177

Ship Hydrodynamics

One can verify this readily by rewriting the Bernoulli equation on z = [ with the pressure now given by this expression. In the absence of a body, one can write down a formal solution of the problem by consulting Wehausen and Laitone (1960). However, their formal solution is only proven to be valid if the pressure distribution is absolutely integrable, and the right-hand side of (5.21) is certainly not so, since the first term includes a term like that in (3.3 la). In fact, the difficulty is much worse than this, because $ 1, has just the y dependence of a solution of the homogeneous problem corresponding to the $2 problem. There are various ways in which one can nevertheless construct a solution of the problem. The one used by Ogilvie and Tuck (1969) is as follows: Let the troublesome part of P ( x , y) in (5.21) have the form PO(X, Y ) = Poble- i v b l - PM,

(5.22a)

where p is a small positive number that will eventually go to zero. Then the corresponding potential, say, 40(x, y , z), can be found as prescribed by Wehausen and Laitone, after which we can allow p + 0. The resulting potential has an 'unusual behavior far away however: As I y I -+ 00, $,(x, y , z )

-

(io/pg)evzpo(x)( -z

+ i 1 y 1 )e-ivlyl.

(5.22b)

This expression represents outgoing waves that have amplitude increasing linearly with distance from the origin. The solution of the i,bz problem will have a similar term, along with terms that are relatively conventional. Such a result is difficult to interpret in the usual way of solving ship/wave problems, but fortunately this is just part of the near-field solution. The only requirement at large I y I in the near field is that the solution should match with an appropriate far-field solution. This is similar to the situation discussed in Section 111,D,3; see, for example, (3.57) and (3.57'). If the above result is to be accepted, we must find the matching far-field expansion. Ogilvie and Tuck (1969) expressed the far-field solution in the following way: Let the potential of a line distribution of sources of density c(x)eio' be written 4 ( x , y , z)eiW'.Then

where o*(k) is the Fourier transform of a ( x ) , and C ( k )is a countour from - 00 to 00 passing under the pole of the inner integrand at - I, and over it at + I o , with lo defined as the positive root of the denominator of the integrand. The above expression has a two-term inner expansion ie"ze-i'lYl[o(x)- ( 2 i o U / g ) ( z - i I y I )o'(x)]

as y

+ 0.

(5.24)

178

T. Francis Ogilvie

The first term matches with the first-order near-field solution, which appears in a form like (3.3la), with a(.) known. Thus ~ ( xis) determined. The second term here is obviously of just the right form to match with the expression in (5.22b). In fact, the matching occurs precisely and automatically: The coefficient po(x) in (5.22b) can be found from the first-order solution and expressed in terms of cr(x), and, when- this has been done, it agrees completely with the second term in (5.24). Thus one can have some confidence that the manner of constructing cpo, as well as its interpretation, is indeed correct. It should be noted that a complete matching of the second-order solution requires that the far-field solution also be obtained to two orders of magnitude. Thus, one may replace cr(x) above by crl(x) and then introduce a second-order source distribution a,(x), which will provide terms to match with the remaining parts of $,, which have been ignored above. We d o not need such results explicitly, however. In principle, then, we can solve for the first two terms in the near-field expansion, and the solution has the behavior that it ought to have. We could even carry out such a solution, but the amount of computation needed would be tremendous, for two reasons: (i) The second-order term depends on the explicit form of the steady-motion potential, as indicated in (5.19b) and (5.20b), and, in fact, it depends on the x derivative of that potential, which is especially difficult to compute, since x is only found section by section, with x appearing as a parameter. (ii) The solution of a problem involving an imposed pressure distribution on the free surface is very complicated to carry out, even if there are no difficulties in principle. Therefore, it is fortunate that we can compute the force and moment on the ship without actually solving for This can be done largely as the result of a theorem proven by Tuck (see Ogilvie and Tuck, 1969): Let $(x, y, z) be a differentiable scalar function. Let v(x, y, z ) be as specified in (5.4), that is, it satisfies (a) V v = 0 in the fluid, (b) V x v = 0 in the fluid, and (c) n v = 0 on So(x, y, z) = 0. Let nj and mj be given by (5.5) and (5.6) f o r j = 1, . . .,6. Let Co denote the line of intersection of So and the plane z = 0, with t denoting a unit vector tangent to C , (positive in the counterclockwise sense when the ship is viewed from above). Then the theorem states that

+,.

-

-

ij[mj4 + nj(v - V$)] so

dS =

-1

-

dlnj#(t x n) v. co

(5.25)

If the ship is wall-sided at the waterline (which we always assume anyway), then t x n = k, the unit vector parallel to the z axis. The first approximation of v(x, y, z) satisfies the condition k v = 0 on the plane

-

179

Ship Hydrodynamics

z = 0, and so the integrand of the line integral is of higher order than any quantity that we have considered so far. Furthermore, for the heave and pitch cases, nj = 0 at the waterline, and so the line integral is identically zero. This theorem is used in the formula for computing force and moment on the ship. Let F j ( t ) correspond to t h e j mode of motion, representing force or moment, as appropriate. Using the Bernoulli equation, we can write

This must be integrated over the hull surface in its instantaneous location. However, we can expand all quantities with respect to points on the mean location by applying the operator (1 a V ...) [see (5.l)l to the quantity in brackets in the integrand. Limiting our attention to the unsteady part of the force, we find that

+

F j ( t )= - p

- +

(1 dSnj{(io$ + U(v - V)$]eiur + g a - kJ So'

= -p

Jid S n j { -[ i o $ , + iot+h2

SO

O(bE)

+ U(v

*

V)$,

+ ...Ie'""+ g a

*

k}.

Y

o ( ~ 7

The last term yields the buoyancy force, which can be computed readily. The most troublesome term is the one involving the gradient operator. This requires differentiation in the x direction, which is awkward, since is determined at each section in turn, and so differentiation must be a strictly numerical process. However, this term is recognized as being identical to one term in the theorem (5.25), and so we can rewrite the force expression Fj(t)= - p

ifd S [ ( n j i w $ , + n j i o $ ,

- Umj$l)e'"r

+ga

k]. (5.26)

SO Thus we have eliminated the need to differentiate $,, but the presence of the mi factors implies the presence of second derivatives of x [see (5.9)], which is equally awkward. However, the terms here involving t,h2 and mj can now be manipulated by Green's theorem. Let us write: = io @'ti and $2 = U C ",ti. Then, from (5.19a) and (5.19b), it is evident that a@,/an = n, and dY,/ a n = mi.When these definitions are used in the above formula, a long series of operations, mostly involving Green's theorem, can be performed, with the result that only appears in the final formulas, and

T. Francis Ogilvie

180

there is no call to differentiate this function. Even the steady-motion potential x vanishes, and we can write $1

=i

4 3 ( < 3

- x55).

so that only the single numerically determined potential (P3 appears. Details will be found in Ogilvie and Tuck (1969) for the heave/pitch case. The result is rather complicated but nevertheless worth writing out. It can be expressed most simply as follows ( 0 2 a j i- iob.. J I + c..)tieiof, JC

Fj(t)=

(5.27)

i

where cji comes from the buoyancy term, and aji and bji are given by (5.28a) (5.28b) (5.28~) (5.28d) (5.28e) Re (2p02U/g)

1

dS@3 ,

f,

(5.28f) (5.283)

bS3= bi0J u\'J

+ Ua\'J + Re

I

J

(2pw2U/g)-f dS@$ , F

+ (l/io)b\Oi = p

dSn@, ,

(5.28h) (5.29a)

0

ai0i + (l/io)b'P,'= p a:'\

.I,

dSx2n@, ,

(5.29b)

+ (l/io)b$O: = a?'\ + (l/io)b\OJ = -

Lo

dSxn@,.

(5.29~)

The integral over F has a special interpretation, suggested by the slash through the integral sign. Basically, it represents an integral over the free surface, which arises when Green's theorem is applied to (5.26) (remember that t,b2 does not satisfy the same free-surface condition as t,b1)¶ but the integrand oscillates with nondiminishing amplitude all the way to infinity. Let

181

Ship Hydrodynamics

-

%(x, Y , 0) &(x)exp( - iv I y I) as 1 y question as follows L

-f, dScp: = c, d x JYO@)dY(@

I -,co. Then we define the integral in

m

- .&Ze-

Z ~ V Y )-

2vi

5

L dx&ze- 2ivyo(x)

0

where y o ( x ) is the half-beam at x. Thus,we subtract from the integrand just enough to remove its troublesome behavior at infinity; this yields an unwanted contribution at the lower limit, y = y&), which is removed by the single-integral term. In the representation of F,(t) in (5.27), the aji coefficients are added-mass coefficients, and the bji coefficients are damping Coefficients.' The manner of writing the results above shows explicitly how the forward speed enters into calculations: The coefficients a, b,, , b,, do not depend on speed at all. The coupling coefficients do depend on speed, but the speed-dependent terms are antisymmetric, as required by a theorem proved by Timman and Newman (1962). The zero-speed results, as given in (5.29), are the same in all of the standard methods of predicting wave loads. The first speed-dependent term in each of the formulas (5.28e)-(5.28h) is also generally used in current practice. However, the integral terms in these formulas are not standard, because they arose from the nonhomogeneity of the free-surface condition on I ) ~ which , is ignored by everyone except Ogilvie and Tuck. It may also be recalled that the body boundary condition on t,h2, (5.19), is usually simplified too, as discussed above following (5.9). However, that simplification has no consequences in the final results just presented, thanks to Tuck's theorem. It is not likely that the above results will ever form the basis of a standard method of computing wave loads on a ship, because the integrals in (5.28e)-(5.28h) are complicated and expensive to compute. Faltinsen (1974) computed them in order to evaluate the seriousness of ignoring them, and he found that they did lead to some improvement in the prediction of the coupling coefficients, but the changes were not very great. His judgments were based on a direct comparison of the predicted coefficients with experimentally measured values. By a private communication, he reports that the predictions of ship motions were worse when the improved coefficients were used in the equations of motion. This indicates that there must be other errors in the prediction procedure, presumably in the prediction of the force caused by incident waves, and that such errors are canceling the errors caused by omitting the integral terms in the coupling coefficients. It seems This is true in the sense that they appear, respectively, in acceleration and velocity terms in the force expression. However, the coupling added-mass coefficients(aiiwith i # j) are involved in a calculation of work done on the ship by the fluid, and the coupling damping coefficients have some reactive force eNects.

182

T. Francis Ogilvie

that the fundamental basis for predicting wave loads and ship motions is not so secure as is often stated. Troesch (1976) has analyzed the problem of a ship in incident waves under essentially the same assumptions used by Ogilvie and Tuck. The results, too, are similar but more complicated, because there are more length scales to be considered. If the incident waves move in the direction p with respect to the x axis, one can expect the diffraction potential to be nearly periodic along the x direction, with effective wave number vo cos /I, where v,, = o i / g , oo being the radian frequency of the waves in a reference frame fixed to the fluid at infinity. Then, just as in the problem of Section III,D,3, the near-field problem is found to be governed by the Helmholtz equation [see Eq. (3.44’)]. Troesch finds two terms in the expansion for the diffraction potential in the near field, and they satisfy free-surface conditions much like (5.20a) and (5.20b). The final result for the force on the ship is similar to the results above; in particular, he finds it necessary to perform an integration over the free surface. Such computations have not been done for the diffraction problem, however.

C. HIGH-SPEEDSLENDER-SHIP THEORY The slender-ship theory of the preceding section was developed by Ogilvie and Tuck (1969) to provide a formal framework for the strip theory of Korvin-Kroukovsky (1955) which had proved to be quite effective in use even if its derivation was largely heuristic. Of course, the key to providing such a framework was the assumption that frequency is “high,’’ in the sense ) . standard methods of predicting wave loads and ship that o = O ( E - ’ / ~The motions, mentioned at the beginning of the last section, were motivated similarly, although the authors have generally been less explicit in stating this. Two defects of the foregoing approach should be noted, however: (1) The steady-motion problem is either avoided completely [by ignoring certain terms in the boundary conditions, (5.19b) and (5.20b)l or the ordinary steady-motion slender-ship theory (as described in Section IV,B) is adopted. The latter has some prominent shortcomings, as discussed in Section IV,C. (2) The foregoing approach is not clearly valid at either high or low speeds, although it seems to work adequately at low to moderate speeds. The second point is worth some elaboration. The form of the solution, especially as presented in (5.28at(5.28h), suggests that the theory is a lowspeed theory, in the sense that the first approximation is the zero-speed solution, all forward-speed effects entering in the second-order terms. But the analysis by Ogilvie and Tuck shows clearly that one must assume that T = o U / g > $; this appears in going from (5.23) to (5.24).The importance of the parameter T in ship-motion problems has been recognized for many

183

Ship Hydrodynamics

years. At T = 4,the group velocity of the unsteady ship-generated waves precisely equals U , the ship speed. For T < $, such waves can propagate out ahead of the ship, whereas, for T > there are no significant ship-generated ) , so waves ahead of the ship.’ In the Ogilvie-Tuck analysis, T = O ( E - ” ~ and it is not enough in practical applications to require simply that o be large. On the other hand, theory is certainly not to be trusted at very high speeds either. The neglect of the U 2 term in (5.18) and the relegating of even the U term to the second-order problem suggest that U cannot be very large (although we cannot know where the limit of validity occurs). Furthermore, if the foregoing theory were carried to a third order of magnitude, many of the quantities in (5.28) would contain terms proportional to U 2 , and these might be very important indeed at high speed. Chapman (1975) has extended his steady-motion analysis, described in Section IV,D,3, to the case of a vertical surface-piercing flat plate in unsteady yaw and sway motion. Although he has not done this in terms of a systematic perturbation analysis, his method is at least plausible and his results for high-speed problems are very impressive. This is the only present-day approach of which we are aware in which the first approximation is not a simple strip theory. For this reason, as well as for the good high-speed predictions that have been produced, it is worth intensive study in the near future. Chapman’s approach is basically very simple. A flat plate of length L and draft T is located in a uniform stream of speed U moving in the positive x direction. The plate undergoes a sway motion, given by t2exp(iot), which is defined to be the lateral displacement of the midpoint (at x = L/2), and a yaw motion, given by t6 exp(iot). The body boundary condition, corresponding to (5.8), is

a,

Because the flat plate makes no disturbance at all in its neutral position, we note that v = i, a fact which greatly simplifies the problem. Chapman satisfies the usual full linearized free-surface condition (5.18) 0 = U2$,,

+ 2iwU1//, - 02$+ g$,

on z = 0.

(5.3 1)

He assumes, in the usual spirit of slender-body theory, that $yy

+ $zz

=0

(5.32)

+ In the analysis of such unsteady forward-motion problems, one finds that many quantities become undefined (infinite) at T = This is a clear boundary between two quite distinct regimes. It is curious to note, then, that experiments do not show any extraordinary phenomena near this mathematically critical condition, at least, not with respect to wave forces and ship motions.

a.

184

T. Francis Ogilvie

in the fluid region. In addition, he requires that the potential satisfy a radiation condition in the crossplanes, a condition which he applies numerically, and he assumes that [cf. (4.27)]

*=o,

x
(5.33)

Chapman has already solved a somewhat similar steady-motion problem, and he reduces the above problem to computing a superposition of such steady-motion solutions. Let the solution of the steady-motion problem, say, f ( x , y, z), satisfy the following conditions

TI aY

=1,

-T
O<x
y=fO

O=

Uyxx+4fi on

fyy

+I22

f=O,

z=O,

= 0,

x < 0.

Then any function of the form Y , 2) = ei"x'u [ A f ( x , y ,

4 + jx d 5 B ( 5 ) f ( x - 5, y, z ) 0

]

(5.34)

satisfies (5.31), (5.32),and (5.33),regardless of how A and B ( x )are chosen. If now the choices A

= b(0)

and

B ( x ) = (d/d~)[b(x)e'""'~]

(5.34)

are made, it can be shown by direct substitution that the condition on the plate, (5.30), is satisfied too, and so the solution is at hand in terms of the solution of the steady-motion problem. Chapman has computed the resulting force and moment on such a plate and compared his predictions with experimental data and with strip-theory predictions. The agreement with experiments is extraordinarily good at high speed, where strip theory fails badly. This theory should fail at low speed, since it is based on a steady-motion solution which includes only the diverging waves of the Kelvin wave system. See the discussion that accompanies (4.32).There appears to be no way to remove this deficiency while retaining the 2-D Laplace equation, although presumably Chapman's solution could be interpreted as a term in an inner expansion, which ought to match an outer expansion, as in the method of matched asymptotic expansions. Perhaps then the rest of the wave system generated by the oscillating plate could be introduced into the near field too. One encounters some difficulty in trying to develop a formalism for Chapman's problem. In order to obtain the 2 - D Laplace equation in the near

Ship Hydrodynamics

185

field, it seems to be necessary to assume that (2.1) is valid, just as in the strip-theory approaches. Then, in order to keep all four terms in (5.3l), one must assume that (i) o = 0(&-’l2) and (ii) either U = O ( E - ~ ’or~ )a/ax = O(E-1’2). The assumption about frequency hardly needs further comment. The choice in (ii) is more difficult, however. The assumption of large U seems attractive, especially in view of the foregoing comments about the validity of Chapman’s theory. However, it leads to difficulty: The freesurface elevation should then be large, in an order-of-magnitude sense, so that the boundary condition there cannot be transferred to the plane z = 0, and so one does not obtain (5.31) at all. The alternative, d/ax = O ( E - ” ~ ) , leads to an interpretation of Chapman’s solution as being valid only in the bow region. Perhaps this is reasonable, especially in view of the interpretation in Section IV,D of Chapman’s steady-motion solution as a local bowregion solution. If this interpretation is correct, it should be possible to match Chapman’s solution to another solution valid along the rest of the ship,. away from the bow, and that solution would probably be the usual strip theory. Finally, it may be noticed that (5.30)contains terms of two different orders of magnitude, just as the right-hand side of (5.8) contained two orders of magnitude. [See the breakdown in (5.19a) and (5.19b).] So it is necessary in principle to find a solution to two orders of magnitude in order to satisfy (5.30). However, it can be shown then that the free-surface condition (5.3 1) is valid for just the lowest-order term; the second-order condition will be nonhomogeneous [rather like (5.20b), but much more complicated]. Thus, if the above formal reasoning is valid, the term U16 should be omitted from (5.30) so that the problem may then be consistent in order-of-magnitude sense. However, from a physical point of view, it seems hard to imagine that such an omission is correct. Chapman’s method will be much more difficult to apply to shiplike bodies, since the body condition (5.30) will have to be applied on a different contour at each cross section, and so the superposition implied by (5.34) will be much more complicated, if indeed it is possible at all. Nevertheless, Chapman’s accomplishment so far has made possible an entirely new outlook on the ship-motion problem, and a promising outlook at that, providing guidance on how to handle those cases for which strip theory is least adequate. REFERENCES ABRAMOWITZ, M., and STEGUN, I. (1964). “Handbook of Mathematical Functions.” US Govt. Printing Office, Washington, D.C. ADACHI,H. (1973). “ On Some Experimental Results of a Ship with Extremely Long Parallel Middle Body,” Rep. 10, No. 4, pp. 159-173. Ship Res. Inst., Mitaka, Tokyo.

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186

ASHLEY, H., and LANDAHL, M. (1965). “Aerodynamics of Wings and Bodies.” Addison-Wesley, Reading, Massachusetts. BABA,E. (1974). Analysis of bow near field of flat ships. J . SOC. Nau. Archit. Jpn. 135, 25-37. BABA,E., and TAKEKUMA, K. (1975). A study on free-surface flow around bow of slowly moving full forms. J. SOC. Nav. Archit. Jpn. 137, 1-21. BOLTON, W. E., and URSELL,F. (1973). The wave force on an infinitely long circular cylinder in an oblique sea. J. Fluid Mech. 57, 241-216. BORODAI, I. K., and NETSVETAYEV, Y. A. (1969). “Ship Motions in Ocean Waves.” Sudostroenie, Leningrad (in Russian). CHAPMAN, R. B. (1975). Numerical solution for hydrodynamic forces on a surface-piercing plate oscillating in yaw and sway. Proc. Zst l n t . Symp. Numer. Hydrodyn., pp. 333-350. David W. Taylor Naval Ship R & D Center, Bethesda, Maryland. CHAPMAN, R. B. (1976). Free-surface effects for yawed surface-piercing plate. J . Ship Res. 20, 125- 136.

CUMMINS, W. E. (1956). The wave resistance of a floating slender body. Ph.D. Thesis, American University, Washington, D.C. M. P. (1972). Two-dimensional free-surface gravity flow past blunt DAGAN,G., and TULIN, bodies. J. Fluid Mech. 51, 529-543. DAOUD,N. (1973). “Force and Moments on Asymmetric and Yawed Bodies on a Free Surface,” Rep. NA 73-2. College of Engineering, University of California, Berkeley. DAOUD,N. (1975). “Potential Flow Near to a Fine Ship’s Bow,” Rep. No. 177. Dep. Nav. Archit. Mar. Eng., University of Michigan, Ann Arbor. 0. M. (1971). Wave forces on a restrained ship in head-sea waves. Ph.D. Thesis, FALTINSEN, University of Michigan, Ann Arbor. [Also: “A Rational Strip Theory of Ship Motions,” Part 11, Rep. No. 113. Dep. Nav. Archit. Mar. Eng., University of Michigan, Ann Arbor (1972).]

FALTINSEN, 0. M. (1974). A numerical investigation of the Ogilvie-Tuck formulas for addedmass and damping coefficients. J . Ship Res. 18, 73-85. FALTINSEN, 0. M., and MICHELSEN, F. C. (1974). Motions of large structures in waves at zero Froude number. Proc. Int. Symp. Dyn. M a r . Vehicles & Struct. Waves, pp. 91-106. Inst. Mech. Engrs., London. FRANK, W. (1967). “Oscillation of Cylinders in or Below the Free Surface of Deep Fluids,” Rep. No. 2375. Nav. Ship Res. & Dev. Cent., Bethesda, Maryland. T. (1959). On the reverse flow theorem concerning.wavemaking theory. Proc. J p n . HANAOKA, Natl. Congr. Appl. Mech., 9th, pp. 223-226. HAVELOCK, T. H.(1923). Studies in wave resistance: Influence of the form of the water-plane section of the ship. Proc. R. SOC. London, Ser. A 103, 571-585. HIRATA,M. H. (1972). “ O n the Steady Turn of a Ship,” Rep. No. 134. Dep. Nav. Archit. Mar. Eng., University of Michigan, Ann Arbor. [Also: (1975). The flow near the bow of a steadily turning ship. J. Fluid Mech. 71, 283-291.1 INUI,T., and KAJITANI,H. (1976). Hull form design, its practice and theoretical background. Proc. Int. Semin. W a v e Resistance, pp. 159-183. SOC.Nav. Archit. Japan, Tokyo. JOOSEN, W. P. A. (1964). Slender-body theory for an oscillating ship at forward speed. Proc. Symp. Nau. Hydrodyn., 5th, ACR-112, pp. 167-183. Office of Naval Research, Washington, D.C. KHASKIND, M. D. (1946). The oscillation of a ship in still water. Izu. Akad. Nauk SSSR, Otd. Tekh. Nauk pp. 23-34; translated in (1953) SOC. Nav. Archit. Mar. Eng., Tech. Res. Bull. Nos. 1-12, pp. 45-60. SOC.Nav. Archit. Mar. Eng., New York. KHASKIND, M. D. (1957). The exciting forces and wetting of ships in waves. Izv. Akad. Nauk SSSR. Otd. Tekh. Nauk pp. 65-79; translation: (1962) David Taylor Model Basin Trans.,

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p. 307. Bethesda, Maryland. KORVIN-KROUKOVSKY, B. V. (1955). Investigation of ship motions in regular waves. SOC.Nav. Archit. M a r . Eng., Trans. 63, 386-435. LAMB,H. (1932). “ Hydrodynamics.” Cambridge Univ. Press, London and New York (also: Dover, New York, 1945). MARUO, H. (1962). Calculation of the wave resistance of ships, the draught of which is as small as the beam. J . SOC.Nau. Archit. Jpn. 112, 21-37. MARUO,H. (1967a). Application of the slender body theory to the longitudinal motion of ships among waves. Bull. Fac. Eng., Yokohama Narl. Uniu. 16, 29-61. MARUO,H. (1967b). High- and low-aspect ratio approximation of planing surfaces. Schiffstechnik 14, 57-64. MARUO,H., and SASAKI, N. (1974). On the wave pressure acting on the surface of an elongated body fixed in head seas. J . SOC.Nav. Archit. Jpn. 136, 34-42. MICHELL, J. H. (1898). The wave resistance of a ship. Phil. M a g . [5] 45, 106-123. MUNK,M. ( 1924).The aerodynamic forces on airship hulls. Natl. Aduis. Comm. Aeronaut., Rep. 184. NEWMAN, J. N. (1959). The damping and wave resistance of a pitching and heaving ship. J . Ship Res. 3:1, 1-19. NEWMAN, J. N. (1962). The exciting forces on fixed bodies in waves. J . Ship Res. 5:1, 34-55. NEWMAN, J. N. (1964). A slender-body theory for ship oscillations in waves. J . Fluid Mech. 18, 602-618. NEWMAN, J. N. (1965).The exciting forces on a moving body in waves. J . Ship Res. 9, 19C199. NEWMAN, J. N. (1970). Applications of slender-body theory in ship hydrodynamics. Annu. Rev. Fluid Mech. 2, 67-94. J. N. (1976). Linearized wave resistance theory. Proc. Jnt. Semin. W a v e Resistance, NEWMAN, pp. 31-43. SOC.Nav. Archit. Japan, Tokyo. NEWMAN, J. N., and TUCK,E. 0. (1964). Current progress in the slender-body theory of ship motions. Proc. Symp. Nav. Hydrodyn., 5th, ACR-112, pp. 129-167. Office of Naval Research, Washington, D.C. J. N., and Wu, T. Y. (1973). A generalized slender-body theory for fish-like forms. J . NEWMAN, Fluid Mech. 57, 613-693. OGILVIE,T. F. (1967). Nonlinear high-Froude-number free-surface problems. J. Eng. Math. 1, 215-235. OGILVIE,T. F. (1968). “Wave Resistance: The Low-speed Limit,” Rep. No. 002. Dep. Nav. Archit. Mar. Eng., University of Michigan, Ann Arbor. OGILVIE, T. F. (1972). The wave generated by a fine ship bow. Proc. Symp. Nau. Hydrodyn., Pth, ACR-203, pp. 1483-1525. Office of Naval Research, Washington, D.C. OGILVIE, T. F. (1973). The Chertock formulas for computing unsteady fluid dynamic force on a body. Z. Angew. Math. Mech. 53, 573-582. OGILVIE, T. F. (1974). Fundamental assumptions in ship-motion theory. Proc. I n t . Symp. Dyn. Mar. Vehicles & Struct. Waves, pp. 135-145. Inst. Mech. Engrs., London. OGILVIE,T. F., and TUCK,E. 0. (1969). “A Rational Strip Theory for Ship Mqtions,” Part 1, Rep. No. 013. Dep. Nav. Archit. Mar. Eng., University of Michigan, Ann Arbor. J. J. (1957). The motion of a ship, as a floating rigid body, in a PETERS.A. S., and STOKER, seaway. Commun. Pure Appl. Math. 10, 399-490. PORTER,W. R. (1960). Pressure Distributions, Added-mass and Damping Coefficients for Cylinders Oscillating in a Free Surface,” Rep. No. 82-16. Inst. Eng. Res., University of California, Berkeley. REED,A. M. (1975). “Wave Making: A Low-speed Slender-body Theory,” Rep. No. 169. Dep. Nav. Archit. Mar. Eng., University of Michigan, Ann Arbor. I‘

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RISPIN,P. P. (1967). A singular perturbation method for nonlinear water waves past an obstacle. Ph.D. Thesis, California Institute of Technology, Pasadena. ROGALLO, R. S. (1969). A trailing-edge correction for subsonic slender-wing theory. Ph.D. Thesis, Stanford University, Stanford, California. 0.M. (1970). Ship motions and sea loads. Soc. Nau. SALVESEN, N., TUCK,E. O., and FALTINSEN, Archit. M a r . Eng., Trans. 78, 25C287. SHEN,Y. T., and OGILVIE, T. F. (1972). Nonlinear hydrodynamic theory for finite-span planing surfaces. J. Ship Res. 16, 3-20. SODING,H. (1969). Eine Modifikation der Streifenmethode. Schiflstechnik 16, 15-18. STOKER, J. J. (1957). “Water Waves.” Wiley (Interscience), New York. TANEDA, S. (1974). Necklace vortices. J. Phys. Soc. Jpn. 36, 298-303. TASAI,F. (1959). On the damping force and added mass of ships heaving and pitching. J. Soc. N a n Archit. Jpn. 105, 47-56. TASAI,F.,and TAKAGI,M. (1969). “Theory and Calculation of Ship Responses in Regular Waves.” SOC.Nav. Archit. Jpn., Tokyo. TIMMAN, R., and NEWMAN, J. N. (1962). The coupled damping coefficients of symmetric ships. J . Ship Res. 5:4, 34-55. TROESCH, A. W. (1976). “The Diffraction Potential for a Slender Ship Moving Through Oblique Waves,” Rep. No. 176. Dep. Nav. Archit. Mar. Eng., University of Michigan, Ann Arbor. TUCK,E. 0. (1963). The steady motion of slender ships. Ph.D. Thesis, Cambridge University. [See also: (1964). A systematic asymptotic expansion procedure for slender ships. J. Ship Res. 8 1 , 15-23.]

TUCK,E. 0. (1966). Shallow water flows past slender bodies. J. Fluid Mech. 26, 8 1-95. TUCK,E. 0. (1967). Sinkage and trim in shallow water of finite width. Schiflstechnik 14,92-94. TUCK,E. 0. (1973). “Low-aspect-ratio Flat-ship Theory,” Rep. No. 144. Dep. Nav. Archit. Mar. Eng., University of Michigan, Ann Arbor. F. (1949a). On the heaving motion of a circular cylinder on the surface of a fluid. Q. J. URSELL, Mech. Appl. Math. 2, 218-231. URSELL, F. (1949b). On the rolling motion of cylinders in the surface of a fluid. Q. J. Mech. Appl. Math. 2, 335-353. URSELL,F. (1953). Short surface waves due to an oscillating immersed body. Proc. R. Soc. London, Ser. A 220, 9C103. URSELL,F. (1960). O n Kelvin’s wave pattern. J . Fluid Mech. S, 418-431. URSELL,F.(1961). The transmission of surface waves under surface obstacles. Proc. Cambridge Philos. SOC. 57, 638-668. URSELL, F. (1962). Slender oscillating ships at zero forward speed. J. Fluid Mech. 19,496-516. URSELL,F. (1968). The expansion of water-wave potentials at great distances. Proc. Cambridge Philos. SOC.64, 81 1-826. URSELL, F.(1975). The refraction of head seas by a long ship. J . Fluid Mech. 67, 689-703. VAN DYKE,M. (1975). Perturbation Methods in Fluid Mechanics” (Annotated ed.). Parabolic Press, Stanford, California. VOSSERS, G. (1960). Some applications of the slender-body theory in ship hydrodynamics. Ph.D. Thesis, Tech, Univ., Delft. WARD,G . N. (1955). “Linearized Theory of Steady High-speed Flow.” Cambridge Univ. Press, London and New York. WEHAUSEN, J. V., and LAITONE, E. V. (1960). Surface waves. In “ Handbuch der Physik ” (S. Fliigge, ed.), Vol. 9, pp. 446-778. Springer-Verlag. Berlin and New York. Wu, T. Y. (1967). A singular perturbation theory for nonlinear free surface problems. l n f . Shipbuilding Prog. 14, 88-97. “

Special Topics in Elastostatics' J. L. ERICKSEN The Johns Hopkins University Baltimore, Maryland

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . , . . . . . . . C. Homogeneous Isotropic Materials . . . . . D. Averages . . . . . . . . . . , . . . . IV. Experiment and Mechanistic Theory . . . . . A. The Arena . . . . . . . . . . . . . . B. Inaccessibility . . . . . . . . . . . . . C. Crystal Lattices . . . . . . . . . . . D. lattice Kinematics . . . . . . . . . . E. Molecular Theory . . . . . . . . . . F. Symmetry-Induced Instabilities . . . . . . References . . . . . . . , . . . . . . . .

. . A. Euler-Lagrange Operators . B. Elastostatic Equations . . 111. Semi-Inverse Methods . . . . A. Kinematics . . . . . . . B. Reduced Equations . . . . 11. Basic Equations

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189 192 192 197 200 200 201 210 . 213 . 220 . 220 . 224 . 229 . 232 . 234 . 231 . 24 1

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I. Introduction As the title suggests, this is far from a comprehensive survey of the subject of elastostatics. It is a personal view of the subject covering some but not all of the topics that interest me. The basic equations of elastostatics are, of course, old and well known, but they can be reformulated in various ways. Having experimented with several 'The research work herein reported was supported by a grant from the National Science Foundation. 189

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possibilities, I have come to prefer a format that, at least for me, makes analyses easier. It emphasizes the fact that, mathematically, elasticity theory has much in common with modern theories of thin elastic shells, plates, or rods; static theories of liquid crystals; and various other local theories associated with the most common types of variational principles. It seems that some properties of this format are not well known to workers in mechanics, so I begin with a summary of those I find useful. I then use it in an attempt to seek out features that are common to most of the semi-inverse methods. Having fairly broad interests and a rather poor memory, I find such unifications helpful, almost necessary; it makes it easier for me to retain some general understanding, even if the details get‘lost. Also, I am not averse to borrowing an idea or bit of analysis from one theory to apply to another. With some adaptation, the analysis of semi-inverse solutions is taken from my similar study (1977) in the theory of Cosserat surfaces. Primarily, my interest in elasticity theory lies in improving the theory underlying the somewhat mystical process whereby we select definite forms of constitutive equations. Rather obviously, inverse and semi-inverse methods play an important role in matching experiment to theory, so it seems desirable to attempt to upgrade these. My view is that the domain of constitutive equations may include subdomains that are, in principle, inaccessible to the experimentist. Of course, the operationalist holds a contrary view. As I have come to see it, it is a matter of mathematics to determine what the theory can or cannot predict. That is, we should not declare inadmissible constitutive equations that seem “ physically unreasonable.” As a novice, I thought differently but, eventually, I came to realize that my interpretation of “prediction” was too naive; I had poorly understood matters relating to stability. As I see it, it is a problem in stability theory to ascertain what limits the experimentally accessible domain. It is somewhat thorny to reduce this notion to problems of definite mathematical form. I describe my attempts, which seem to me somewhat instructive, but possibly incomplete. Gradually, it has become clear that elasticity theory can predict effects that we d o not commonly think of as being associated with the adjective “elastic.” In such cases, we should, I think, let elasticity theory enter into free competition with other theories capable of describing the effect at hand. There is a simple, classical example which serves to illustrate my views, as well as some of the subtleties which are to be expected: the equilibrium theory of the van der Waals’ fluid. Here, by commonly accepted stability calculations, we infer that part of the domain is not accessible to experiment, part is easily accessible, and part is accessible only if sufficient care is taken to minimize disturbances. With a proper accounting for stability, the simple predictions seem quite sensible, despite the “ wild ” behavior occurring in the inaccessible part. For solids, it is trickier to formulate physically appropriate

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stability criteria, and thus to classify similarly the different parts of the domain of constitutive functions. We do not evade the problem by discarding ‘‘unreasonable” constitutive. equations. A criterion is also needed for this, and no one seems to please all workers. As I have indicated (1975a), rather detailed studies of elastic bars are feasible. This work, combined with that of Dafermos (1969) illustrates the familiar fact that stability predictions obtaining from energy methods and dynamical studies sometimes disagree. Another approach which can yield definite constitutive equations involves use of molecular theory. From the fact that the constitutive equation of the van der Waals’ fluid is so derivable, it is clear that this method is not restricted to the experimentally accessible domain. For solids, one of the neatest and cleanest is, in essence, the old theory for crystals set down by Cauchy (1829). It, or very similar alternatives, provides the basis for modern attempts to calculate the strength of perfect crystals, such as are summarized by Hill (1975); from the nature of the model, it seems rather evident that it should be capable of predicting the onset of plasticity. Older estimates of such strength made very crude use of the model, so crude that the relation to elasticity theory was obscured. I find it illuminating and consider it fruitful to ascertain what predictions of molecular theory are relatively independent of the fine details of the model, such as the molecular force law assumed, then to study what these imply at the continuum level. In essence, this is the approach adopted by Hill (1975). I have another ax to grind: a constitutive equation produced from a plausible model is, by one interpretation, physically reasonable. If this be granted, most proposed restrictions on constitutive equations are unreasonable. To my way of thinking, one of the most interesting features of this theory is, almost universally, overlooked. As is to be expected from these remarks, some space will be devoted to discussion of this model. With some important topics, it is relatively easy to locate good surveys in the literature; one can always add a few new references or quibble about a value judgment. Thus, for example, there is the article of Truesdell and No11 (1965) which does cover much of the nonlinear theory of elastostatics. The article is a bit weak on elastic stability theory, but Knops and Wilkes (1973) do much to fill this gap. Similarly, Gurtin (1972) covers much of the classical linear theory, etc. In brief, my goal is to concentrate on some of the more neglected areas and to fill some of the gaps I find. The reader will, I think, be hard pressed to find a comparable treatment of semi-inverse solutions anywhere else. He will find classical molecular theories discussed briefly and rather well by Stakgold (1949);there is a ponderous literature on the subject. However, he is among those who overlook just those points that I find most interesting. I do not mean to pick on him. In this matter, he is certainly not alone, perhaps I am. He, as an amateur in the area, reveals a rather deep

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understanding of the subject, deeper than that exhibited by some experts. In this context, the remarks of Hill (1975) are worth reading. I cover only a little of the theory, just enough to make clear issues that interest me. 11. Basic Equations

A. EULER-LAGRANGE OPERATORS In elasticity theory, we encounter two kinds of activities. One involves physical interpretation, either in setting up a problem or in interpreting results of analyses. Here it can be convenient to introduce geometric paraphernalia such as base vectors or metric tensors. Another kind involves mathematical analyses of the basic equations. Here, our view is that such paraphernalia can serve as excess baggage, cluttering up the analyses and obscuring the basic mathematical structure. We prefer to store the baggage when it is not needed. To effect this, we simply remember that, at bottom, we are dealing with differential operators of the well-studied Euler-Lagrange form or, if you prefer, with corresponding variational statements. In a wellrounded treatment, we should probe deeply into the description of loading devices or environments involved in the construction of the idealistic isolated systems. Insofar as is feasible, we ignore these issues. We begin by discussing some features of such operators which we have found useful but which tend to be omitted from books on the calculus of variations. With independent variables X K (K = 1, . .., N), dependent variables x i (i = 1, . . ., n), and a Lagrangian W of the type W = W(x'; x i K ;

xL),

(2.1)

the operator in question is

d i= (aw/axi)- (awlaxi,K), , (2.2) where commas denote what might be called the total partial derivative, taking into account the fact that x is considered as a function of X.Throughout, commas will have this interpretation. Often, the identity

[ W d$

- x i M(d W/dx[ K ) ] ,K = bi xi, M -k

(aW / a x M )

(2-3) proves useful. In the latter term, we refer as usual to the derivative holding x i and xi,K fixed. A happy feature of these operators is that they retain their form under rather general transformations. Among other things, this obviates the necessity for introducing metric tensors, covariant differentiations, etc. For example, we may introduce new independent variables Yr (r= 1, . . ., N) and new independent variables y" (a= 1, . . ., n) by locally

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invertible transformations of the form y = Y(X, X),

Y = Y(x, X).

In essence, we think of the action as taking place in one large smooth manifold, rather than being associated with two smaller manifolds, as is the more common view. We denote by colons partial derivatives of these functions or their inverses; so, for example,

Via such a transformation, each mapping x(X) is to induce a locally invertible map Y = Y(X),

(2.6)

with YfK X K A

= d:,

and, consequently, a map Y = Y(Y).

Here we adhere to the common sloppy but convenient practice of using the same symbol for different functions, when it does not cause confusion. It is then easily seen that we can generate another Lagrangian of the same general type as indicated by

W = WJ = W(ya;y s r ; YA),

(2.9) (2.10)

There is then the new operator

I ,= (aW/dya) - (aW/aya,r), r .

(2.11)

What is less obvious is that the two operators are simply related. Specifically, we have I, =Aidi, (2.12) di =

xza,

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Of course, it is possible to verify these relations by brute force calculation, employing the chain rule and the relations between the quantities involved. However, we find it easier to follow the lead of Emmy Noether (1918), who treated transformations in some ways more elaborate. Carlson and Shield (1969) have used much the same technjque in elasticity theory and generalizations of it, for more general forms of W than we use, but less general transformations. Consider a one-parameter family of mappings x = x ( X , E ) and the associated integral I(&)=

1 W dX,

dX = dX' ... d X N ,

' R

(2.16) where R is an arbitrary fixed region, R(E)is its image under the mapping (2.6). With the usual rules of the calculus of variations, we calculate that

1

61 = I ~ O =) 8,dxj d x

I,

+ (a w/axi,

R

K)

dxj d s ,

.

(2.17)

If dR be given parametrically by X K = XK(u', . . . , u N - ' ) ,

dsK = k E K K ...1 KN-l(ax~l/aU ...1 () a x K y a u N - 1 ) du,

(2.18)

being the usual permutation symbol with E' ... = 1, and the sign being selected so that - d S , points into R . In mechanics, it is rather common to adjust the definition by a factor depending on the determinant of a metric tensor, and similarly to juggle the other factor to keep the integrand as we have it, so we here discard one piece of luggage. Here 6 has the usual meaning, E

so 6y"

= y:i

6x',

SY' = Y ; ax'. It is convenient to introduce the operator

(2.19)

(2.20)

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Now, by the alternative route, we calculate that

We now map a(0) back to R, then employ (2.19), (2.20), and the transformation (2.22)

d S r = J-'XKr dSK This yields

&,By 6x' dX +

61 =

6xiXKr dSK. (2.23)

'R

We now compare (2.17) and (223), regard R and 6x' as arbitrary, and conclude that (2.12) holds and, as a bonus,

awiax',,

= [ J - ' @ ' Y ~ ~ (aw/dyn,r)By]X:r. +

(2.24)

Of course, this too can be verified by direct calculation. Clearly, (2.12), obtains from a simple interchange of arguments, so it follows from the same reasoning. We leave to the reader the verification of (2.15). We now record without proof special cases of some results established by Noether (1918). Recently, their relevance to elasticity theory has been emphasized by Knowles and Sternberg (1972) as well as Fletcher (1975). Texts in the calculus of variations tend to ignore them. There are exceptions, e.g., Gelfand and Fomin (1963), Rund (1966), or Lovelock and Rund (1975). Consider an rn-parameter Lie group of transformations

x, 5), Y = Y(x, x, S), Y(X. x,0)= x, Y(x, x, 0 ) = x, Y = Y(X,

(2.25)

ta(a= 1, . . ., rn) denoting the parameters. Set

Suppose also that W ,the transform of W by any of these, is the same function as W. We then say that W is an invariant function under the group.

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The theorem is that such a W satisfies the m identities, bi

Di + [(aw/axi*K)Df+ wc,"], K

0.

(2.27)

From this arose the notion that invariance gives rise to conservation laws ; B = 0 gives rise to vanishing of divergences. For typical classical mechanical theories, W is invariant under the operations corresponding to rigid translations, normally a three-parameter group. The corresponding conservation law represents balance of forces. It is also invariant under the threeparameter group of rigid rotations, which gives rise to balance of moments. In older theories, where the form of these conservation laws is known, one can verify these assertions by direct calculation. With newer abstract theories, the idea can be used to obtain formulas for stress tensors or couple stress tensors, once one decides what transformations are to represent these groups, as is discussed by Rivlin (1968) or Truesdell (1966), among others. The Cosserats (1909) systematically exploited the idea. For constrained materials, for example, incompressible materials, it is common to employ the method of Lagrange multipliers. The above analysis then carries over with a slight modification; we add to W the constraint functions, multiplied by arbitrary functions, then proceed as if there were no constraints. As is discussed by Young (1969), there is some reason to be wary of the method in general. In elasticity theory, the usage seems not to have led to any trouble. Much of what we d o can be adapted to constrained materials, but we shall not consider them explicitly. With some caution, one can apply the formalism to elastodynamics and other dynamical theories associated with Hamilton's principle. If one interprets Was the action density and treats the time as an additional coordinate, the basic differential operator is of the Euler-Lagrange form. However, under Galilean transformations, the action density is not an invariant function, because of the noninvariance of kinetic energy. Rather, it is a semiinvariant function, by which we mean in general that

iv = W(y=;yly,;

YA)

+ v,

(2.28)

where V has the property that the Euler-Lagrange operator based on it vanishes identically; in this sense, V is a null Lagrangian. In some static theories, a similar complication arises in transformations describing certain types of material symmetries, for example, in the work of Wang (1965) on subfluids. Noether realized that sometimes, but not always, the differential identities (2.27) imply the invariance of W ,and discussed the matter in some detail (1918). For Lagrangians of the general type discussed here, it is feasible to characterize the set of null Lagrangians, as is discussed by Edelen (1962). In the context of elasticity theory, relations between the invariance of the stres and energy constitutive equations are discussed in some detail by

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Wang and Truesdell (1973, Chap. 111). Some of these considerations are of importance when one wishes to exploit certain material symmetries in elasticity theory. However, for simplicity, we here avoid these subtler issues, considering only the invariant functions.

B. ELASTOSTATIC EQUATIONS Concerning the theory of elastostatics, we take as known the form of the equations introduced by Kirchoff (1852). Here n = N = 3, and we begin by considering x and X as rectangular Cartesian coordinates, the latter being material coordinates. Here and in much of the following, we drop indices, using boldface notations to denote the sets of numbers indicated. What is implied by these words about coordinates is that the group of translations and rotations can then be represented by the transformations JJ

= RX

+ C,

Y = x, where R and c are constants, arbitrary except for the restrictions

(3.1)

R - ' = RT, det R = 1. (3.2) From the standpoint of analysis, this is the only important implication of these words. For (3.1) to apply, it is in fact unimportant that the material coordinates be Cartesian. Of course, W is to be invariant under this group. Using (2.28), we then obtain two sets of identities. The translation subgroup yields the rather trivial relation

t' + (aw p x , K ) ,

=

a w/ax = 0,

(3.3)

while the rotation subgroup yields

+

x h d [x A (dw/ax,K ) ] , K = X I K A (aw/dx. K ) 0, (3-4) a way of stating that the Cauchy stress tensor is symmetric. The equations in question are (awlax,K),K+f=f- & = 0 , (3.5) f being interpreted as the body force per unit reference volume. Also, the traditional measure of the force acting on the integration element dS is ti dS = ( d W / d ~ d~S,, ~, )

(3-6) dS being the usual scalar measure of reference area. Here, some geometry sneaks in, for ease of interpretation. We could evade it by replacing dS by du' du2, writing t dS = t^ du' du2. Integration of (3.3) gives the traditional

J . L Ericksen

198 balance of forces

I R t dS

+ 1 f dX = 0.

(3.7)

'R

Also, (3.4) generates the usual balance of moments

xhtdS+ \xhfdX=o. 'R

Observations like this have lent credence to the notion that balance of forces and moments are derivable from translational and rotational invariance in other types of theories. Of course, Galilean invariance can be restated as the condition that, for all rotations R,

W(X,, X) = W(Rx,, X).

(3.9) In practice, it is not uncommon to interchange independent and dependent variables, a transformation of the type 9

9

y=x,

Y=x,

(3.10)

which will serve to illustrate methods just covered. Using (2.9), (2.lo), (2.12), and (2.13), we calculate that IdetI)Xfh)II = IdetllxtKII W = J W = W(XK;X"), J =

2,

= - Jxi,

bi.

(3.11)

Of course, f must be transformed in the same way as d to keep the equations balanced, so (3.12) f~= - J X i , K f i , and the new equations are

I, = (awlax,) - (aw/ax:i),i=fK The identity (2.3), applied to W ,yields ti, J . 1 = F",XKj

.

=f~xKj = -Jfi,

(3.13)

(3.14)

the term on the right being identifiable as the negative of the force per unit present volume. Here ti

= w s;

- (aW/axKi)xfj

(3.15)

would then appear to be the Cauchy stress tensor. Using (2.22) to transform the surface element, with (2.14) and (2.24), we find that ti

dS

=

(aW/axt,) dSK

= t{

dSj

= t{JJ-'xKj dSK

(3.16)

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199

which confirms this, dS, being interpretable as the usual vector element of area in the present configuration. With J # 0, (3.14) in the present configuration is equivalent to (3.13), and the former is more often used. Equations (3.13)-(3.15) d o occur in the work of Shield (1967). Also, if we like, we can use (3.11)* and the identity

aJlaxfj = J X :

(3.17)

to write t! = -

Jxya w/axfj),

(3.18)

a prescription due to Hamel (1912, pp. 573-575). Application of the identity (2.3) to W is made by Knowles and Sternberg (1972) in the direct proof of their theorem 4.1. The identity is also useful in treating forces acting on singularities, as is discussed by Eshelby (1956), for example. A variant follows trivially from (3.14)-(3.15). A more commonly used prescription is ti = J X :

K(aw/axi,K ) ,

(3.19)

often with J replaced by the ratio of present to reference mass densities. Another common practice is to introduce curvilinear coordinates by transformations of the type y

y = Y(X),

= Y(X),

(3.20)

but not to introduce the Jacobian factor in the transformation of W . Such a factor is needed to get the proper transformation of the differential operators but, for these special transformations, one can introduce a material metric tensor and covariant differentiation to produce an equivalent operation. Here is more excess baggage, which, we feel, is better stored, when one is doing mathematical analysis. Of course, for some material symmetries, introduction of a material metric tensor can facilitate description of the forms of W that are appropriate. Here, the introduction of a spatial metric tensor gap has a little more merit. In Cartesian form, it is well known that W reduces to a function of X K and the right Cauchy-Green tensor C.With the special transformations (3.20), we have .

CKL= x',K x :

. L

= qasf r Y S A

Y'K . YpL

= CrA y,rK ypL

9

(3.21)

so explicit dependence on the curvilinear coordinates y" occurs only through this dependence on the metric tensor. The Euler-Lagrange operator then

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J . L. Ericksen

involves derivatives of this metric tensor, which can be replaced by Christoffel symbols. By following this through, one can see how covariant differentiation can be used to rearrange the equations to put them in equivalent, more popular forms. Our experience is that it is simpler to bear in mind (3.2 1) and use the Euler-Lagrange operators. Since we have no occasion to use such coordinates, we shall not belabor the details. In discussing semiinverse methods, we shall employ yet another type of special case of the transformations (2.4). 111. Semi-Inverse Methods

A. KINEMATICS

Generally, a semi-inverse method is one that reduces the basic equations to equations involving fewer independent or dependent variables, or both, for a limited set of solutions. Commonly, this involves exploiting some invariance of the equations. It seems probable that, by better developing the underlying group theory, one could make the search for such methods more routine. Here, we take a small step toward this goal by exploring what can be done with a simple but popular kind of group. We here concentrate on reducing the number of independent variables. We follow customary practice; the size and shape of bodies and the domain of functions involved is not specified in advance. We will generate formal procedures that work, subject to some tacit assumptions on such domains. Said differently, we aim for schema, not precise theory. In fact, some of the important questions of matters topological and of smoothness are far from trivial. In our view, it is better to prepare a rough guide map before tackling these deeper questions. Suppose that we have an energy function W referred to rectangular Cartesian coordinates x and possibly curvilinear material coordinates Y,with the property that W does not depend explicitly on Y 3 .Then, plausibly, we may expect some cases where the corresponding differential operator involves only the remaining independent variables. If Y2 is also missing, we foresee possible reduction to ordinary differential equations, and if all Yfare missing, we might reduce some problems to algebra. The more likely possibilities occur when the arguments upon which W depends are, within symmetry transformations, functions of Y' and Y 2 , or Y' only, or constants, respectively. Formally, W being independent of Y3 means that it is invariant under the continuous group Y 3 -+ Y 3 + const. That this point is not completely devoid of subtlety can be seen by considering Y3 as one of the angular coordinates in the familiar spherical coordinate systems; the group

20 1

Special Topics in Elastostatics

may or may not be compact. Bearing in mind the Galilean invariance of W , (3.9), we are led to consider the possibility of locally one-to-one mappings x(Y) such that

x,r=RPr, (4.1) where R is a rotation matrix which may depend on Y, while Pr depends at most on Y' and Y2. Implicit here is the notion that W need not have more symmetry than we have recognized. Later, we return to this point. There is the obvious restriction 0 # detllx ,r ( 1

= P 1 A ~2

* ~3

.

(4.2)

There is then the purely kinematical problem of characterizing such deformations. To solve it, we set

R,r= Rar where the matrices

(4.3)

9

are, necessarily, skew symmetric, (4.4)

Q;=-Ctr.

For convenience, we introduce index labels a, b taking on values 1 and 2. The integrability conditions x ,

= x ,A,- then reduce to

+ a2 P1 = P 2 . 1 + a1 P2 0 3 Pa = ~ 3 . + a a a ~3 .

P1.2

7

(4.5) (4.6)

From (4.4), (4.5), and (4.6), it follows that

P r = P r( Ya)

fir

= Qr( Ya),

(4.7)

P r = PAY') a a r = a r ( Y 1 ) ,

(4.8)

Pr = const * Rr = const.

(4.9)

To verify this, one considers (4.4)-(4.6) as linear equations for determining the unknowns 0,. It is a straightforward exercise to show that if (4.2)holds, they are uniquely determined by the remaining quantities, from which the conclusions are immediate. The exercise is made easier by noting that (4.4) implies the existence of equivalent axial vectors m y , such that, for every vector v, arV=WyAV,

and (4.2) implies that the Pr form a basis, i.e., we can write mr=A$pA,

where the components A$ are unique. We omit remaining details.

(4.10)

J. L.Ericksen

202

Using (4.3), the integrability conditions R , K L = R , LK then reduce to Q1.2

+ Q2Q1= Q3Qa

Q2,1+

QlQ,

+ QaQ3

= Q3,a

9

(4.11) (4.12)

9

in all three cases; there are the obvious simplifications, if (4.8) or (4.9) apply. Suppose first that (4.7) holds. With it, (4.11) can be viewed as integrability conditions, locally sufficient for the existence of a rotation matrix R( Y"), such that (4.13) Then (4.14)

is a rotation matrix. Further,

R,, = RQ, R - '

- RQ,

R-'

= 0,

so R depends at most on Y 3 . We write

RLEQ

(4.15)

Q T = - Q .

Of course Q depends at most on Y 3 . On the other hand, differentiation of (4.14) yields

RQ

= RR3R -

= RRQ,

R-

I.

With R independent of Y 3 ,it follows from this and (4.7) that SL is independent of Y 3 ,so

R = const.

(4.16)

Then solutions of (4.15) can be presented in the form

R =R,P~~,

(4.17)

where Ro is an arbitrary constant rotation matrix. Changing it amounts to applying a trivial Galilean transformation so, with no real loss of generality, we set

~~=i=,R=pr~.

(4.18)

Were we not interested in the subcases (4.8) and (4.9), it would be simplest to absorb R in Pr, effectively setting R = 1. As a compromise, we set (4.19) (4.20) and realize that the obvious analogs of (4.3h(4.6) apply equally well to R'

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203

and qr . These analogs are sufficiently simple so that it is relatively easy to integrate (4.19).Omitting details, we obtain mappings which, within a trivial rigid translation, can be represented in the form

x = R[y(Y") - b + aY3e] + b = R(y - b)

+ aY3e + b,

with (4.22)

Qa=Y,a,

q3 = R(y - b) + ae.

(4.23)

Here, the scalar a and vector b are essentially arbitrary constants, the latter being retained for later convenience. The constant vector is restricted by the condition that it be a unit vector

e.e=l.

(4.24)

If IR = 0, it is otherwise arbitrary, except for the restriction which follows from (4.2), viz.,

-

(4.25)

Re = e o i L e = 0.

(4.26)

91 A q 2 q3 # 0. If R # 0, e must be the axis of rotation, i.e., Of course, these relations are trivially satisfied if R (4.21), we have

x( Y', Y2, 0) = y( Y O ) .

= 0.

From (4.18) and (4.27)

This arbitrary surface then represents the deformed configuration of the material surface Y3 = 0. By translating and rotating it as indicated by (4.21), we generate the three-dimensional configuration. Theory accompanying typical static experiments, designed to help determine the form of W , invariably involves such deformations; the experimentist ignores parts of his specimens which, in his judgment, deform in a different fashion. In linear elasticity theory, for homogeneous isotropic materials, the famous theories of St.-Venant for torsion, bending, and simple tension or compression introduced early examples of such deformations. His theory of flexure seems not to fit the mold. It is interesting to note that, again excepting the flexure solutions, these have a certain status as minimum energy solutions, as is discussed by Sternberg and Knowles (1966). Bogy (1967) discusses the " optimal " solution for flexure of a circular cylinder, which is quite elaborate. Of course, with the classical uniqueness theorems, one must compare solutions with different surface tractions, but suitably matched resultants. In

204

J . L. Ericksen

St.-Venant's theory of prisms, the flexure solution is needed to construct solutions having arbitrarily given resultant force and moment on a cross section. Of course, for this purpose, one could replace it by various other solutions. That other solutions, with resultants matched to those of a St.Venant solution, tend to approach the latter, as we move away from the ends, is clear from the rigorous version of St.-Venant's principle established by Toupin (1969, for example. Here the flexure solution is not excluded. However, various solutions approach each other, so the St.-Venant solutions are here not particularly distinctive. It is rather obvious that extension of these theorems to larger deformations is nontrivial. For a fixed, small initial strain, we hardly expect the elementary solution for simple compression to have such a favored status, if the dimensions of the body and type of loading are such as to favor buckling. If one employs the corresponding equations of small deformations superposed on (possibly) large, one quickly sees why Toupin's proof does not carry over to cases involving compressive loads. With linearized theory, it would seem feasible to do more to clarify these matters. Theorems such as are covered by Thompson (1969) do have some relevance. We now turn to the analysis of (4.8). It implies the existence of a rotation matrix R ( Y 1 ) such that R = Ra,. (4.28) Proceeding as we did to obtain (4.17), we can write R = BR, and infer that fi = R,%iyz = eiiY2R1,

(4.29)

(4.30)

where

iz = R, S%R;

= const,

(4.31)

R, being an arbitrary constant rotation matrix. It can be absorbed in R, so, with no loss of generality, we take R, = 1. It then follows from (4.30) that

fin= afi,

fi = 8 y 2 .

(4.32)

Using (4.3), (4.14), (4.29), and (4.32), we calculate that R, = R- 'nR,

n, = R - 'OR, while (4.12) yields n2n3 =

From (4.33) and (4.34),

aja,.

nn = m-i,

(4.33) (4.34) -1.351

Special Topics in Elastostatics

205

or, alternatively, the axial vectors equivalent to these two matrices must be parallel. If S2 # 0, a is then proportional to R, so, from (4.32),

fin = nfi,

'

(4.36)

and, trivially, (4.36) holds if a= 0. Since R depends only on Y', we now absorb it in pr, which amounts to setting

-

R = 1.

(4.37)

Then, from (4.20), (4.22), (4.23), (4.29), and (4.37), we must have

(4.38)

qa = Y . a = fiPa(Y1), q3 = n ( y - b)

+ ae = $p3(

Yl),

(4.39)

where, of course, R( Y') is given by (4.32). Integrability conditions follow from (4.5) or (4.6) or from appropriate combinations of derivatives of (4.38) and (4.39). They are Pi

= QPl,

Pi

= nP1,

(4.40) n P 2 = fiP3 . It is now an easy matter to obtain y by integration. Omitting details, we find that solutions fall into one of three classes. In all cases, bear in mind the restriction (4.2). First, we have

R=fi=O, y = z(Y1) + cfY2,

f * f = 1, P1 = z', P2 = cf,

(4.41)

p3 = ae,

wherein z( Y l ) describes an arbitrary curve, c and f being arbitrary constants. Second, we have Q=O#iZ, iZe = o o f i e = e,

y - b = fi[z(Y') - c]

P1 = z', pz = n ( z - c ) p3

= ae,

+ deYZ + c,

+ de, (4.42)

J . L Ericksen

206

c and d being arbitrary constants. Finally, there are the cases where Q

*

# 0,

Re = e, y - b = 8(z - b )

+ cY2e,

P1 = z',

+ ce, p3 = Ck(z - b ) + ae, p2 = a(z - b )

c being an arbitrary constant. Here, constants in z,arranging that

(4.43)

0 = 0 is included. We have absorbed

y( Y', 0 ) = z( Y').

(4.44) Clearly, in all cases, we generate the three-dimensional configuration by taking an arbitrarily deformed material curve Y 2 = Y 3 = 0 and suitably rotating and translating it. Roughly, if we can ensure that equilibrium obtains for the curve, it should obtain for the entire body, by symmetry. Hence, equilibrium equations should reduce to equations for the curve. Later, our task will be to make this loose idea a bit more precise. Finally, there is the last category, given by (4.9). Any student of mechanics knows that a rigid motion is a homogeneous deformation. By a linear transformation of material coordinates, one can arrange that p i = 8 ; . Thus each reader knows a proof that (4.9) implies that = const.

(4.45)

Thus, we have at hand characterizations of all of the deformations that we guessed to be likely. The next step in a semi-inverse method is to produce conditions that limit the arbitrary functions, here labeled y( YO)or z( Y ' ) . We then leave the realm of pure kinematics and enter the world of mechanics. For any of the deformations given by (4.21), it is not difficult to show that there exists a one-parameter family of rotation matrices R(k)and translation vectors c(k) such that X( Y O ,

Y 3 + k ) = R(k)x(Y O , Y 3 )+ c(k),

(4.46)

k being the parameter. Specifically, we have R(k) = R(k) = P, c(k) = kae + [l - R(k)]b.

(4.47)

Conversely, one can show' that any deformation satisfying relations of the

' I am indebted to R. Muncaster for showing me a proof.

Special Topics in Elastostatics

207

type (4.46) is representable in the form (4.21). A priori, it did not seem completely obvious that our hypothesis was equivalent to (4.46) and, for later considerations, in Section III,C, our hypotheses seem to us more natural. In any event, (4.46) provides a definition of analogs of the invariant functions” introduced by Michal (1951) in his study of applications of group theory to partial differential equations. To begin to apply reasoning like this, one needs characterizations of such invariant functions.” Possibly, some such routine might be used to similarly reduce the number of dependent variables, but we shall not pursue this. While we make some headway toward tying together various semi-inverse methods, we do not pretend that our treatment is exhaustive. “

“

B. REDUCED EQUATIONS Consider (4.21), for the moment fixing the parameters a, b, e etc., and forgetting that y is there considered to be a function of Ya only. It then generates a transformation of the form (2.4), of a special type, i.e.,

x =R(Y3)(y - b)

+ a Y 3 e + b,

x = Y.

(5.1)

For configurations y = y(Y), now depending on all material coordinates, it follows easily that -

x,a = R Y , ~ , X,

= R[n(y

-

b)

+ y, + ae].

(5.2)

Thus, with W presumed independent of Y3 and Galilean invariant, we see that W(x,a; x . 3 Y’) = W(y,a; 9; Y’), 9

= O(y - b) + y , 3 + ae.

(5.3)

J = 1,

(5.4)

w = W Y - b; Y , # ; y . 3 ; Yb),

(5.5)

q

Also, from (2.10) and (5.1),

so, from (2.9), where W is given by the right-hand side of (5.3),. Clearly, the transformations introduce no explicit dependence on Y3, even though some such dependence appears in the arguments in the left-hand side of (5.3),. From (2.13) and (5. l),

208

J. L. Ericksen

we have so, from (2.12),

8 = R-’e,

(5.7)

with B and 8 denoting, respectively, the Euler-Lagrange operators based on W and W. Then, using ( 3 4 , the new equilibrium equations are -

-

r = R-’f.

(5.8) We now make the semi-inverse assumption that y = y( I“) and, for consistency, require that =

T = r(Y“).

(5.9)

Of course, the usual case, f = 0, is included. Within some limits, so also is the usual estimate of gravitational force. For the latter, the reference mass, per unit “coordinate volume” dY, should be independent of Y 3 and, if fi # 1, this force should be in the direction of the axis of rotation of R. One might bear in mind d’Alembert’s principle; we are not too far from applications to elastodynamics. With these assumptions, the equations to be satisfied are

8 = (aw p y ) - (awpY,

= T.

(5.10)

These, then, are the basic equations for cases associated with (4.7). To obtain equations applying to the more special cases indicated by (4.8), we can begin with (5.10) and similarly transform it. Suppose, for example, that we are interested in the cases covered by (4.43). We then similarly treat (4.43) as a transformation, temporarily dropping the restriction that z depend only on Y’. We first note that, because of the Galilean invariance of W , we have, for any rotation R such that RR=nR, w[Y,a; Q(Y - b) + Y . 3

Re=e,

(5.1 1 )

+ aeYb] = W R Y , , ; n R ( y - b) + R y , , + ae, Yb].

(5.12)

Then, for this subset of rotations, (5.5) and (5.12) yield ~ R ( -Y b ) ; R y , a ; R y , , ; Ya] =

W(y - b; Y , ” ;y . 3 ; Y“).

(5.13)

From (4.36) and (4.43), R is included in this subset. Of course, we may apply this when y , = 0 as is presumed in (5.10). Also, we note that, because of (4.26), we have (5.14) n ( y - b + l e ) = n ( y - b),

Special Topics in Elastostatics

209

where I is an arbitrary scalar. Thus, from (5.3) and (5.5), W(y - b

+ I e ; y,"; Y , ~ p) ; = W(y - b; Y,";

Y . ~ Yb). ;

(5.15)

With (4.43)3, we then have, with y , now zero, but z now allowed to depend on Y' and Y z , X

Y.1 y,,

= Rz,,,

=a[& - b) + z ,+~ce].

(5.16)

Here we have used (4.43), and (4.32). We now employ (5.13) with R = R, (5.15), (4.43), , and (5.16) to obtain W(y - b; y , ' ; y , , ; Y") = W ( z - b; z * ~n ;( z - b) + z,,

+ ce; Y"). (5.17)

We then apply the general transformation theory, again noting that the relevant Jacobian satisfies (5.4). The new Lagrangian is

W(Z - b; 2 . 1 ;

(5.18)

Y") = W ,

2.2;

with W given by the right-hand side of (5.17). The Euler-Lagrange operator transforms by the analog of (5.7), so the new equations are = k-q= f,

2=

(5.19)

2 being the Euler-Lagrange operator based on W. It is easily seen that, if W does not depend explicitly on Yz or Y3, neither will W or W. With this assumption, and the further specialization that

I = f( Y l ) ,

(5.20)

we can make the semi-inverse assumption inherent in (4.43), z = z( Yl), these functions being governed by the ordinary differential equations

2 = (awl&) - (aw/azi),= t

(5.21)

It is easily established that W still enjoys invariance analogous to (5.13) and (5.15), i.e.,

@R(z

-b

+ Ae); Rz,,;Rz,,; Y"]

=

W ( z - b; z , ~z,,; ;

YO),

(5.22)

where R is any of the rotations satisfying (5.1 1). With this invariance under a continuous group, we have the associated identities indicated by (2.28). A calculation shows that these become

[k + (aW/8zf),] * e = 0, {(z - b) A 8 + [(z - b) A (di@/az')]'}

-e

(5.23) = 0.

(5.24)

2 10

J . L Ericksen

Clearly, these can help ease the integration of the ordinary differential equations (5.21), in special cases. If W is also independent of Y’, and 8 = 0, the corresponding identity generates the usual energy integral,” which can be read off from (2.3). Of course, there are analogous differential identities associated with (5.10). Also, it seems obvious that the analysis of (4.41) and (4.42) is only slightly different from that of (4.43), so we omit the details. It seems pointless to belabor the special case (4.45); if W does not depend explicitly on any of the coordinates, we have such solutions when f = 0, and everyone is familiar with some of the features of homogeneous deformations and algebraic problems relating to them. Equations (5.10) bear some resemblance to equations arising in modern membrane theories of shells, if one takes the trouble to put the latter in corresponding Euler-Lagrange form. Rather obviously, one can simply borrow our analysis to generate some cases where the equations reduce to ordinary differential equations. However, the corresponding Lagrangian enjoys more invariance than does W ,and the requirements on membrane deformations are milder, so one might miss some possibilities. “

C. HOMOGENEOUS ISOTROPIC MATERIALS The preceding analysis applies to materials that need not be isotropic and need have only a very limited kind of symmetry. However, it might help bring things into focus if we indicate how we find the coordinatizations where W is independent of Y 3 or Y3 and Y 2 ,for the familiar homogeneous, isotropic materials. If we require, as in Section III,B, that x represent rectangular Cartesian coordinates, Y denoting unspecified,curvilinearmaterial coordinates, we are then interested in Lagrangians of the form W = W(1,11, III)G1”,

(6.1)

where G = det//GrA)/, Gr,(Y) being the material metric tensor. Here I, 11, and I11 represent principal invariants or equivalent invariants of the tensor C given by (3.21), e.g., I = CrAGrA.

What is important for us is that they depend only on x , r and GrA or, equivalently, GrA. We shall not belabor (6.1), this form being discussed in

Special Topics in Elastostatics

21 1

most expositions of nonlinear elasticity theory. Recall from (2.16) that we obtain the energy by integrating W dY

=

WG”’ dY = W dV,

(6.2) where d V denotes the reference volume element used by many writers. Said differently, G’’2 automatically takes care of the Jacobian factor, under the rather restricted transformations here implied. Obviously, if we want every such W not to depend explicitly on Y3, we want t o choose the curvilinear coordinates so that GrA is independent of Y3, etc. We recall that, if X denotes rectangular Cartesian coordinates, with

x = X(Y) describing the transformation to the transformation to curvilinear, then

G rA = XtrXKA . (6.3) It is then obviously sufficient that G,, be independent of Y3 to have

X ,r = R(Y)Pr( Y’, Y2),

(6.4) where R is a rotation matrix. A routine application of the polar decomposition theorem or other known theorems shows that (6.4) is also necessary. In Section III,A, we characterized all such mappings, and, obviously, there are infinitely many possibilities. Of course, the overworked polar and spherical coordinates occur among those and have some pleasant special features, for example, orthogonality. Not all measures of simplicity are invariant under a change of observers, so we leave it to the reader to pick out what he considers to be the simpler possibilities. There is a more subtle point that might escape attention. In the argument leading to (4.1), we made no allowance for the additional symmetry (isotropy) which here obtains. With it, not all of P r matters. That is, if

P r = rAg$(Y), and if 99 is the representation of a rotation, viz.,

(6.5)

9%@%GAe= Gro (6.6) then W will cancel out of W and W. Effectively, we have assumed W independent of Y3, so we have characterized only these special cases. Even then, we have merely indicated possibilities, not covered them in detail. It is interesting to note that, with Y taken as rectangular Cartesian coordinates, there are nontrivial deformations with rA = const or, equivalently, with I, 11, and 111 all constant. At one time, I (1954) guessed the contrary, despite an example that was under my nose. Fosdick (1966) caught the error and, since then, a literature involving these has grown. That which relates to the prob9

J. L. Ericksen

212

lem I (1954) considered is summarized by Marris (1975). Such deformations are also encountered in the interesting, offbeat study of certain phase transformations of isotropic solids by Varley and Day (1966). Complete characterization of cases indicated by (6.5) would then seem to be an unsolved problem. Of course, the problem can be restated as the requirement that I, 11, and 111 depend at most on two curvilinear coordinates, being thus functionally dependent. Of course, the representation (6.1) derives from the fact that W is invariant under the group represented in rectangular Cartesian material coordinates by y = x,

(6.7)

Y=RX+c,

R and c being any constants such that

R - ' = RT,

det R

=

1.

(6.8)

Here, material and spatial coordinates are subject to the same kind of transformations. One W then generates another by interchanging interpretations of the two sets of coordinates. This idea is exploited by Adkins (1958) and Shield (1967).Of course, for this six-parameter group, (2.28) yields six identities. The most interesting consequence is that AKL =

(dW/dxi, K)xi,L = A L K .

(6.9)

This result is a trivial consequence of the Truesdell-No11 (1965) equation (85.14) and other equations occurring in the literature. Of course, it is an analog of (3.4). For some classroom illustrations, we have found it easier to use these than to employ (6.l), which does, of course, imply them. Consider, for example, a common simplistic description of a uniaxial stress experiment, presuming homogeneous deformation, for a homogeneous material. The presumption is that 0 0

01

(6.10) O O T

where T denotes the usual force per unit underformed area. Equation (3.4) quickly gives restrictions on the homogeneous deformations which apply to anisotropic as well as isotropic materials. From (6.9) follow additional restrictions implied by isotropy. Since we were a bit critical about the introduction of material metric

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213

tensors, it seems in order to comment on the apparent change of heart. Invariance of W supplies some group, intimately related to the governing equations. In the sense of Felix Klein’s Erlanger Program, a group generates a geometry. In our view, this intrinsic geometry is what is important, and there is no harm in exploiting it. Sometimes it “leaves invariant” some “metric tensors,” sometimes not. Actually, it commonly leaves invariant infinitely many, or none. For the case at hand, any two are related by a scalar factor of proportionality. If none is left invariant, it seems unwise to try to force one into the description. This is the case for the elastic crystals to be discussed later, for example, as well as for elastic fluids or subfluids. It is easier to get around some of the technical difficulties involved in developing a coordinate-free treatment if one introduces a Euclidean geometry or a geometry that is, in certain respects, much like this. This seems to me a poor reason for carrying the excess baggage; it is not likely that everyone will agree.

D. AVERAGES Typically, use of a semi-inverse method restricts our ability to specify detailed boundary conditions or other side conditions. We may retain the ability to prescribe certain averages, typically some resultant forces or moments, and some of the .details. Alternatively, in matching theory to typical static experiments, the typical boundary value problem is rarely appropriate. Usually, we know from experiment something about the deformation on free surfaces, but elsewhere know something about resultants only. From one point of view, this is the raison d’bre for the semi-inverse methods. Of course, it was St.-Venant who recognized the issue and provided a way out of the difficulty which is practical, although not devoid of subtlety. It was, in our opinion, his greatest achievement, a landmark in the theory of elasticity. We consider the general case (4.21), presuming that (5.10)is to be satisfied in some domain D ;we are happy to restrict D so that the surface implied by (4.27) can be covered by one coordinate chart, with (4.21) valid in a domain R of the form R = D x [0, Y 3 ] , (7.1) for Y 3 sufficiently small, all mappings and domains nice enough to permit formal analysis. After seeing the forest, one might wish to inspect more closely the trees. For any constant value of Y 3 ,we then have a surface S( Y 3 ) given by x = x(Y“, Y 3 ) ,

Y” E D.

(7-2)

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J. L. Ericksen

On this surface, the resultant force F and moment M about x = b are given by the prescriptions (7.3)

and the overall balance of forces and moments must hold for R, viz.,

In these relations dSr is, of course, to be calculated as indicated by (2.18), outward directed or, in (7.3) and (7.4), pointing in the direction of increasing Y3. Actually, for use with (5.10), we really want the equivalent relations in terms of the functions W and variables y, so, obviously we should apply the corresponding transformation to the stress tensor, using (2.24). This yields

a wpx ,

= R aw p y ,

.

(7.7)

Of course, we are interested in cases where Y , =~ 0, but aW/ay,, need not then vanish and will not, in general. In (7.2) and (7.3), we can use Y' and Y2 as surface coordinates. Bearing in mind (4.18), the first gives

F(Y3)=

- aw 1. R-dY'

dYZ= RF(0).

(7.8)

'D

Similarly, employing (4.21) and (4.26), we find that

M ( Y 3 ) = R ] (y - b D

aw + aY3e)~---dY' dY.3

dY2

+ a Y 3 e A F(0)] = RM(0) + aY3e A F ( Y3).

= R[M(O)

(7.9)

We here used the identity

(Ra) A (Rb) = R(a A b),

(7.10)

for all vectors a and b, and any rotation R. Said differently, if

4 Y 3 ) = M(Y3) - a Y 3 e ~ F ( Y 3 ) denotes the moment calculated relative to the variable point x

(7.11) =b

+ aY3e,

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we have the neater relation (7.12)

A ( Y 3 )= R&(O).

Now, we use (7.7) and (7.8) to rewrite (7.5) as

F ( Y 3 )- F(0) +

5

r3

dY3

0

+ lo

i?w

(.

Y3

R-ds,

aY,a

'PD

d Y 3 I D R f d Y ' d Y 2 = 0, (7.13)

where ds, represents the integration element for aD. If we use (4.18) and (7.8), differentiate (7.13) with respect to Y 3 , and evaluate at Y 3 = 0, we get

n F ( 0 ) + @ = 0,

aw

a=

\~

-dsa+

Y d Y 2I,

(7.14)

'D

'8Day.a

reducing balance of forces to conditions applying directly to (5.10). From (7.7), the relation of

(a W I ~ ,Y d s a a)

to surface tractions seems clear enough, so we are in a position to formulate sensible boundary conditions. Rather clearly, it is feasible to specify or partly specify F(0) in setting a problem. By entirely similar reasoning, one can derive the analogous balance of moments from (7.6), which reduces to M'(0) + 'I'= 0,

+ [ (y - b ) r \ f d Y '

dY2.

(7.15)

'D

From (4.26) and (7.9),

M'(0) = nM(0) + ae A F(0).

(7.16)

Rather frequently, in practice, the "lateral boundary" aD x [0, Y 3 ] is considered free and body forces are neglected, implying that @ = 'f' = 0, in which case these relations merely give the rather obvious limitations on F and M. We now turn to energetics. Clearly, for the deformations considered,

E=

1 W d Y ' dY2 = \ W d Y ' dYZ

' D

(7.17)

' D

is independent of Y 3 ;we designed the deformations to make W independent of Y 3 .Thus E can be thought of as energy in R, divided by Y 3 ,an average of

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the energy. We now consider as variable the parameters occurring in (4.21), the energy function W remaining fixed. That is, we fix the material, but change the deformation, as is done in typical experiments. It is convenient to reduce some of the constraints on parameters by introducing the axial vector o equivalent to R and writing

o = me,

(7.18)

as is required by (4.26). We could eliminate the constraint

e-e=1

(7.19)

by replacing e by ae, but prefer not to exclude the case where a = 0. Then, from (5.3) and (5.5), with y , = 0,

W(Y - b; y.0; Yb) = W(y,a; q, Yb)x q = w e r \ ( y - b)

+ ae.

(7.20)

Thus E is a function of the several parameters and a functional of the functions y( Ya) over the domain D ; we denote this by

E

= E(a, 0, b,

e, YD).

(7.21)

Of course, F(O), M(O), etc., will then depend on the parameters. Using (7.17), the Gateaux differential is of the form aE 6E = -6a

aE aE + -66w + -. aa am ab

aE 6b + -. 6e de (7.22)

assuming D is not varied. Of course, one must bear in mind (7.19) and the consequent ambiguity in aE/ae, a familiar type of problem. Let us first explore the notion that translational invariance of E corresponds to balance of forces. It is first necessary to decide how to represent a translation in the variables used. From (4.27), if we translate x, we should similarly translate y. It is then easily seen from (4.21) that we should similarly translate b. Nothing else should change, so the only nonzero variations are

6y = 6b = const,

(7.23)

the constant being arbitrary. As is clear from a glance at the left-hand side of (7.20),, the energy will not change. It then follows easily from (7.22) and (5.10) that

(aE/db)

+ Q, = 0,

(7.24)

which should represent balance of forces, Q, being given by (7.14).It is of the

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217

same form as (7.14), and will agree if

aE/ab = RF(0).

(7.25)

One can verify this by a direct calculation but, from the two analyses given, the conclusion is inescapable. Similarly, one can argue that nonzero variations describing an infinitesimal rotation are of the form 6y = b ~ y , 6e = h A e ,

6b = h A b ,

(7.26)

where C;, is an arbitrary constant vector. We anticipate that the condition that 6E = 0 for these will correspond to balance of moments. By calculation, one gets a condition comparable to (7.15), the two agreeing if

e A (aE/ae)= M'(0) = e A [oM(O)

+ aF(O)].

(7.27)

Thus

a E p e = oM(0)

+ aF(0) + Le.

(7.28)

To verify (7.25) or (7.28) directly, one needs to note that, from (5.3) and (7.20)~

a wlay,

=

a wps.

(7.29)

We now observe that, from (7.8), (7.9), (7.17), (7.20), and (7.29),

- IDaga w d Y 1 dY2 e - F(O), aEaw dY2 e - M(0). - e - j i y - b)r\-dY' ao as aE -= e aa

=

=

(7.30) (7.31)

Using (7.25), (7.28), (7.30), and (7.31), we obtain

-

+ M(0) - S(oe) B S ( y - b ) dY' dY2 + a w

6E = F(0) 6(ae) *

S ( y - b ) ds, . (7.32)

Now suppose that there are no body forces and that the lateral sides are free, so that the last two terms vanish, and assume that o # 0, a type of situation rather commonly encountered. It then follows from (7.14)-(7.16) that F(0) and M(0) must be parallel to e,

F(0) = f e , M(0) = me,

(7.33)

and (7.32) simplifies to

6E = @ a

+ m6u.

(7.34)

Of course, underlying this are nontrivial assumptions concerning existence,

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J . L Ericksen

smoothness, and uniqueness of such solutions. In itself, nonuniqueness is not devastating, if one can stay on a branch varying smoothly with the parameters. There are the obvious integrability conditions

ayaw = amlaa.

(7.35)

Typically, in such situations, the experimentist can measure a, o,the functions f ( a , w ) and m(a, w ) ; know the shape of the domain D; and, on aD, observe the functions y. He will then attempt to learn something about the form of W , employing guesses about the deformation within D, or approximations that seem to him reasonable. If one looks critically at such enterprises, one sees some of the deepest, most difficult, and least studied problems in elasticity theory. Not infrequently, it is decided that one has reached the limits of applicability of elasticity theory when the deformation departs from the symmetry assumed, or the implied smoothness breaks down, unless breakdown can be attributed to a familiar elastic buckling phenomenon. Perhaps, my study (1975a) of bars illustrates some of the subtleties that might arise. It is not terribly easy to formulate tractable problems which will supply sounder bases for such enterprises, but it does not seem hopeless to make a little more progress. It seems obvious that one can, by entirely similar methods, discuss averages appropriate for use with (5.21). One can assume that D is of the form (7.36) where a and jl are constants and employ similar limiting procedures. Also, one can estimate how far we have gotten, or what we might have missed, by taking St.-Venant’s theory of torsion, superposed on his theory of tension, and putting it through these paces. We leave to the reader these exercises. In certain cases, simplification occurs if W also admits a discrete invariance group. Typically, one would like to have a derivative of W , with respect to some argument, vanish when theargument vanishes. Often, this is implied by some symmetry with respect to reflections. No doubt, the reader has his own list of illustrative examples. Of course, given any transformation leaving W invariant, we can calculate how it affects derivatives; it hardly seems worthwhile to belabor this. There is a point that is a bit more subtle. Often, we feed in the requirement that a constitutive equation be invariant under some group. Sometimes, this implies that it is invariant under a larger group. One of the more classical examples in linear theory is commonly stated in more physical terms; a cubic crystal is optically isotropic. That is, a symmetric second-order tensor that is invariant under the group considered appropriate for cubic crystals is, necessarily, invariant under the full orthogonal group. From the viewpoint of inverse or semi-inverse methods, bigger groups are better. Actually, it is

Special Topics in Elastostatics

219

not entirely clear how to formulate the problem of determining what is the maximal group which leaves a specified set of energy functions invariant or semi-invariant. For this, one should consider at least the set of transformations summarized in (2.4). As is mentioned there, the work of Noether dealt with more general transformations. Of course she accounted for all possibilities . . . or did she? With respect to the application of group theoretic methods to partial differential equations, I have digested rather little of what is known. What I have done here is to attempt to organize the art, as it is practiced in elasticity theory, indicating how the practice can be improved. In pondering the general transformation theory, the thought comes to mind that existing theories of material symmetry are, perhaps, overly special. There is, we think, some food for thought here, but there are subtleties. The simplest case occurs in the one-dimensional theory of elastic bars, where n = N = 1, and the simplest subcase occurs when W is the energy associated with the linear theory. With the usual idea that x and X refer to the same rectangular Cartesian coordinate system, we then have

2 W = E(x‘ - 1)2,

X‘

= dx/dX,

(7.37)

where E is Young’s modulus. With a # 0, b and c arbitrary constants, consider the transformations

+ ( * a - 1)aX + b, Y = + a 2 X + c. y = ux

A calculation shows that y’ = d y / d Y = ~ u - ’ ( x ‘- 1

*

(7.38)

a),

(7.39)

J = a-’,

which gives

2 w

= E(y’ - l ) Z ,

so W is invariant under this group. At first sight, we might include these as material symmetry transformations. On the other hand, there is the kinematic restriction that x’ > 0 and not all the transformations are such that x’ > O o y ’ > 0. To guarantee this, we need * a > 0,

a 2 T 1 > 0,

(7.40)

as is easily seen by letting x’ -+ co and x’ + 0, respectively. On further thought, we really d o not expect linear theory to apply over the whole kinematically admissible domain, and a good material symmetry transformation should keep us in the subdomain where the theory applies. This

220

J. L. Ericksen

leads to some further specialization, which depends somewhat on what one takes as the subdomain. In other words, the domain of W plays a role that is not unimportant. Also, physically, there is room for argument as to what is the domain, as is illustrated here. On subtleties of this kind, the literature is a bit fuzzy. For problem solving, particularly of the semi-inverse type, there is no real harm in considering transformations not preserving domains. Often, deformations can be pared to fit the domain of W, by restricting parameters, adjusting the shape of bodies, etc. With other fish to fry, we content ourselves with this rather bare outline of a theory of semi-inverse solutions. It would take a much longer memoir to elaborate details which are within easy grasp and to say a bit about what has been learned from special cases. Green and Adkins (1960) devote quite a bit of space to what they call “general solutions for isotropic and anisotropic materials; they also cover theory of finite plane strain. In the process they provide a variety of illustrative examples. We have not made the effort to carefully pore over these and more recent examples in fine detail, looking for ways to improve the general format. Thus it is entirely possible that we have missed a trick. Inverse solutions are even nicer when they exist. For unconstrained, homogeneous materials of the more solid varieties, with the particular form of W left unspecified, the only possibilities are homogeneous deformations. For a more precise statement, we refer the reader to Shield (1971), who improved my older analysis (1955). Sometimes, for constrained materials, there are inhomogeneous deformations which qualify. Truesdell and No11 (1965, pp. 171-219) cover then-known examples for incompressible materials, Wang and Truesdell (1973,Chapters IV and V) covering some newer examples. Kao and Pipkin (1972), for example, give an interesting analysis of a highly constrained material, applying to fiber-reinforced materials. References to more recent work on this topic are given by Pipkin and Sanchez (1974). Also, there is a book on this topic, written by Spencer (1972). Deformations of the kind here characterized pop up in such studies, for somewhat similar reasons. A general view of the subject does not enable us to dispense with the study of the special cases which provide so much enrichment. It can promote efficiency of mining and provide a different perspective. As a matter of perspective, we remind the reader that there is a 328-page memoir on torsion and bending of linearly elastic materials which is better balanced, considering the time. We refer to the famous work of St.-Venant (1856). ”

IV. Experiment rrtld Mechanistic ’Iheory

A. THEARENA Here, our aim is not to promote specific forms of the function W which might be useful for numerical calculation of static solutions. Rather, we attempt to recognize and explore a bit more of the theory that underlies the

Special Topics in Elastostatics

22 1

selection process. As we have presented the equations, this function depends on the choice of coordinates, although, except for such ambiguities as are represented by (2.28), we know how to transform it. One might think to avoid the difficulty by going to a coordinate-free treatment. However, as this is usually done, the form of W depends on the choice of reference configuration, which is similarly ambiguous ; there is a certain isomorphism between the two ambiguities. Usually, we find some way to eliminate this problem and to fix a group under which W is invariant or, perhaps, semiinvariant, which helps cut down the cases that need to be considered. It is not this phase of the selection process on which we wish to concentrate, although some of the discussion will touch upon it. Already, we have discussed some ramifications. For simplicity, we restrict our attention to the materials that we commonly regard as homogeneous. In practice, the selection process involves some variety of lines of thought, and some, like the method of divine revelation, seem not to lend themselves to mathematization. Of course, the classical linear theory illustrates one common approach; by some formally systematic perturbation procedure, we arrive at a function that is more or less definite, then employ a few experiments to eliminate such arbitrariness as might remain. In various such schemes, we encounter a curious but easily understood phenomenon; at each state of approximation, the approximating form may not share the invariance exhibited by W. Later, we discuss a rather subtle example of this kind. It is not always necessary to employ such perturbations. This was first established in the pioneering work of Rivlin and his colleagues, who did much to generate a revival of interest in nonlinear elasticity theory. The experimental work is summarized by Green and Adkins (1960); it seems fair to say that, in large measure, the book owes its existence to this pioneering effort, although various other writers contributed substantially to the development. This is but one of many examples of the influence. The experimentation is also covered by Bell (1973, pp. 734-741); Truesdell and No11 (1965, pp. 171-197) cover the theory and experiment. Sometimes we forget that, by either route, we determine W only over a limited domain. Via perturbation theory, something should remain quite small and there is always something that restricts deformations in the experiments. Rather unconsciously, we extend the domain of empirical functions by analytic continuation until we encounter some natural barrier. We might as well face the fact that, in practice, such continuation is one of the crutches we use in selecting definite constitutive functions. There is a rough check we should attempt to make. Assume, for the moment, that experimentation is inherently limited to a domain smaller than that covered by the extrapolation. If, from theory, we can predict the boundary of the smaller domain, we have some reason to be a bit more confident of the extrapolation or, at least, we see the matter in a different light. For the van der Waals’ fluid, for example, it is

222

J . L. Ericksen

generally agreed that we have long known how to do certain calculations which accomplish this, presuming that, in some ranges, theory accurately conforms to experiment. For solids, even the formulation of such problems involves deeper difficulties. By rather plausible arguments, we can deduce some conditions that, necessarily, are satisfied in the experimentally accessible domain. The criterion is good enough to predict what is commonly regarded as the experimentally inaccessible domain for the van der Waals’ fluid, but this case is too degenerate to provide much of a test. If one thinks about the nonlinear elastic analysis of common experiments, say the simple compression test, one begins to see that something else is needed, but it is less than trivial to determine just what it is. Questions arise concerning existence and uniqueness of simplistic solutions, concerning questions of a familiar type in elastic stability theory, and concerning questions as to appropriate forms of St.-Venant’s principle. For simple types of stresses in isotropic, incompressible materials of the neo-Hookean or Mooney-Rivlin variety, Rivlin (1948a,b) made early studies of invertibility, presuming homogeneous deformation. He includes some discussion of stability, although the analyses are less than complete. Rather similar types of invertibility problems are discussed by Moon and Truesdell (1974) and Truesdell and Moon (1975) for isotropic materials, who attempt to correlate some invertibility conditions with adscititious inequalities. It appears that the oft-heard phrase, more work will be required,” applies to this topic. This is not quite the same thing as what is involved in typical bifurcation studies, where one looks for smooth dependence of deformations on loads and related branching problems: it is not obvious that one branch need cross any other, for example. Similarly, there have been various studies of stability, that shed light, including bifurcation analyses. For example, for incompressible, isotropic materials, newer studies such as are summarized by Sawyers and Rivlin (1974) indicate that, for simple equilibrium configurations, it is feasible to isolate conditions critical for stability and to give insight as to the nature of instabilities, without assuming special forms of the constitutive equations. Older, pertinent studies are covered by Knops and Wilkes (1973). It is perhaps relevant to point out that there is a monograph by Knops and Payne (1971) dealing just with uniqueness theorems in linear elasticity theory; here one can find prototypes of problems arising in bifurcation studies. Questions relating to some form of St.-Venant’s principle have received little attention except for work in the classical linear theory ;it would be desirable to generalize such analyses to the linearized theory of small deformations superposed on large. Our knowledge of what W must do to avoid bifurcation, etc., is improving, but it will take more hard work to sort out the pieces and fit them together. Here, we have glossed over one point. Even traditional studies, such as are encountered in Euler’s theory of the Elastica, make clear “

”

“

223

Special Topics in Elastostatics

that “elastic” buckling occurs and that the buckling deformation is not so simplistic. Photographs of less-classical types of buckling are presented by Beatty and Hook (1968), with corresponding data and discussion of thenavailable theory. The experimentalist who is seriously interested in determining the form of W will scrap data obtained after buckling occurs, judging from what I have seen of such activities. He might well shift to a specimen of different dimensions, but his need for a St.-Venant’s principle imposes some limits on this, and another instability might stymie him. In brief, there are pragmatic limits set on the experimentally accessible domain by the habits of the practitioners and limitations of their apparatus. There is always the possibility that an ingenious person, with better equipment, might extend the experimentally accessible domain. For example, no one has yet made experiments with the entire boundary displacement fully controlled, and theory suggests that this is most likely to promote stability. Of course, our criterion is designed in an attempt to decide where the most ingenious experimentalist must, perforce, give up. It seems impossible to prove that no genius could outwit us. Rather obviously, the theorist could and perhaps should assume a set of ground rules in accord with current practices, and, hopefully, determine what parts of the domain are, currently, inaccessible. At this time, I have implied three variations on the Hauptproblem proposed by Truesdell (1956). As originally stated, the question is “Welches ist die Klasse der Funktionen C, die als Formanderungsarbeitsdichte eines vollkommen elastischen Stoffes dienen durfen?”’ First, my general experience tells me that it is neither wise nor fruitful to impose any restrictions upon the constitutive equations, in general. In proving any particular theorem, there is some necessity to introduce some limitations, and I believe in the value of rigorous theorems. I merely propose adoption of the mathematician’s criterion: the weaker the hypothesis, the better the theorem. Without sharing it, I have some understanding of the viewpoint ,of the operationalist. To him, inaccessible parts of the domain do not exist. Whether his domain is to be viewed as “currently accessible,” or whether it can be stretched to include “accessible in principle” is an issue that is not for me to decide, since I am not of the faith. If we leave this option open, we thus obtain two more possible interpretations. Thus, in all, we have here three interpretations of the problem. By my first interpretation of the problem, it is, as far as I am concerned, solved. With the third interpretation, I see no obvious shortcomings of the criterion to be discussed; perhaps it is a solution. With some effort, we might come to some better understanding of the second interpretation as stability theory improves. Here, it will take more effort to formulate the problems to be solved. Possibly, still other interpretations may spring to the mind of some readers, but I am not sure what they might be. “

’

”

“What is the class of functions Z, which can serve as strain energy densities, for perfectly elastic materials?”

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J . L Ericksen

When experiment fails us, we can turn to some form of mechanistic theory. For homogeneous materials, this is most likely to be molecular theory of some form. In one sense, the current efforts to calculate the strength of perfect crystals is in this category. We are learning more about the manufacture of better crystals, but do not yet know how to attain perfection; so the ideal material is not to be had. Of course, one encounters comparisons with experiments on imperfect crystals, employing intuitive judgments as to the importance of the defects; one must somehow decide what intermolecular force law is best. Work of this kind is summarized by Hill (1975). Roughly, the idea is to use molecular theory to calculate W. The molecular model suggests that the onset of plasticity or, perhaps, different instabilities should be predictable. It is then likely that some stability condition, independent of the size or shape of the specimen, should sometimes fail. Various forms of the criterion have been proposed, and finite shear or tensile strengths calculated; it is less than trivial to formulate the problem. Seemingly, what is desired is a criterion that would be in agreement with experiments of a more or less conventional nature, such as are now conducted. We are then back, I think, to the problem of understanding what is currently accessible.” By habit of thought, the experimentalist often gives up the idea that elasticity theory applies, whenever he sees that certain phenomena, which he identifies with the word plasticity,” become important. In a nutshell, the most classical molecular theory of perfect crystals implies that they have a certain type of material symmetry which, in itself, implies the existence of certain types of instabilities. One implication is that certain parts of the domain of Ware, inherently, inaccessible to static experiment. Another is that permanent deformation can occur. If one looks at commonly used approximate methods for calculating W, one can see that the approximating forms do not exhibit this symmetry, although instabilities can and d o survive. The aspect of symmetry which is important here is without influence in the range of infinitesimal deformations envisaged in the classical linear theory. In macroscopic studies of finite deformations of crystals, the common procedure has been to make the obvious extrapolation of symmetry assumptions; the additional symmetry is overlooked or ignored. We shall attempt to clear the air, as best we can, concentrating on conclusions that seem to us the firmest. Lack of space and energy inhibit us from giving a critique of the various local inequalities that have been proposed. “

“

B. INACCESSIBILITY Let us consider the problem of using static experiments to determine the function W. We presume that selection of coordinates is somehow stan-

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225

dardized, with x taken as the usual rectangular Cartesian coordinates. For simplicity, we restrict our attention to cases covered by the common presumption of homogeneity; for some selections of material coordinates X, W does not depend explicitly on these. For similar reasons, we exclude constrained materials. With the commonly understood ground rules, the experimentalist is free to employ specimens of different sizes or shapes. To determine the value of W at a particular value of the argument x , K , he must arrange to get some specimen to accept this deformation not necessarily everywhere, but at least somewhere within the specimen. Some cheating is permitted. Given the Galilean invariance of Wand, perhaps, plausible mate~ by the rial symmetry assumptions, we agree that values of x , differing implied transformations will rise to the same value of W , so values of x . arrange themselves in equivalence classes ; it suffices to observe one representative in each equivalence class. To make such an observation, we agree, I think, that his specimen must be stable with respect to at least the smallest disturbances. To explore this, we need some criterion for stability. We shall use the energy method, but in a way that evades common objections. We consider that, in predicting sufficient conditions for stability, it can be unreliable: it is a matter of experience that nonequilibrium phenomena can produce instabilities which, in principle, cannot be predicted without knowing what constitutive equations then apply. My early efforts (1966a,b) seem to have spurred various studies which lend credence to the notion that, interpreted as a necessary condition for stability, it is more reliable. Coleman (1973) covers more recent work of this type. He also cites an older work by Duhem which I had overlooked. I am not unaware of sufficiency conditions such as are discussed by Coleman (1973) and numerous earlier writers, but find it hard to construct a norm or semimetric for which such conditions are plausible, for nonlinear elastodynamics or nonlinear thermoelasticity theory. It was my view in 1966, as now, that the ideas of Liapounov, etc., do not work out well for these. Perhaps I should have said so. Perhaps someone will enlighten me. There is a persistent rumor that nonlinear elastodynamic solutions will never settle down to equilibrium. I do not find this obvious; such equations are notorious for their tendency to develop shock waves, which can dissipate energy. Indeed, the difficulty in selecting norms is, in part, correlated with this predilection. There is another possibility, illustrated by the following anecdote. A machinist made a clock gong for me. He was tempted to throw it out, after holding it in his hand and rapping it with a hammer. Being good, he spent a sleepless night. The next day, he clamped it in a steel vise, and changed his mind. There is some tendency for sophisticated elasticians to similarly overlook dissipation associated with loading devices. The energy method does suffer from a certain vagueness. It is not very

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certain what function space of virtual displacements is most appropriate. Should these satisfy just the kinematic boundary conditions, as is often alleged? In certain rather special cases, dynamical theory suggests that one should also respect traction conditions. In various calculations, one finds that this restriction sneaks in. Also, there is room for debate over the question of how much smoothness tp require. I (1966a) did acknowledge such difficulties, but still do not know how to cope with them. If one assumes that some definite type of dynamical theory applies out of equilibrium, one can, in principle but almost never in practice, settle the issue, by proving requisite theorems on the existence and regularity of dynamical solutions. Instead of entangling ourselves by getting into these difficulties, we proceed naively, exercising a little caution. Simple logic reveals that one simple observation is relevant. If we are after reliable necessary conditions, it is best to take the function space as small as is feasible. If we seek sufficient conditions, we should, on the contrary, enlarge it as much as we can. With such factors in mind, I think that one conclusion can be made rather firm. At a value %, of x , where the Hadamard condition a2w

(x,,)aiaiA.A, 2 0,

V

a, A,

fails, no equilibrium experiment can yield the value of W. This is one of the conditions mentioned by Truesdell (1956) in discussing his Hauptproblern. Here, we merely attempt to make clearer its physical significance. We remark, without proving it, that this relation is invariant under the usual Galilean and material symmetry transformations. The reader might find it amusing to explore more general transformations. The arguments favoring (9.1) begin with some assumptions on the energies associated with the loading devices which might be employed. First, the experimentalist must employ some shape of specimen, corresponding to a domain D.There is then the energy associated with this body, given by the usual prescription,

Usually, he will employ devices exerting loads on the boundary. With an ideally hard device, he can control surface displacements at will. With a soft device, or, realistically, with hard devices which are less than ideal, there is associated an energy E , ; there is room for debate on how best to describe it. For such devices, our assumption is that, if the deformation does not change in the neighborhood of aD, then neither does E , . Schematically, we make the relatively safe assumption that Ax = 0

on and near d D

A E , = 0.

(9.3)

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Of course, “near” means “in some neighborhood.” This requirement is met, for example, by the rather general types of energy functionals considered by Sewell (1965, 1967), Batra (1972), or Gurtin and Murdoch (1975). Indeed, for these, we could replace on and near by ‘’on in (9.2). Conceivably, the experimentalist might attempt to use some field acting in the interior of D, with associated energy E 3 . If this is necessary to effect stability, he must somehow disentangle this energy from the total energy to determine W , and this will be hard if E 3 depends on deformation in a manner too complicated. With this in mind, it seems to me a reasonable guess that E , is of the form “

E, =

[

”

”

U(X, X) dX.

(9.4)

‘D

Thus we assume that the total energy E of the isolated system is of the form E

= El

+ E 2 + E , = j,’W + U ) d X + E z .

(9.5)

To pursue the energy method, we must evaluate this for an actual equilibrium deformation x(X) and some virtual deformations y = x(X)

+ u(X),

u E Y,

(9.6)

or disturbance, Y being whatever function space we consider appropriate. Of course, it must include u = 0. Roughly, the idea is as follows. According to the energy criterion for stability, we must have u denoting the usual virtual displacement

AE = E(y) - E(x) = E(x

+ U) - E ( x ) 2 0,

V

uE

9. (9.7)

We restrict our attention to u such that u=O on and near Then, for such u, (9.5) and (9.7) give

A

aD.

(9.8)

I

(W+ U)dX20.

(9.9)

With various mild restrictions on Y and the smoothness of the functions W and U , one can determine necessary conditions that (9.9) hold when (9.8) holds. We should say by what norm we measure smallness. In most stability studies made in elasticity theory, infinitesimal disturbances means that the left-hand side of (9.9) can be replaced by a functional quadratic in u. It is hard to see how this can be justified unless ‘‘infinitesimal” means that I ui I and, perhaps, I uiI are everywhere small. Various writers have deduced that (9.8) and (9.9) imply (9.1) after making this approximation, sometimes with“

”

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J. L Ericksen

out justifying it. Those interested in the more complete story might consult the derivation sketched by Morrey (1966, OM), who also covers a stronger inequality, necessary for stability with respect to finite disturbances. We refer the reader to his treatment for discussion of technical requirements. One of the alternative derivations is given by Wang and Truesdell (1973, pp. 533-535). If one studies the proofs, one can see that rather little is required for the result to obtain. Employing the elementary rules of logic, we now conclude that, iffor some a and A,

ax:L aiaJA,AL< 0,

a 2 w. aXi,,

(9.10)

in some part of the domain of W, then this part is experimentally inaccessible. With slight variations. I am repeating an argument that I (1975b) have published before; it was published in a language (Russian) that may deter some readers. There was a considerable lag between submission and publication, so it has influenced my thoughts for a longer time. Also, I there mentioned a view which I elaborated elsewhere (197%). Inequality (9.1) is, as is well known, and discussed by Truesdell and Noll (1965, pp. 267-272) equivalent to the condition that, at the state considered, all acoustic speeds be real. If it fails, some have visions of complex exponential solutions of linearized equations growing exponentially. Some reason that this implies instability. My (197%) note points out that linearized elasticity equations with constant coefficients necessarily permit solutions growing exponentially in time, when (9.1) applies. R. D. Mindlin informs me that he was aware of this and has covered it, in a different way, in his lectures. Thus, the reasoning seems to me unsound, although I believe the end result. There is the additional factor that viscoelastic or thermal effects, etc., can influence wave propagation, without vitiating (9.10) as a condition for inaccessibility, as it is reasoned here. I have played with alternatives, finding no other that seems to me as firm. Implications of the strict version of (9.1) with respect to existence and regularity of solutions of the linearized equations, for displacement boundary value problems, are summarized by Truesdell and No11 (1965, @8). It appears that (9.1) is close to sufficient to get well-behaved solutions for such problems; no stronger local inequality is suggested by such analyses. For the nonlinear equations, much the same implications appear in the more recent work of Ball (1977). Herein lies one of the hints that accessibility is most likely when the surface displacement is controlled. There is a sizable literature on inequalities considered by some to be plausible. Summaries are given by Truesdell and No11 (1,965,pp. 125-133,142-147, 153-171,246-260,278-284,319-324,332-335), Hill (1970,1975), Wang and Truesdell (1973, Chap. 111) and Krawietz (1975). Of course, adoption of a particular surface energy functional will permit one to derive conditions that

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229

are likely to be stronger; it is a more definite problem in stability theory which is likely to lead to some other conditions. As applied to van der Waal's or other simple fluids, (9.9) yields, as inaccessible parts, just those subsets, if any, where pressure decreases with increasing mass density. I do not think that elasticity theory could or should forecast much else for a constitutive equation of the type implied. Of course, it is possible to design a reasonably smooth constitutive equation which, in part of its domain, looks like that appropriate for a solid while, in another part, looks like that appropriate for a fluid. Presumably, this could accommodate fluid-solid phase transformations, as the van der Waal's equation accommodates different fluid phases. We do.not find in the literature a discussion of this possibility. There are in the literature constitutive equations of empirical origin for which (9.10) holds, in part of their domain. For one such, the Blatz-Ko material, Knowles and Sternberg (1975) determine where (9.10) holds, and discuss some implications ; analytic continuation is used to extend the domain of W. We here have a case where, sometimes, the governing equations are hyperbolic, and they discuss the nature of characteristics. Equations discussed by Varley and Day (1966)also admit real characteristics. In this respect, there is a certain similarity with the theory of fiber-reinforced composites ; the equations for these materials also are sometimes hyperbolic. For the latter, Pipkin and Sanchez (1974) study existence theory for conventional boundary value problems. It is at least amusing to ponder what might be the relation between the tendency of sometimes hyperbolic equations to admit solutions less than perfectly smooth and the commonly observed discontinuous deformations in solids, such as are involved in Liider's lines, Savart-Masson steps, etc. Observations of such phenomena are discussed by Bell (1973, pp. 41-44,220-230,449-457,474-478,570-578,649-666,690-716. C. CRYSTAL LATTICES From the viewpoint of crystallography, a crystal has a periodic structure. The crystallographer might view an inhomogeneously deformed crystal as a crystal, if the inhomogeneity is negligible over distances very large compared to atomic spacing, but it is hard to make such notions precise. What is repeated periodically is either an atom or cluster of atoms. We take the most simplistic view, that it is an atom, and that these are subject to central forces. Also, we assume that all atoms are alike. This puts us back with Cauchy (1829). It will be obvious to the crystallographer that, at that time, the quality of X-ray equipment, etc., left something to be desired, so Cauchy might be forgiven for not knowing, say, that there are hexagonal closepacked crystals which d o not fit his picture.

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J . L. Ericksen

His theory led to the clear-cut and famous Cauchy relations. It is now rather generally agreed that, for most crystals, these are contradicted by experiment. Surely, you say, modern-day workers are aware of the flaws of this theory and evade them. Workers interested in large deformations of crystals d o not vary much from this old theory, for simple, pragmatic reasons. If they attempt to generalize the ideas set down by Cauchy, in almost any conceivable way, they soon find that they encounter some complications of rather major proportions. The common exception is the diatomic crystal which happens to form a simple lattice with the atoms alternating, NaCI, for example. For infinitesimal deformations, prospects are a bit less bleak; linear theory is always a bit easier. Actually, Cauchy’s treatment is restricted to infinitesimal deformations. However, it is not difficult to remove this restriction. One way of evading the Cauchy relations is to retain the central force assumption, but to replace Cauchy’s atoms by clusters of atoms. In essence, this is the model of Kelvin and Born,+ the best-known evasion ; there are cases where this procedure gives the Cauchy relations. Poincare (1892) saw a test of the relations as a test of the central force assumption; at least some modem workers agree. This is not the place to go into the twentieth century ideas concerning this problem. Quantum mechanics has changed our views concerning the billiard balls, as is discussed by Seitz (1940), for example. As a matter of definition, a countably infinite set of identical mass points form a crystal lattice, in the sense of Cauchy, if there exist three linearly independent constant lattice vectors hl, h,, and h, , such that the positions x, (n = 1, 2, . . .) of all the atoms are representable in the form X, = M i

ha,

(10.1)

where the M i represent sets of positive or negative integers, any such set giving the position of some atom. For simplicity, we have omitted an arbitrary constant vector; one atom is placed at the original of coordinates. It follows easily that another set of lattice vectors h, describes the same lattice, provided that

ha = Nf:h, ,

(10.2)

where the Ng are any integers such that detllN;)) = f 1.

(10.3)

As is discussed by Love (1927), the Born model is a rather obvious generalization of a model worked out by Kelvin, who did realize its potential for evading the relations. References to older work on this topic are given by Love. A clear, brief exposition of this type of theory is given by Stakgold (1949).

Special Topics in Elastostatics

23 1

Such matrices represent a group G which, in the sense indicated, leaves the lattice invariant. The group G is neither finite nor compact. However, it follows from a theorem in algebra that it has a finite set of generators which, according to Hua and Reiner (1949), can be taken as

,

-1 0 0 0 1 011 . I I o o 1

1 1 01 0 1 011, 0 0 1

110 0 1 1 0 0’1 ,

(10.4)

Also, they note that the first two generate the subgroup consisting of

matrices with positive determinants. For elasticity theory, there is no loss in generality in replacing G by this subgroup. Of course, (1O.4), reflects the geometrically obvious fact that certain finite movements of a shearing type will map a lattice onto itself. Loosely, this implies that crystals have finite strength; of course, this is the basic idea that underlies the rough “billiardball model calculations which proponents of dislocation theory commonly use to justify the need for considering lattice imperfections. The well-known crystallographic groups are related to G , but different. It can happen that, for certain choices of lattice vectors and for certain choices of N : , ”

h, = N : h,

= Qh,

,

(10.5)

Q representing some orthogonal matrix. Some writers require it to be a rotation matrix; it seems that the implied dispute will forever recur, in different forms, whatever be the evidence. For applications to elasticity theory, the difference is inconsequential. With ha fixed, such Q form a group called the point group. Similarly, given a point group, there are sets of lattice vectors associated with it, those compatible with (10.5). Each such set generates a lattice via (10.1); we take one atom and translate it in ail the ways permitted by (10.1). Such a translation group is called a space group. It is not our purpose to elaborate the features of the point or space groups; it is hard to find any discussion of elasticity theory pertaining to crystals which does not discuss at least the point groups. Elementary derivations of these groups are given by Seitz (1934, 1935a,b, 1936a,b), for example. Often, some crystal lattices are described in a slightly different way, which is not so obviously equivalent. For example, consider the body-centered cubic lattice. Often, we start with three orthogonal lattice vectors k, of the same length, indicating the basic cube

k, * k, kl

*

= 0,

a f b,

kl = k2 * k2 = k3 * k3 > 0.

(10.6)

J . L Ericksen

232

If we apply (10.1)to these lattice vectors, we get the atoms at the corners of the elemental cube, but miss those in the center. We can incorporate these in the lattice by writing their positions y. in the form yn = M: k,

+ $(kl + k z + k3).

(10.7)

The problem then is to check whether there is a different choice of lattice vectors ha such that (10.1) produces exactly the same set of positions. One finds that there is; one can take

+ kz + k 3 , 2hz = k1 + kz k 3 ,

2h1 = k l

2h3 = k , - kz

k,

= hz

+h3,

-

kz = h, - h 3 ,

+ k3,

k3 = h, - h , ,

(10.8)

for example. On the other hand, the hexagonal close-packed lattice provides an example of one not representable in the form (10.1). It is what is sometimes called a multilattice. Multilattices can be viewed as generated by taking one lattice, translating it a few times to get a set of interpenetrating lattices; one can put different atoms on each of the lattices. Born's theory can cope with these, Cauchy's theory being limited to the special cases summarized in (10.1). The larger group G clearly applies to all lattices of the form (10.1) and, with some technical modifications, to multilattices. Roughly, it sees all crystals as having the same symmetry, the point and space groups picking out subtypes bearing familiar names. I). LATTICE KINEMATICS

Clearly, if in (lO.l), we apply any invertible linear transformation to the lattice vectors, (11.1) h, -+ fi, = Lh, , det L # 0, we generate another lattice described by in = M:: fi, = Lx, .

(1 1.2)

With all atoms alike, this will, in some cases, be indistinguishable from the old lattice. This occurs when there exist integers N : , as indicated in (10.2), satisfying (10.3), with (11.3) Lh, = Nf:h, . The set of all such matrices provides another representation of the group G or, in older language, a conjugate group; the particular matrices depend on

Special Topics in Elastostatics

233

the selection of the lattice vectors ha. Clearly, any two lattices can be brought into coincidence by applying some linear transformation to one and then translating it. It is convenient to introduce the reciprocal lattice vectors ha, satisfying hashb=&,

h a Q h a = 1.

(U.4)

In passing from the atomistic to the continuum analog, it is customary to pick one lattice configuration, equipped with definite lattice vectors Ha,as a reference. In the model of Cauchy, we d o the obvious. In the continuum view, a homogeneous deformation acts as a linear transformation, with positive determinant, on material line elements. Simplistically, material line elements are relative positions of atoms, so we apply the same linear transformation to the lattice vectors, thus correlating atomic movement with gross movement. As is not uncommon in molecular theory which aims at predicting macroscopic constitutive equations, we have decided what type of macroscopic theory is to be sought. It is elasticity theory, and, for this, it will suffice to consider homogeneous deformations. Of course, various intuitive judgments underlie such decisions. Here, for example, it is a common notion that interactions to be accommodated are of quite short range. Formalizing these notions, we interpret the coordinates x and X as referring to the same rectangular Cartesian coordinate system, with x = FX,

Fa = x

~ K ,

(1 1.5)

or, atomistically, X, =

FX,

.

(11.6)

Then, consistently, we pick among the new sets of lattice vectors, that given by (11.7) ha = F H a o F = haQ Ha. Here, for ease of interpretation, we do introduce a “material geometry,” in the customary fashion. These configurations have one very nice feature, as they relate to central force laws. Where any atom is pulled by another, there is yet another atom pulling it just as hard in the opposite direction. Thus, glossing over any convergence problems, each atom in the infinite lattice automatically has zero resultant force acting on it. If one complicates the configuration, as Kelvin did, one must, in general, solve a difficult problem in mechanics to determine how some atoms can move to maintain equilibrium. When this problem is not rendered trivial by simple symmetries, the temptation to

234

J . L Ericksen

linearize has proved irresistible, so far. Of course, for a finite crystal, the configurations assumed above would, in general, leave unbalanced forces on atoms, particularly near boundaries. The usual view is that, with short-range interactions, we can cover this separately, as a surface effect. After all, most of crystal physics survives by ignoring this problem. From the viewpoint of elasticity theory, it is one problem to get the differential equations, another to get the boundary conditions or, perhaps, side conditions of a different kind.

E. MOLECULAR THEORY With central forces exerted on each other by identical atoms, everything follows once one has selected the pair potential function cp = cpw,

(12.1)

r being identified with distance between pairs of points. Cauchy's (1829) approach was to calculate the stress tensor, as a function of deformation. This has some value, as a pedagogical device. By the time a student masters the calculation, he will, I think, understand rather well what Cauchy had in mind by stress. Commonly, cp is normalized to approach zero as r -+ 00, rather rapidly for short-range interactions. Normally, it is assumed to be defined and smooth for all positive values of r ; one can envisage atoms as having a finite radius, in which case the domain of cp is bounded below, in the obvious way. In concept, Cauchy eschewed the latter complication which, rather obviously, can lead to some packing problems. Most later writers replace the calculation of stress tensor by the calculation of the scalar energy function; the results are 'consistent. We follow the crowd. Immediately, we face one problem. From either an atomistic or macroscopic point of view, the energy of an infinite crystal is likely to be infinite. To avoid this, we apportion the energy among the atoms, to get an energy per unit atom E . If p denotes the (constant) mass density in the reference configuration then, for a finite reference volume V,, large enough to contain many atoms, we should have, to a good approximation, pVR = Nm,

where m is the atomic mass, N the number of atoms in V,. Equating energies, we should have WVR = N E ,

or W = pEfm.

(12.2)

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235

We now digress, to estimate very roughly what short-range interaction means. The usual practice is to carry out the sum over the closer neighbors of the atom in question, ignoring the rest. If the spherical shell between the distances r and r + Ar contains N atoms, where N is relatively large, we should have “

Nm

%

pp(4nr3/3)[(1

+ Ar/r)3 - 11,

”

(12.3)

where pp is the (constant)density in the configuration considered. For r 2 R , where R is large, we should be able to take Ar/r small, keeping N large enough so, as above Nm

%

4nppr2Ar,

r 2 R.

(12.4)

Typically, we have in mind potentials which vary quite slowly when r is large, so the contribution to mE from atoms in the shell should be about Nmcp(r) z 4nppcp(F)r2Ar,

r 2 R,

(12.5)

+

where Frepresents any value between rand r Ar, if R is large enough. Thus the sum over atoms with distance greater than R is roughly estimated by the integral

1

.m

mE = 4npp

cp(r)r2 dr.

(12.6)

‘ R

The obvious condition for convergence is cp=0(f3)

for r + m ,

(12.7)

so cp must tend to zero quite rapidly for E to be well approximated by a reasonable number of terms in the finite sum. Of course, with the development of computers, the definition of reasonable number does change with time. Even so, computation is laborious since one must estimate E for some variety of lattice configurations to get an estimate of the energy function. It is not our intention to belabor the computational problems, but it seems desirable to have some idea of what types of potentials are included. Cauchy (1829) similarly replaced sums by integrals and was criticized for it, primarily because of false impressions that the Cauchy relations were somehow the result of this approximation. It does have some status as an approximation and the practice continues. For example, Born and Furth (1940)employ this approximation to estimate contributions from all but nearest neighbors. Those who worry about “admissibility of constitutive equations might ponder the uniaxial stress-strain curve which they graph. The work of Macmillan and Kelly (1972a,b) exemplifies calculations which are now feasible. Of course, potentials used in practice taper to zero somewhat more rapidly ”

236

J . L. Ericksen

than is indicated by (12.7), but the series seem not to be well approximated by a small number of terms. What is involved in principle is the infinite sum (12.8)

ri = x,

- x, ,

(12.9)

where x, are representable in the form (9.1). We associate E with the atom located at the origin, so the M: run over all the integers except M: = 0. The question of how atoms are to be numbered is at our discretion; changing it does not affect E. As is clear from (10.1) and (12.8), for a fixed choice of the pair potential function, E will depend on the lattice configuration, but only through the lattice vectors, E = E(h,).

(12.10)

If we apply a transformation of the type (10.2), it will change the lattice vectors but not the lattice. The effect is to renumber atoms, which does not affect E. Thus, E(N!:hb) = E(ha),

(12.11)

N!: being any set of integers satisfying (10.3). We note that, generally, finite sums approximating the infinite series d o not enjoy this property. Neither d o the integral approximations mentioned earlier, at least as they have been calculated. Thus one type of invariance is lost in the common approximations. In essence, it is (12.11) which endows crystals with certain instabilities, somewhat like those considered in our study (1975b) of elastic bars. Since cp depends only on distance, we clearly have E(Qh,)

=

(12.12)

E(h,),

where Q denotes any orthogonal matrix, Q-1

(12.13)

= QT.

-

If you prefer, E depends on h, only through the scalar products h, hb. Of course, approximating sums share this invariance, since it applies term by term. If we now introduce deformation as indicated by (11.5)-(11.7), and use (12.2), plus (12.10)-(12.12), we have W ( x I K= ) W(F) = pE(h,)/m = pE(FH,)/m = pE(QNf: F

HJh

(12.14)

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237

summarizing the prescription of and predicted invariances of W . Further, from (3.21), (11.6), and (12.9) X,

-

X, =

(FX,) * (FX,) = CKLX;IKX;IL,

(12.15)

so, rather automatically, W is obtained as a function of C.If one inserts this in (12.8) and uses (12.12), one gets the famous Cauchy relations, viz.,

acKL

acMN

a 2 w aCKM

(12.16)

acLN’

presuming, of course, that W is written so that these derivatives have the symmetries obviously implied by the symmetry C K L= C L K . Each term, hence approximating sums conform to (12.16). What is involved in (12.14) reflects Galilean invariance and the periodicity of the lattice. It does not seem unreasonable that (12.14) should apply, rather independent of the nature of the molecular theory for perfect crystals, as long as it leads to elasticity theory. On the other hand, (12.16) obtains for this molecular model, but not for others. The first study of implications of (12.14) is, it seems, in my paper (1970b). This is rather curious, since much the same idea has long been used in the “billiard ball” estimates of strength. Elsewhere, I (1970b) have discussed plausibility arguments favoring the notion that the symmetry implied by (12.14) also follows from molecular theory applying to multilattices. There is a presumption that elasticity theory similarly applies to them, which here becomes a more subtle issue. Of course, one can forget the molecular theory and attempt to construct energy functions that satisfy (12.14) and/or (12.16), or to otherwise study their implications. We prefer this approach. By private communication, R. Hill informs me that he and his student, G. Parry, are engaged in an effort to construct such functions; Parry (1976) has written one paper on this. Constructions producing some solutions of (12.16) are given by Hill (1975); one can take an arbitrary function of the type

w = W(C,, VKVL),

( 12.17)

where V is any fixed vector, and use linearity of (12.16) to superpose functions of this kind. Of course, this is close to what is done in the molecular calculations. Parry (1975) aims at constructing constitutive equations consistent with (12.14). We refer the reader to his paper for details.

INSTABILITIES F. SYMMETRY-INDUCED In general, if the domain of W is large enough and, if it there satisfies (12.14), there will be parts that are not accessible to experiment. More

J . L. Ericksen

238

precisely, there are parts where (9.10) must hold. If W depends only on h, h h 2 * h, cc det F, describing in this sense a simple fluid, (12.14) is satisfied, but (9.10) need not be satisfied in any part of its domain. To exclude such cases and to keep analysis fairly easy, we introduce the following hypotheses : (a) There exists some set of integers ma and nb (a, b = 1, 2, 3) such that a2w

aXtKax!,

a i d A K A L> 0

at

F = 1,

d A i = 0,

(13.1) (13.2)

a' = m a H i , A,

= n,H",

(13.3)

(b) The domain of W includes the deformations given by

F'K -- 8'; + y a ' A , ,

o
(13.4)

From (11.4), applied to the reference lattice vectors, (13.2) and (13.3) imply that the integers must be chosen so that aiAi = manbHtHb,= mana= 0,

(13.5)

wherein, as usual, the summation convention applies. In turn, (13.5) implies that det IJNt11 = 1,

A$ = 8 :

+ manb,

(13.6)

giving us a set of integers satisfying (10.3), eligible to be used in (12.14). With everything but y fixed, W reduces to a function of y , (13.7)

(13.8) (13.9) Thus, (13.1) implies that

f"(0)> 0.

(13.10)

Using (11.4), (13.4), and (13.5), we verify that

+ ymbHbn,,

(13.11)

+ l)H,K = N:FL(y)H,K,

(13.12)

FL(y)Hf = H: F;(y

Special Topics in Elastostatics

239

where the N i are given by (13.6). From (12.14), (13.7), and (13.12), we have

f ( ; . + 1) = f ( y ) ,

0 Iy I 1,

so the derivatives satisfy f’(7

+ 1) = f ’ ( r ) ,

and, in particular, (13.13) Obviously, permanent deformation can occur, F( 1) is indistinguishable from F(0) in the eyes of the crystal. With (13.10), (13.13) implies thatf”(y) cannot be nonnegative everywhere in the interval [0, 11. Thus there must exist y* such that f”(y*) < 0, 0 < y* < 1. (13.14)

f ’( 1) =f’(0).

We now compare (13.14), (13.9), and (9.10) to infer that under the assumptions ( a )and (b), the symmetry inherent in (12.14) implies that thedeformation F(y*) is not accessible t o experiment. A slightly weaker version of this theorem occurs in one of my earlier papers (1970b). Elsewhere, I (1973) showed that for a configuration to be stable with respect to large disturbances, it is necessary that the Cauchy stress reduce to a hydrostatic pressure. There is a thought here which seems to me intriguing. Here, elasticity theory tells us that, if we induce large disturbances in a loaded crystalline solid, we should expect to see shear stresses relax, if the loading device will permit it. Actually, there have been numerous observations of an effect somewhat like this called the Blaha effect. Here, usually in a hard device, shear stresses are diminished by vibrating a plastically deformed specimen. Within the practical limits, the bigger the amplitude of vibration, the greater the effect; complete relaxation of shear stresses seems not to have been accomplished. Varying frequency over a relatively large range has little or no effect. Typical observations of this softening effect are reported by Langenecker (1963), for example. There seems to be no adequate theory. Contrarily, there are cases where vibration induces hardening, as is discussed by Schmid (1972). Because of the peculiar nature of the group G, most of it exerts no influence on the theory of infinitesimal deformations. Roughly, if F corresponds to an infinitesimal strain, its transform by an element of G will not, in general. Exceptions occur when the lattice vectors at F = 1 admit point groups; these map zero strain into zero strain. In the infinitesimal theory, it suffices to consider such point groups and Galilean invariance, in the traditional manner. In fact, we can prescribe Win some neighborhood of F = 1, arbitrarily except for these restrictions. We forego detailed discussion of the technical points here involved. The reader who understands the theory of material symmetry should have no trouble in filling in the gaps. If we push

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J . L. Ericksen

too far, we begin to feel the influence of other elements of G: we have seen this type of situation before, in the theory of multiply periodic functions. This suggests one way to proceed. We calculate the fundamental domain, a maximal neighborhood over which W can be prescribed rather freely, a problem in kinematics. It is here where the more conventional symmetry considerations, such as are summarized by Rivlin (1966, @XVIII),are relevant. If we d o molecular calculations, it is for deformations here that the calculation is important. Phenomenologically, it is here and only here that we have latitude. Either way, G will define W elsewhere. Parry (1975) discusses characterizations of a type of fundamental domain. To get back to our stability problem, we infer that no unfamiliar symmetry considerations prevent (13.1) from being satisfied for all vectors a and A, at F = 1. By the same token, it emphasizes the need for assumption (b). If we prescribe W over less than a fundamental domain, then extend the domain with G, we can get W defined over a set which need not even be connected. Roughly, (b) indicates that we can continuously vary F from one value to its image under a nontrivial element of G,following a simple path, and stay in the domain of W. One might modify the assumption a bit, but some assumption of this kind seems necessary. In one sense, nonlinear theory simplifies the theory of crystal symmetry, by replacing the various crystallographic groups by one, the group G. If we see an unloaded crystal take up a cubic configuration, we might plausibly infer that W there has at least a relative minimum. Clearly, not all energy functions have this property, so we can use the nature of minima as a basis for classification. If we only wish to study the bottoms of these energy wells, such a classification seems apt. In essence, this is what we do with the classical infinitesimal theory, and some of the perturbations about it. There is the danger that easy perturbation methods will blind us to possibilities which would become apparent, if we were to take a broader view. In most studies of crystal elasticity, the “easy perturbation borrows the theory of material symmetry from the theory of infinitesimal deformations. As we have noted, this view has its place, but a broader view reveals quite a different picture. Hopefully, we have made clear that elastic stability theory entails more than the buckling phenomena which arise from another easy perturbation. Certainly, we have not covered all such evidence, attempting more to dispel prejudice against violating familiar inequalities. The simple old workhouse, the neo-Hookean material, predicts a kind of fracture instability, as was shown some years ago by Gent and Lindley (1958). The recent study of elastic rods by Antman and Carbone (1976) makes it plausible that, without violating (9.I), the three-dimensional theory can predict some variety of “anelastic” phenomena, including a type of hysteresis. Indeed, for a time, we seriously considered the possibility of devoting the entire article to ”

Special Topics in Elastostatics

24 1

elaboration of such questions. There is good reason to become more aware of blinding effects of words like ‘‘anelasticity or ‘‘ plasticity.” In closing, we suggest that readers try out on other calculations the rather classical formulation of the basic equations used here. Our experience is that it eases various types of analyses in shell theory, rod theory, or liquid crystal theory, as well as in three-dimensional elasticity theory. ”

REFERENCES ADKINS,J. E. (1958). A reciprocal property of the finite plane strain equations. J. Mech. Phys. Solids 6, 267-275. ANTMAN,S. S., and CARBONE, E. R. (1977). Shear and necking instabilities in nonlinear elasticity. J. Elasticity (To be published.) BALL,J. M. (1977). Convexity conditions and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal. (To be published.) BATRA,R. C. (1972). On non-classical boundary conditions. Arch. Rarion. Mech. Anal. 48, 163-191. BEATTY, M.F., and HOOK,D. E. (1968). Some experiments on the stability of circular rubber bars under end thrust. I n t . J. Solids Strucr. 4, 623-635. BELL,J. F. (1973). The experimental foundations of solid mechanics. I n “Handbuch der Physik” (S. Fliigge and C. Truesdell, ed.), Vol. 6A, Part 1, pp. 1-813. Springer-Verlag, Berlin and New York. BOGY,D. B. (1967). An “optimal” solution of Saint-Venant’s flexure problem for a circular cylinder. Trans. A S M E , Ser. E 34, 175-183. BORN,M.,and FORTH,R.(1940). The stability ofcrystal lattices. 111. An attempt to calculate the tensile strength of a cubic lattice by purely static considerations. Proc. Cambridge Philos. SOC. 36, 454-465. CARLSON, D. E., and SHIELD,R. T. (1969). Inverse deformations results for elastic materials. Z. Angew. M a t h . Phys. 20, 261-263. CAUCHY, A.-L. (1829). Sur les equations diflerentielles dequilibre ou de mouvement pour un systeme de points materiels sollicites par de forces &attraction ou de repulsion mutuelle. I n “Exercisede Mathematique,” Vol. 4, reprinted in “Ouevres Completes d’Augustin Cauchy,” Ser. 2, Vol. 9, pp. 162-173 (1891) Gauthier-Villars, Paris. COLEMAN, B. D. (1973). The energy criterion for stability in continuum thermodynamics. Rend. Semin. M a t . Fis. M i l a n o 43, 85-99. COSSERAT, E., and COSSERAT, F. (1909). “ Theorie des corps deformables.” Hermann, Paris. DAFERMOS, C. M. (1969). The mixed initial-boundary value problem for the equations of nonlinear one-dimensional viscoelasticity. J. Differ. Equat. 6, 7 1-86. EDELEN,D. G. M. (1962). The null set of the Euler-Lagrange operator. Arch. Ration. Mech. Anal. 11, 117-121. ERICKSEN, J. L. (1954). Deformations possible in every isotropic incompressible perfectly elastic body. Z. Angew. M a t h . Phys. 5, 466-486. .I.L. (1955). Deformations possible in every compressible, perfectly elastic material. ERICKSEN, J. M a t h . Phys. 34, 126-128. ERICKSEN, J. L. (1966a). A thermo-kinetic view of elastic stability theory. I n r . J. Solids Srruct. 3, 573-580. ERICKSEN, J. L. (1966b). Thermoelastic stability. Proc. N a t l . Congr. Appl. Mech., 5th. 1966. pp. 187-193. ERICKSEN,J. L. (1970a). 0 CHMMeTPHH KpHCTannOB (nepeBOn B. A. UaMHHOfi). In

242 “np0611eMbI

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(n. kl.

CenOB and Leningrad. ERICKSEN, J. L. (1970b). Nonlinear elasticity of diatomic crystals. Int. J . Solids Struct. 6, 951-957. ERICKSEN, J. L. (1973). Loading devices and stability of equilibrium. In “Nonlinear Elasticity,” pp. 161-173. Academic Press, New York. ERICKSEN, J. L. (1975a). Equilibrium of bars. J. Elasticity 5, 191-202. ERICKSEN, J. L. (1975b) HeKOTOpbIe IIpO6neMbI YCTO&WiBOCTH B HenHHefiHOfi TeOpHA YIIpYrOCTH. In “YCIIeXH MeXaHHKH nel$OpMHpyeMbIX CwB” (kl. n. BO ~ O BHand Y A. JI. ronbne~sefi3ep,eds.), pp. 559-565. “Nauka,” Moscow. ERICKSEN, J. L. (1975~). Complex exponential solutions of linear elasticity equations. J. Elasticity 3, 65-71. ERICKSEN, J. L. (1977). Simpler problems for elastic Cosserat surfaces. J. Elasticity 7 , 1-11. ESHELBY, J. D. (1956). The continuum theory of lattice defects. Solid State Phys. 3, 79-144. FLETCHER, D. C. (1975). “Conservation Laws in Linear Elastodynamics,” Tech. Rep. No. 33. California Institute of Technology, Pasadena. FOSDICK, R. L. (1966). Remarks on compatibility. I n “Modern Developments in the Mechanics of Continua” (S. Eskinazi, ed.), pp. 109-127. Academic Press, New York. I. M., and FOMIN, S. V. (1963). “Calculus of Variations” (R. A. Silverman, ed. and GELFAND, transl.), Rev. Engl. transl. Prentice-Hall, Englewood Cliffs, New Jersey. P. B. (1958). Internal rupture of bonded rubber cylinders in tension. GENT,A. N., and LINDLEY, Proc. R . SOC. London, Ser. A 249, 195-205. GREEN,A. E., and ADKINS,J. E. (1960). “ Large Elastic Deformations and Nonlinear Continuum Mechanics,” Chapter X. Oxford Univ. Press (Clarendon), London. GURTIN,M. E. (1972).The linear theory ofelasticity. I n “Handbuch der Physik” (S. Fliigge and C. Truesdell, ed.), Vol. 6A, Part 2, pp. 1-296. Springer-Verlag, Berlin and New York. GURTIN,M. E., and MURWCH,I. (1975).A continuum theory ofelastic material surfaces. Arch. Ration. Mech. Anal. 57, 291-323. HAMEL,G. (1912). “Elementare Mechanik,” pp. 573-575. Teubner, Leipzig. HILL,R. (1970). Constitutive inequalities for isotropic elastic solids under finite strain. Proc. R. SOC.London, Ser. A 314, 457-472. HILL, R. (1975). On the elasticity and stability of perfect crystals at finite strain. Math. Proc. Cambridge Philos. SOC.71, 225-240. HUA,L. K., and REINER,I. (1949). On the generators of the symplectic modular group. Trans. Am. Math. SOC.65,415-426. KAO,B.-C., and PIPKIN, A. C. (1972). Finite buckling of fiber-reinforced columns. Acta Mech. 13, 265-280. KIRCHOFF, G . (1852). Ueber die Gleichungen des Gleichgewichts eines elastischen Korpers bei nicht unendlich kleinen Verschiebungen seiner Theile. Sitzungsber. Akad. Wiss. Wien, Math.-Naturwiss. Kl. 9, 162-773. L. E. (1971).Uniqueness theorems in linear elasticity theory. Springer KNOPS,R. J., and PAYNE, Tracts Nat. Philos. 19, 1-130. KNOPS,R. J., and WILKES,E. W. (1973). Theory of elastic stability. In “Handbuch der Physik” (S. Fliigge and C. Truesdell, ed.), Vol. 6A, Part 3, pp. 125-302. Springer-Verlag, Berlin and New York. KNOWLES,J. K., and STERNBERG, E. (1972). On a class of conservation laws in linearized and finite elastostatics. Arch. Ration. Mech. Anal. 44,187-21 1. E. (1975). On the ellipticity of the equations of nonlinear KNOWLES,J. K., and STERNBERG, elastostatics. J. Elasticity 5, 341-362. KRAWIETZ, A. (1975). A comprehensive constitutive inequality in finite elastic strain. Arch. Ration. Mech. Anal. 58, 127-149. MeXaHAKA

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LANGENECKER, B. (1963). Effect of sonic and ultrasonic radiation on safety factors of rockets and missiles. A I A A J. 1, 8&83. LOVE,A. E. H. (1927). “A Treatise on the Mathematical Theory of Elasticity,” 4th ed., Note B. Cambridge Univ. Press, London and New York. LOVELOCK, D., and RUND,H. (1975). “ Tensors, Differential Forms and Variational Principles.” Wiley (Interscience), New York. MACMILLAN, N. H., and KELLY,A. (1972a). The mechanical properties of perfect crystals. I. The ideal strength. Proc. R. Soc. London, Ser. A 330, 291-308. MACMILLAN, N. H., and KELLY,A. (1972b). 11. The stability and mode of fracture of highly stressed ideal crystals. Proc. R. Soc. London, Ser. A 330, 309-317. MARRIS,A. E. (1975). Universal deformations in incompressible isotropic elastic materials. J. Elasticity 5, 111-128. MICHAL,A. D. (1951). Differential invariants and invariant partial differential equations under continuous transformations in normed linear spaces. Proc. Natl. Acad. Sci. U.S.A. 37, 623-627. C. (1974). Interpretation of adscititious inequalities through the MOON,H., and TRUESDELL, effects pure shear stress produces upon an isotropic elastic solid. Arch. Ration.Mech. Anal. 55, 1-17. MORREY, C . B., Jr. (1966). “ Multiple Integrals in the Calculus of Variations.” Springer Publ., New York. NOETHER,E. (1918). Invariante Variationsprobleme. Nachr. Akad. Wiss. Goettingen, Math.Phys. Kl., 2 pp. 235-257. PARRY, G. P. (1976). On the elasticity of monatomic crystals. Math. Proc. Camb. Phil. Soc. 80, 189-211. PIPKIN, A. C., and SANCHEZ, V. M. (1974). Existence of solutions of plane traction problems for ideal composites. SIAM J. Appl. Math. 26, 213-220. POINCARB, H. (1892). LeGons sur la theorie de I’elasticite.” Georges Carre, Paris. RIVLIN,R. S. (1948a). Large elastic deformations of isotropic materials. Philos. Trans. R. Soc. London,Ser. A 240,459-508. RIVLIN, R. S. (1948b). A uniqueness theorem in the theory of highly elastic materials. Proc. Cambridge Philos. SOC.44, 595-597. RIVLIN,R. S. (1966). The fundamental equations of nonlinear continuum mechanics. In “Dynamics of Fluids and Plasmas” (S. I. Pai, ed.), pp. 83-126. Academic Press, New York. RIVLIN,R. S. (1968). The formulation of theories in generalized continuum mechanics and their physical significance. 1st. Naz. Alta Mat., Symp. Math. 1, 357-373. RUND,H. (1966). “The Hamilton-Jacobi Theory in the Calculus of Variations.” Van NostrandReinhold, Princeton, New Jersey. ST.-VENANT, A,-J.-C. B. de (1956). Memoire sur la torsion des prismes. Mem. Diuers Savants Acad. Sci. Paris 14, 233-560. K., and RIVLIN,R. S. (1974). Bifurcation conditions for a thick elastic plate under SAWYERS, thrust. I n t . J . Solids Struct. 10, 483-501. E. (1972).Basic phenomena of the influence of ultrasound on material properties. Proc. SCHMID, Int. Symp. High-Power Ultrason., lst, 1970, pp. 6-10. SEITZ,F. (1934). A matrix-algebraic development of the crystallographic groups. I. Z. Kristallogr., Kristallgeom., Kristallphys., Kristallchem. 88, 433-459. SEITZ,F. (1935a). 11. Z. Kristallogr., Kriitallgeom., Kristallphys., Kristallchem. 90,289-313. SEITZ,F. (1935b). 111. Z. Kristallogr., Kristallgeom., Kristallphys., Kristallchem. 91, 336-366. SEITZ,F. (1936a). IV. Z. Kristallogr., Kristallgeom., Kristallphys., Kristallchem. 94, 1W130. SEITZ,F. (1936b). On the reduction of space groups. Ann. Math. 37, 17-28. SEITZ,F.(1940). “The Modern Theory of Solids.” McGraw-Hill, New York.

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SEWELL, M. J. (1965). On the calculation of potential functions defined on curved boundaries. Proc. R. SOC.London, Ser. A 286,402-4 11. SEWELL,M. J. (1967). O n configuration-dependent loading. Arch. Ration. Mech. Anal. 23, 327-351. SHIELD,R. T. (1967). Inverse deformation results in finite elasticity. 2. Angew. Math. Phys. 18, 4w500. SHIELD,R. T. (1971). Deformations possible in every compressible isotropic perfectly elastic material. J. Elasticity 1, 91-92. A. J. M. (1972). Deformations of Fibre-reinforced Materials.” Oxford Univ. Press, SPENCER, London and New York. STAKGOLD, I. (1949).The Cauchy relations in a molecular theory of elasticity. Q. Appl. Math. 8, 169- 186. STERNBERG, E., and KNOWLES, J. K. (1966). Minimum energy characterizations of St.-Venant’s solution to the relaxed Saint-Venant Problem. Arch. Ration. Mech. Anal. 21, 89-107. THOMPSON, J. L. (1969). Some existence theorems for the traction boundary value problems of linearized elastostatics. Arch. Ration. Mech. Anal. 32, 369-399. TOUPIN,R. A. (1965). Saint-Venant’s principle. Arch. Ration. Mech. Anal. 18, 83-96. TRUESDELL, C. (1956). Das Ungeloste Hauptproblem der endlichen Elastizitats-theorie. 2. Angew. Math. Mech. 36, 97-103. TRUESDELL, C. (1966). “Six Lectures on Modern Natural Philosophy,” Lect. 11. Springer Publ., New York. C., and MOON,H. (1975). Inequalities sufficient to ensure semi-invertibility of TRUESDELL, isotropic functions. J. Elasticity 5, 183-190. TRUESDELL, C., and NOLL,W. (1965). The nonlinear field theories of mechanics. In ” Handbuch der Physik” (S. Fliigge, ed.), Vol. 3, Part 3, pp. 1-590. Springer-Verlag, Berlin and New York. VARLEY, E., and DAY,A. (1966). Equilibrium phases of elastic materials at uniform temperature and pressure. Arch. Ration. Mech. Anal. 22, 253-269. WANG,C.-C. (1965). A general theory of subfluids. Arch. Ration. Mech. Anal. 20, 1-40. WANG,C.-C., and TRUESDELL, C. (1973). “Introduction to Rational Elasticity.” Noordhoff Int., Groningen, Leyden. YOUNG, L. C. (1969). “Lectures on the Calculus of Variations and Optimal Control Theory,” Vol. 11. Saunders, Philadelphia, Pennsylvania. “

On Nonlinear Parametric Excitation Problems C. S. HSU Department of Mechanical Engineering University of California. Berkeley. California

I . Introduction . . . . . . . . . . . . . . . . . . . . . . . I1. Asymptotic Analysis for Weakly Nonlinear Systems . . . . . . . . . A. Nonlinear Systems with Single Degree of Freedom . . . . B. Parametrically Excited Hanging String in Fluid . . . . . . . I11. Analysis by Difference Equations . . . . . . . . . . . . . . A. Discrete Time Formulation . . . . . . . . . . . . . . . B. Periodic Solutions and Their Local Stability . . . . . . . . . . . C. Bifurcation and Birth of New Periodic Solutions . . . . . D. Backward Images of Periodic Points . . . . . . . . . . . . IV. Second Order Difference Systems . . . . . . . . . . . . . . A. Singular Points of Second Order Linear Difference Systems . . . B. Nonlinear Second Order Systems . . . . . . . . . . . . . C. Separatrices of a Saddle Point . . . . . . . . . . . . . . V. Global Regions of Asymptotic Stability . . . . . . . . . . . . . A . Regions of Asymptotic Stability for One-One Second Order Systems B. Regions of Asymptotic Stability for One-One Higher Order Systems C. Regions of Asymptotic Stability for Systems Not One-One . . . . VI . Impulsive Parametric Excitation . . . . . . . . . . . . . . . . A. Impulsively and Parametrically Excited Nonlinear Systems . . . . B. Linear Systems and Their Stability and Responses . . . . . . . VII . An Example: A Hinged Bar Subjected to a Periodic Impact Load . . . A . Governing Differential and Difference Equations . . . . . . . . B. A Damped but Elastically Unrestrained Bar . . . . . . . . . . C. An Elastically Restrained Hinged Bar . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . .

245 241 241 252 251 258 259 261 265 266 266 270 213 216 216 280 280 283 283 284 286 286 288 296 298

.

I Introduction One of the most interesting areas in the theory of vibrations is that of parametric excitation problems . Systems under parametric excitation can 245

246

C . S. Hsu

exhibit extremely complex behavior patterns and stability character. The unravelling of this complexity is not only a mathematically challenging problem but also has great practical significance. It is, therefore, not surprising that a great deal of work has been devoted to this class of problems. Mathematically one is led to the study of differential equations with time dependent coefficients. Some of the books covering this topic are by Cesari (1970), Hale (1963), Shtokalo (1961), and Yakubovich and Starzhinskii (1975). A basic reference on parametric excitation of elastic bodies is by Bolotin (1964). Other general sources of references in mechanics are by Evan-Iwanowski (1965, 1969), Mettler (1967), Hayashi (1964), and Schmidt (1975). Being essentially based upon the theory of differential equations with variable coefficients, the theory of parametric excitation has, of course, applications in many fields of science and engineering. In mechanics, besides the classical problems of strings, columns, and plates under axial or in-plane oscillatory loads, the parametric excitation problems of shells have attracted increasing attention in recent years. Some of the interest is connected with the dynamic snap-through instability phenomenon. For references on this topic the reader is referred to an article by Hsu (1974a). Some of the more recent contributions to parametric excitation of shells are by Tani (1974, 1976) and Kalnins (1974). More recent papers dealing with beams and columns are by Tso (1968), Ghobarah and Tso (1972), Iwatsubo et d.(1974), and Elmaraghy and Tabarrok (1975). Examples of investigation of parametric excitation in other areas of mechanics are studies by Mote (1968) and Rhodes (1971) for moving strings and bands, by Black and McTernan (1968) for rotating shafts, by Nishikawa and Willems (1969) for satellite attitude stability problems, by'Friedmann and Silverthorn (1974, 1975)for helicopter rotor blade vibration problems, by Houben (1970) for excitation problems of piston engines, and by Bohr, and Herrmann (1974) and Paidoussis and Sundararajan (1975) for pipes conveying fluid with variable flow rate. These citations are to indicate typical applications; they are, of course, far from a complete listing. Most of the studies on parametric excitation deal with linear systems. While a linear analysis does provide us with important information, it by itself is inadequate in many instances. Particularly, when the linearized system is unstable, the linear theory predicts an exponential growth for the motion. However, as soon as an appreciable growth has taken place the nonlinearity will come into play and modify the growth. In order to determine the response of the system under these circumstances a nonlinear analysis is required. Perhaps because even the linearized differential equations with variable coefficients are difficult to analyze, nonlinear parametric excitation problems have not been treated extensively. Besides the problems

Nonlinear Parametric Excitation Problems

247

investigated in the books by Bogoliubov and Mitropolsky (1961) and by Kauderer (1958) we cite here the work by Weidenhammer (1956), Somerset and Evan-Iwanowski (1967), Somerset (1967), Tondl (1970), Haight and King (1971), Evan-Iwanowski et al. (1971), Schwarz (1970), Tso and Asmis (1970), Dugundji and Chhatpar (1970), Ghobarah (1972), Dugundji and Mukhopadhyay (1973), Hagedorn (1968, 1969, 1970a,b), Hsu (1974b, 1975a,b),and Troger and Hsu (1977). In these papers one can also find other references. Most of these articles treat individual problems on a more or less ad hoc basis. It is not our intention here to provide a review of these diversified studies. Rather, we shall discuss in this paper two special topics of nonlinear parametric excitation. One is a review of the results obtained from an asymptotic analysis of weakly nonlinear second order systems. Since for many mechanical systems treatments in terms of normal modes can be effective even for nonlinear problems, we believe that this approach provides engineers with a tool to extract useful information about the response of a system. The second topic is a study of nonlinear parametric excitation problems through the theory of nonlinear difference equations. Here one fortunately finds that when the excitation is periodic and impulsive, the analysis can be carried out without approximation. Such an exact analysis gives us a much more definitive understanding and appreciation of various nonlinear phenomena, such as bifurcation and limit cycle responses. The results obtained in this manner could also serve as a guide to other more complex problems of similar nature. In all cases treated in this paper the parametric excitation is assumed to be periodic in character. 11. Asymptotic Analysis for Weakly Nonlinear Systems

A. NONLINEAR SYSTEMS WITH

SINGLE

DEGREE OF FREEDOM

In this section we examine the response amplitude of a nonlinear system when the linearized system indicates instability. We consider a second order system, or a system with a single degree of freedom, subjected to harmonic parametric excitation

x +c,x

+ c,/x/a sgn x + (6 + E cos t ) ( x + b x 3 ) = 0,

(2.1) where an overhead dot denotes differentiation with respect to time t, c1 is the viscous damping constant, c, the coefficient of a nonlinear damping term proportional to ath power of the velocity with a 2 1, E the strength of the parametric excitation, and p the coefficient of the cubic term in displacement. Such a system with cubic nonlinear displacement term has been discussed by Kauderer (1958). A nonlinear damping proportional to x2X was

C. S. Hsu

248

investigated by Bolotin (1964) and Bogoliubov and Mitropolsky (1961). Our main purpose for presenting this problem is to show that simple formulas are available to evaluate the response amplitude. Moreover, since this system includes the important case of quadratic velocity damping these simple formulas have practical engineering applications. For the analysis of this system we follow essentially the work of Hagedorn (1970b) and Hsu (1975a). When damping is absent and B = 0, one has in (2.1) the Mathieu equation. For that linear case there are instability regions in the &E plane near E = 0 and 6 = (r1/2)~,n = 1, 2, . . . . We now seek to determine how the nonlinearities will limit the growth of the motion in the instability regions. We assume that c,, ca , E , and B are small and of the order of a small parameter p. Let c1 = pc1,

c, = pi?,,

& = p-,

B = pp,

6

=

(n/2)'

+ y,

y = p?.

(2.2) In terms of the barred parameters, (2.1) takes the form j;-

+ pc1 x + ptc. I x 1" sgn x + [(n/2)' + pjj + pz cos t](x + j$x3)

= 0.

(2.3) Adopting the asymptotic method of Bogoliubov-Mitropolsky (1961), we set

+ fl] + p d l ) ( ~8,, t) + p2d2)(u,8, t) + -.., a = p ~ ( ye)u+, 0) + . . ., 4 = p B ( l ) ( a ,0) + p W ' ) ( a , 0) + . . . .

x = u cos[(n/2)t

p ~ ~ ( 2 ) ( a ,

1. Response in the First Instability Region (n

=

(2.4) (2.5) (2.6)

1)

The most important instability region for the Mathieu equation is at n = 1. Carrying out the analysis to the first approximation, we can determine A") and @') and hence find (Hsu, 1975a)

+ E sin 2%- c,bl(a/2)"-'], B = +[2y + E cos 2%+ +pa2],

u = (a/2)[-c,

(2.7)

(2.8) where b, is a mathematical constant whose magnitude is a function of a. For the case a = 2, b, = fn. As we are interested in the eventual steady state motion, we look for stationary amplitude a* and phase angle 8* by setting u = 0 and 8 = 0 in (2.7) and (2.8). Ignoring the trivial solution a* = 0, we are led, after eliminating O*, to [27 + @(a*)']'

+ [cl + c,(a*/2)"-'bll2

= 0.

(2.9) This is a simple equation relating the limit cycle response amplitude to the - E'

Nonlinear Parametric Excitation Problems

249

FIG. 1. Surfaces of the response amplitude a* in the a*-y-e space for various cases. From Hsu (1975a).

five parameters of the system y. E , cl, c,, and b. Among the solutions of (2.9) only the real and positive ones are of interest to us. It is also informative to examine (2.9)in the light of special cases and to exhibit the results in graphical form as in Fig. 1: Case A : c1 = c, = = 0. There does not exist nontrivial stationary 7 I I E ) / 2 the response is unbounded. response. For - le1/2 I Case B : c, = p = 0. There is again no stationary responses. For - (E' - ~ : ) " ~ I / 2y I (E' - c:)"'/2 the response is unbounded. This case shows that linear damping does not limit the growth of the motion due to instability. The limiting factors must come from nonlinearities. Case C : c1 = B = 0. The response amplitude is given by

c,b1(a*/2)"-' = (

-4 ~ ~ ) " ~ .

E ~

(2.10)

250

C . S. Hsu

In Fig. 1C the limiting effect of the nonlinear damping can now be clearly seen. For the important case of quadratic velocity damping (a = 2) one has (2.11) a* = (3X/4C2)(E2- 4y2)1/2. Case D : /? = 0. The stationary amplitude is shown by Fig. 1D and for the case a = 2 is given by a* = (3X/4C2)[(E2 - 4y2)”2- CI].

(2.12)

The surface shown in Fig. 1D is the same one as in Fig. IC but lowered by an amount equal to cl. Case E : c1 = c, = 0. The stationary amplitude is given by (2.13) T 2y). There are two sheets for the amplitude surface as shown in Fig. 1E. When /? < 0, the sheets bend to the right instead. Case F : c, = 0. The stationary amplitude is given by @(a*)’ = f( E

d a*

1

-

+-“E 2 - c

y

271.

(2.14)

There are again two sheets. After having found the stationary solutions from (2.7) and (2.8), one can also examine the stability of the solutions. Such an analysis shows that solutions in Figs. 1C and 1D are stable. In Figs. 1E and 1F solutions are stable on the upper sheets but unstable on the lower sheets. Quantitatively (2.9) is also shown in Figs. 2 and 3 for the case c1 = 0, a = 2, and /? = -4. A second order approximation to the stationary response has also been carried out (Hsu, 1975a). Moreover, both the first and second order approximations have been compared with results obtained from direct numerical integration of the differential equations for the cases c1 = 0, a = 2, /? = 0, and E = $, and f. For E = the agreement is excellent. For E = which represents a very strong parametric excitation the agreement is very good for the second order approximation and is acceptable for the first order result. These comparisons lend confidence to the simple formulas (2.9)-(2.14). 2. Response in the Second Instability Region ( n = 2) The stationary response amplitude in the second instability region has been obtained for systems with c1 = 0, a = 2, and /? = 0. Here a second order approximation is necessary and it yields (Hsu, 1975a) (2.15) With regard to how the stationary amplitude increases with the strength of

Nonlinear Parametric Excitation Problems

B.

h

25 1

Y

a

FIG.2. Response amplitude as a function of ;I/ I E I for different values of c 2 / I E I”*, and as a function of y/c: for different values of E / c ~From . Hsu (1975a).

the parametric excitation, it is interesting to compare (2.11) and (2.15) for the first and second instability regions. Take 6 = 4 and 6 = 1. Each of these values corresponds to the center of an instability region. One finds 1

6=4’ 6 = 1,

cza* c2a* =

3n

IE~,

(2.16)

(~n)’/zcz.

(2.17)

=-

4

32

Thus, at the middle of the first instability region the amplitude increases linearly with E , but at the middle of the second instability region it increases according to E’.

C.S. Hsu

252

1

FIG. 3. Response amplitude as a function of E / I y I for different values of c 2 / I y Hsu (1975a).

11’2.

From

Another interesting feature of the analytic results is that when p = 0, c2 and a* appear together as one parameter. In other words, if other parameters are fixed, the response amplitude is inversely proportional to the coefficient of quadratic velocity damping. This is obviously a significant and useful result so far as applications are concerned.

B. PARAMETRICALLY EXCITED HANGING STRINGIN FLUID 1. Partial Diferential Equation with a Periodic Coejjicient As an application of the results presented in Section II,A we consider the motion of a heavy and inextensible hanging string in fluid when the suspen-

Nonlinear Parametric Excitation Problems

253

sion point moves up and down periodically. This is a basic problem in the theory of vibrations and it also has practical implications because of the increasing use of devices like drill strings and long conveying pipes in ocean engineering. With the support point oscillating vertically, a purely up-and-down motion of the string as a whole is always a possible one. However, the problem being one of parametric excitation, one can expect that when the frequency and amplitude of the motion of the support satisfy certain conditions the vertical and straight configuration will become unstable and transverse vibration can be excited into being. When this happens it is important to determine the amplitude of the eventual steady state response. Consider a string of uniform cross section and material composition along the length. The string is assumed to be entirely submerged in a fluid medium, and the fluid resistance is taken to satisfy a velocity square law. The governing equation for the transverse vibration of such a string is given by

where fis the physical time, w o / g is mass per unit length of the string proper, w , / g denotes all the additional mass per unit length which is also being accelerated in the transverse direction, pr the mass density of the fluid, do an appropriate frontal width of the string, CD the drag coefficient appropriate for the cross section at hand, w1 the weight per unit length of the string in fluid, and w o / g also the mass per unit length being accelerated longitudinally. In (2.18), z ( g is the motion of the support point of the string and will be taken to be

z ( g = zo cos(27r/T,)t= zo cos o,t,

(2.19)

where zo, T , , and o,are, respectively, the amplitude, period, and circular frequency. The axis of the coordinate x points upward with the origin placed at the rest position of the lower end of the string. The length of the string is 1; hence the upper end is at x = 1. Equation (2.18) is a nonlinear partial differential equation with a periodic coefficient. Only nonlinear damping is included here. Thus, we are only seeking to determine the limiting effect of velocity square damping on the nonlinear steady state response. Implicit here is that the transverse motion is not too large so that it is not necessary to include the geometrical nonlinearity of the motion. It is also assumed here that the motion being excited is of such a character that the member may be adequately treated as a string with zero flexural rigidity. For a further discussion on the validity of these assumptions the reader is referred to Hsu (1975b).

C . S. Hsu

254

2. Analysis of the Linearized System Consider first the linearized system by ignoring the nonlinear damping term. Let us express y in terms of the normal modes of a hanging string with its upper end stationary, so that (2.20) where

x , ( x )= J,,(j0.nx1’2/11’z).

(2.21)

Here J o is Bessel’s function of the first kicS 3 order zero and jo,,is the nth zero of J o . Associated with the nth mode X , ( x ) is the nth natural circular frequency o,and period T, given by (2.22) where P1

= Wl/WO

7

Pz = wz/wo.

(2.23)

Substituting (2.20) in (2.18) and making use of the orthogonality condition among modes, one finds that for each mode the motion is governed by

(dzqn/dtz)+ (6,

+

cos t)q, = 0,

E,

n = 1, 2, . . .,

(2.24)

where t = - 2a t -

(2.25)

T,

En = -(;)2($J

Equation (2.24) is the Mathieu equation. A well-known result is that the trivial solution q, = 0 is unstable when

+ - 3I

(2.26) < 6, < + f I E” I . Equation (2.26) represents, to the first order of smallness of E , , the most important instability region near 6 = 4 (Stoker, 1950). Making use of (2.22), (2.23), and (2.25), we may express the instability condition (2.26) as follows E,

1

(2.27)

Here R may be taken as a measure of excitation frequency and

r

as a

Nonlinear Parametric Excitation Problems

255

measure of the strength of the parametric excitation. Equation (2.27) can be used to check whether instability exists and, if it exists, which mode is involved. By using (2.27) one can construct instability regions in the R-T plane, one region for each mode. Figure 4 is such a diagram with the instability regions shaded and the unstable mode number associated with each region labeled accordingly. It is simple to use this diagram. For a given

5

n

FIG.4. Instability regions in the Q-rplane. Numbers 1-7 represent the mode number n in the different regions. From Hsu (1975b).

string one computes o1by (2.22). When the excitation (2.19) is known one calculates R and by (2.28). If the calculated point (R, r)falls in one of the instability regions, then the vertical and straight configuration of the string is unstable and the particular mode of transverse vibration associated with that region will be parametrically excited. 3. Steady State Response

Assume now that the excitation z ( i ) is such that a particular Nth mode is being parametrically excited. In that case the motion of the string is given by

(2.29) where qN(t)grows with time according to the linear theory. As the amplitude increases, the nonlinear damping comes into play to limit the growth. It is reasonable to assume that the mode shape X,(X) will remain adequate to describe the spatial shape of the string during the nonlinear growth of the Nth mode as long as the motion does not become too large. Substituting

C . S. Hsu

256

(2.29) in (2.18), applying the Galerkin method by first multiplying the equation by X,(X) and then integrating with respect to x from 0 to l, and rearranging, one obtains a nonlinear governing equation for q N

1

%l=o,

(2.30)

where (2.31) with 1

B ~ .= J ; ( j o , N ) ,

B3, N

=2

(. IJXjo,

N

~ I S)

ds7

(2.32)

‘0

Here J , is the Bessel function of order one. B2, and B 3 , are mathematical constants and some of them have been tabulated by Hsu (1975b). Equation (2.30) is seen to be of the form (2.1) with c 1 = 0, u = 2, and p = 0. Hence the response amplitude formula (2.11) is immediately applicable, leading to aN

=

(3n/4C2, N)[&; - 4(6N - 51 )211 / 2 .

(2.33)

This formula may be put in a more explicit form as follows

where

The quantity uN is a dimensionless amplitude measured in units of the frontal width d o . Besides the purely mathematical constantsj,. 1 , j o ,N , B 2 ,N , and B3. the right-hand side of (2.34).depends only upon R and r. Therefore, for a given excitation the response amplitude uN is inversely propor~ , may be called an effective damping coefJicient (Hsu, tional to c ~ ( which 1975b). One notes that c ~ (is ~a product ) of two factors. One is CD,the drag coefficient which depends upon the shape of the cross section. The other factor rM is proportional to the ratio of the mass of displaced fluid to the mass of the string that is involved in the transverse vibration. This parameter rMmay be called the displaced mass ratio. From a practical point of view it is of interest to note that the amplitude is inversely proportional to this ratio. Consider, for example, two fluid media, water and air. The corresponding

Nonlinear Parametric Excitation Problems

251

displaced mass ratios differ by a factor of order lo3.The response amplitude can then be expected to differ by the same factor in the inverse way. Formula (2.34) is shown in Fig. 5 for = 0.2 and 0.4.

R FIG. 5. Response amplitude c2,E,aNvs. R curves with

r = 0.4;lower curve, r = 0.2. From Hsu (1975b).

r = 0.2

and 0.4.Upper curve,

As an example of application, consider a steel cable of length 300 m suspended in air. In this case p2 = 0 and PI may be taken to be equal to 1. Suppose zo = 0.4 m, T, = 4.016 sec. Then the linear analysis shows that the third mode will be parametrically excited. Assuming now that the steel cable has a circular cross section with a diameter do = 0.025 m and taking CD = 1.2, one finds from (2.34) a 3 = 34.15 m. Thus, a support motion with an amplitude 0.4 m parametrically excites the third mode of the cable to an amplitude equal to 34.15 m. For the details of calculation involved in this example the reader is referred to Hsu (1975b).

111. Analysis by Difference Equations

In Section I1 we have reviewed an asymptotic method of analyzing certain nonlinear parametric excitation problems. It is seen through an example to be a very effective one in some instances. Nevertheless, the method merely treats some limited aspects of the nonlinear problem, such as steady state responses and their stability. In this and the following sections we present an

C . S. Hsu

258

entirely different approach which hopefully might shed more light on the overall behavior of nonlinear systems under parametric excitation. Here our object is not to search for numerical results of practical problems but rather to gain more insight into the nonlinear features special to this class of systems.

TIMEFORMULATION A. DISCRETE Instead of looking for the continuous time history of motion of a dynamical system, one could seek for the state of the system at a sequence of discrete time instants. This approach is a particularly appropriate one if the system is subjected to a periodic excitation. Consider a dynamical system governed by x ( t ) = F(t, x ( t ) ) , (3.1) where x is a real-valued N vector and F is a real-valued vector function periodic in t of period 1 but in general nonlinear in x. Suppose now that we are able to express the state of the system at the end of a period in terms of the state at the beginning of the period. In that case we can recast the problem in the form of a system of difference equations,

+

x(n 1) = G(x(n)), n an integer, (3.2) where G is a real-valued vector function of x(H). For the purpose of this paper we shall assume G to be continuously differentiable. When the dynamical system is viewed in the context of (3.2) the state vector x is only defined at the integer time instants. The task is to study the solutions of (3.2). After having ascertained the behavior of the system (3.2), one can then go back to (3.1) to obtain the continuous time history of the original system. In general, for a given F, G is difficult to determine or it can only be determined approximately. There are, however, instances where the nonlinear function G can be evaluated exactly. For these cases the new approach then allows us to study the nonlinear problem in a rather complete and clear-cut manner. In the mathematical theory of dynamical systems this discrete time mapping approach may be traced back to Poincare (1881) and Birkhoff (1920). In more recent years the development along this line has taken the form of the theory of dzfleomorphisms; see for instance the work by Amol'd (1963), Smale (1967), Markus (1971), and Takens (1973). The theory of diffeomorphisms is, however, founded on topology and modern differential geometry. Perhaps because of its very abstract nature, it does not seem to be readily available to engineers and, as a consequence, has not been applied to many concrete mechanics problems. In what follows we shall endeavor to carry out a theoretical development of this difference-equation approach in the spirit of the classical theory of nonlinear oscillations. We hope that via this

259

Nonlinear Parametric Excitation Problems

route we will be able to discuss various special features of this class of nonlinear problems more explicitly and in more physical terms.

B. PERIODIC SOLUTIONS AND THEIR LOCALSTABILITY The function G of (3.2) may be viewed as a mapping; (3.2) states that G maps ~ ( n to ) x ( n 1). We denote the component functions of G by g j , j = 1, 2, ..., N . In component form (3.2) becomes

+

xj(n

+ 1) = gj(x,(n),x2(n),. . ., x N ( n ) ) ,

j = 1, 2, .. ., N .

(3.3) The mapping G applied k times, k being an integer, will be denoted by Gk, with Go understood to be an identity mapping. Starting with an initial point x(O), (3.2) maps out a sequence of points Gk(x(0)),k = 1, 2, . . . . This sequence of ordered points will be called a discrete trajectory, or simply trajectory, of the system with initial point x(0). For reasons which will be made clear later sometimes it is important not to look at the complete discrete trajectory in its entirety at once, but to follow points after every K forward steps. Thus, sometimes we consider a sequence of points Gk(x(0))with k = J j K . Here K is a given positive integer, J is a positive integer having one of the values 1,2, . . ., K , and j is a running index j = 0, 1,2, . . . . We call this sequence the Jth branch of a K t h order discrete trajectory. For example, the first branch of a Kth order trajectory consists of points x(l), x(K l), x(2K l), . . ., and the second branch consists of points x(2), x(K 2), x(2K 2), . . ., and so forth. While this nomenclature is somewhat cumbersome, it is descriptive of the usage we have in mind later. We shall also have occasions to use the sequences of points {x(n), n = 0, 2, 4, . . .} and {x(n), n = 1,3, . . .}. These will be called, respectively, the euen and the odd branches of the discrete trajectory. A dynamic equilibrium state, or a fixed point, x* of the difference system (3.2) satisfies

+

+ +

+ +

G(x*). (3.4) There may also exist a solution to (3.2) which consists of a sequence of K distinct points x*(j),j = 1, 2, . . ., K , such that X* =

x*(m

+ 1) = Gm(x*(l)), x*( 1 ) = GK(x*(1)).

m = 1, 2, .. ., K

-

1,

(3.5) Such a solution is a periodic solution of period K . Since we will refer to periodic solutions of this kind time and again, it is convenient to adopt an abbreviated name. We shall call a periodic solution of period K as a P - K solution and any of its elements x*(j), j = 1, 2, . . ., K , a periodic point of

C . S . Hsu

260

period K or, in abbreviation, a P-K point. When viewed in this broader framework, dynamic equilibrium states are simply P-1 points. A remark with regard to (3.5) may be in order here. If one merely solves for the last equation of ( 3 . 9 , then one obtains as real roots of that equation not only all the P-K points but also all the P-L points with L being an integer factor of K. Having determined a P-K solution, its local stability character may be studied in the following manner. Here, stability or instability will be taken in the sense of Liapunov. For the sake of definiteness, let us take x*(l) of the P-K solution under examination as a typical point. To study the local stability consider small perturbations near x*(l) and then follow the Kth branch of the Kth order perturbed discrete trajectory. Let the perturbation vector be denoted by 6 so that x(0) = x*(l) + 5(0), x(mK) = x*(l)

+ g(m),

m = 1, 2, . ...

(3.6)

The perturbed state x((m -t l) K ) satisfies x((m

+ 1)K) = G"(x(mK)),

m = 0, 1, 2, . ...

(3.7)

Substitution of (3.7) into (3.6) leads to x*(l)

+ g(m + 1) = GK(x*(l)+ c ( m ) ) .

(3.8)

Expanding the right-hand side of (3.8) around x*(l) and recalling that x*( 1 ) = GK(x*(l)), one obtains immediately S(m + 1 ) = H W ) + P(S(m)b (3.9) where H is a constant matrix, H((rn) the total linear part, and P(c(m)) is the remaining nonlinear part. The mapping G will be assumed to be such that P satisfies the condition (3.10)

Here we take the norm of a vector to be its Euclidean norm. If we denote the Jacobian matrix of a vector function v(x) with respect to x by J(v(x), x), then the matrix H in (3.9) is with elements hij

= [a(GK(~))i/d~jIx=x*(l) 9

(3.12)

where (G"(x)), denotes the ith component of the vector function C"(x).

Nonlinear Parametric Excitation Problems

26 1

Recalling x ( K ) = G ( x ( K - l)), . . ., (3.13) 4 2 )=W l ) ) and using properties of the Jacobian of a composite function, it is not difficult to show that H may also be put in the form = [J(G(x), x ) ] ~ = x*(K)[J(~(~), x)]~= x*(K-

1)

... [J(G(x), 4 I x = * , u .

(3.14) Consider now first the linear system of (3.9) by deleting the nonlinear term. The stability of the trivial solution 5 ( m ) = 0 for the linear case is entirely determined by the matrix H. For a discussion of the stability criteria the reader is referred to books on linear difference equations or books on sampled-data systems (Jury, 1964).We summarize below some of the results which are useful for our purpose. ( 1 ) The trivial solution of the linear system is asymptotically stable if and only if all the eigenvalues of H have absolute values less than one. (2) The trivial solution is unstable if there is one eigenvalue of H that has absolute value larger than one. (3) If there are eigenvalues of H with absolute values equal to one but they are all distinct and all other eigenvalues have absolute values less than one, then the trivial solution is stable but not asymptotically stable. (4) If ,Ii is a multiple eigenvalue with absolute value equal to one and if all the other eigenvalues have either absolute values less than one or have absolute values equal to one but all distinct, then the trivial solution is stable if the Jordan canonical form associated with the multiple eigenvalue ,Ii is diagonal, otherwise the solution is unstable. With regard to the eigenvalues of H, it might be worthwhile to recall that if a matrix is the product of several matrices, then the eigenvalues of the matrix are unchanged by permuting the factor matrices provided that the cyclic order is preserved. This means that different H s of (3.1 1 ) evaluated at different P-K points of a P - K solution will all have the same eigenvalues and, therefore, all the P - K points will have the same stability character. This is, of course, as it should be. When there is stability but not asymptotic stability we have the critical cases. Except for the critical cases the local stability character carries over when one goes from the linearized system to the nonlinear one.

c.

BIFURCATION A N D

BIRTHOF NEWPERIODIC

SOLUTIONS

1. P - K Solution into P - K Solutions One of the most interesting phenomena for nonlinear systems is that of bifurcation. The nonlinear difference systems are no exceptions. Consider a

C. S. Hsu

262

difference system which depends upon a parameter s

+

~ ( n 1) = G ( x ( ~ )s). ;

(3.15)

Let x*(j; s), j = 1, 2, . . . , K, be the elements of a P-K solution for a given value of s. One may pose the following question: As one changes s, at what value of s, say sb if any, a bifurcation from this P-K solution will take place so that new periodic solutions are brought into existence? It is obvious that any of the P-K points is a point of intersection of the following N hypersurfaces of dimension N - 1

S,(x; s)

x, - (GK(x;s)), = 0,

i = 1, 2, ..., N.

(3.16)

We shall assume that the nonlinear function G(x; s) is such that these surfaces are smooth at any point of intersection. Consider first an isolated point of intersection. In that case the surfaces intersect “transversely” and the gradient vectors to these N hypersurfaces at the point of intersection form an independent set. A bifurcation is possible when there are multiple roots for the system of equations (3.16). A necessary condition for this is that the intersection of the surfaces is not transverse or, equivalently, the gradient vectors to the surfaces at the point of intersection are no longer linearly independent. This leads to the condition -, ,0~,) det(I - J(GK(x;s), x ) ) ~ = ~ * ( -

(3.17)

where I is a unit matrix and x*(l; s) is taken to be the point of intersection. In view of (3.11), one may also express this condition as det(I - H(s)),= a(1 , s )

= 0.

(3.18)

Since det(1 - H(s)) = (1 - A N ) ( l - A N - l ) ... (1 - A1),

(3.19)

where A,, A 2 , . . ., AN are the eigenvalues of H(s), (3.18) is seen to be equivalent to requiring one of the eigenvalues of H(s) be equal to one. Thus, as one varies s one can expect bifurcation from this P-K solution into additional P-K solutions when one of the eigenvalues of H calculated for the original P-K solution moves across the value of 1. One notes here that the condition discussed above is a pointwise condition along the s axis. The direction of change of s is not immediately involved in the consideration. A bifurcation can take place when s increases across sb or when it decreases across it. In the latter case a reversal of bifurcation phenomenon takes place as s increases, i.e., merging of several P-K solutions back into a single P-K solution. In (3.17) x*(l; s) is taken as the point of intersection. Based upon the discussion given near the end of Section III,B, it is evident that any element

Nonlinear Parametric Excitation Problems

263

of the P-K solution may be used. If the bifurcation condition is met at one of the P-K points then it is met at all the other P-K points of the same P-K solution. This is a consistent result and also one to be expected. 2. Orientation of the Set of Gradient Vectors Condition (3.17) or (3.18) can be given additional geometrical interpretation which will be found useful later. When the hypersurfaces Si(x;s) = 0 intersect transversely, the N gradient vectors to the surfaces form a linearly independent set and this set can be assigned an orientation. The gradient vector to Si(x;s) has a S i / a x j as its jth component. We now define the orientation of the set as positive or negative according to whether det(dSi/dxj)= det(1 - J(GK(x;s), x)) 2 0.

(3.20)

Condition (3.17) implies that when the set of gradient vectors at a P-K point changes its orientation, then a bifurcation may take place. By (3.19) this orientation is also given by the sign of (1 - AN)(l - A N - ... (1 - Al). Let us examine this expression more closely. There are N factors, one for each eigenvalue. The complex eigenvalues appear in a conjugate pair, say A j = a p i and Aj+ = a - pi. The corresponding factors obviously form a product that is positive

+

(l-Aj)(l-Aj+l)=(l

(3.21)

-a)2+p>o.

If a P-K point is asymptotically stable, all the eigenvalues have absolute values less than one. The factor (1 - A) associated with each real eigenvalue is then positive. Combining this result with (3.21), one finds that at an asymptotically stable P-K point the orientation of the set of gradient vectors must be positive. At a point of bifurcation the orientation of the gradient vectors is not defined.

.

3. P-K Solution into P-2K Solution Next, we examine the possibility of bifurcation from a P-K solution into a P-2K solution. A P-2K point is a point of intersection of the hypersurfaces

Si(x;s)

xi - ( G 2 K ( s~);) ~= 0,

i = 1, 2,

..., N.

(3.22)

The P-K point from which this bifurcation takes place is, of course, also periodic of period 2K and, therefore, also a point of intersection of these surfaces (3.22). Evidently a condition for this new kind of bifurcation is again that the set of gradient vectors at the P-K point, say x * ( l ; s), should be no longer linearly independent. This leads to s), det(J - J(GZK(x;

x*(1; s)

- 0.

(3.23)

C. S. Hsu

264

Since x*( 1 ; s) = G'(x*( 1 ; s); s), (3.23) may be written as (3.24)

det(1 - J2(GK(x;s), x ) ) , = , * , ~ :=~ 0, )

or, after using (3.11), det(1 - H2(s)) = det(1 - H(s)) . det(1

+ H(s)) = 0.

(3.25)

The condition det(1 - H(s)) = 0 leads to bifurcation from P-K to P-K. Hence, a condition for bifurcation from a P-K solution to a P-2K solution is det(1

+ H(s)) = (1 + &)(l +

... (1 + A l )

= 0.

(3.26)

This is evidently equivalent to requiring one of the eigenvalues of H(s) to have the value - 1. Thus, as one varies s,when one of the eigenvalues of H(s) associated with the P-K solution changes across the value - 1, then a bifurcation from this P-K solution into a P-2K solution is possible. One can again examine the set of gradient vectors to the surfaces (3.22) [not (3.16)] at the P-K point concerned. The orientation of this set may be defined as before and is to be governed by the sign of - A;-

det(1 - H2(s))= (1 - A:)(l

... (1 - A:).

(3.27)

By the same reasoning one finds that if the P-K solution is asymptotically stable then the orientation of the set of gradient vectors at any of its P-K points is positive. When a bifurcation from this P-K solution into a P-2K solution takes place, then the orientation of gradient vectors at the P-K points will, in general, change sign.

4. P-K Solution into P-MK Solution

In a similar manner one can investigate the condition for a P-K solution bifurcate into a P-MK solution, where M is an integer. Generalizing the treatment given above for M = 1 and 2, one finds the condition to be to

det(1 - HM(s))= (1 - #)(1 -

... (1 - Ay) = 0.

(3.28)

Condition (3.28) requires one of the eigenvalues of H(s) associated with the P-K solution to satisfy

ny = 1

(3.29)

or

A j = exp(2npi/M),

p = 1, 2, . . ., M - 1.

(3.30)

Here p = M is deleted because it leads to bifurcation into other P-K solutions. As a matter of fact, all p values for which p / M is not an irreducible

Nonlinear Parametric Excitation Problems

265

rational number should be deleted. Let us assume M = M’Q and p = p‘Q where M‘, Q, and p’ are positive integers such that p’/M‘ is an irreducible rational number. Then a combination of such p and M will merely lead to bifurcation into a P-M’K solution instead of a P-MK solution. Consider now a unit circle in a complex plane with its center at the origin. One may locate the eigenvalues of H associated with a P-K solution on this complex plane. When one varies the parameter s, one obtains N loci of the eigenvalues. Whenever one of the loci meets the unit circle at an polar angle equal to 2n times an irreducible rational number p/M, then there is a possibility of bifurcation into a P-MK solution. One also notes here that the bifurcation conditions require one of the eigenvalues to take on an absolute value equal to unity. Recalling that stability and instability of a P-K solution also depend upon the absolute values of the eigenvalues, one can expect that if a bifurcation takes place and if before bifurcation the P-K solution is stable, then after bifurcation the original P-K solution will, in general, have one or more eigenvalues with absolute values larger than one and become unstable.

5. Birth of New Periodic Solutions Not from Bifurcation Besides the phenomenon of bifurcation which can bring in periodic solutions, new periodic solutions can also appear suddenly as the parameter s varies. Geometrically, this corresponds to the occurrence of an intersection of the surfaces (3.16) at a certain threshold value of s, say s E . When this intersection first takes place at s = s E , it cannot be a transverse one and the set of gradient vectors cannot be linearly independent. This again leads to the condition (3.18) or (3.19). Consequently, when a new P-K solbtion comes into being at s = sE, the corresponding H(s) matrix must have one eigenvalue equal to unity. D. BACKWARD IMAGESOF PERIODIC POINTS

The mapping G of (3.2) is naturally assumed to be single valued. However, for most physical problems one cannot assume that the backward mapping is also single valued. This limits the applicability of some of the results obtained from the theory of diffeomorphisms. The possibility of the mapping being not one-to-one introduces complexity to the global behavior of the system. Here we discuss one aspect of the complication. Consider a P-K solution with elements x * ( j ) , j = 1, 2, . . ., K. Starting with any particular P-K point x * ( j ) , the forward trajectory merely covers the whole set x*(j l), . . . , x * ( K ) ,x*( l), . . . , x * ( j ) repeatedly. If the mapping is not single valued backward, then there may be backward images of x*(j) other than the

+

266

C . S. Hsu

point x*(j - 1). For convenience, these points will be referred to as points one-step removed from x*(j). The set of all points one-step removed from x*(j), j = 1, 2, . . ., K will be called the set of points one-step removed from that P-K solution. Each point one-step removed from x*(j) has, in turn, its own image points under a backward mapping. These will be called points 2-step removed from x*(j). In this manner one can find points R-step removed from x*(j), R being a positive integer. The union of the sets of points R-step removed from x*(j),j = 1, 2, . . ., K is called the set of points R-step removed from the P-K solution. That set has the property that starting with any point in the set the system becomes locked into that P-K solution after R steps of forward mapping. One is cautioned not to confuse this set with points on a forward discrete trajectory moving toward a P-K point; the existence of the former requires the backward mapping not single valued while the latter does not. In any event, on a forward discrete trajectory moving toward a P-K point it usually takes infinite number of steps to get to that point. IV. Second Order Difference Systems

When the systems are of second order all the system properties described in Section I11 can be discussed in a more detailed manner and be given attractive geometrical interpretation (Panov, 1956; Miller, 1968; Yee, 1975). Second order systems are also significant in their own right, because for many nonlinear Mechanical problems where one particular mode is dominant the problem can often be reduced approximately to the study of a second order system. A. SINGULAR POINTS OF SECOND ORDER LINEAR DIFFERENCE SYSTEMS Consider a linear second order difference dynamical system

where H with elements hi,, i, j = 1, 2, is a real constant matrix. For this system x,(n) = x2(n)= 0 is an equilibrium state and will be referred to in this section as a singular point. In order for (0, 0) to be an isolated singular point, H is assumed to satisfy det(H) # 0 and det(H - I) # 0. With regard to the discrete trajectory discussed in Section III,B, we shall have a special need here for the concept of even and odd branches of discrete trajectories.

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Let I, and I 2 be the eigenvalues of H. If they are real and distinct, then there exists a nonsingular real linear transformation from (xl, x2) to (tl,t2) which transforms (4.1) to

If I, = I, = I and the Jordan canonical form is diagonal, (4.2) still applies with i,, = I, = I. If I, = I, = I and the Jordan canonical form is not diagonal, then there exists a nonsingular real linear transformation that leads to

t,(n + 1) = It,(.), t,(n + 1) = vt,(n) + X , ( n ) ,

(4.3)

where v is a nonzero constant which may be made as small as we wish. If I, and I, are complex, they are complex and conjugate. Let them be a pi. For this case there exists a real nonsingular linear transformation from ( x , , x2) to (u,, u,) which transforms (4.1) to

+ 1) = au,(n) - bu,(n), u,(n + 1) = aul(n) + au,(n).

u,(n

(4.4) The solutions to these difference equations can be easily found. Consider first I, # I, and both real. Let the initial point be (tl(0), <,(O)). Then by (4.2) the solution is given by

tlb) = Gt1(0),

= 4t,(O).

(4.5) If I, = I, = I and the Jordan form is diagonal, the solution is still given by (4.5) but with I, and I, replaced by I. If I, = 2, = I and the Jordan form is not diagonal, the solution is given by

t , ( n ) = I"t1(0), t,(n) = vnni"-'t,(0) + I"tZ(0). (4.6) If 2 , and I, are complex, we introduce polar coordinates r and 8 so that u, = r

cos 8,

u, = r

sin 8,

(4.7)

and also express I, and A, in the form

1, = pe'",

I 2 = pe-'q.

The solution is then given by

r(n) = p"r(O), 8(n) = e(0) + ncp. (4.9) Having obtained the solutions, the geometrical character of the trajectories can be studied and the singular point may be classified in a systematic way, very much like in the case of second order differential equations.

268

C. S . Hsu

1. I , # 1 2 , both real, I, > 1, I, > 1. By eliminating n from (4.2), one finds that the trajectory points are on the curve

(4.10) Here as well as in subsequent discussions, In of a quantity will be understood to mean the principal value. In the present case a. is obviously positive. The singular point is called a node. By (4.2) it is evident that in the 5,-t2 plane the discrete trajectory remains in the same quadrant as the initial point ((,(OX <,(O)). To signify this property the singular point is called a node of the first kind. Moreover, A, and I , both being larger than one, the forward discrete trajectory moves away from the origin. Hence, the singular point is called an unstable node of t h e j r s t kind. 2. I , # I , , both real, 0 < I , < 1, 0 < I 2 < 1. One can show that the discrete trajectory lies on (4.10), that a0 is again positive, and that in the 5,-r2 plane the discrete trajectory remains in the same quadrant as the initial point. However, now I , and I 2 both being less than one, the forward discrete trajectory approaches the origin. Therefore, the singular point is called a stable node of t h e j r s t kind. 3. A, # 1 2 , b o t h r e a l , 1 1 > 1 a n d O < 1 , < 1 , 0 r O < I , < 1 a n d I , > 1 . 1 n this case the discrete trajectory is still on (4.10) except that a,, is now negative. The singular point is called a saddle point and is unstable. The discrete trajectory remains in the same quadrant as the initial point in the 5,-t2 plane; hence, the singular point is called a saddle point of t h e j r s t kind. 4. I , # A 2 , both real but at least one of them negative. The behavior of the discrete trajectories is now different. The consecutive solution points lie in different quadrants of the 5,-t2 plane, in two neighboring quadrants if one of the A's is negative and in two opposite quadrants if both A's are negative. Here it is desirable to consider the even and odd branches of a trajectory separately; each lies in one quadrant entirely. Singular points having nearby trajectories of this nature is called the second kind. It can be shown that the odd and even branches are of such a nature that it is appropriate to call the singular point a node of the second kind if (In l12/)/(1n[ I lI) > 0 and a saddle point of the second kind if (In 1 I , I )/(ln I1 I ) < 0. A saddle point of the second kind is unstable whi!e a node of the second kind may be stable or unstable depending upon whether both I I , I and 1 I, I are smaller than or larger than one. 5. I, and I 2 complex and conjugate. By (4.9) the discrete trajectory lies on a spiral curve r(n) = r(0)pt@n)-W)l/w.

(4.11)

If p > 1 the forward trajectory spirals outward and the singular point is an

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unstable spiral point. If p < 1 the trajectory spirals in and the singular point is a stable spiral point. If 1, and A, are complex and p = 1, then r(n) = r ( 0 ) and the singular point is called a center. 6. In a similar manner we can treat the case l1 = ;L2 = A. I f 1 > 0 but not equal to one, the singular point is a node of the first kind. If 1 < 0 but not equal to negative one, then the singular point is a node q f t h e second kind. If A = - 1, it is a center.

The dependence of the character of a singular point upon the system parameters may be expressed in another way. Let A = trace H,

B

H.

= det

(4.12)

Then the A-B parameter plane may be divided into a number of regions according to the character of the associated singular point. Figure 6 shows such a diagram.

'1 I

SADDLE OF THE SECOND KIND

UNSTABLE

/

SECOND KIND

+ -2

-1 SADDLE OF THE SECOND KIND

-2

t

-.

I

-3

SADDLE

(

Y;RI:i! UNSTABLE NODE OF THE

STABLE SPIRAI

I

T-

UNSTABLE

-.SPTRAI .

B

"-'

3

UNSTABLE NODE OF THE SECOND

SAD THE SECOND

FIG.6. Dependence of the character of a periodic solution on A and B.

So far as the question of stability is concerned, the results may also be summarized as follows: If A 1 # A, or A 1 = 1, = 1and the Jordan canonical form is diagonal, the singular point is stable if and only if A 1 and 1, both have modulus less or equal to one. If A1 = 1, = A and the Jordan canonical

C . S. Hsu

270

form is not diagonal, the singular point is stable if and only if A has modulus less than one. The singular point is asymptotically stable if and only if both 2, and A, have modulus less than one. In term of A and B of (4.12) we have the well-known result (Cheng, 1959)that the singular point is asymptotically stable if and only if IBI<1,

1-A+B>O,

l+A+B>O.

(4.13)

+

The three lines B = 1, 1 - A + B = 0 and 1 + A B = 0 are marked in Fig. 6. The region of asymptotic stability is the triangle inside these three straight lines. It is useful to note here that on CD, A1 and A, are complex conjugate with modulus equal to I, on 1 - A B = 0 one of the A's has a value equal to 1, and on 1 A + B = 0 one of the A's is equal to - 1.

+

+

B. NONLINEAR SECOND ORDERSYSTEMS 1. Singular Points and Periodic Solutions Next, we consider second order nonlinear systems x l ( n + 1) = h l l x l ( n ) + h , z x , ( n ) + P l ( x l ( n ) ,xzb)), x2(n + 1) = h z , x , ( n ) + h,,x,(n)

+ P z ( x , ( n ) ,x2(n)),

(4.14)

where P , and P , represent the nonlinear part of the system and are assumed to satisfy the condition (3.1). With regard to the classification of the singular point (0, 0) of this nonlinear system, one can follow the same kind of trajectory analysis and define the singularity accordingly. On this point there is a theorem (Panov, 1956) which states that if (0, 0) of (4.1) is a spiral point, a node, or a saddle point of the,first or second kind, then (0, 0) of (4.14) remains to be a spiral point, a node, or a saddle point of the first or second kind, respectively. It should be pointed here that while the analysis given in Section IV,A with regard to the character of singular point and trajectories nearby is explicitly done for an equilibrium point, it is, of course, applicable to any P-K point. It is only necessary to identify (4.14) and (4.1) with (3.9) and its linearized counterpart. So far as the nearby discrete trajectories are concerned, one should look at every Kth trajectory point, i.e., at a branch of the Kth order discrete trajectory. 2. Bifurcation and Geometric Visualization The conditions for bifurcation developed in Section II1,C can be applied to second order systems. Again, because of the simpler nature of second order systems more elaboration can be made. For a P-K solution one can

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271

first evaluate A and B of the associated matrix H according to (4.12). This locates a point in the A-B plane (see Fig. 6 ) . As one varies the parameter s, one obtains a locus in this plane. For second order systems the conditions (3.18) and (3.26) can be easily shown to be equivalent to 1 - A + B = 0 and 1 + A B = 0, respectively. Hence, whenever the locus of the system meets the straight line 1 - A B = 0, a potential bifurcation into other P-K solutions exist. Similarly, when the locus meets the straight line 1 A B = 0, a bifurcation into a P-2K solution is possible. Bifurcation into a P-MK solution is possible if the locus meets the line B = 1 at

+

+

+ +

A = 2 C O S ( ~ T C ~ / M p) , = 1, 2, . .., M - 1,

(4.15)

where p and M are relative primes. It is interesting to observe here that if for the P-K solution under examination the value of B, which is equal to det(H), is different from 1, then the locus cannot meet the line B = 1 and no bifurcation into P-MK solutions with M > 2 is possible. Later in an example we shall see this important and useful facet manifested clearly. So far as the hypersurfaces (3.16) or (3.22) are concerned, they become merely two curves in the x1-x2 plane for second order systems. Geometrical properties, such as transverse intersection and linearly independency of gradient vectors, are now easy to depict. Many stability results can be gleaned simply by geometric observation. Consider, for example, the orientation of the gradient vectors. Previously in Section III,C, it is shown that if a P-K point is asymptotically stable then the orientation of the gradient vectors must be positive. We can now show that for second order systems if the orientation is negative then the P-K solution is an unstable one. This conclusion follows immediately from the fact that a negative orientation means det(1 - H ) < 0. This, in turn, means 1 - A B < 0. By Fig. 6 this condition leads to only unstable singular points. Next, consider the process of bifurcation. Figure 7A shows a typical situation before bifurcation. There is a transverse intersection of the curves S , = 0 and S , = 0 at point A , , say a P-K point. Let v1 and v2 be, respectively, the gradient vectors to the two curves at A , . When the P-K solution is asymptotically stable the set of v1 and v, has a positive orientation. This is the.case shown in Fig. 7A. Consider now a bifurcation from this P-K solution to other P-K solutions. At bifurcation the curves intersect with a common tangency. The vectors v1 and v, then coincide in direction as shown in Fig. 7B and they are not linearly independent. Immediately beyond bifurcation the gradient vectors at A , change their orientation to negative as indicated in Fig. 7C. The old P-K solution necessarily becomes unstable. Let us examine next the two newly bifurcated P-K points A and A , . As can be seen in Fig. 7C, the orientation of the gradient vectors is positive at both points. Refer now to Fig. 6. Noting that the bifurcation takes place at a point

+

C . S. Hsu

272

(A)

(B)

(E)

(F)

FIG.7. Geometrical visualization of some processes leading to birth of new periodic solutions.

on ED (point D excluded) and that the region below ED has a positively oriented set of gradient vectors and represents stable singular points, one concludes immediately that the new P-K points A, and A 2 are asymptotically stable ones. On this point see also Hsu (1976). Similarly, by examining the orientation of the gradient vectors to the curves of (3.22) and making use of Fig. 6, one can show that if a bifurcation into a P-2K solution takes place from an asymptotically stable P-K solution, then after bifurcation the new P-2K solution is asymptotically stable but the original P-K solution becomes unstable. This method of orientation test of the gradient vectors can also be used to study the stability of a new pair of P-K solutions suddenly born at s = s B . Figure 7D shows that S , = 0 and S2 = 0 have no intersection. In Fig. 7E, s = sB and the curves intersect with a common tangency. At that instance, as discussed in Section III,C, one of the eigenvalues of H associated with the point of intersection has the value 1. One also notes that the system at s = sB is represented by a point on the line 1 - A + B = 0 in Fig. 6. For s slightly larger than sB there will be, in general, two points of intersection as shown in Fig. 7F. The orientations of the two sets of gradient vectors at A, and A, are evidently opposite in sign. This means that as s increases from sB onward

Nonlinear Parametric Excitation Problems

273

there will be two branches leading away from a point on the line 1 - A + B = 0 to the two opposite sides of the line. From this observation it follows that at most only one of the two new P-K solutions can be stable. It is entirely possible that both will be unstable. OF A SADDLE POINT C. SEPARATRICES

1. Definition

The trajectories of a difference system are by their very nature discrete ones. It is, however, possible to define certain continuous curves which are informative. Separatrices are curves of this kind. Refer now to (4.2) and consider, for definiteness, the case A 1 > 1 and 0 < 1, < 1. The t1axis has the distinction that it passes through the saddle point (0, 0) and that every point on this line will have its subsequent discrete trajectory entirely on this line. The 5, axis has the same property. These lines are called the separatrices of the saddle point and they can be mapped from the t1-t2plane back to the x l - x , plane. For nonlinear difference systems we define a separatrix of a saddle point as a continuous curve in the x l - x , plane having the following properties (Yee, 1975; Hsu et al., 1977a): a. It is a collection of even or both the even and odd trajectories. b. All the initial points of these trajectories are on this curve. c. One of the end points of this curve is the saddle point. In the immediate vicinity of the saddle point these separatrices should have the same slopes as the separatrices of the linearized system. If we attach to each separatrix a direction of forward flow of the discrete trajectories involved, then there are two separatrices approaching the saddle point and two leaving. 2. Mutual Winding of the Separatrices With regard to the separatrices for nonlinear systems, there is a very important and also very interesting feature. This feature is best discussed by looking at an example. Consider x l ( n + 1) = x l ( n ) + 2[xl(n)I2, x2(n

+ 1) = - x l ( n ) .

(4.16)

It is easily shown that this system has a center at (0,0) and a saddle point S at (1, - 1). As a better example, we should use one where there is a stable

2 74

C . S . Hsu

spiral point instead of a center. However, in order to make the example as simple as possible we elect to use (4.16). Plotted in Fig. 8 are a separatrix G’ approaching S and a separatrix G” leaving S . The curve G’ is seen to have a definite direction as it approaches S , but in its backward trace it is found to wind around G“,a separatrix leaving S . As the backward trace approaches S , the winding becomes more violent and, moreover, does not seem to have an end. In Fig. 8 we also show a small beginning portion of G“ leaving S . If we

FIG.8. Winding of the separatrix G‘ around the separatrix G”. From Hsu et a/. (1977a).

continue G” forward, then it is found to come back again toward S but the curve becomes an oscillatory one winding around G‘. Again, the oscillation becomes more intense as G” approaches S in its forward direction and, again, it seems to be a never ending process. We have examined a number of specific nonlinear systems and have always found this mutual winding of the separatrices present. In some cases this mutual winding is a very localized affair but in other cases, such as the one shown in Fig. 8, it extends to a substantial distance away from the saddle point. Also, in some cases the backward trace of G winds more or less around G”,but in other cases where

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275

the second singular point is not a center but a strongly stable spiral point G’ may oscillate about a different curve. This will be seen in a later example. This feature of mutual winding of the separatrices seemed to be first indicated by Arnol’d (1963). It has also been discussed by Laslett (1958) and Gumowski and Mira (1972). Moser has mentioned (Laslett, 1958) that the phenomenon is connected with the nonexistence of integral for the system. Besides the mathematical interest, the phenomenon has a most serious consequence with regard to the global behavior of the system. Consider again Fig. 8. Based upon the concept of separatrix, one might expect that the shaded region “inside” G’ is a global region of stability for the singular point (0,0) in the sense that points initially in the region will have their trajectories remaining in the region. However, since the winding is a never ending one and, moreover, becomes more and more violent, it is impossible even numerically to determine the entire G’. Therefore, there is n o way to delineate the region “inside” G‘ completely. In Fig. 8 the point P o seems to be located in the shaded region, but a backward continuation of G’ beyond what is shown in that figure may very well put Po in the unshaded region. One recalls here that in the theory of second order autonomous differential systems, the separatrices often serve to delineate regions of stability. For difference systems, on account of the complication just discussed, one suspects that separatrices cannot be useful for that purpose to the same extent. One does recognize, however, that in all likelihood the separatrices, even partially determined ones, will form a barrier beyond which the region of stability cannot go. 3. Primitive Separatrices of Negatiue Orders Another useful concept is concerned with the backward images of the separatrices. Consider a P-K point x * ( j ) which is a saddle point. Through this point S pass four separatrices. As we have discussed in Section IILD, this P-K point may have points R-step removed from it, R being a positive integer. Let us designate a point R-step removed from this saddle point S by S - , . Then it is obvious that any separatrix G leaving or approaching S will have R-step removed backward image curves G - leaving or approaching S - , . A separatrix of this kind will be called a primifire separatrix ofnegatire order -R (Yee, 1975; Hsu et al., 1977a). The discrete trajectory with an initial point on S-, will consist of the following points: x(0) is on S - , , x( I ) on a certain S- + x(2) on a certain S-,+ 2 , . . . , and x(n), n 2 R, will stay on a corresponding separatrix through S either going away or approaching S as the case may be. Thus, when there is an unstable P-K solution having saddle points as its elements, there are not only separatrices through all the P-K points but also primitive separatrices of various negative orders distributed throughout the phase plane. In principle, these separatrices taken

,

,

276

C. S. Hsu

together should govern the global behavior of the system, but because of the feature discussed in Section IV,C,2, it is not easy to take advantage of this global property.

V. Global Regions of Asymptotic Stability

When a P-K point is asymptotically stable, it is most desirable to be able to determine a meaningful global region of stability around that point in the state space, meaningful in the sense that the region is not too small to be of no practical value. By the discussion given in Section IV,C,2, separatrices cannot be easily used for this purpose. We offer below a different approach for this task. We shall first discuss how the method is developed for second order systems and then indicate how the theorems can be extended to higher order ones.

A. REGIONSOF ASYMPTOTIC STABILITY FOR ONE-ONE SECOND ORDERSYSTEMS First we study systems of (3.2) for which there is only one point x(n)which is mapped forward into a given point x(n + 1). In other words, the backward mapping G - is also single valued. We refer to these as one-one systems. 1. Regions of Stability for P-I Solutions

Consider a one-one second order system having a mapping G and an asymptotically stable P-1 solution. Without loss of generality we take the origin of the phase plane to be located at this P-1 point. The governing equation for the trajectories is then in the form of (4.14) where the coefficient matrix of the linear system has both eigenvalues with absolute values less than 1. Theorem I . l f (0,0 )is an asymptotically stable P-I point, then there exists a closed curve Co of sujj’iciently small size and encircling (0,0 ) which is mapped by G into a closed curve C , lying entirely inside Co . We shall present here only the proof for the case where the P-1 point is a stable spiral point. The proof for a node of the first or second kind follows the same line of analysis. According to (4.4), for stable spiral point there is a real nonsingular linear transformation from (xl, x2) to ( u , , u 2 ) which transforms (4.14) to u,(n + 1) = w ( n ) - Bu2(n)+ P;(u,(n),~ 2 ( n ) ) , u2(n + 1) = Bu,(n) + w ( n ) + p;(ul(n), ~ 2 ( n ) ) ,

(5.1)

Nonlinear Parumetric E.ucitcition Problems

277

where P‘, and P i are the nonlinear functions P I and P 2 transformed into functions of u I and u 2 . The spiral point being stable, one has 1)’ = a2

+ p’

< 1.

(5.2)

Let us define some two dimensional vectors as follows

P’ = ( P : , P;).

u(n) = ( u l ( n ) ,u2(n)),

5 = (au1(n)- D42(n)- P . l ( . ) + au2(n)).

(5.3)

Then by (5.1) one can show that

D 2 [u:(n + 1) =

+ u:(n + l)] - [u:(n) + u:(n)]

- ( I - p 2 ) 1 1 u ( n ) / I ’ + 2 1 ~ . P I + llP112

-

- ( I - P2)l14n)l12+ 2pllu(n)l/ IIP’II + IlP’1I2.

(5.4)

A condition like (3.10) on P implies that for each E > 0 there exists a 6 > 0 such that I(PII < EIJu(n)llfor llu(n)ll < 6. Applying this to (5.4), one obtains

D 5 -(I = (p

- p2)/l~(n)l+ 1 22PEllu(n)l12+ E21/~(n)l12

+ + l)(p + E

E -

I ) ~ ~ u ( n ) / ~ ’ whenever ,

/lu(n)ll < 6. ( 5 . 5 )

Consider now a circle C , in the ul-u2 plane centered at (0, 0) and having a radius r, equal to IIu(n)ll. Then the corresponding curve C , after one-step forward mapping will lie completely inside Co if we can show (p

+ + I)(p+ E

E

- 1) < 0.

(5.6)

Evidently, for a given p ( < 1) we can choose E < 1 - p so that (5.6) is satisfied. For this E there exists a corresponding 6. The curve C , will be entirely inside Co if we choose r, of Co to be equal to this value of 6. As the transformation from (x,, x2) to (ul, u 2 ) is a linear one, disjointed C , and C , in the u,-u2 plane are transformed into disjointed curves in the x1-x2 plane. Theorem 2. If the diflerence dynamical system is one-one and f a closed curve Co encircling (0,0 ) is mapped b y G into a curve C , which lies entirely inside C , , then the curves Cj, j = . . . , - 2, - 1,0, 1,2, . . . obtained from Co b y mapping Q’form a nested set of disjointed curves with Cj+, lying entirely inside C j . The proof is by contradiction. Assume C2 is not entirely inside C , . In that case there must be a point Q of C , which is either (i) on C , , or (ii) outside C,. For the case (i), this point Q considered as a point on C2 should have its backward image on C , . On the other hand, this point being also on C1must have its backward image on Co . As Co and C , are disjoint, this implies that Q has two distinct backward image points. This contradicts the one-one

278

C . S. H s u

requirement. Similarly, for the case (ii) this point Q considered as a point on C, has a backward image on C,. However, as a point outside C, it must have a backward image outside Co . This again leads to two distinct image points under G - I , contradicting the one-one assumption. Consequently, no point on C , can be on C , or outside C,. C, is, therefore, entirely inside C,. In a similar manner one can show C- to be entirely outside Co . Theorem 2 then follows from induction. Theorems 1 and 2 taken together lead to: Theorem 3. For a one-one difference dynamical system there exist around an asymptotically stable singular point a nested set of disjointed curves C,, j an integer, such that C j + is the image of C, under G and that C j + is inside C,. A region of asymptotic stability of a stable singular point may be defined as a region of initial points whose trajectories converge to that singular point eventually. It is evident that the region inside any C j discussed above one cltains C - cc . qualifies as such a region. Extending j backward to -The region inside C- is then the total region of asymptotic stability of that singular point. A method of constructing global regions of stability is, therefore, as follows: (1) Study the local stability of the asymptotically stable singular point by determining the eigenvalues I 1 and I , . (ii) Pick an E so that (5.6) or some other similar inequality (for stable nodes) is satisfied. (iii) Examine the nonlinear part in order to determine the appropriate value of 6 to be used for (5.5). (iv) In the transformed uI-u2 (or plane take a circle centered at (0,O) with radius ro equal to 6. The closed curve in the x l - x z plane corresponding to this circle is then taken to be as C,, . (v) Map C,, backward to obtain C - C- .. . . In practical application one does not have to follow the above procedure. Any closed curve encircling (0,O) whose backward image lies entirely outside of it qualifies as C, . Theorem 2 guarantees that C j will form a nested set. As an example we consider the system x l ( n 1) = (1 - o ) x z ( n ) (2 - 2 0 r ~ ~ ) [ x , ( n ) ] ~ , x z ( n 1) = - ( I - o ) x l ( n ) . (5.7) The backward mapping can be shown to be single valued. The system is, therefore, one-one. It has a stable spiral point at (0, 0) and a saddle point at (1, - (1 - 0)).Figure 9 is for the case with CJ = 0.1. For a detailed discussion of this figure the reader is referred to the paper by Hsu et al. (1977a). We merely wish to point out here that C- 2 2 is a region of asymptotic stability of substantial size. Also shown in the figure is G , a separatrix going into the saddle point. It is interesting to see G’ to act as a “glove” surrounding the highly oscillatory curve C- 2 2 . This figure also clearly shows that even for a simple system such as (5.7) the region of stability can be of very complex shape. For example, in Fig. 9 points PI,P , , P 3 , and P , are all near to each 07

,,

+ +

+

+

279

Nonlinear Parametric Excitation Problems

FIG.9. Regions of asymptotic stability for the system (5.7) with r~ = 0.1 and the bounding separatrix G . From Hsu et a/. (1977a).

other. Yet P I and P , are in the stability region, while P 3 and P , are definitely not. 2. Regions of Stability for P-K Solutions

In the last section the singular point is taken to be a P-1 point. If the singular point is in fact a P-K point, the three theorems are still valid except that C j + l should be understood to be the image curve of C j under the mapping GK.Let the P-K point be the Jth element x * ( J ) of a P-K solution. Then in constructing a region of stability around x*(J)we should follow one branch of the Kth order trajectory. For a region around x*(J 1) follow the next branch. It is obvious that a region of stability for an asymptotically stable P-K solution will consist of K separate component regions, one around each P-K point.

+

280

c. s. Hsu B. REGIONSOF ASYMPTOTIC STABILITY FOR ONE-ONE HIGHER ORDERSYSTEMS

Consider now a one-one difference system of order higher than 2. Let x * ( J )be the J t h element of an asymptotically stable P-K solution. Then the Jth branch of the Kth order trajectory near x * ( J ) is governed by an equation of the type (3.9). The matrix H will have all its eigenvalues with absolute values less than 1. Moreover, a real nonsingular linear transformation exists and it transforms the linear part of (3.9) into a real canonical form analogous to (4.2), (4.3), or (4.4); see, for instance, the book by Coddington and Levinson (1955). One can then follow the same line of attack to show that in the transformed space there exists a sufficiently small " sphere " encircling the P-K point whose forward image under mapping GKlies entirely inside the sphere. In this manner one can prove that Theorems 1-3 of Section V,A are also valid for higher order one-one systems. C j are now understood to be hypersurfaces of dimension N - 1. C. REGIONSOF ASYMPTOTIC STABILITY FOR SYSTEMS NOT ONE-ONE For systems that are not one-one, the method presented in Section V,A needs modifications. There are several things to consider. First, the multiplicity of the backward image points may vary from point to point. The state space may be divided into a number of regions such that in each region all the points have the same number of backward images. It is possible to have a region in which the points have no backward mapping. Such a region may be appropriately called a " one-stage region " in the sense that the system can only appear in this region once and that if the initial point is not in the region the system cannot get into it in the process of forward mapping. The second thing to observe is that for a system which is not one-one a P-K point may have backward image points other than the periodic points of the same solution. This has been discussed in Section II1,D. To simplify further discussion let us consider an asymptotically stable P-1 point at F , . Let us assume that F , has two backward images and designate them as F - , ( A )and F - , ( B ) (Fig. 10). Here we use the letters A and B to differentiate the two branches of the backward mapping. F , being a P-1 point, one of F - , ( A ) and F - , ( B ) is F , itself. Let that be F - , ( A ) . F - , ( B ) is then a point one-step removed from F , . F - ,(B), in turn, has its own backward images. Let there be two again. They can be designated as F - *(B, A) and F - @, B) and they are points 2-step removed from F,. Continuing in this manner, one can locate points 3-step and 4-step removed from Fo , and so forth. Next, let us turn to the construction of regions of asymptotic stabilit]. Let C, be a closed surface around F, whose backward mapping C - lies entirely

Nonlinear Parametric Excitation Problems

28 1

FIG. 10. Backward images of an asymptotically stable periodic point F , and various backward images of C, .

outside Co . We designate this C - as C - , ( A ) for a reason which will be clear shortly. The regions inside Co and C - , ( A ) are regions of asymptotic stability. However, when Co is mapped backward to C - ,(A), it generates at the same time a closed surface around F _ , ( B ) . This surface will be designated as C - , ( B ) and is shown in Fig. 10. The region inside C - , ( B ) is now a satellite region of asymptotic stability for F , because every point inside C - , ( B ) will be mapped to inside Co after one forward step and subsequently will follow a discrete trajectory inside Co and approach F , eventually. If we map backward one step further, we obtain C - , ( A , A ) around F , and C - , ( A , B) around F - , ( B ) from C - l(A), while from C - , ( B ) we obtain C - , ( B , A) around F - , ( B , A ) and C - , ( B , B) around F - , ( B , B). Two more satellite regions have been generated. In this manner we can construct other C - surfaces. In the process we generate new satellite regions of stability and at the same time enlarge the old satellites and the main region around F , . The enlargement of the regions may eventually cause the satellites to join in with the main region at various stages of backward mapping. To exhibit the features discussed above let us examine the system xl(n x2(n

+ 1) = 0.9X,(n) + 2[x1(n)]’, + 1) = -0.9x1(n) + 0.5xZ(n).

(5.8)

282

C. S. Hsu

-0.6

4

-0.2

0

0: 2

0:4

FIG.11. A region of asymptotic stability inside C - 19 for the system (5.8) and the bounding primitive separatrices of various negative orders. From Hsu et al. (1977a).

This system is not one-one. It can be easily shown that the area above Lo in Fig. 11 is a “one-stage region and every point in the area below Lo has two backward image points. The system has a stable spiral point F , at (0, 0) and a saddle point So (not shown) at (1.310, - 2.358). Both have their respective R-step removed backward image points. In Fig. 11 global regions of stability are shown. At step number - 15 there are a main region around (0, 0) and a satellite inside the curve C- 5(A, . . ., A, B). Other satellites are outside the confine of the figure. At step number - 19, four satellites have joined the main region along L- L- 2 ( A ) ,L-3 ( A , A ) , and L - , ( A , A , A ) . In Fig. 11 Go ”

283

Nonlinear Parametric Excitation Problems

is an approaching separatrix to the saddle point So.Various primitive separatrices of negative orders, G- l(B), G- z ( B , A ) , G- 3(B, A , A ) , etc., associated with Go are also shown. They are seen to form an outer barrier for the regions of asymptotic stability. For further discussion of the figure the reader is referred to the paper by Hsu et al. (1977a).

VI. Impulsive Parametric Excitation

As mentioned at the beginning of Section III,A, for nonlinear parametric excitation problems it is usually difficult to determine the mapping G needed in the difference equation formulation (Pun, 1973; Bernussou et al., 1976). Often G can only be evaluated by an approximate method and it is not always easy to estimate the consequences of the approximation. There is, however, one class of problems for which G can be determined exactly. This class allows us to study various nonlinear phenomena in a context devoid of any uncertainties due to approximation. This is the subject of this section. A. IMPULSIVELY A N D PARAMETRICALLY EXCITEDNONLINEAR SYSTEMS Consider a nonlinear system M

MY

X

+ Dy + Ky + 1f'"'(y) 1 6(t - T m - J ) m= 1

= 0,

(6.1 )

J * - X

where y is an N' vector, M, D, K are N' x N' constant matrices, f'"'(y) is a vector-valued nonlinear function of y. The instant t , at which the mth impulsive parametric excitation takes place is assumed to satisfy the following ordering 0I t , < t 2 < ... < t M < 1. (6.2) Thus, the parametric excitation is periodic of period 1 and is consisted of M impulsive actions within one period. The strength of the rnth impulsive excitation is governed by f(,). Because of the impulsive excitation term in the equation one can expect the velocity y to be discontinuous but the displacement y continuous at t = t,. Integrating (6.1), one finds y ( t , + ) - y ( t , - ) = -M-lf(m)(y(t,,,)).

(6.3)

Here we use + and - behind t , to denote the instant just after and the instant just before t , . Between impulses (6.1) is a linear equation. If we denote 1" = x z

9

4't

= XNfCt

3

(6.4)

C. S. Hsu

284 (6.1) may be written as

where

Denoting @ as the fundamental matrix of the linear equation X = Ax with @(O) = I and letting x(0) be the initial state, we find we can express the solution of (6.5) as follows

0 I t < t,,

x(t) = @(t)x(O), X(t, - ) = @(t,)x(O), X(t,

+)

=X(t,

-)

+ g'l'(x(t,

-)),

x(t) = @ ( f - t,)x(t, +),

t,
x(t) = @ ( t - t M ) X ( t M +),

tM

< t I 1.

(6.7)

The solution can be continued period after period. Telescoping the equations of (6.7) together, we can express the state at the end of a period in terms of the state at the beginning of the period. This leads to the task of determining the state at t = 0, 1,2, . . . and to the nonlinear difference equation (3.2). Once G has been found, the whole development presented in Sections 111, IV, and V can be brought to bear on the problem to obtain in a systematic way the response of the system. Sometimes it is convenient to have t , = 0. In that case ~ ( n will ) understood to be x(n -).

B. LINEARSYSTEMS A N D THEIRSTABILITY A N D RESPONSES Consider the case where f'")(y) is linear in y f(m)

= fbm) + B(m)y.

In that case (6.1) is reduced to a linear equation

= -

M ..

rr

m= 1

j= -m

C fbm) 1 s(t - t ,

-j).

Nonlinear Parametric Excitation Problems

285

If P;), rn = 1, 2, . . . , M , are all zero vectors, then the system is a homogeneous one. For systems of this kind the stability of the trivial state y = 0 has been studied by Hsu (1972). Closed form exact stability criteria have been obtained for several problems. For instance, consider a system with a single degree of freedom and having only one impulse in each period M l l j l +D,ljll

+

I

Kll +Bll

z

1 6(t-t, j=-a

I

- j ) y , =O.

(6.10)

The stability of the trivial solution is entirely determined by the stability chart of Fig. 12 where the parameters h, a, and p are

4 CT

3

A p=02

-I

0 -I

-2 -3 -4

FIG. 12. Stability chart for a single degree-of-freedom system with a single parametric impulse in a period. From Hsu (1972).

If there are two equal and opposite and equally spaced impulses in each period so that the equation becomes

286

C. S . Hsu

the stability of the trivial solution is given by Fig. 13. In both figures the shaded areas are regions of stability for p = 0. For stability charts for other similar problems including systems with two degrees of freedom, the reader is referred to the paper by Hsu (1972) and Hsu and Cheng (1973). It is of interest to note certain features of Figs. 12 and 13 which resemble those of the Mathieu equation.

FIG. 13. Stability chart for a single degree-of-freedom system with two parametric impulses in a period. From Hsu (1972).

If fl$'", m = 1,2,. . . , M , are not all zero, the system is under both parametric and forcing excitation. The response of such a system has been studied by Hsu and Cheng (1974).

VII. An Example: A Hinged Bar Subjected to a Periodic Impact Load A. GOVERNING DIFFERENTIAL AND DIFFERENCE EQUATIONS Hsu et al. (1977b) have considered the nonlinear plane motion problem of a rigid bar hinged at one end and subjected to a periodic impact load at the other end. Let us denote the hinged point by A and the moment of inertia of the bar about A by I , . Let the bar be restrained at the hinged end by a linear rotational spring of modulus k and by a linear rotational damper with a damping constant b. Let cp be the angular position of the bar with cp = 0 denoting the natural equilibrium position of the bar. The bar is subjected to a periodic impact load Po at the free end. The load has a fixed direction

287

Nonlinear Parametric Excitation Problems

which coincides with the direction cp = 0. If 1 is the distance between the hinged point and the point of application of the load and ro is the period of the impact load, the equation of motion is given by d2cp IAT + b -dcp dt dt

+ kcp +

m

1 d(F-jro)

Pol

j= -m

I

sin cp = 0,

(7.1)

where f is the physical time. The problem is seen to be one of the kind discussed in Section VI. Introducing -

t

= t/ro

,

p

u = l'olTO/IA,

X I

krillA,

= br0/2IA ,

0;=

=~p,

x 2 = dq/dt

one can rewrite (7.1) in the form of (6.5) with M

=

1, t ,

=0

=

4,

and

0

(7.3)

- u sin x 1

The fundamental matrix @(t)with @ ( O )

cos wt @(t)= e - p t [-(w

=

I is 1 . -sin wt

' + P sin wt -

+ $;sin

wt

w

cos w t

-

-sin P ' wt w

where 0 2 = w;

(7.2)

- p2.

I

9

(7.4)

(7.5) Here the reader is alerted to note that u and p used here are different from those used in Section VI,B. The nonlinear mapping G can now be obtained by following (6.7). This yields

+ 1) = s l ( x l ( n ) ?x2(n)) w

x2(n + 1) = g2(x,(n), X

h ) )

+ (C -;sjx2(n)J. where E

= eKp,

C

= cos w,

S = sin w.

C . S. Hsu

288

The Jacobian matrix J(G(x), x ) is given by P c + -s w

J(G(x), X )

as - -cos w

XI,

=E

(7.8) w

B. A DAMPEDBUT ELASTICALLY UNRESTRAINED BAR Consider first the case where the spring is absent, k takes the form

x2(n

+ 1) = g,(x,(n), x 2 ( n ) )

-e-’”a

= 0. In

sin x l ( n )

this case (7.6)

+ e-+x,(n),

(7.9)

and (7.8) becomes

For this case without spring, the parameter a may be taken positive. A negative a means an impact loading pointing to the negative direction of the 4 = 0 axis. In that case we can use + 1 = + 7c as the angular displacement. In terms of 4, the equation of motion will be of the same form as (7.1) but with a positive. 1. Periodic Solutions and Their Stability

Having obtained G in explicit form, one can then follow the analysis given in Section III,B, to find various periodic solution and to study their stability. We present some of the results below. a. P-1 Solutions. The P-1 solutions are given by x:( 1) = rnz, x t ( 1) = 0,

rn an integer,

(7.11)

With regard to the stability, the solution with rn odd and a > 0 is always

Nonlinear Parametric Excitation Problems

289

unstable. For even m and with p > 0 the solution is

When p

=0

asymptotically stable for

0 < a < 4p Coth p,

unstable for

a > 4p Coth p.

(7.12)

and m even, the solution is stable for

O
unstable for

a > 4.

(7.13)

This solution will be designated by the symbol (P-1, m). In Fig. 14 which is for p = 0, (P-1, 0) is on the a-axis. It is stable up to a = 4, at B,. In this figure and subsequent ones stable periodic solutions are indicated by solid lines and unstable ones by dotted lines. In Fig. 15 the case p = 0.171 is shown. Here (P-1, 0) is stable for a up to 4.13074 at B,. b. P-2 Solutions. There are two types of P-2 solution. For the Type I solutions, x:( 1) is determined from 2p Coth p [2mn - 2x:(l)]

+ a sin x:(l)

= 0,

m an integer, (7.14)

and x:(2), xf(l), and xf(2) are given by xT(2) = 2mn - x:(l), xf(1) = -xf(2) = [ 4 ~ e - ~ ” / ( 1e-‘”)][x:(l)

- mn]. (7.15) This solution will be designated by (P-2, I, m). When p > 0 it is asymptotically stable when 0 < a cos x:( 1) < 4p Coth p

and unstable when a cos x: < 0

or

a cos x:( 1) > 4p Coth p.

(7.16) (7.17)

Stability condition for the case p = 0 is easily deduced from (7.16). The solution (P-2, I, 0) is shown in Figs. 14 and 15. In both cases it is seen to be bifurcated from the (P-1, 0) solution at B, and it becomes unstable beyond points B , and B 3 .There are two branches of (P-2, I, 0) starting from B,. Since a P-2 solution has two vector elements, either branch can be taken to be x:( 1) and the corresponding point on the other branch is then x:(2); i.e., the two branches taken together make up this P-2 solution. The Type I1 of the P-2 solution has x:(l) determined by a sin x:( 1) = 2mnp Coth p,

m odd,

(7.18)

x f (1) = - xf(2) = 2nmpe- ’”/( 1 - e-’”)

(7.19)

and x:(2) = x:(l) - mn,

m odd,

C. S. Hsu

290

I

(P- 3, II , O ) 3E/

.FF

\ ___-_-_--_----(P-l,-l)

FIG. 14. Various P-K solutions for the case wo = 0, p

=

0. From Hsu et al. (1977b).

This solution to be designated as (P-2,11, m)is shown in Figs. 14 and 15 with rn = ? 1. If we take a point on branch A, as x:(l) then the corresponding point in branch A , gives xT(2) with rn = 1. If branch A, is taken to be xT( 1) then branch A, gives xT(2) with m = - 1. In a similar way branches A, and A, form another P-2 solution of Type 11. Type I1 solutions are seen to be bifurcated from (P-2, I, 0) solution at B , and B , . They are stable at first but become unstable beyond CI =

+

+

[2p/(1 - e-2P)][n2(1 e - 2 P ) 2 2(1 + e-4P)]1'2.

(7.20)

Nonlinear Parametric Excitation Problems

29 1

- -7l-FIG. 15.

Various P-K solutions for the case oo= 0, p = 0.17~.From Hsu et al. (1977b).

At this value of solutions.

c1

the (P-2, 11, k 1) solution bifurcates into stable P-4

c . P-3 Solutions. For periodic solutions of higher periods we shall not present here any detail formulas or analysis except to offer a few curves of P-3 solutions. There are again several types. Some of them are shown in Fig. 14 for p = 0 and in Fig. 16 for p = .002n. In Fig. 14 we wish to single out the (P-3, I, 0) and (P-3,111,0) solutions which are seen to be bifurcated from the (P-1, 0) solution at c1 = 3. There are four P-3 solutions represented

292

C . S. Hsu

FIG. 16. P-3 solution curves in the x:(l) - a plane for the case oo= 0,p Hsu er a/. (1977b).

= 0.00271. From

by these bifurcated branches; some of the branches are actually double ones. Of the four bifurcated solutions two are unstable, and the other two are stable at first but become unstable at a = 4.0965. At this point bifurcation into P-6 solution takes place. For the case p = 0.002n these branches deform to the curves as shown in Fig. 16. Here there is no longer bifurcation. The solutions appear suddenly at a = 3.328. There are, however, still two unstable solutions represented, respectively, by the group of C , , C 2 , C3 branches and the group of C , , C 5 ,C 6 ,and two stable solutions represented

Nonlinear Parametric Excitation Problems

293

by G I , G , , G 3 and G4, G5,G 6 , respectively. These stable solutions are now asymptotically stable, but they become unstable at u = 4.098. 2. Bifurcation

Two special features of bifurcation are worth mentioning here. First we have observed that (P-1,O)bifurcates into (P-2, I, 0) which in turn bifurcates into (P-2, 11, 1). These solutions then bifurcate into P-4 solutions. As u increases the bifurcation process continues and P-8, P-16, . . . solutions are brought into existence. Moreover, new bifurcations take place at smaller and smaller increments of u. For instance, for the case p = 0.0272 there is an asymptotically stable P-2 solution at a = 6.6000, an asymptotically stable P-4 solution at u = 6.6300, an asymptotically stable P-8 solution at u = 6.6430, an asymptotically stable P-16 solution at u = 6.6457, and an asymptotically stable P-32 solution at u = 6.6460. This feature is similar to the behavior of difference systems discussed by May (1974) and by Hsu and Yee (1975). The second feature concerns the bifurcation from the (P-1, 0) solution to a P-M solution, M # 1. We have seen that in the case p = 0 bifurcation to P-2 takes place at u = 4, to P-3 at u = 3. As a matter of fact, using (3.30)for this problem, one can easily show that bifurcation to a P-M solution takes place at u = 2[ 1 - cos(272p/M)],

(7.21)

where p = 1, 2, . . ., M - 1 but is relative prime to M. However if p # 0, the determinant of H is different from 1 and the two eigenvaluescannot be of the form (3.30).Therefore, bifurcation into P-M solution with M # 2 will not be possible. This is exhibited by the disappearance of bifurcation at u = 3 in Fig. 16. 3. Global Regions of Asymptotic Stability

Consider the case u = 3.5 and p = 0.0027~.From Fig. 16 we find that x:( 1 ) = x?( 1) = 0 is an asymptotically stable configuration. However, besides this solution there are also four P-3 solutions among which two are unstable and two are asymptotically stable. Each P-3 solution having 3 P-3 points, there are altogether 12 P-3 points. These are shown in Fig. 17. Viewed in the context of difference system (7.9), A l , A2, A3 associated with an unstable P-3 solution turn out to be saddle points. So are Bl, B2,B3. C1, C2, C3 and D1,02,03 associated with the stable P-3 solutions turn out to be stable spiral points. In Fig. 17 we have also filled in the trajectories between the P-3 points.

C . S . Hsu

294

i I I I

I I

Ir

I

I

I I I I

I I

L B1 C

t-'

FIG. 17. Continuous trajectories of the P-3 solutions for the case wo = 0, a = 3.5, p = 0.002rr. From Hsu ef a/. (1977b).

Thus, each closed path linking three P-3 points represents the continuous trajectory of a periodic solution of the original system (7.1); it is periodic after every third impact. Consider the system now again in the context of difference equations. When there are more than one asymptotically stable periodic solution such as the case we are now considering, it is important to determine the global regions of stability for each stable periodic point. For the present problem we again find the phenomenon of mutual winding of the separatrices. For example, a separatrix leaving A1 is seen to approach B2. However, as it

Nonlinear Parametric Excitation Problems

295

approaches B2, it oscillates about a separatrix leaving B2. On account of this we adopt the method discussed in Section V,A to determine the global regions of stability. In Fig. 18 regions determined in this manner are shown The three islands around D 1 , 0 2 , 0 3 are for the case a = 3.5 and p = 0.0027~. regions of asymptotic stability for the P-3 solution having 01, 02, 0 3 as its periodic points. They should be appreciated in the following manner. If the

FIG. 18. Global regions of asymptotic stability of the P-1 and P-3 solutions for the case a = 3.5 and p = 0.00271.

system starts with an initial point anywhere inside the island around 01, the next forward step will take the system to the inside of the island around 02, the second forward step to the inside of the island around 03, and the third step will bring the system back to the inside of the island around D l again. Or, if one only examines every third step, then the first, second, and third branches of the third order discrete trajectory stay respectively within each island. Moreover these branches approach asymptotically 01,0 2 , and 0 3 , respectively. The same picture is true for the three islands around C1. C2, and C3.

C. S. Hsu

296

There is also a large region of asymptotic stability around the origin. If the system starts with an initial point anywhere inside this region, then the discrete trajectory will remain in this region and approaches (0,O) asymptotically. This main region around (0,O)is seen to have six arms which tend to wrap around the six islands. The islands and the main region shown are quite close to each other, but they lead to two different types of motion. For instance, if the system starts at Q1,it will eventually approach (0, 0), the equilibrium state. However, if it starts from Q2, then it will eventually settle down into a P-3 solution whose continuous trajectory in the phase plan will be the closed path Dl-D2-D3-D1 in Fig. 17. C. AN ELASTICALLY RESTRAINED HINGEDBAR Consider now the case when the linear spring is present. In that case oo # 0 and (7.6) should be used as the governing difference equations. Periodic solutions and their stability can again be studied in a systematic 3 $

I

xi(l)/n

-

2

1

y(F-2) y

(P-1, I I )

- --

/I

(P-1, I /

-1 --__ \

k‘

-

*5

/I

\

~

n

)m II

-3

(P-2)

I

A 1< w ;+ -(&- - _ _ _

0

-2

I

I

i

p=o

1i

----

I

1

STABLE

UNSTABLE

I

;

i et al.

way. We present here merely Figs. 19, 20, and 21 to indicate the general feature. Figure 19 is for a = 3 and p = 0, Fig. 20 for a = 3 and p = 0.1, and Fig. 21 for u = - 3 and p = 0.1. For a detailed discussion of the figures and

Nonlinear Parametric Excitation Problems

In

,

297

ASYMPTOTICALLY STABLE ---- UNSTABLE

..\ L 1.5

-3

i

FIG.20. P-1and P-2 solutions as functions of w for a = 3 and p = 0.1. From Hsu (1977b).

et

al.

other periodic solutions the reader is referred to the paper by Hsu et al. (1977b). To a large extent the response of the nonlinear system may be ascertained by examining these periodic solution curves. As an example, let us look at Fig. 21 which is for a = - 3 and ,u = 0.1. A negative a means an impact load which is “destabilizing” so far as the equilibrium state Cp( = x l ) = 0 is concerned. In that case if the spring is weak the equilibrium state can be expected to be unstable. This is indeed the case as is shown in Fig. 21. For w / a in the interval &Do the system will not remain nearby x1 = x2 = 0 even if it starts there. Rather, the system will asymptotically approach a (P-1, 11) solution on the solid curve bifurcated from D o . The continuous trajectory in the phase plane corresponding to this solution is a closed curve having one vertical jump. If w/7c is increased so it lies in the interval Do-C1, the equilibrium state is now an asymptotically stable one and the continuous trajectory will eventually approach the origin of the phase plane. If w/7c lies in the interval C,-C, , the equilibrium state again becomes unstable and the system will eventually settle down into a P-2 solution on the solid ovalshaped line C1-C2. In the phase plane the continuous trajectory of this P-2 solution is a closed curve going around the origin once and having two vertical jumps on each side of the origin. Examined in this manner one can find the asymptotic behavior of the system as a function of w. It is perhaps of

298

C . S. Hsu

El p =0 . 1

- ASYIPTOT ICALLY ----

STABLE UNSTABLE

FIG. 21. P-1 and P-2 solutions as functions of o for a = -3 and p = 0.1. From Hsu et al. (1977b).

interest to point out that a P-2 solution on the oval-shaped solid curve bifurcated from C , has a continuous trajectory in the phase plane which again has two vertical jumps but goes around the origin three times. ACKNOWLEDGMENTS Some of the results presented here were obtained in the course of research supported by a grant from the National Science Foundation and some during the tenure of an appointment in the Miller Institute for Basic Research in Science at the University of California, Berkeley. The assistance of Dr. W. H. Cheng in preparing some of the figures is also gratefully acknowledged. REFERENCES ARNOL’D,V. I. (1963). Small denominators and problems of stability of motion in classical and celestial mechanics. Russ. Math. Suru. 18, 85-191. J., LIU,H., and MIRA,C. (1976). On non-linear periodic differential equations and BERNUSSOU, associated point mappings. Application to stability problems. Int. J . Non-Linear Mech. 11, 1-10, BIRKHOFF, G. D. (1920). Surface transformations and their dynamical applications. Acta Math. 43, 1-119. BLACK,H. F., and MCTERNAN, A. J. (1968). Vibration of a rotating asymmetric shaft supported in asymmetric bearings. J . Mech. Eng. Sci. 10, 252-261.

Nonlinear Parametric Excitation Problems

299

BOGOLIUBOV, N. N., and MITROPOLSKY, Y. A. (1961). “Asymptotic Methods in the Theory of Nonlinear Oscillations.” Gordon 8t Breach, New York. BOHN,M. P., and HERRMANN, G. (1974). The dynamic behavior of articulated pipes conveying fluid with periodic flow rate. J. Appl. Mech. 41, 55-62. BOLOTIN, V. V. (1964). “The Dynamic Stability of Elastic Systems.” Holden-Day, San Francisco, California. CESARI,L. (1970). “Asymptotic Behavior and Stability Problems in Ordinary Differential Equations.” Springer-Verlag, Berlin and New York. CHENG,D. K. (1959). “Analysis of Linear Systems.” Addison-Wesley, Reading, Massachusetts. CODDINGTON, E. A,, and LEVINSON, N. (1955). “Theory of Ordinary Differential Equations.” McGraw-Hill, New York. J., and CHHATPAR, C. K. (1970). Dynamic stability of a pendulum under parametric DUGUNDJI, excitation. Rev. Roum. Sci. Tech., Ser. Mec. Appl. 15, 741-763. J., and MUKHOPADHYAY, V. (1973). Lateral bending-torsion vibrations of a thin DUGUNDJI, beam under parametric excitation. J . Appl. Mech. 40,693-698. ELMARAGHY, R., and TABARROK, B. (1975). On the dynamic stability of an axially oscillating beam. J. Franklin Inst. 300,25-39. EVAN-IWANOWSKI, R. M. (1965). On the parametric response of structures. Appl. Mech. Rev. 18, 699-702. EVAN-IWANOWSKI, R. M. (1969). Nonstationary vibrations of mechanical systems. Appl. Mech. Rev. 22, 213-217. EVAN-IWANOWSKI, R. M. SANFORD, W. F., and KEHAGIOGLOU, T. (1971). Nonstationary parametric response of a nonlinear column. Dev. Theor. Appl. Mech., Proc. Southeast. Con& 5th, 1970 pp. 715-744. FRIEDMANN, P., and SILVERTHORN, L. J. (1974). Aeroelastic stability of periodic systems with application to rotor blade flutter. A I A A J. 12, 1559-1565. FRIEDMANN, P., and SILVERTHORN, L. J. (1975). Aeroelastic stability of coupled flap-lag motion of hingeless helicopter blades at arbitrary advance ratios. J. Sound Vib. 39, 409-428. GHOBARAH, A. A. (1972). Dynamic stability of monosymmetrical thin-walled structures. J . Appl. Mech. 39, 1055-1059. GHOBARAH, A. A,, and Tso, W. K. (1972). Parametric stability of thin-walled beams of open section. J. Appl. Mech. 39, 201-206. GUMOWSKI, I., and MIRA,C. (1972). Sur la distribution des cycles d‘une recurrence ou transformation ponctuelle, conservative du deuxieme ordre. C. R . Hebd. Seances Acad. Sci., Ser. A 274, 1271-1274. HAGEDORN, P. (1968). Zum Instabilitatsbereich erster Ordnung der Mathieugleichung mit quadratischer Dampfung. Z. Angew. Math. Mech. 48, T256-T260. HAGEDORN, P. (1969). Kombinationsresonanz und Instabilitatsbereiche zweiter Art bei parametererregten Schwingungen mit nichtlinearer Dampfung. Ing.-Arch. 38, 80-96. P. (1970a). Uber Kombinationsresonanz bei parametererregten Systemen mit CouHAGEDORN, lombscher Dampfung. 2. Angew. Math. Mech. 50, T228-T23 1. HAGEWRN, P. (1970b). Die Mathieu-gleichung mit nichtlinearen Dampfungs-und Riickstellgliedern. 2. Angew. Math. Mech. 50, 321-324. HAIGHT,E. C., and KING, W. W. (1971). Stability of parametrically excited vibrations of an elastic rod. Dev. Theor. Appl. Mech., Proc. Southeast. Conf., 5th, 1970 pp. 677-714. HALE,J. K. (1963). “Oscillations in Nonlinear Systems.” McGraw-Hill, New York. HAYASHI, C. (1964). “Nonlinear Oscillations in Physical Systems.” McGraw-Hill, New York. HOUBEN, H. (1970). Einfluss des Antriebs in Schwingungssystemen mitt Parametererregung. 2. Angew. Math. Mech. 50, T231-T234. Hsu, C. S. (1972). Impulsive parametric excitation: Theory. J . Appl. Mech. 39, 551-559.

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Hsu, C. S. (1974a). On parametric excitation and snap-through stability problems of shells. In “Thin-Shell Structures: Theory, Experiment, and Design” (Y. C. Fung and E. E. Sechler, eds.), pp. 103-131. Prentice-Hall, Englewood Cliffs, New Jersey. Hsu, C. S. (1974b). Some simple exact periodic responses for a nonlinear system under parametric excitation. J. Appl. Mech. 41, 1135-1137. Hsu, C. S. ( 1975a). Limit cycle oscillations of parametrically excited second-order nonlinear systems. J . Appl. Mech. 42, 176-182. Hsu, C. S. (1975b). The response of a parametrically excited hanging string in fluid. J. Sound Vib. 39, 305-3 16. Hsu, C. S. (1976). Bifurcation in the theory of nonlinear difference synamical systems. I n “The Volume in Honor of Dr. Th. Vogel” (to be published). Hsu, C. S., and CHENG,W. G. (1973). Applications of the theory of impulsive parametric excitation and new treatments of general parametric excitation problems. J . Appl. Mech. 40,78-86. Hsu, C. S., and CHENG,W.H. (1974). Steady-state response of a dynamical system under combined parametric and forcing excitations. J . Appl. Mech. 41, 371-378. Hsu, C. S., and YEE, H. C. (1975). Behavior of dynamical systems governed by a simple nonlinear difference equation. J. Appl. Mech. 42, 870-876. Hsu, C. S., YEE, H. C. and CHENG,W. H. (1977a). Determination of global regions of asymptotic stability for difference dynamical systems. J . Appl. Mech. 44, 147-153. Hsu, C. S., YEE,H. C., and CHENC,W. H. (1977b). Steady-state response of a nonlinear system under impulsive periodic parametric excitation. J. Sound Vib. 50, 95-1 16. IWATSU~O, T., SUGIYAMA, Y., and OGINO,S. (1974). Simple and combination resonances of columns under periodic axial loads. J. Sound Vib. 33, 211-221. JURY,E. I. (1964). “Theory and Application of the Z-Transform Method.” Wiley, New York. KALNINS, A. (1974). Dynamic buckling of axisymmetric shells. J. Appl. Mech. 41, 1063-1068. KAUDERER, H. (1958) “Nichtlineare Mechanik.” Springer-Verlag, Berlin and New York. LASLETT, L. J. (1958). “Long-term Stability for Particle Orbits,” Rep. NYO-1480-101. Courant Inst. Math. Sci., New York University, New York. MARKUS,L. (1971). ‘‘ Lectures in Differentiable Dynamics,” Reg. Conf. Ser. Math., No. 3. Am. Math. Soc.,Providence, Rhode Island. MAY,R. M. (1974). Biological populations with nonoverlapping generations: stable points, stable cycles, and chaos. Science 186,645-647. MEITLER,E. (1%7). Stability and vibration problems of mechanical systems under harmonic excitation. In “Dynamic Stability of Structures” (G. Herrmann, ed.), pp. 169-188. Pergamon, Oxford. MILLER,K. S. (1968). “Linear Difference Equations.” Benjamin, New York. MOTE,C. D., Jr. (1%8). Dynamic stability of an axially moving band. J. Franklin Inst. 285, 329-346. NISHIKAWA, Y.,and WILLEMS, P. Y. (1969). A method for stability investigation of a periodic dynamic system with many degrees of freedom. J. Franklin Inst. 287, 143-157. PAIWUSSIS,M. P., and SUNDARARAJAN, C. (1975). Parametric and combination resonances of a pipe conveying pulsating fluid. J . Appl. Mech. 42, 78C784. PANOV, A. M. (1956). Behavior of the trajectories of a system of finite difference equations in the neighborhood of a singular point. Uch. Zap. Ural. Gos. Univ. 19, 89-99. H. (1881). Sur les courbes definies par les equations differentielles. J. Math. Pure POINCARB, Appl. Ser. [3] 7, 375-422. PUN, L. (1973). Initial conditioned solutions of a second-order nonlinear conservative differential equation with a periodically varying coefficient. J. Franklin Inst. 295, 193-216.

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RHODES,J. E.,Jr. (1971). Parametric self-excitation of a belt into transverse vibration. J. Appl. Mech. 37, 1055-1060. SCHMIDT, G. (1975). “ Parametererregte Schwingungen.” VEB Dtsch. Verlag Wiss., Berlin. H. R. (1970). Uber periodische Losungen einer nichtlinearen Differentialgleichung SCHWARZ, mit periodischen Koeffizienten. 2. Angew. M a t h . M e c h . 50, T23-T25. I. Z. (1961). “Linear Differential Equations with Variable Coefficients.” Gordon & SHTOKALO, Breach, New York. SMALE, S. (1967). Differentiable dynamical systems. Bull. Am. M a t h . SOC. 73, 747-817. I. H. (1967). Transition mechanisms attendant to large amplitude parametric vibraSOMERSET, tions of rectangular plates. Trans. A S M E , Ser. B 89, 619-626. SOMERSET, J. H., and EVAN-IWANOWSKI, R. M. (1967). Influence of non-linear inertia on the parametric response of rectangular plates. I n t . J. Non-Linear Mech. 2, 2 17-232. J. J. (1950). “Nonlinear Vibrations.” Wiley (Interscience), New York. STOKER, TAKEN$ F. (1973). “ Introduction to Global Analysis,” Commun. Math. Inst., Rijksuniversiteit Utrecht. TANI, J. (1974).Dynamic instability of truncated conical shells under periodic axial loads. Inr. J. Solids Strucr. 10, 169-176. TANI,J. (1976). Influence of deformations prior to instability on the dynamic instability of conical shells under periodic axial load. J. Appl. Mech. 43, 87-91. TONDL,A. (1970). “Domains of Attraction for Non-linear Systems,” Monographs and Memoranda. No. 8. Natl. Res. Inst. Mach. Des., Bkhovice. H., and Hsu, C. S. (1977). Response of a non-linear system under combined parametric TROGER, and forcing excitation. J. Appl. Mech. 44, 179-181. Tso, W. K. (1968). Parametric torsional stability of a bar under axial excitation. J. Appl. M e c h . 35, 13-19. TSO, W. K.,and ASMIS, K. G. (1970). Parametric excitation of a pendulum with bilinear hysteresis. J. Appl. Mech. 37, 106-1068. WEIDENHAMMER, F. (1956). Das Stabilitatsverhalten der nichtlinearen Biegeschwingungen des axial pulsirend belasten Stabes. 1ng.-Arch. 24, 53-68. YAKUBOVICH,V. A., and STARZHINSKII, V. M. (1975). ‘‘ Linear Differential Equations with Periodic Coefficients.” Wiley, New York. YEE,H. C. (1975). A study of two-dimensional nonlinear difference systems and their applications. Ph.D. Dissertation, University of California, Berkeley.

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Foundations of the Theory of Surface Waves in Anisotropic Elastic Materials P. CHADWICK AND G . D . SMITH School of Mathematics and Physics University of East Anglia Norwich. England

I . Introduction . . . . . . . . . . . . . . I1. Algebraic Preliminaries . . . . . . . . . . A . Notation . . . . . . . . . . . . . . B. Inner and Tensor Products . . . . . . C. Special Tensors . . . . . . . . . . . . D . Representations and Decompositions . . E. Vector and Triple Products . . . . . . 111. Elasticity Tensors . . . . . . . . . . . A . Elastic Constants and Acoustical Tensors B. Elasticity Tensors in a Reference Plane . C. Differential Relations . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . . . . . IV. The Fundamental Eigenvalue Problem . . . . . A . Eigenvalues and Eigenvectors of N(q) . . . . B. Completeness and Spectral Relations . . . . C. Orientation Dependence of p a , k a , and qa . . D . Integral Representations . . . . . . . . . E. Degeneracy of the Eigenvalue Problem . . . V. Plane Elastostatics . . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . . . . .

. . . . . . . .

. . . . . . . . 314 . . . . . . . . 315 . . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . . . . . . .

. . . .

A . General Solution . . . . . . . . . . . . . . . . . . . . . B. A Stationary Line Singularity . . . . . . . . . . . . . . . . C. Properties of the Solution . . . . . . . . . . . . . . . . . .

VI . A Uniformly Moving Line Singularity . . . . A. Modification of the Elastostatic Solution . . B. The Limiting Speed 6 . . . . . . . . . C. Possible Modes of Breakdown of the Solution D. Properties of the Tensors S,( v ) and S, ( u ) . . E. Illustrative Example . Hexagonal Symmetry .

. . . . . . . . . . . . . . . . . . . .

. . . . VII . Elastic Surface Waves. Basic Analysis . . . . . A . General Surface Wave Solution . . . . . .

. . . . . .

304 306 307 308 309 309 310 310 311 312 314

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

317 317 318 321 325 325 327 328 332 333 335 339 342 345

. . . . . 347 . 348

B. Free Surface Waves . . . . . . . . . . . . . . . . . . . . 350 C. The Dislocation-Surface Wave Analogy . . . . . . . . . . . . 352 D. The Surface Impedance Tensor . . . . . . . . . . . . . . . . 353 303

304

P . Chadwick and G. D . Smith

VllI. The Uniqueness and Related Properties of Free Surface Waves . . . A. Uniqueness . . . . . . . . . . . . . . . . . . . . . . . B. Calculation of the Speed of Propagation . . . . . . . . . . . C. Calculation of the Polarization Vector . . . . . . . . . . . . IX. The Existence of Free Surface Waves . . . . . . . . . . . . . . A. Formulation of the Existence Problem . . . . . . . . . . . . B. Behavior of A(u) in the Transonic Limit . . . . . . . . . . . C. Exceptional Transonic States . . . . . . . . . . . . . . . D. Summary of Conclusions . . . . . . . . . . . . . . . . . E. Hexagonal Symmetry . . . . . . . . . . . . . . . . . . F. An Asymptotic Lemma . . . . . . . . . . . . . . . . . . X. Supplementary Topics . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . .

. 355 356 . 357 . 358 . 359 . 360 . 361 . 366 . 368 . 369 . 370 373 374

I. Introduction Rapid progress has been made in recent years in the investigation of basic aspects of the theory of surface waves in anisotropic elastic media. In particular, solutions have been found to the fundamental problems csncerning the existence and uniqueness of free surface waves' and in this respect it can be claimed that a complete theory is now in existence. An exposition of the subject is therefore timely, but we have also been motivated in writing this article by another consideration. The outstanding new ideas have of late come from a rather unexpected quarter, namely, the theory of dislocations, and results of major significance may consequently lie hidden from specialists in the dynamics of elastic materials who do not habitually attend to developments in this fieId. The possibility of oversight is not reduced by the dependence of the main contributions on a multitude of sources mostly located in the literature of solid state physics, much of the precursory material having no obvious connection with elastic surface waves. An analogy between the solutions of the equations of linearized elastodynamics representing, on the one hand, a line dislocation in uniform motion through an unbounded anisotropic body and, on the other, a progressive surface wave in a semi-infinite body of the same composition was first recognized by A. N. Stroh in 1962 and subsequently extended and powerfully exploited by D. M. Barnett, J. Lothe, K. Malen, and their coworkers during the period 1971-1974. We provide in the later sections a

' That is, waves propagating in the presence of a traction-free boundary.

Surface Waves in Anisotropic Elastic Materials

305

connected account of the theory of elastic surface waves which has resulted from the work of these authors. Having first in mind the needs of readers interested in the theory of elastic waves, we include no more of the analysis of line dislocations than is necessary to make the presentation self-contained and comprehensible. It should be noted, however, that the methods devised by Barnett, Lothe, and others have led to the solution of important problems in dislocation mechanics as well as in the theory of elastic surface waves. An assessment of these techniques from the standpoint of the study of line defects in anisotropic elastic materials has recently been made by Hirth (1976). The prevailing tone of the article is theoretical with the accent firmly on basic mathematical questions. Since our primary concern is with anisotropic elastic materials in general we make little mention of the many papers devoted to surface wave propagation in elastic media exhibiting specific types of crystalline symmetry. A concise survey of these investigations is contained in the book by Musgrave (1970, Chapter 12)and additional references are given by Chadwick (1976). Again, while accepting that the careful scrutiny of numerical results is essential to a rounded view of our subject, we make no attempt to discuss computational procedures or to summarize the large amount of detailed information on the properties of elastic surface waves supplied by computer studies. The review by Farnell (1970) remains the best available introduction to this topic. The last of our intentional omissions is a conspectus of the widening range of technological innovations calling for some theoretical understanding of surface wave effects in crystalline materials. Numerous applications of surface waves are described in the book by Viktorov (1967) and articles by Smith (1970), White (1970), and de Klerk (1972). Each of the succeeding sections except the last begins with an outline of its contents, so it suffices here to indicate the structure of the article and the progression of the salient results. The subject matter divides naturally into three parts, consisting of Sections 11-IV, V and VI, and VII-IX. The closing section touches briefly on some cognate topics and problems meriting further research. Sections 11-IV are devoted to essential preliminaries, largely algebraic in character, culminating, in Section IV, in a detailed treatment of the eigenvalue problem for the fundamental elasticity tensor N(cp).This discussion is the mainspring of the overall development. The acute observation, due to Barnett and Lothe (1973), that the tensors determining the properties of free elastic surface waves stem from the mean value of N(cp) over its domain leads to integral representations and associated relationships [Eqs. (4.3 1 (4.34)] which eventually provide the means for solving the existence and uniqueness problems.

306

P. Chadwick and G. D. Smith

Sections V and VI deal in turn with stationary and uniformly translating line singularities in an anisotropic elastic body, the passage from the elastostatic to the elastodynamic solution being achieved by a simple modification of the fundamental elasticity tensor with some accompanying reinterpretations. Section VI contains a wealth of ideas and results which, in accordance with the analogy mentioned earlier, carry over in toto to elastic surface waves. In particular, we meet here the crucial notion of a limiting speed, marking the transition from subsonic to supersonic behavior of the motion excited by a moving line source. The implications of this concept are much more elaborate in the present context than in acoustics, owing to the anisotropy of the ambient medium. It transpires, in fact, that there are six possible types of transonic states, the controlling factor being the number and the nature of the contacts between the slowness surface of the material and a tangent line orthogonal to the plane of motion of the source. Arriving at length at our main theme, we are able, in Sections VII-IX, to develop the theory of elastic surface waves quickly and in depth. The basic analysis of progressive surface waves of arbitrary form is set out in Section VII, and Sections VIII and IX are concerned with the uniqueness and existence of free surface waves. The property of uniqueness is established in Section VIII without restricting the elastic constants of the transmitting material, but the investigation of existence made in Section JX requires the strain energy to be positive definite. In general terms, the existence of a unique free surface wave is found to represent the normal state of affairs, but a motion of this kind may fail to exist when the transonic state meets certain requirements. The criteria obtained for distinguishing between these possibilities are quite simple and they show that nonexistence is likely to occur only when the boundary of the half-space and the direction of propagation are aligned with a plane and an axis of material symmetry. This conclusion is supported by numerical results. Two broad insights into the character of elastic surface waves confirmed by the present study are: first, that a free surface wave is intrinsically a subsonic phenomenon, and, second, that the set of directions on a particular anisotropic elastic half-space in which such waves can travel is determined by the slowness surface of the material.

11. Algebraic Preliminaries

For the most part direct notation for vectors and tensors is employed in this article, but two factors force departures from standard usage. First, many of the vector and tensor quantities entering the theory are complex-

307

Surface Waves in Anisotropic Elastic Materials

valued, and, second, we need to consider vectors and tensors in six dimensions as well as the usual three. In this section we lay down a suitable scheme of notation and formulate the rules of vector and tensor algebra underlying the subsequent calculations. A. NOTATION

We shall be concerned extensively with vectors in a three-dimensional unitary (i.e., complex inner product) space C3, with tensors in L(C3)(the set of all linear transformations from C3 into itself), and with vectors and tensors in C6 and L(C6),C6 being regarded as the six-dimensional unitary space obtained by forming the direct sum C3 @ C 3 (see, for example, Halmos, 1958, Sections 18 and 19). Vectors in C 3 and tensors in L ( C 3 ) (referred to also as 3-vectors and 3-tensors) are denoted by bold Latin minuscules and bold Latin majuscules, respectively. Bold Greek minuscules are used for vectors in C6 (6-vectors) and bold sans serif Latin majuscules for tensors in L(C6)(6-tensors). Every vector in C6 has a unique representation as an ordered pair of vectors in C3 and we use the notation a = (a, b)

(2.1) to describe this situation. Every 6-tensor is similarly made up of four 3tensors: the statement A=

(: :)

means that the action of A E L(C6)on an arbitrary vector a is specified by Aa = (A,a A, b, A, a A, b).

+

=

(a, b) in C6

+

(2.3) It is often convenient to carry out manipulations involving 6-vectors and 6-tensors in matrix format. Thus the right-hand side of Eq. (2.3) can be set out as the matrix product

(: :)(;) If =

(BB:

:>

is a second 6-tensor, a double application of the rule (2.3) gives ABa

=

((A,B, ('43

Bl

+ A, B3)a + (A,B, + A, B4)b, + A4 B3)a + (A3 B, + A4 B'db),

308

P . Chadwick and G . D. Smith

from which it is seen that

with matrix multiplication again implied.

B. INNERAND TENSORPRODUCTS The inner product (a, b) of two vectors in C3 has the standard properties

(a, b) = F-4 a), hal

+ a2 a 2

(2.6)

b) = al(a19b) + a2(a2,b),

(2.7)

(a, a) 2 0; (a, a ) = 0 if and only if a = 0, (2.8) where a1 and a2 are arbitrary complex numbers and an overbar denotes a complex conjugate. Frequent use is made of the property (Aa, b) = (a, ATb),

(2.9)

the superscript T signifying transposition. The tensor product a @ b is defined by

(a @ b)u = (u, b)a

V

u

E

C3,

(2.10)

and we note that

(a @ b)T= 6 @ a,

(a @ b)(c @ d) = (c, b)(a @ d),

(2.11)

and A(a 0 b) = (Aa) 0 b, a

(a 0 b)A = a 0 (ATb).

(2.12)

The inner and tensor products induced on C6 are defined as follows. Let (a, b) and y = (c, d) be arbitrary 6-vectors. Then

=

+ (b,d)

(2.13)

a@c a@d ( b @ c b@d

(2.14)

(a, Y) = (a, c)

and a@y=

It can easily be verified that the inner product (2.13) has the properties (2.6b(2.9) and that, in conformity with the definition (2.10), (a@yk=(P,y)a

v PEP.

(2.15)

Surface Waves in Anisotropic Elastic Materials

309

C. SPECIAL TENSORS The zero tensors in L(C3)and L(C6)are denoted by 0 and 0,respectively, and the corresponding identity tensors by I and 1. The decompositions of 0 and I into their constituent 3-tensors are plainly

I

0 0

O=(o

I=(o

0).

0 1).

(2.16)

The special 6-tensors J and K are defined by J=(

0

-I

),

1

K=(;

:>,

0

(2.17)

and we take note of the relations J2=

-1,

JT=

J-'=

K2 = I,

-J;

KT = K - ' = K,

K(a, b) = .

J(a, b) = (b, -a);

(2.18) (2.19)

D. REPRESENTATIONS AND DECOMPOSITIONS Let e = { e l ,e, , e3} be a real orthonormal basis of C3. The components relative to e of an arbitrary 3-vector a and an arbitrary 3-tensor A are in turn+ ai= (a, ei)

and

Aij = ( A e j ,e,).

(2.20)

The representations of a and A in terms of e are accordingly a = a, ep = (a, e,)e, ,

A = A,, ep0 eq = (Ae, , e,)e, 0 e, . (2.21)

The inner product (a, b) of two 3-vectors a and b with components a, and bi relative to e is a, 6, and the components of the tensor product a @ b are a, S j . Given an arbitrary 6-vector a,define a = (a,( e , , O>)e,,

b = (a, (0, e,))e,.

(2.22)

Suppose that a = (a', b). Then, by appeal to Eqs. (2.13) and (2.21)', a = (a', ep)ep= a',

b = (b, ep)ep= b,

and Eqs. (2.22) are seen to effect the decomposition of a 6-vector into its constituent 3-vectors. A similar argument based on Eqs. (2.13) and (2.21), 'Latin subscripts take the values 1, 2, 3. The summation convention applies over the repeated subscripts p , q, r, and s, but not over repetitions of the subscript i.

P . Chadwick and G. D. Smith

3 10

(2.23)

E. VECTORAND TRIPLE PRODUCTS The vector product . A . of pairs of vectors in C 3 is defined (to within an ambiguity of sign) by the following axioms.

b A a = -aAb, (a,al

+ a2 a2)

A

b = a,(a,

A

b)

(2.24)

+ a2(a2

A

b),

(a, X)= 0, (a A b, a A b) = (a, a)(b, b) - (a, b)(&%),

(2.25) (226) (2.27)

a, and a2 again being arbitrary complex numbers. The scalar triple product

[ . , . , . ] of triads of 3-vectors is then given by ~

(2.28)

[a, b, c] = (a, b ~ c ) .

The determinant det A of an arbitrary 3-tensor A can be introduced invariantly through the statement (det A)[u, v, w] = [Au, Av, Awl

V

u, v, w E C 3 ,

(2.29)

and we record for later use the identity

[a, b, c]’

= det(a @ %

+ b 0 6 + c 0 C)

(2.30)

which can be deduced from (2.29) with the use of Eqs. (2.10), (2.25), (2.26), and (2.28). 111. Elasticity Tensors

In this section we construct from the set of constants specifying the response to infinitesimal strain of an anisotropic elastic material certain real 3-tensors and 6-tensors central to the later theory. The advantages of introducing a six-dimensional formalism into the study of plane problems of anisotropic elasticity were first expounded by Stroh (1962, Section 4) who acknowledged previous work by Gurney. Ingebrigtsen and Tonning (1969,

Surface Waves in Anisotropic Elastic Materials

311

Sections I1 and 111), seemingly unaware of these earlier findings, arrived at a closely similar formalism, and the crucial 6-tensor N(cp), referred to here as the fundamental elasticity tensor, originated in their paper. Subsequent developments by Malen and Lothe (1970, Section 2.1) and Barnett and Lothe (1973, Sections 2 and 3) have guided the treatment given in Sections III,B and II1,C.

A. ELASTIC CONSTANTS AND ACOUSTICAL TENSORS We are concerned in this article with an elastic body that is anisotropic in relation to a stress-free configuration N . No restriction is placed on the symmetry of the material, but the existence of a strain-energy function is presumed. The components Cijkl relative to the basis e of the fourth-order linear elasticity tensor in N therefore obey the symmetry relations (3.1) the number of independent elastic constants being in general twenty-one. The linear elasticities are assumed to satisfy the strong ellipticity condition Cjikf

=

Cklij

= Cijkl

>

Cpqrsapbqarbs >0 V nonzero real a, b E C 3 , (3.2) and, for a certain purpose, to be disclosed in due course, we impose the strong convexity condition

CpqrsSpqSrs >0

V

nonzero real symmetric S E L(C3). (3.3)

Strong convexity, which is equivalent to positive definiteness of the strainenergy function, is a more stringent limitation on the linear elasticities of the material than strong ellipticity. For an isotropic material, for instance, with Lame constants 1,p, Cijkl = 16ijdkl + p(Bik6jl + hifdjk),

(3.4) and necessary and sufficient conditions for the satisfaction of (3.2) and (3.3) are A+2p>O,

(3.5)

p>o,

and (3.6) respectively. Clearly, (3.6) (3.5), but the reverse implication does not hold. Let S denote the set of all real unit vectors in C3. The acoustical tensor Q(n) and the associated acoustical tensor R(m, n ) are real 3-tensors defined by the component relations 1++p>o,

p>o,

Q i j ( n ) = Cpiqjnpnq

V

n

Rij(m,n ) = Cpiqjmpnq

V

orthogonal

E

S,

m, n E S.

(3.7)

312

P . Chadwick and G. D. Smith

Equation (3.1), and the strong ellipticity condition (3.2) ensure that Q(n) is symmetric and positive definite, and therefore invertible. In general R(m, n) is not symmetric and the relation (n, R(m, n)m) = (m, Q(n)m) > 0,

(3.8) derived from Eqs. (3.7) and (3.1), affirms that R(m, n) can never be spherical (i.e., a scalar multiple of I). B. ELASTICITY TENSORS IN

A

REFERENCE PLANE

Because of the two-dimensional character of the elastostatic and elastodynamic fields studied in Sections V-IX it is appropriate to confine the directions involved in the definitions (3.7) to a fixed plane R, hereafter termed the reference plane. The real orthonormal basis e is so chosen that el and e, define fixed directions in f2, and we introduce a second orthogonal pair of real unit vectors in R given by m

= sin cpe, - cos cpe,

,

n

= cos cpe,

+ sin cpe, ,

(3.9) and therefore oriented relative to el and e, in the manner shown in Fig. 1.

FIG. 1. Orientation in the reference plane R of the orthogonal unit vectors m and n.

The triad {m, n, e3}evidently furnishes a real orthonormal basis of C3 similar to e. Acoustical tensors in the reference plane are denoted by Q(cp)

= Q(n) =

Q(cos 'pel + sin cpe,),

R(cp) = R(m, n) = R(sin cpe, - cos cpe, , cos cpe,

+ sin cpe,).

(3.10)

313

Surface Waves in Anisotropic Elastic Materials

The periodicity properties Q(cp

+ n) = Q(V),

R(cp + 3.)

=

-RT(~),

R(v

+ n) = R(v), (3.1 1 )

follow from the definitions (3.10) and (3.7), as d o the useful identities Clilj

= Qij(e1) = Qij(o),

C2i2j = Qij(e2) = Qij(&L Clizj =

Rij(el, e2)= Rij(3n)'

CZilj = Rij(e2

,e l ) =

(3.12)

-Rij(O),

and Q ( q )= cos2 cpQ(0) - sin cp cos cp(R(0)

+ RT(0)} + sin2 cpQ(+n),

+

R(cp) = sin cp cos cp(Q(0)- Q ( f n ) } cos2 cpR(0) - sin' cpRT(0). (3.13)

We remark here, once and for all, that the eigenvalues and eigenvectors of Q(cp)are single-valued and piecewise twice continuously differentiable in any closed interval of length 2n. The acoustical tensors in R are now used to construct two real 6-tensors M(cp) and N(cp). First we set (3.14)

and observe that since det M(cp) = det Q(cp) > 0, M(cp)is invertible. In fact (3.15)

The fundamental elasticity tensor N(cp) is defined by N(cp) = M-'(cp

+ 3n)JM(cp),

(3.16)

J being the constant 6-tensor given by Eq. (2.17)l. The existence of N-'(cp) is assured by the invertibility of M(cp) and J, and the periodicity relations

M(v + n) = M(v),

N(cp + 9.)

= - N-'(cp),

N(cp

+ n) = N(cp),

(3.17)

follow from the application of Eqs. (3.11) and (2.18), to the definitions (3.14) and (3.16). The decomposition of N(cp), supplied by Eqs. (3.14)-(3.16) and

314

P . Chadwick and G . D. Smith

(3.11)2, is (3.18) where

C. DIFFERENTIAL RELATIONS Differentiation of Eqs. (3.13) with respect to cp gives

Q'(v)= -R(v) + R(q + h)= - R(v) R'(P) = Q(cP)- Q(CP + in),

- RT(v),

(3.20)

and it then follows from the definition (3.14), with the use of (2.17), and (3. l q l , that

+ $a) = - J(M(v) - Mtv + in)}. (3.21) On premultiplying each side of Eq. (3.16) by M(cp + in) and differentiating M'(q)= - M'(v

with respect to cp we next obtain, with the aid of (3.21) and (2.18)', J(M(44 - M(cp + in)j"v)

+ Wcp + *)N'(cp)

= M(cp) - M(cp + *).

(3.22) It is then an easy matter to confirm, with appeal to Eqs. (3.16) and (2.18),, that N(cp) satisfies the differential equation N'(q)

=

- I - N2(cp)

(3.23)

(Barnett and Lothe, 1973, Section 3). In the next section we go on to derive further properties of the fundamental elasticity tensor N(cp).It should be noted, however, at this juncture, that the results given above and those that now follow are valid so long as the 3-tensors Q(cp)and R(q) satisfy the relations (3.11) and (3.13) and, in addition, Q(cp)is symmetric and invertible. IV. The Fundamental Eigenvalue Problem

The feature of the fundamental elasticity tensor N(cp) of greatest moment in relation to the plane elastostatics and steady elastodynamics of anisotro-

Surface Waves i n Anisotropic Elastic Materials

315

pic elastic bodies is the associated eigenvalue problem. The study of this problem has been pursued somewhat haphazardly in the literature, the principal contributors being Stroh (1958, Section 3; 1962, Section 4), Ingebrigtsen and Tonning (1969, Section III), Malen and Lothe (1970, Section 2), Braekhus and Lothe (1971, Section 2), Nishioka and Lothe (1972a, Section 2), Barnett and Lothe (1973, Sections 2-4), and Lothe and Barnett (1976a, Section IV). We attempt here to provide a unified account of the problem embracing all that is presently known. Basic properties of the eigenvalues and eigenvectors are developed in Sections IV,A and B for a fixed value of the orientation angle cp. The dependence of the eigenvalues and eigenvectors on cp is then determined in Section IV,C and these results lead in Section IV,D to integral representations of certain 3-tensors that figure prominently in Sections V-IX. Lastly, in Section IV,E, we examine the degenerate cases which arise when N(cp) has repeated eigenvalues.

A. EIGENVALUES A N D EIGENVECTORS OF N ( p ) Let p be an arbitrary real number and let 0 = tan- ' p (-$n < 8 < $71). Then, using Eqs. (3.14)-(3.16), (2.17)1, and (3.11)2, det{N(cp)- PI1 = det[(

Q- '(cp + RT(cp)Q-'(cp

= {det Q(cp

+ &)I-

= sec6 0(det Q(cp

+ 14.

0 I

).4

R(cp) - PQ(V + pRT(cp)- Q(v)

>(

')I

---PI

det[p2Q(cp + &) - ~{R(cp)+ RT(cp))+ Q(cp)I

+ in)}-' det Q(cp + 0).

(4.1) The expression (4.1) is strictly positive on account of the positive definiteness of the acoustical tensor, so N(cp) has no real eigenvalue (cf. Eshelby et al., 1953, p. 259). Since N ( q ) is a real tensor this means that its eigenvalues p,(cp)' form three pairs of complex conjugates. When no statement is made to the contrary it is assumed henceforth that, for all values of cp, the eigenvalues p,(cp) are distinct,this being the usual situation for elastic materials displaying genuine anisotropy. It then follows (see, for example, Pease, 1965, pp. 75-76) that there are associated eigenvectors t,(cp) which are linearly independent and therefore form a basis of C6. From this point to the end of Section IV,C we simplify the notation by not Where no indication is made to the contrary, Greek subscripts take the values 1, 2, . . . , 6. Summation is understood to apply over the repeated subscripts x and p , but not over repetitions of the subscript a. +

3 16

P. Chadwick and G.D.Smith

stating explicitly the dependence on cp of the fundamental tensor and its eigenvalues and eigenvectors. The vectors qardefined by

constitute a second basis of C6 and, as we now prove, q, is a reciprocal eigenvector of N with associated eigenvalue p a . From the decompositions (2.17), and (3.18) of K and N it can easily be verified that (4.3)

NTK = KN. Thus, using the definition (4.2) and the eigen-relation N6.Z = P a 5,

(4.4)

9

we find that

NTqa = NTK5a = KN5,

= pol K5, = pa q a ,

(4.5)

as claimed. It can be proved in the usual way that the eigenvectors and reciprocal eigenvectors are pairwise orthogonal. Forming the inner product with qs of each side of Eq. (4.4) and calling on (2.9) and (4.5) we obtain (N5, tla) = (Pa 9

=

5 , 7

tip) = P,(L ‘Is) 9

(5, NTtls)= 7

(5a9

Pfl tls)

= P&

9

tl&.

(4.6)

Since pa # ps when a # /? it follows that (5a,

tip) = 0

when a #

P.

(4.7)

Suppose that, for some a,(go,q,) = 0. Since the vectors 6.: span C6 there are scalars b, such that q, = b,c,. Hence, with reference to Eqs. (4.2), (2.18), , and (4.7), (tla

7

tla) =

( 5 , 9

5,)

=

(bx 5 x 9

tla)

= bz(5,

7

q a ) = 0,

implying, via (2.8), that 6. = 0. This conclusion is false, however, since 5, is a member of a basis. Thus (I&, qa)# 0 and, bearing in mind that each eigenvector is arbitrary to the extent of a complex multiplier, we can so normalize the C s and q,’s that Eq. (4.7) takes on the extended form ( 5 . 9

tip) = 6,s .

(4.8)

It is apparent from Eqs. (4.4) and (4.5) that eigenvectors and reciprocal eigenvectors associated with a pair of complex conjugate eigenvalues are also complex conjugates. For future convenience we adopt the following

Surface Waves in Anisotropic Elastic Materials

317

system of labeling in which the subscripts 1, 2, 3 are assigned to the eigenvalues with positive imaginary parts. (4.9)

B. COMPLETENESS AND SPECTRAL RELATIONS

en

Let u be an arbitrary vector in C6 and let a, be the representation of u in terms of the normalized eigenvectors. Then, invoking the orthogonality relation (4.8) and Eq. (2.15), (4.10) (a,tla) = azi5, t l a ) = a u and (4.11) a= tl,K* = (5, 0 t l , b 3

(a9

In view of the arbitrariness of a there follows first the completeness relation

5,o tl, = 1,

(4.12)

and then, with the use of Eq. (4.4), the spectral representation of N N = NI = N(5,6tln)= (NS,)Otl,

= Pn

5,otl,.

(4.13)

The eigenvector 5. can be decomposed into a pair of 3-vectors by applying Eqs. (2.22) and the decomposition of qu is then supplied by Eqs. (4.2) and (2.19), . We write these results as 5u

= (aa

9

t l u = (1,

la),

(4.14)

9

Decomposition of the completeness and spectral formulas with the aid of Eqs. (2.14), (2.16),, and (3.18) then yields the relations a, 0 a, = 0,

I,@

T,

= 0,

a, @ T, = I,

(4.15)

and

01, ,

N3 = P , I, 01,. (4.16) N, = p , a, 0 a,, Equations (4.15) and (4.16) are referred to in the literature cited earlier as closure relations and sum rules, respectively (see Nishioka and Lothe, 1972a, Section 2). Nl

= p n a,

C. ORIENTATION DEPENDENCE OF pa,

, AND q,

On differentiating with respect to ~0 the eigen-relation (4.4) and utilizing the differential equation (3.23) satisfied by N we obtain

+ 1 + ~ , 2 K a= (N - pa 1)Ch

3

(4.17)

318

P . Chadwick and G . D. Smith

primes, as before, denoting derivatives. The result of forming the inner product of each side of this equation with q, and applying Eqs. (4.8) (with p = a), (2.9), and (4.5) is p'

=

-1 - p za

(4.18)

)

and we then see, from Eq. (4.17), that 6; and 6, are eigenvectors of N associated with the same eigenvalue p , . This means that 6; is a scalar multiple of 6,; & = k, 6, , say. The general solution of this differential equation is (4.19) proving that 6, is a scalar multiple of a constant vector which, of course, is also an eigenvector of N associated with p a . Without loss of generality the eigenvectors 6, can therefore be taken to be independent of orientation in the reference plane R and, by an exactly similar argument, the reciprocal eigenvectors q, may be shown to have the same property. In contrast, the eigenvalues p , satisfy the same differential equation as the fundamental elasticity tensor N itself. These conclusions, due to Barnett and Lothe (1973, Section 3), extend to any semisimple tensor defined on the unit circle whose angular derivative is a polynomial in the tensor itself.

D. INTEGRALREPRESENTATIONS The general solution of the differential equation (4.18) ist P,(cp) = tan(+,

where

-

(4.20)

cpx

4, is a complex constant, and a second integration yields jorpp,(cp)dcp = log(cos cp

+ tan +,

sin cp).

(4.21)

+

Let 6, = X, i$,. Then Eq. (4.20) and the argument of the logarithm in (4.21) become sin{2(x, - cp)} pu(cp) = cos{2(~,- cp)}

'

+ i sinh 2$, + cosh 2$,

(4.22)

If p,(cp) = ki for some value of cp, then p,(cp) = k i for all cp. Equation (4.25) below thus continues to hold.

Surface Waves in Anisotropic Elastic Materials

3 19

and

cos cp

+ ra sin cp + is,

(4.23)

sin cp,

with ra =

sin 21, cos 21, + cosh 2$, ’

s,

sinh 2t,ha . cos 21, cosh 2$,

= ~-

+

(4.24)

Equation (4.22) reveals that, for all values of cp, Im p,(cp) has the same sign as the real constant t,hu. From (4.23) it can easily be deduced that as cp increases from 0 to 2n the point representing the complex number cos cp + tan +u sin cp makes one circuit of an ellipse in the complex plane, the sense of description being positive when s, > 0 and negative when s, < 0. Since s, has the same sign as and hence Im p,(cp), the logarithm in Eq. (4.21) changes by 2ni sgn Im p,(cp) and we reach the important result

+,,

2n

i, P,(V) d~

= 2ni(

*

)a

(4.25)

9

also due to Barnett and Lothe (1973, Section 3). Here the abbreviated notahas been introduced for sgn Im p,(cp); by virtue of the convention tion ( (4.9), 1 (+)a

=

-1

when when

a = 1, 2, 3, a = 4,

5, 6.

(4.26)

We now return to the eigen-relation (4.4) and integrate each side over the interval 0 cp 5 2n. Using Eq. (4.25) and remembering that the eigenvectors 5, are independent of orientation we obtain %a

= i(+

)a

5,,

(4.27)

where 2x

S = ( 2 ~ ) - Jo N(cp) dcp.

(4.28)

Equation (4.27) tells us that the t a ’ s furnish a complete set of eigenvectors of the real 6-tensor S with associated eigenvalues ( f), i, and it is clear from Eq. (4.2) that the q,’s form a complete set of reciprocal eigenvectors of S. Thus S is a semisimple tensor which, following the same steps as in (4.13), can be given the spectral representation S = i(

*)n

5, O ‘1,.

(4.29)

Equation (4.28) shows that S can be interpreted as the mean value of the fundamental elasticity tensor N(cp) over its domain, the interval 0 5 cp I 2n.

320

P. Chadwick and G. D. Smith

When S is decomposed into its constituent 3-tensors it consequently has the same structure as N ( q ) and we write, in conformity with Eq. (3.18), (4.30) S2 and S3 being symmetric. On decomposing the 6-vectors and 6-tensors in Eqs. (4.27)-(4.29) with the aid of Eqs.(4.14), (4.30), (3.18), (3.19), and (2.14) we obtain from (4.27)

S1aa+S21a=i(f)aaa, S3aa+ST1a=i(f)bla,

(4.31)

and from (4.28) and (4.29)

S1 = i(+),

a,

@T,

-Iff'* Q-'(q)R'(cp) d q ,

= -C1

(4.32)

-42

xi 2

~ ~ = ~ : = i ( ~ ) , a , ~ a , = n ~- -l1j( c p ) d q ,

(4.33)

-4 2

S3 = S: = i( +), 1, @I, = x - l

R(q)Q-'(q)R'(q) d q

- %Q@)+ Q(&)).

(4.34)

In (4.32) to (4.34) the periodicity of N ( q ) ,expressed by Eq. (3.17), ,has been used to halve the interval of integration and modify the integrands, and in (4.34) the integration of Q ( q )has been effected with the aid of Eq. (3.13)1. Equations (4.32)-(4.34), first obtained in this way by Barnett and Lothe (1973, Section 4), may be regarded as the pivotal results of the entire text. They provide representations as definite integrals of tensors which could otherwise be calculated only by solving in full the sextic eigenvalue problem for N. The integrals are analytically as well as cornputationally expedient and form the basis for the investigation of the existence of elastic surface waves described in Section IX. Equations (4.31) also yield relations of cardinal importance in the theory of surface waves. We observe that since the strong ellipticity condition (3.2) guarantees that the acoustical tensor Q ( q )is positive definite, the positive definiteness, and hence the invertibility, of S2 is an immediate consequence of Eq. (4.33)3. It is proved in Section V,C,3, by an indirect method, that S3 is negative definite subject to the strong convexity condition (3.3). In conclusion we notice the further property

s2 = - I

(4.35)

which follows from Eq. (4.27), or alternatively from (4.29) and the completeness relation (4.12). On decomposing this result with the help of

32 1

Surface' Waves in Anisotropic Elastic Materials Eqs. (4.30) and (2.16), we obtain

s: + s2 s3 + I = 0,

s 1

sz + s 2 s: = 0,

s 3 s1

+ s: s3 = 0. (4.36)

In view of the invertibility of S2,noted above, Eq. (4.36), can be rewritten as s-1s 2 1

- - (Si'Sl)*.

(4.37)

E. DEGENERACY OF THE EIGENVALUE PROBLEM

So far in this section it has been supposed that the eigenvalues p,(cp) of the fundamental elasticity tensor N(q) are distinct. If, exceptionally, there is at least one multiple eigenvalue, two types of degeneracy may occur. 1. Semisimple Degeneracy In this case N ( q ) is a semisimple tensor which means that there continue to exist sextuples 6, and q, of linearly independent eigenvectors and reciprocal eigenvectors. As asserted at the end of Section IV,C, the situation regarding the orientation dependence of these vectors and the eigenvalues p , ( q ) is unchanged and the derivation of Eq. (4.18) plainly remains valid. However, a different argument is required thereafter. Suppose, for definiteness, that p 1 has multiplicity 2 and that and k2 are linearly independent associated eigenvectors. Then Eqs. (4.17) and (4.18) jointly imply that 5; and 5; are expressible as linear combinations of and

c1

el

5 2

5; = a t 1 + K 2 ,

5; =

+dg2.

(4.38)

The vector r t 1 + sk2 also belongs to the eigenspace of p1 and can be made independent of orientation by choosing the coefficients r and s to satisfy the ordinary differential equations r'

=

-ar - cs,

s' = - br - ds.

(4.39)

The solutions (rl, sl) and ( r 2 ,s2) of Eqs. (4.39) which satisfy the subsidiary conditions r,(O) = 1,

sl(0) = 0;

r2(0)= 0,

s2(0)= 1,

(4.40)

can easily be shown to have the property

(4.41)

c2

Hence the eigenvectors rl g 1 + s1 and r2 c1 + s2 k2 are linearly independent as well as orientation-free. Similar reasoning shows that there also exist

P . Chadwick and G . D. Smith

322

two linearly independent orientation-free reciprocal eigenvectors associated with pl, and the method clearly extends to an eigenspace of dimension 3. The results of Section 1V.D now follow without modification. 2. Non-Semisimple Degeneracy

When N ( q ) is not a semisimple tensor its eigenvectors no longer span C6 and generalized eigenvectors must be introduced in order to form a basis (Pease, 1965, Chapter 111). An important instance of non-semisimple degeneracy lies at hand in the case of an isotropic elastic material, and we now discuss this example as an illustration of the general state of affairs. The linear elastic moduli of an isotropic material are specified by Eq. (3.4) and the expressions for the acoustical tensors derived from Eqs. (3.7) and (3.10) are Q(q)= pI

+ (A + p)n 0 n,

R ( q ) = Am 0 n

+ pn 0 m,

(4.42)

with the unit vectors m and n given by (3.9). The inversion of Eq. (4.42)1 gives (4.43)

-

0

0

l/p

0 0

0 0

0

0

0

0

0

0

0

0

-w 0+ 2 4 -441

-

+ p)/(A + 2p)

-1

-p

0 0 -1 0

0

+2 4 0 -A@ + 2p) l/(A

0 0

0 0 1/P

0

0 0 (4.45)

The eigenvalues of N(O), calculated from (4.45), are i and -i, each with multiplicity 3. Thus, observing the convention (4.9) and bearing in mind the footnote on p. 318, we can write p 1 = p 2 = p 3 = i,

p4 = p s = p6 = -i,

(4.46)

for all values of cp. It now transpires, with further reference to (4.45), that

323

Surface Waves in Anisotropic Elastic Materials

N(0) has four eigenvectors, 5,, k2,k4, k 5 ,and two generalized eigenvectors 5,, 56, belonging to the chains t,, 5, and g 5 , 56 each of length 2 (Pease, 1965, pp. 76-78). These vectors satisfy the relations N(0)Ga = i 5 a = Sa

Sa+3

and the components of

a = 1,

5

2,

N(0)53 = i 5 3

52

9

a = 192, 3,

3

(4.47)

5,, t2,and 5, relative to the basis (4.44) are, in turn,

+(1 + i ) p - ”’(0, 0,i, 0,0, - p ) ,

y( 1, i, 0,2ip,

- 2p, O),

(4.48)

y ( - i ~ , - K , 0, p, -ip, 0).

Here y = {8p(1 - v ) } - ” ~ ,

K = 4(3 -

(4.49)

4~),

+

where v = @/(A p ) is the Poisson’s ratio of the material and suitable normalizing constants have been introduced (cf. Nishioka and Lothe, 1972b, Section 2). The sa’s given by (4.48) and Eq. (4.47), are easily shown to be linearly independent. The relations satisfied by the reciprocal eigenvectors ql, q 3 , q4, q 6 and the generalized reciprocal eigenvectors q, ,q5 belonging to the inverted chains q3, q2 and q6, q5 are a = 1, 3,

NT(0)qa= - iq. , qm+3

= iia

NT(0)q2= - iq2

+ q,

,

a = 1, 2, 37

9

(4.50)

(Pease, 1965, pp. 89-91). On account of Eqs. (4.47) and (4.3) a solution of (4.50) is provided by ‘11

= KC,,

‘12

=

KE3

tl4

= Kc4 3

‘I5

= K56

q3 = 9

KS2

tl6 = Kc5 ,

(4.51)

and the invertibility of K ensures that the qa’s, like the are linearly independent. A final check confirms that the two sextets of vectors defined by (4.48), (4.47), , and (4.51) are orthogonal to one another in the sense of Eq. (4.8). The expressions for the tensors S,, S,, and S, found by substituting from Eqs. (4.42), (4.43), and (3.9) into the integral representations (4.32), , (4.33), , and (4.34), and performing the integrations are

Sl = -f{(l - 2v)/(1 - v))(elo e2 - e2 S,

= p-’[i{(3

- 4v)/(l - v))(el0 el

oel),

+ e, O e,) + e30 e,],

S , = -p{(l -v)-’(e,@e, + e , O e , ) + e , ~ e , ) .

(4.52) 44.53) (4.54)

If, instead, we treat the ta’s calculated above as genuine eigenvectors, extract

324

P . Chadwick and G . D. Smith

the constituent 3-vectors a, and I, defined in Eq. (4.14)1,and calculate S1, S2, and S3 algebraically from Eqs. (4.32),, (4.33), , and (4.34), , exactly the same results are obtained and Eqs. (4.31) are found to hold. We have thus demonstrated, for the special case of an isotropic elastic material, that the decisive formulas (4.31)-(4.34), appropriately reinterpreted, remain valid even when the fundamental eigenvalue problem is subject to non-semisimple degeneracy. We now go on to show that this conclusion is true generally, the method of proof being due to Lothe and Barnett (1976a, Section IV). For simplicity we consider once more the situation in which there are four eigenvectors cl, 5,, 5, , g5 and two generalized eigenvectors g3, 5 6 belonging to the chains t2,g3 and g 5 , k 6 . Equation (4.18)applies to the repeated eigenvalue p 2 and since the eigenspace associated with p2 contains only the one authentic eigenvector k2, we are justified by the steps leading to Eq. (4.19)in taking to be independent of orientation. The generalized eigenvectors must be presumed to vary with cp, whence, for the chain g 2 ,

c2

c3,

N(cpK2 = P2(cpK2

3

N(cpK3(cp) = P,(cpKdcp)

+52.

(4.55)

On differentiating Eq. (4.55)2with respect to cp, employing the differential Eqs. (3.23) and (4.18),and simplifying with the aid of Eqs. (4.55) we find that N(cpK3cp)

= P,(cpK>(cp)

+ 2P,(rP)52.

(4.56)

With reference to Eq. (4.55), we infer from this result that (4.57) 2P2(cp)63(cp) + fi((Pk2 where B is a disposable function. The connectiont 2p2(cp)= -p;(cp)/p;(cp), supplied by Eq. (4.18), enables us to integrate (4.57) in the form c3(cp) =

Pi(cpK3(cp) = Pi(OK3(0)

9

+ jSB(cp)P;(44

dV52

7

(4.58)

0

and at this point we may exercise our freedom over the choice of /? by discarding the final integral. Equations and (4.58) can then be combined to give N((p)53(0)

= PI('?k3(O)

+ P;(cp){P;(o))-

'52

9

(4.59)

whereupon integration over the interval 0 I cp I 2n leads to the relation S s 3 ( 0 ) = i53(0), (4.60) appeal being made to Eqs. (4.25) and (4.28)and to the single-valuedness of p2(cp). A similar procedure, applied to the chain t 5 ,g6, delivers the analo-

' The possibility of p2 being equal to i is disregarded here.

Surface Waves in Anisotropic Elastic Materials

325

gous result (4.61) It is now evident that Eq. (4.27) continues to hold with the generalized eigenvectors evaluated at cp = 0' replacing the eigenvectors that have disappeared as a result of the degeneracy. Since g3(O)and C6(O) form with the four eigenvectors a basis of C 6 , Eqs. (4.29)-(4.37) follow as before.

V. Plane Elastostatics

We turn now to the study of small static deformations from the natural state N of an infinite body composed of the anisotropic elastic material specified in Section II1,A. The deformations to be considered are plane in the sense that the displacement, and hence the stress, is the same in all planes parallel to a reference plane R The general solution of this class of problems was first obtained by Eshelby et al. (1953) and applied by them to the determination of the displacement and stress fields due to a straight dislocation acted on by a concentrated body force. Extensions to the analysis of such a line singularity were subsequently made by Stroh (1958, Section 4; 1962, Section 6). In Sections V,A and B the main results of these authors are rederived within the framework of the fundamental eigenvalue problem. We then discuss in Section V,C some particular properties of the solution for a line singularity, notably the calculation of the total strain energy, originally carried out in a different manner by Stroh (1958, Section 4). By this route we are led to important information about the 3-tensors S , and S3 introduced in Section IV,D. A. GENERAL SOLUTION As in Section III,B we select a real orthonormal basis e = {el, e 2 ,e3}with e3 normal to the reference plane R. The associated rectangular Cartesian coordinates are denoted by xl,x2,x3 and the comma notation is employed for partial derivatives with respect to these variables. In relation to the basis e the constitutive and equilibrium equations are in turn

The vectors g3 and g6 in our earlier discussion of isotropic degeneracy are, in the present notation, t3(0)and c6(0).

326

P . Chadwick and G. D. Smith

and

Tpi, p = 0, (5.2 1 where ui and T j are the components of the displacement vector u and the stress tensor T and it is assumed that no distributed body forces act. The plane property of the deformation relative to R means that the displacement components ui,and hence, from Eq. (5.1), the stress components T j , are functions of x1 and x2 only. Equation (5.2) accordingly reduces to Tli, 1 + T2i, 2 = 0. It can easily be verified that the stress vector t, defined by XI

ti(X1, ~ 2 =) J”

a2) d t -

~2i(53

a1

J

(5.3)

xz

Tli(X1r a) d ~ ,

(5.4)

a2

where a, and a2 are arbitrary real constants, generates the stress compoand T2i through the relations nents TIi

T1i = - t i , 2 T2i = ti, 1, (5.5) and thereby ensures that the equilibrium equation (5.3) is satisfied. In addition the resultant force per unit width on the surface swept out by translating parallel to e3 a simple arc A in the reference plane R is given byt 3

PI

(5.6)

A

(Eshelby et al., 1953, Section 3), and the strain-energy density by $Cpqrs(Up, q

+ uq, p ) ( u , s + us. r ) = H t p , 1 u p , 2 - t p , 2 u p , 1 ).

(5.7)

The elimination of the stress components between Eqs. (5.5) and (5.1) yields two relations among the first partial derivatives of u and t which can be written in direct notation as

R(O)u, I - Q($)u,

2

+ t , 1 = 0,

- Q ( o ) ~ , -i R ( + x ) u , ~ - t , 2 = 0 , use being made of Eqs. (3.12). Introducing the 6-vector

(5.8)

(u, t>, (5.9) made up from the displacement and stress vectors, and utilizing the definitions (2.17), and (3.14) we can express these relations in the sextic form 0 =

JM(O)O,1 - M(+n)O,2

(5.10)

= 0.

Finally, on premultiplying each term by M-

(4~)and

recalling the

’ That is, the variation oft between the end points as A is positively described.

Surface Waves in Anisotropic Elastic Materials

327

definition (3.16) we obtain N(O)o,, - 0 , 2 = 0 .

(5.1 1)

Equation (5.11) exhibits the fundamental role of the 6-tensor N ( q ) in plane linearized elastostatics. Since the eigenvalues p,(O) of N(0) are complex, the first-order system of linear partial differential equations for the components of o presented by (5.11) is elliptic in type. The general solution is

=f& + Pn(O)X2)5,,

(5.12)

where thef,’s are arbitrary functions (cf. Eshelby et al., 1953, Section 4). The eigenvectors 5, of N ( q ) are taken to be normalized, and we revert to our earlier assumption that the eigenvalues p&) are distinct. B. A STATIONARY LINESINGULARITY The general solution (5.12) is now used to derive the displacement and stress fields due to a line singularity occupying the x 3 axis. The singularity is characterized as a dislocation line which is also the site of a concentrated body force. The Burgers vector b of the dislocation, assumed constant but not restricted in orientation, is, by definition, the change in displacement accompanying one circuit in the positive sense of a simple closed curve C situated in the reference plane R and threaded by the singularity. It follows from (5.6) that the localized body force f per unit length, likewise supposed constant, is equal to the associated change in the stress vector t. Hence, introducing the 6-vector

B = (b,

f>,

(5.13)

representing the strength of the singularity, and bearing in mind the definition (5.9) we obtain the condition

B =[4c.

(5.14)

The displacement distribution in R evidently has a branch point at the origin, but the stress, and hence the displacement gradient, must be singlevalued and decay toward zero with increasing distance from the singularity. These requirements are met by choosing the functions fa in the general solution (5.12) to be

fn(z)= (2ni)-’c, log z,

(5.15)

where the c,’s are disposable complex constants (Eshelby et a!., 1953, Section

328

P. Chadwick and G . D . Smith

5). The displacement-stress vector is then given by

o = (27ri)-'c, log(x,

+ px(0)x2Kx,

(5.16)

and it remains to fix the c,'s with reference to Eq. (5.14). The argument used in Section IV,D in deriving the integral formula (4.25) shows that as the point (xl, x2) in R makes a positive circuit of the unit circle x: = 1, the point in the complex plane representing x1 p,(O)x, traces x: out an ellipse, the sense of description being positive if Im p,(O) > 0 and negative if Im ~ ~ (<00.)We therefore deduce from Eq. (5.16), using (5.14) and (4.26), that

+

+

B=

(5.17)

(f)nC,5.,

and on forming the inner product of each side with the reciprocal eigenvector q, and appealing to the orthogonality relation (4.8) we find that (5.18) 11,) (cf. Stroh, 1962, Section 6). Combining Eqs. (5.16) and (5.18) and recalling the algebraic relation (2.15), we can express the solution for a stationary line singularity in the form c, = (k

M

B

9

a = FB,

(5.19)

F(X1, x2) = (2ni)- '(k). log(x1 + Pn(0)X2K, 0 tl, .

(5.20)

where The properties (4.9) of the eigenvalues and eigenvectors of N ( q ) ensure that the 6-tensor F is real. C . PROPERTIES OF THE SOLUTION

1. The Displacement Gradient and Stress Fields In order to determine the components of displacement gradient and stress the partial derivatives of the 6-tensor F must be found. On differentiating Eq. (5.20) with respect to x1 and involving the orthogonality relation (4.8) we obtain F, 1

=

(2ni)-'(k)p(xi

+ ~,(0)~2)-'(Sp

9

q&=Otlp

(5.21) '{(XI + Prr(0)X2)F'5, 0 tlJN+-) p 5 p 0 s p > . In view of the spectral representations (4.13) and (4.29) of N ( q ) and S it follows that =-

+

F,1 = - ( 2 ~ ) - ' { ~I1

~2

N(O))-'S,

(5.22)

Surface Waves in Anisotropic Elastic Materials

329

and the partial differential equation (5.11) provides the companion result

+ x2 N(O))-'S. (5.23) Since N(0) has no real eigenvalues the 6-tensor x1 I + x2 N(0) is invertible F,2= N(O)F, = -(27~)-lN(O){x~ I

provided that x, and x2 are not both zero. Equations (5.22) and (5.23) are therefore valid at all points of the reference plane R except the origin. The nonzero components of displacement gradient and the components of the stress vectors acting on the coordinate planes orthogonal to R are the components of the 6-vectors p = (u, 1, u , ~ )

and

t=

(T'e,, TTe2)= ( - t , 2 , t . l). (5.24)

A simple calculation, drawing on Eqs. (2.19) and (5.9), gives p+t=o,,-Jo,2,

p-~=JKo,l+Ko,2,

(5.25)

and with the aid of the solution (5.19) and the formulas (5.22) and (5.23) we find that

+ JK - (J - K)N(O)}{x, I + t = -(47~)-'{l - JK - (J + K)N(O)}(x, I +

p = -(47~)-'{l

~2

N(O)}-'Sfl,

~2

N(O)}-'SP. (5.26)

The tensor S in Eqs. (5.26) can be determined either algebraically from Eq. (4.29) or by integration from (4.28). As pointed out in Section IV,D, the latter method is computationally the more convenient since the solution of the fundamental eigenvalue problem is then circumvented. An alternative integral representation of the displacement gradient components ui. has been derived by Barnett and Swanger (1971, Section 2) and subsequently simplified by Barnett and Lothe (1973, Section 4; see also Asaro ez al., 1973, Sections 2 and 3; Mura, 1975). The method of Barnett and Swanger, which couples an integral formula of Mura (1963, Section 2) for the displacement gradient field of a dislocation loop with the Fourier integral representation of the elastostatic Green's tensor for an infinite region, provides an alternative path to the key results (4.32) to (4.34).' The relationship of the Green's tensor to the solution (5.19) and (5.20) is touched on in Section V,C,4. 2. The Total Strain Energy The property of the line singularity which bears most directly on the theory of elastic surface waves is the total strain energy E per unit length, found by integrating the energy density (5.7). As is well known (see, for Historically, this means of representing the tensors S , , S,, and S, [as defined by Eqs. (4.32),, (4.33), , and (4.34),] in integral form preceded the direct approach of Barnett and Lothe (1973) described in Section IV,D.

330

P . Chadwick and G . D. Smith

example, Lardner, 1974, p. 73; Steeds, 1973), a convergent result is obtained only if the region of integration in the reference plane R is limited to a circular annulus by the imposition of inner and outer cutoff radii r and R. Thus 1 {(o,2 E = 4 r 4 (x12 + x 2 2 ) 1 1 2 j R

j-j-

7

h.1)

- (w.1,

Jo.2))

d x , dx2

9

(5.27)

where the definitions (5.9) and (2.17)1 have been used to write the second expression in (5.7) in terms of o. To calculate E from Eq. (5.27) we first substitute for the partial derivatives of o from Eqs. (5.19), (5.22), and (5.23) and next transform to plane polar coordinates (w, 8). The integration with respect to a can be carried out immediately to yield E = (167~')-'(SO, ESP) ln(R/r),

(5.28)

where 271

E=

j

CT(6)(NT(0)J- JN(O)}C(B)do,

C(8) = {COS81 + sin 8N(O)}-'

0

(5.29) The integral (5.29) is evaluated by putting the 6-tensor C(8) into its spectral form, with the aid of Eqs. (4.12) and (4.13), and applying the formula ~02n{cos 8

+ ~ ~ (sin0 6}-'{cos ) 8 + ps(0) sin 8}-'

d8

*

*

(5.30) '{( )Q - ( (Stroh, 1962, Section 7). With reference to Eqs. (2.12), (4.4), (4.12), and (4.29) we can express the outcome as = 27Ci{pQ(o) - ps(')}-

E = 21~( STJ - JS).

(5.31)

Returning to Eq. (5.28) and remembering the relations (2.9), (2.18)3, and (4.35), we then arrive at the final result E

=

(47~)- (4,SS)ln(W),

(5.32)

or, using the decompositions (2.17)', (4.30), and (5.13),

E = (41~)-'( - (b, S3 b)

+ (f, Szf)} ln(R/r).

(5.33)

We observe that there is no strain energy of interaction between the line dislocation and the concentrated body force (cf. Eshelby, 1956, Section 5a). If the linear elasticity tensor of the material satisfies the strong convexity

33 1

Surface Waves in Anisotropic Elastic Materials

condition (3.3) the first of the expressions (5.7) for the strain-energy density is necessarily positive and it follows that E > 0 for all nonzero real 6-vectors a.Hence, from Eq. (5.33), S3 is negative definite and S2 positive definite. In fact, as already proved in Section IV,D, the weaker condition (3.2) suffices for the positive definiteness of S 2 and considerations relating to a line singularity need not be invoked in establishing this property. No direct proof of the negative definiteness of S3has yet been given, and the special case of an isotropic elastic material serves to demonstrate that the strong ellipticity condition (3.2) is not stringent enough. The intervals to which the Poisson's ratio v is confined by the strong ellipticity and strong convexity conditions are found, with the aid of the inequalities (3.5) and (3.6),to be v < v > 1, and - 1 < v < respectively. Inspection of Eq. (4.54) duly confirms that the first of these restrictions does not guarantee the negativity of the eigenvalues of S3.

4,

4,

3. The Energy of a Stationary Line Dislocation The total strain energy per unit length of a stationary line dislocation, obtained from Eq. (5.33) by taking the concentrated body force f to be zero, is E

=

-(4n)-'(b,

S3b) ln(R/r).

(5.34)

The integral representation

-&(h {Q(o)+ Q(tn)P))W / r ) 2

(5.35)

provided by Eq. (4.34), , was first obtained, in a less transparent form, by Barnett and Swanger (1971, Section 2), using the technique mentioned at the end of Section V,C,l. In the case of an isotropic elastic material the explicit formula for E found by substituting the expression (4.54) for S3 into Eq. (5.34) yields well-known formulas for the energies of edge and screw dislocations (see, for example, Lardner, 1974, pp. 72 and 78). 4. The Infinite Region Green's Tensor

In the absence of a dislocation the total strain energy due to a concentrated body force is given by Eq. (5.33) as

E = (4n)-'(f, S2 f ) In(R/r)

(5.36)

per unit length. This result suggests a close connection between S2 and the

332

P . Chadwick and G . D . Smith

infinite region Green's tensor G(x, x') of the anisotropic elastic material under consideration, and on looking back to the integral formula (4.33), for S 2 , we identify in the right-hand member the definite integral appearing in a representation of G(x, x') first obtained by Synge (1957). Thus G(x, x') = { 1 6 d ( ~ x', x - x ' ) ) " ' ~ S ~ ,

(5.37)

where the reference plane for the determination of Szis the plane orthogonal to the vector x - x' joining the source to the observer. Equation (4.33), furnishes the alternative representation G(x, x') = (16n2(x - x', x - x')}"''i(

+)n

a, @an,

(5.38)

due to Malen (1971). Again it is understood that the eigenvalue problem from which the 3-vectors a, are derived is referred to the plane orthogonal to x - X'.+

Equation (5.37), in combination with (4.53),reproduces the familiar formula of Kelvin for the infinite region elastostatic Green's tensor of an isotropic material (see, for example, Lardner, 1974, p. 53). VI. A Uniformly Moving Line Singularity

It was first recognized by Stroh (1962) that the analysis of a uniformly translating line singularity supplies a link between the plane elastostatic problem studied in Section V and the theory of elastic surface waves. We describe in Section VI,A the modifications of the static solution needed to accommodate steady motion of the singularity, and it quickly emerges that the range of translation speeds for which the amended solution is valid has an upper bound. Mathematically this limiting speed is associated with the merging of one or more conjugate pairs of complex eigenvalues of the modified fundamental elasticity tensor into one or more repeated real eigenvalues, with the consequent departure from ellipticity of the governing system of partial differential equations. In physical terms the limiting speed marks the transition from subsonic to supersonic motion. At the transonic stage there appear discontinuities in the displacement field corresponding to the emission of one or more elastic body waves and, in normal circumstances, the divergence of the Lagrangian function of the moving singularity. The Green's function for a semi-infinite anisotropic elastic body, subjected to point loading on its plane boundary but otherwise traction-free, has been considered by Barnett and Lothe (1975a, Section 7). The expression for the surface displacement obtained in their paper involves the tensors S, and S , , regarded now as functions of an angle specifying the orientation of the reference plane. +

Surface Waves in Anisotropic Elastic Materials

333

This behavior is discussed in Section VI,B and we then present in Section VI,C a simple geometrical interpretation of the transonic condition in terms of the slowness surface of the elastic material bearing the singularity. The involvement of the slowness surface in this way leads naturally to a sixfold classification of the possible transonic states. In Section VI,D we establish crucial properties of the 3-tensors S,(u) and S,(u), which are the modified forms of S2 and S 3 .Lastly, to illustrate some main features of the preceding developments, we consider in Section VI,E the special case of a hexagonal material having its zonal axis normal to the reference plane. A. MODIFICATION OF THE ELASTOSTATIC SOLUTION

The line singularity introduced in Section V is now assumed to travel through the ambient anisotropic elastic medium, moving parallel to itself in the plane x2 = 0 with constant speed u. The resulting displacement and stress fields are of the form and T(x, - ut, x~), (6.1) and in place of the equilibrium equation (5.3) we have the equation of motion

u(xl - ut, x2)

+

Tli,1 T,i, = p P U i / d t 2 , (6.2) where p is the uniform density of the material in the natural state N. Distributed body forces are again supposed absent. The relation (6.31 for the velocity components, afforded by (6.l),, allows us to deduce from Eq. (6.2) the existence of a stress vector t(x, - ut, x2) such that ui

= a u i p t = -uui,

1

Tli + puu, = -ti, 2 , TZi= ti,1. (6-4) These equations replace (5.5), and the concentrated body force acting on unit length of the singularity is again equal to the variation o f t round a simple closed curve in the reference plane R encircling the singularity (Stroh, 1962, Section 2). Furthermore, on multiplying each side of Eq. (6.2) by ui, summing on the repeated index, and using Eqs. (6.3) and (6.4) we find, after some manipulation, that the right-hand side of Eq. (5.7) now represents the negative of the Lagrangian per unit volume, i.e., the strain-energy density minus the kinetic energy density +pup up (cf. Malen and Lothe, 1970, Section 4.3). The relations between the partial derivatives of u(xl - ut, x2) and t(xl - ur, x2) obtained by using Eqs. (6.4), with (6.3), to eliminate the stress components from the constitutive equations (5.1) are of the same form as

334

P. Chadwick and G . D.Smith

Eqs. (5.8), but with the acoustical tensors Q(cp)and R(q) replaced byt and

Q(cp,u ) = Q(cp)- pu2 (30s’ cpI

R(cp, u ) = R(cp) - p d sin cp cos cpI. (6.5) The recipes (3.14) and (3.16), applied to Q(q, u ) and R(q, u), yield the modified 6-tensors M(cp, u ) and N(p, u), and we follow systematically the practice of indicating explicitly the dependence on u of quantities derived from the modified acoustical tensors (6.5).Thus the eigenvalues of N(cp,v) are denoted by p,(cp, u) and associated normalized eigenvectors by S.(u). The converted forms of the basic partial differential system (5.11)and the general solution (5.12) are in turn N(0, 0 ) W . I - 0 . 2 = 0

(6.6)

and

- ut, x,),

o(xl - ut, x2) =
+ P,(O,

t(x,

- ut, x,))

(6.7) it being assumed, as in Section V,A, that the eigenvalues are distinct. To summarize: the principal differences between the solutions appropriate to stationary and steadily moving line singularities in an elastic body concern (a) the stress relations, where Eqs. (5.5) give way to (6.4), (b) the fundamental elasticity tensor, modified to N(cp, u ) from N(cp) (= N(cp, 0)), and (c) the meaning of the right-hand member of (5.7) which changes from the strain-energy density to minus the density of the Lagrangian function. It can easily be verified, with reference to the definitions (6.5), that Q(q,u ) is symmetric and that the properties (3.11) and (3.13) of Q(cp)and R(cp) apply also to the modified acousticad tensors Q(cp,u ) and R(p, u). Moreover, Q(cp,u ) has the same eigenvectorsas Q(cp), and its eigenvalues are smaller by pu’ cos’ cp than the eigenvalues of Q(cp) which, on account of the strong ellipticity condition (3.2), are positive. Hence, for any given value of cp, there is an upper bound to the values of u for which Q(p, u) is invertible. For all nonnegative values of u falling below this bound we are evidently justified by the closing remark of Section I11 in extending to N(cp, u ) the theory of the fundamental eigenvalue problem developed in Section IV. In referring henceforth to results in Sections I11 and IV derived from the modified acoustical tensors we adopt the device of placing an asterisk on the equation number. For example, the modified spectral representation = f h 1

N(cp9

V)

= P,(cp,

%&)

.)x2

- ut)s,(u),

0 %(.)

is cited as Eq. (4.13)*. + It is found initially that Q(0) is replaced by Q(0) - pu21 in Eqs. (5.8) while the other three tensors are unaltered. On carrying this change into Eqs. (3.13) we obtain (6.5).

Surface Waves in Anisotropic Elastic Materials

On setting cp

= 0,

335

8 = cp in Eq. (4.1)* we find, with the use of (6.5j1, that

det{N(O, u ) - tan cpl}

= sec6 cp{det

Q(+z)}-l det Q(q, 0 ) .

(6.8) This identity shows that if Q(cp, u ) is singular, then tan cp is a real eigenvalue of N(0, tl), the 6-tensor controlling the central differential equation (6.6).The validity of the general solution (6.7) is accordingly restricted to the range 0 I v < i, of translation speeds, where i, is the least of the above-mentioned upper bounds. The significance of the limiting speed i, and the manner in which the solution breaks down when u = i are discussed in Sections VI,B and C . Meanwhile it should be noted that, apart from the remark about the concentrated body force in the sentence following equation (6.4), all the results of the present section apply to any plane motion of an anisotropic elastic body excited by uniformly moving surface loads and concentrated extrinsic forces.

B. THELIMITINGSPEEDi, 1. Elastic Body Waves

In preparation for the further study of the translating line singularity we review here some basic facts about the propagation of body waves in anisotropic elastic materials. The behavior of such waves is determined by the acoustical tensor Q(n),the real unit vector n defining the wave normal. Since Q(n) is symmetric and, through the strong ellipticity condition, positive definite, it has three positive real eigenvalues and an orthonormal triad of real eigenvectors. These adjuncts of Q(n) correspond to the three body waves which the material can transmit in the arbitrary direction n, the eigenvalues specifying the speeds of propagation and the eigenvectors the acoustical axes, that is, the directions in which the wave amplitudes can lie. When attention is confined to directions of propagation in the reference plane R, the operative acoustical tensor is Q(cp). The speeds c,(cp)of the three body waves which can travel in the direction associated with the orientation angle cp (see Fig. 1) are then the positive real roots of the secular equation det{Q(cp) - pc2(cp)I}= 0, and we adopt the convention Cl(cp)

CI(cp)

5

c3(cp)

(6.9) (6.10)

in labeling these wave speeds. The associated unit eigenvectors of Q(q) are denoted by qi(cp). Thus

Q(v)Qi(q)= P ~ (q)Qi(q Z ).

(6.11)

336

P . Chadwick and G. D. Smith

2. Dejnition and Mathematical Sign$cance of 6 It now follows from Eq. (6.5), that the eigenvalues of the modified acoustical tensor Q(cp,u ) are P{CZ(cp)

(6.12)

- u2 cos2 cp},

and that the qi(cp)’s are eigenvectors of Q(cp,u). As u increases from zero, Q(q, u ) first becomes singular when u = cl(cp) I sec cp 1. The critical speed ir is therefore defined by

ir =

min

cl(cp) sec cp

(6.13)

- n/2 < (p < n/2

(cf. Barnett et al., 1973b, Section 5), the restriction of the range of cp being warranted by the periodic property (3.11), of the acoustical tensor. The value’ 4 of cp at which the minimum is attained is referred to as the critical angle. Thus

6 = c1(4) sec 4

(6.14)

and (Wcp){c1(cp)sec cp) = 0,

(d2/dcp2){Cl(cp) sec cp) > 0

at cp

=

4. (6.15)

For each speed u in the interval [0, ir), Q(cp,u ) is positive definite for all values of cp and Eq. (6.8) affirms that N(0, u ) has no real eigenvalues. The partial differential system (6.6) is accordingly elliptic and the general solution (6.7) holds good. As u 6and cp -,q? we deduce from Eq. (6.8) that onet of the conjugate pairs of complex eigenvalues of N(0, u ) approaches tan @ which is hence a repeated real eigenvalue of N(0, C). Specifically, recalling the convention (4.9)*, pl(O, b) = p4(0, b) = tan

4.

(6.16)

In the limit u 7 b, marking the breakdown of the solution, the elliptic property of the governing system (6.6) is therefore lost. 3. The Limiting Body Wave At speeds of translation below (above) the limiting value the motion of the singularity may be aptly termed subsonic (supersonic),respectively. We refer to the intervening situation, when u = 6, as the transonic state.

’

If there are two or three such angles the notation qjl, d2 or dl, q j 2 , q j 3 is used. For simplicity we suppose throughout Section VI,B that there is a single critical angle and that N(0, i) has just one pair of (coincident) real eigenvalues: in the terminology introduced later, the transonic state is assumed to be of Type 1.

Surface Waves in Anisotropic Elastic Materials

337

In order to gain further insight into the nature of the transition we utilize the eigen-relation

"0,

U ) W ) =

PU(O9

45.(u)

(6.17)

which, by virtue of equation (3.16)*, can be rewritten as JM(O, u ) L ( u ) = ~ ~ (u 0) M, ( h u ) S u ( ~ ) .

(6.18)

On decomposing with the aid of Eqs. (2.17)1, (3.14)*, and (4.14): and making use also of (3.1 1)2 and (6.5), we obtain

{ - ~ ~ (u)Q(&) 0 , + R(O)}a,(u) + IAu) = 0, { - Q(0) + ~ ( 0u)RT(0) , + pu21}a,(u) - ~ ~ (u )0~ ,,( u= ) 0, (6.19) whence, by rearranging, [pa'(O, ~ ) Q ( $ R )- ~ ~ (u){R(O) 0 ,

(1

+ Pa'(0, V ) ) L ( ~ ) = [

P m

+ RT(0)} + Q(0)- pu21]aa(v)= 0,

UPT@)

- ~ ~ (u){Q(o) 0 ,

- Q($R)

- pv2I) - R(0)IaAu).

(6.20) On setting a = 1 (or a = 4) in Eqs. (6.20) then proceeding to the transonic limit u 1 6 we conclude, with the help of Eqs. (6.16), (3.13), and (6.5), that

Q(@,b)al(6)

= 0,

11(6) = -R(@, 6)al(b).

(6.21)

The first of these relations states that al(6) is an eigenvector of Q(@,6) corresponding to the zero eigenvalue arising, in the limit u 6, cp + @, from the i = 1 member of the set (6.12).We already know, however, that q 1 @ )is a unit eigenvector of Q(@,ij) associated with this particular eigenvalue. Hence

(6.22) al(6) = a&) = a s l ( @ ) , where, without loss of generality, the nonzero scalar multiplier a can be taken to be real. Equation (6.21), then yields (6.23) 11(6) = 14(6) = -aR(@, 6 ) q l ( @ ) (cf. Barnett and Lothe, 1974a, Section 4). From Eqs. (6.14), (6.16), and (6.22) we now see that in the transonic state the R = 1 and R = 4 terms in the displacement distribution supplied by the general solution (6.7) take the form f(sec @{xl cos @

+ x2 sin @ - c l( @)t} ) q l( @).

(6.24)

This expression represents a plane body wave traveling with speed cl(@) in the direction defined by the critical angle @. The breakdown of the solution

338

P . Chadwick and G . D. Smith

is therefore characterized by the appearance of a limiting body waue and since, in the case of a moving line singularity, the function f in (6.24) is logarithmic [see Eq. (6.45) below] the associated displacement is discontinuous. This behavior was first demonstrated, at the present level of generality, by Stroh (1962, Section 6) whose paper should be consulted for further details and references to earlier work. From Eqs. (6.3) and (6.4), the component of the energy flux vector j = Tv in the x2 direction is given by (6.25) j 2 = (i, e2) = - ofp, up, For the limiting body wave, reference to Eq. (6.7) shows that j 2 is proportional to (a,@), Il(ij)). We now prove that

(al(;),

=

- a 2 ( q l ( @ ) ,R@, ;)q,(@)) = 0,

(6.26)

thus verifying that the flux of energy associated with the limiting body wave is everywhere parallel to the plane of motion of the line singularity (cf. Stroh, 1962, Section 5). The argument proceeds from the differentiation with respect to p of the eigen-relation (6.1 1) (cf. Barnett and Lothe, 1974a, Section 3). On forming the inner product with q i ( p )of each term in the differentiated equation and making use of Eq. (3.20), and (6.11) itself we find that

(qi(p),R ( p h i ( p ) )= -Pci((P)(d/&)ci((P).

(6.27)

Hence, employing the definition (6.5), and setting i = 1,

(41((P),R(4b

Uhl(Cp))= --P

cos db’ - c : ( d sec2 d sin cp

+ c~(cP)(~&Kc~(cP)sec ~ ~ 1 1 - (6.28) It is plain, from Eqs. (6.14) and (6.15)1, that the right-hand side of (6.28) is zero in the transonic limit u ij, p + @, and we then reach Eqs. (6.26) on appealing to (6.22) and (6.23). 4. Behauior of the Lagrangian Function It may be anticipated that the development in the transonic state of a singular surface in the displacement field will be accompanied by an unbounded increase in the strain energy of the motion. We examine briefly the modified form of Eq. (5.33) which reads

-Yu) = ( 4 ~ )I{- - (b, S3(u)b)+ (f, S,(u)f)} ln(R/r).

(6.29)

Here L(u) is the Lagrangian function per unit length of the singularity and the 3-tensors S,(u) and S,(u) have the integral representations (4.33): and S,(U) = II-

R(v, u)Q-’(p?~)R*(v,v ) d p - ${(a@)+ Q(&) - PU’I},

(6.30)

Surface Waues in Anisotropic Elastic Materials

339

the latter obtained from Eq. (4.34): with the use of the definition (6.5),.The integrals in Eqs. (4.33); and (6.30) obviously exist if Q(cp, u ) is invertible for all values of and this requirement again entails the limitation on the allowable range of values of u set by Eq. (6.13). In the transonic limit u i, the integrals will normally diverge and, for all choices of the Burgers vector b and the concentrated line force f in Eq. (6.29), -L(u) will then tend to infinity. The related conclusion that the total energy is unbounded in the transonic state was first reached, by different reasoning, by Stroh (1962, Section 7) and Teutonic0 (1962, Section 111). A careful study of the behavior of S,(u) in the limit u 7 i, is made in Section IX,B.

r

C . POSSIBLE MODESOF BREAKDOWN OF THE SOLUTION 1. The Slowness Section

We now enlarge upon our earlier account of elastic body waves by introducing the slowness section of the material, arrived at by the following construction. On the radius vector in the reference plane defined by the unit vector n (see Fig. 1) mark off the three points distant c; '(q)from the origin. As cp traverses an interval of length 2n these points trace out three simple closed curves in R which, because of the periodicity of the acoustical tensor [Eq. (3.1 l)'] are centrosymmetric with respect to the origin. The slowness sectiont so formed is a curve of degree 6, which means that no straight line is R can cut it in more than six points. The constituent curves of the section may intersect one another, but if the innermost branch, associated with the fastest body wave speed c3(cp),is disjoint it is necessarily convex. The branch corresponding to the slowest speed cl(cp) defines the silhouette of the slowness section and is referred to as the outer profile.

2. Geometrical Interpretation of the Transonic State From the foregoing details it is evident that, for -fn < cp < fn, c ; '(cp) cos cp is the projection on the half-line cp = 0 of a point on the outer profile of the slowness section situated in the right half of R According to the definition (6.13)the limiting slowness 6-l is the absolute maximum of the set of such projections. Thus if we visualize a line L parallel to the x2 axis approaching the slowness section from the right the first contact which L makes with the outer profile fixes the limiting speed i, and the critical angle (or angles) at which a body wave is emitted in the transonic state. The slowness section may be viewed as the intersection of the slowness surface of the material with the reference plane R. The quoted properties of the slowness section thus follow from the general theory of the slowness surface (Duff, 1960, Section 2 ; Musgrave, 1970, Chap ters 6 and 7).

340

P . Chadwick and G. D. Smith

The energy flux vector of an elastic body wave is aligned with the normal to the slowness surface at the point representing the wave (Schouten, 1954; Synge, 1956b, Section 4). The fact that the normal to the slowness section at a point of contact with L is parallel to the x 1 axis therefore reflects the property, proved earlier, that a limiting body wave transmits no energy in the x 2 direction. It can be seen from Eqs. (6.8), (6.5),, and (6.9) that the intersection of the movable line L with the slowness section determines the six eigenvalues of N(0, u), and this observation also accords with our previous findings. When L is tangent once to the outer profile and touches neither of the other branches of the slowness section, the only real intersection is a double point corresponding to the coincident real eigenvalues (6.16). When L lies to the right of this position there is no real intersection, implying that all the eigenvalues are complex. In general, owing to the sextic character of the slowness section, L can touch the outer profile once, twice, or three times, and two or all three of the constituent branches may be tangent to one another at a point of contact. We can therefore distinguish six possible modes of breakdown of the solution (6.7) (cf. Malen, 1970, Section V; Barnett and Lothe, 1974, Sections 2 and 5). Descriptions of the associated transonic states now follow and illustrative diagrams appear in Fig. 2. 3. Classijcation of Possible Transonic States a. Type I . L touches the outer profile once, at a point belonging to just one branch of the slowness section. Thus there is a single critical angle @ and

(6.31) < c2(@) 5 c3(4). The form of the outer profile near cp = @ (see Fig. 2a) clearly bears out the behavior of c,(cp) described by Eqs. (6.14) and (6.15). One limiting body wave is excited in this case. Its speed of propagation is cl(@) and its amplitude is aligned with the unit vector ql(@). b. Type 2 . L touches the outer profile once, at a point belonging to two branches of the slowness section (Fig. 2b). As for a transonic state of Type 1, there is only one critical angle and Eqs. (6.14) and (6.15) again apply. Instead of (6.31) we have c1(@)

Cl(4)

= c2(4) < c 3 ( @ ) ,

(6.32)

indicating that two limiting body waves with equal speeds now emerge.' Their amplitudes are polarized in the orthogonal directions defined by ql(@) and q2(@).

' The number of real eigenvalues of N(O, i) is 2L, where L is the number of limiting body waves. The number of distinct real eigenvalues equals the number of critical angles.

Surface Waves in Anisotropic Elastic Materials

(e)

34 1

(f)

FIG.2. Slowness sections illustrating the six possible types of transonic states associated with the breakdown of the solution for a uniformly moving line dislocation.

342

P. Chadwick and G . D. Smith

c. Type 3. L touches the outer profile once, at a point belonging to all three branches of the slowness section (Fig. 2c). Equations (6.14) and (6.15) continue to hold, but now Cl(@)

(6.33)

= c2(@)= c 3 ( @ ) .

Three limiting body waves with equal speeds therefore originate. Since Q(@) is spherical their amplitudes can lie in any three mutually orthogonal directions. d . Type 4 . L touches the outer profile twice, each point of contact belonging to a single branch of the slowness section (Fig. 2d). There are two critical angles, @ 1 and G 2 , and

6 = C l ( @ A ) sec @ A

9

Cl(@A)

< CZ(@A) 5 c 3(@A)3

A = 1, 2. (6.34)

Equations (6.15) also hold at both cp = C1 and cp = @ 2 . Two limiting body waves arise. Their speeds are ~ ~ (and 4 c1(G2) ~ ) and their amplitudes are aligned with the unit vectors ql(@l)and ql(@2),respectively. e. Type 5 . L touches the outer profile twice, one point of contact belonging to a single branch of the slowness section and the other to two branches (Fig. 2e). As for a state of Type 4 there are two critical angles. Equation (6.34), again applies and (6.15) holds at cp = C1 and cp = 4 2 .In place of (6.34), we have (6.35) C l ( @ Z ) = c2(@2) < c3(@2)? c l ( @ l ) < c 2 ( @ l ) 5 c3(@1)9 pointing to the appearance of three limiting body waves, two with equal speeds. The polarizations of the amplitudes are specified by q1(G1)and the orthogonal pair q1(@J and q2(4j2). f. Type 6 . L touches the outer profile three times, each point of contact belonging to just one branch of the slowness section (Fig. 2f). There are three critical angles and the conditions (6.15) and (6.34) hold good for them all. Once more three limiting body waves are created, but their speeds are now distinct. The unit vectors defining the polarizations of the body wave amp1itudes are ql(@l)?q 1 ( @ 2 ) 9 and q 1 ( @ 3 ) . None of the aforementioned possibilities can be excluded on direct physical grounds, but transonic states of Types 3, 5, and 6 could be realized only by imposing on the elastic constants Cijkrconditions which no actual crystalline solid is likely to meet. The interesting types are therefore 1,2, and 4, the others being conceivable but artificial.

D. PROPERTIES OF THE TENSORS S2(u) AND S3(u) The behavior of the real symmetric 3-tensors S 2 ( u ) and S3(u)over the range 0 5 u < ir of translation speeds for which they are defined is of crucial importance to the theory of elastic surface waves. It is already known, from

Surface Waves in Anisotropic Elastic Materials

343

Section V,C,2, that the strong ellipticity condition (3.2) guarantees the positive definiteness of S,(O) ( = S,), and that S,(O) ( 3S,) is negative definite subject to the more stringent strong convexity condition (3.3).The situation regarding S,(u) is straightforward: the strong ellipticity condition secures the positive definiteness of the modified acoustical tensor Q(q, u ) for all 0 I v < ;I,whence, from the integral representation (4.33)3, S,(u) is positive definite throughout the subsonic interval. The investigation of S,(u) is a matter of much greater difficulty and we content ourselves at this stage by proving that the derivative S;(u) = (d/du)S,(u) is positive definite in the open interval (0, i).As in the earlier work on S,(O), the method of proof is indirect, being based on the formula C(U) = 2u- 'T(u).

(6.36)

Here

L(u) = (4n2)-'A(u) ln(R/r), with A(u) = (b, nS,(u)b), (6.37) is the Lagrangian function per unit length of a uniformly moving line dislocation [obtained by annulling the concentrated body force in Eq. (6.29)] and T(u) is the total kinetic energy per unit length. The right-hand side of Eq. (6.36)is a positive function of u in (0, d) and it follows from (6.37) that (b, S;(u)b) > 0 for all values of v in this interval and for all choices of the Burgers vector b, thus confirming the positive definiteness of S;(u). It is interesting to note that this argument involves no restriction on the elastic constants of the material. We describe the main steps leading to Eq. (6.36)+and then indicate how the supporting details can be filled in. The first step is to enter the representation of S3(u) obtained from Eq. (4.34)T into Eq. (6.37) and differentiate with respect to u. The result can be set out as L(u) =

--(+-I{(

+)&(o),

b)(I'Ju),b)

+ (+- ) p ( l p ( 4b)(l;(u), b)l

ww?

(6.38)

and the relationf

is next used to bring this equation into the form

'

The approach followed here is due to Malen and Lothe (1970, Section 4). An alternative method, using a variational argument, has been given by Beltz et al. (1968, Section 2). Summations over the repeated Greek subscripts n and p which do not extend over the complete range 1, 2, , . ., 6 are shown explicitly.

P. Chadwick and G . D. Smith

344

The derivation of (6.36) is concluded by verifying that the right-hand side of this equation is equal to 2u-'T(u). It remains to indicate the origin of Eq. (6.39)and to sketch the calculation of T(u). 1. Calculation of &(u) This is broadly similar to the treatment of orientation dependence in Section IV,C. We differentiate with respect to u the eigen-relation (6.17) using the result (d/du)N(O,U ) = p ~K( - J)

(6.41)

which follows from Eqs. (3.18)*, (3.19)*, and (6.5). Forming inner products with q a ( u ) leads to

wherewith the differentiated eigen-relation reduces to

Equation (6.39) is one of the relations between 3-vectors obtained on decomposing (6.44).

2. Calculation of T(u) The displacement field associated with a uniformly translating line dislocation is given by the modified forms of Eqs. (5.19) and (5.20) as u = (27ri)-l( k), log(xl

+ p,(O,

u)xz - ut)(ln(u),b)a,(u).

(6.45)

The total kinetic energy per unit length is found by integrating over the annular region of R given by rz I (xl - uc)' + XI I RZ and this is accomplished by making the change of variables x1 - ut = a cos 0, x z = a sin 0 and utilizing the formula (5.30)

fp(du/dt, d u / d t )

Surface Waues in Anisotropic Elastic Materials

345

(cf. Stroh, 1962, Section 7). In due order the result turns out to be $I times the right-hand side of Eq. (6.40).

E. ILLUSTRATIVE EXAMPLE. HEXAGONAL SYMMETRY Explicit evaluation of the tensors S,(u), S2(v), and S 3 ( v ) is possible, through the integral representations (4.32)f, (4.33):, and (6.30),when the slowness section consists of concentric circles. This is the case when the elastic material under consideration has hexagonal symmetry and the zonal axis lies normal to the reference plane R. The components of the linear elasticity tensor are, in this instance, given in , cq4 by terms of five independent elastic constants c11, c 3 3 ,ci z, ~ 1 3 and cijkl

+ ?$(c11 - c 1 2 ) ( b i k b j l + b i l b j k )

= clZ8ijSkl

+ (c13 +m

- C12)(bijsk3’13

4 4

+ 6klsi3bj3)

- c11 + c12)

(bikbj.3613

+ bilbj36k3 + 6jlbi3sk3 + 6jk6i3b13)

+ (cll+ c33 - 4c44 - 2 c 1 3 ) b i 3 6 j 3 b k 3 b 1 3

.

(6.46)

From the definitions (3.7) and (3.10) we find that Q(q)= p ( V : n 0 n

+ V: m 0 m + V: e3 0 e3),

R ( c p ) = p { ( V t- 2 V i ) m @ n + V i n o m } ,

(6.47)

where VI =

(cll/p)l’z,

V2

= {f(C,i - c12)/p)1’2,

V3 = (c44/p)1’2, (6.48)

and the unit vectors m and n are given by Eqs. (3.9). Inspection of the acoustical tensor Q ( q )shows that, for all directions of propagation n in Ft, V,, V, ,and V, are the body wave speeds and n, m, and e3 the unit vectors specifying the polarizations of the respective amplitudes. The body wave with speed Vl is therefore longitudinal and the other two body waves are transverse. The branches of the slowness section are concentric circles of radii V ; V ; and V ; The strong ellipticity condition (3.2) ensures that the body wave speeds are real and positive, but places no restriction on their relative magnitudes. However, the strong convexity condition ( 3 . 3 ) requires that V2 < V,, leaving for consideration the following possibilities.

’, ’,

’ Here we observe the convention (6.10).

’.

P . Chadwick and G . D . Smith

346

Case A(i) V, < V, s V,,

c3 = V3 . (6.49)

i.e., c1 = V 2 ,

c2 = Vl,

V, I V3 < V,,

i.e., c1 = V,,

c2 = V, ,

c3 = V,. (6.50)

V3 I Vz < Vl,

i.e., c 1 = V , ,

c, = V,,

c3 = V,. (6.51)

Case A(ii)

Case B

If V2 # V 3 , then in all three cases the transonic state is of Type 1 in the classification given in Section VI,C,3. The limiting speed 1s given by in cases A(i) and A(ii), in case B,

(6.52)

and the single critical angle is equal to zero. If the transverse wave speeds are equal, cases A(ii) and B become identical and the common transonic state is of Type 2. In case A(i), however, there is no change of type. On calculating the modified acoustical tensors from Eqs. (6.5) and (6.47), noting that Q-'(cp,

u ) = p - ' { ( V : - uz cos2 cp)-'n@n

+ (V: - uz cos'

+ (VZ - u2 cos2 q ) - ' m @ m

9)- le3 @ e3},

(6.53)

and substituting into Eqs. (4.32)f, (4.33)3, and (6.30) we arrive at expressions for s,(~), s,(~), and S3(u) containing integrals which can all be evaluated by appeal to the formula

(6.54) The end results can be written ast S,(u) = [I - 2 ( u / ~ , ) - ~ {-1 rl(u)r2(u)}]{r;'(u)e1o e2 - r; '(u)e2@I el>, (6.55)

' A check on these calculations is provided by Eqs. (4.36)*.

Surface Waues in Anisotropic Elastic Materials

347

where (6.58)

ri(u) = (1 - (u/Q’}~/’

and F(x) = x-’{(2 - x)’ - 4(1 - x)”’(l - ( V ~ / “ ) ’ X ) ~ ’ ’ } .

(6.59)

Formulas equivalent to Eq. (6.57) have been given by Barnett and Lothe (1974a, Appendix 1). From (6.55k(6.57) and the associated forms Sl = Sl(0)= Sz = S,(O)

- ( Vz/v,)’(el o e2 - e2 8 el),

=$p-’(~;’

+ Viz)(el

el

+ e2

(6.60) ez) + p - ’ ~ ; ’ e , o e3 , (6.61)

s3= s,(o)= - ~ ~ ( V ~ / V ~ ) ’ ( V-: v$)(elo el + e2o e2) - pV: e3 o e3 ,

(6.62)

it can be verified that the general properties established in Sections VI,D and V,C,2 are substantiated here. The case of an isotropic elastic material is contained in the present example, Eq. (6.46) reducing to (3.4) under the replacements c11

= c33 = I

+ 2p,

C1’ = c13 = I ,

= p.

(6.63)

(p/p)”’,

(6.64)

c44

Equations (6.48) then yield the relations (Vz/Vl)’

= $(l - 2v)/(l - v),

V, =

V3 =

among the wave speeds (v being Poisson’s ratio) and it is evident that the transonic state is now of Type 2. The eigenvalue problem for N ( q , u ) turns out to be degenerate in the isotropic case, but the non-semisimple degeneracy encountered when u = 0 (and discussed in Section IV,E,2) ameliorates to a state of semisimple degeneracy when 0 .c u < V, . In accordance with the general result proved in Section IV,E,l, N(q, u ) is found to have a complete set of orientation-free eigenvectors which, when used to calculate S1(u), S,(u), and S,(u) via Eqs. (4.27) and (4.30), reproduce Eqs. (6.55)-(6.57) as simplified by the relations (6.64). On making the corresponding reductions in Eqs. (6.60)-(6.62) we recover the earlier formulas (4.52)-(4.54). VII. Elastic Surface Waves. Basic Analysis

The prerequisites are now to hand for constructing a general and reasonably complete theory of surface waves in a semi-infinite anisotropic elastic DOC;: The seminal ideas in this analysis, contributed by Stroh (1962, Section

348

P . Chadwick and G . D. Smith

8) and by Malen and Lothe (1971), respectively, are: first, that the appropriate solution of the equations of linearized elastodynamics, in common with the solution for a uniformly moving line dislocation, stems from the general result (6.7), and second, that the tensor S,(u) governing the Lagrangian of a translating line dislocation also determines the speed of propagation of a free surface wave. We start, in Section VII,A, by showing that a surface wave of arbitrary form and polarization can be made to travel, at any subsonic speed, through an anisotropic elastic half-space by applying to the plane boundary of the body a suitably distributed traveling load. The possibility of transmitting a free surface wave is then tantamount to the existence of a subsonic speed us at which the surface tractions vanish. This viewpoint, originated by Ingebrigtsen and Tonning (1969) and advanced by Lothe and Barnett (1976a),is pursued in Sections VII,B and D. Alternative forms of the secular equation specifying the free wave speed us are derived and additional results are obtained in Section VI1,B in preparation for the study of basic surface wave properties in Sections VIII and IX. A final appraisal of the analogy between moving line dislocations and free surface waves appears in Section VI1,C.

A. GENERAL SURFACE WAVESOLUTION We are concerned from now on with an anisotropic elastic body for which the stress-free configuration N is of semi-infinite extent. The boundary of this configuration is identified with the plane x2 = 0 in relation to the system of rectangular Cartesian coordinates xi introduced in Section V,A, the x2 axis being directed into N. Supposing body forces to be absent, we consider a small amplitude disturbance of the half-space traveling in the x1 direction and producing a distribution of displacement which does not vary with x3 and decays toward zero as x2 -+ 00. Such a plane surface waue is subject neither to damping nor dispersion so we can properly speak of a speed of propagation u independent of the form of the motion. The displacement and stress fields associated with a plane surface wave are of the form (6.1). Hence, on the basis of the concluding remark in Section VI,A, the requisite solution of the equations of linearized elastodynamics is included in the general form (6.7), and its validity is limited to the subsonic range 0 < u < i, of propagation speeds defined in Section V1,B. The mathematical analogy between moving line dislocations and plane elastic surface waves, first elicited by Stroh (1962, Section 8), is thus at once apparent. The assignable functionsf, in Eq. (6.7), must be chosen in such a way as to ensure that, for all values of x1 and t, u(xI - ut, x2) 3 0 and t(x, - ut, x2) + 0 as x2 -+ 00. It is natural to represent the motion as a superposition of

Surface Waues in Anisotropic Elastic Materials

349

harmonic waves and to this end we set

f b ( 4

I 5 i O,

=‘

yo.

a = 1,2,3,

U ( k ) exp(ikz) dk,

0

5, 6. (7.1) By the convention (4.9)*, the eigenvalues p4(0, u ) , ps(O, u), and p,(O, u ) have negative imaginary parts when 0 I u < ij and the moduli of the corresponding exponentials, a = 4, 5, 6, exp{ik(xl p,(O, u)x2 - ut)}, = 4,

+

increase without bound as x 2 -+co. For this reason only the first three eigenvalues can be admitted into a solution of the desired type. The result of combining Eqs. (6.7), and (7.1) can be expressed in the alternative forms

1 yn 1 3

=

n=l

+~

U ( k ) exp(ik(xl

0

~ (u)x2 0 , - ut)} dk

(7.2)

m

=

U ( k )exp{ik(x, - ut)}
(7.31

where 3


0).

I(kx2 u ) > =

C yn n= 1

exp(ipn(0, ~)kx2f.

(7.4)

On evaluating the solution (7.3) at the boundary x2 = 0 we obtain


5

m

n

~ ( kexp{ik(x, ) - u t ) ) dk(a(0, u), 1(0, u)>,

where, from the definition (7.4), 3

3

40,

0)

=

C y n a,@), n= 1

I(0,u ) =

C Yn ln(u). n=l

(7.6)

It is seen from Eq. (7.5) that the real and imaginary parts of the complex 3-vector a(0, u ) define the plane to which the paths of surface particles are confined: a(0, u) is accordingly named the polarization vector. Supposing for the moment that the vectors al(u), a2(u),and a3(u)are linearly independent, it is possible, by a suitable choice of the complex amplitude coefficientsy, ,to give the surface wave any specified polarization. Moreover, by the use of Fourier’s integral theorem the function U ( k ) ,which evidently determines the form of the surface displacement, can be made to portray a given wave profile. The complex 3-vector l(0, u ) then represents the directional aspect of

350

P. Chadwick and G. D. Smith

the distribution of surface tractions which must be applied at x2 = 0 in order to maintain the motion. The linear independence of a,(u), a2(u), and a3(u) can be proved, by contradiction, with the aid of the important relations

s,(v)aQ(v)+ Sz(u)k(u) = iaa(u) S,(v)aQ(o) + S:(u)la(u) = iIQ(u)

I

tl =

1, 2, 3,

(7.7)

derived from Eqs. (4.31)* and (4.26). If the aQ(u)’s are linearly dependent they have a nontrivial linear combination which is zero, and Eq. (7.7)1shows that the same linear combination of the IQ(u)’sbelongs to the null space of S2(u). However, the real symmetric 3-tensor S2(u) is known, from Section VI,D, to be positive definite, so this particular combination of the Ia(u)’s must be zero. But then, from Eq. (4.14):, {l(v), g2(u), and S3(u) are linearly dependent, which is false as these vectors are members of a basis of C6 (Lothe and Barnett, 1976a, Section 111). Because of the linear independence of a,(u), a2(u),and a3(u), it follows from Eqs. (7.6), and (7.2) that the polarization vector can vanish only when the elastic body as a whole is undisturbed from its natural state N. In other words, it is impossible to propagate a surface wave in an elastic half-space whose surface is rigidly clamped. This conclusion has been reached, in a different way, by Currie (1974, Section 5).

B. FREESURFACE WAVES The possibility to which surface wave studies are usually directed is the existence of afree waue, capable of propagating when no tractions act at x2 = 0. Suppose that there is such a wave and that it travels with speed u s . Then the stress components TZi(x,- ut, x2) must vanish at x2 = 0, implying, through Eqs. (6.4), , (7.5), and (7.6)2, that

We now make further use of Eqs. (7.7) which, in view of the definitions (7.6), continue to hold when aQ(u)is replaced by a(0, u ) and la(u) by I(0, u). On setting u = us in the resulting equations and incorporating the boundary condition (7.8) we find that the polarization vector of the free surface wave satisfies the relations

S,(us)a(O,us) = ia(0, us), S3(0s)a(09us) = 0. (7.9) Since a(0, us) # 0, Eq. (7.9)2 tells us that the wave speed us is a root of the

Surface Waves in Anisotropic Elastic Materials

35 1

secular equation

det S,(u)

= 0.

(7.10)

Conversely, suppose that there is a speed us in the subsonic range 0 < u < I.?for which Eq. (7.10) holds. Then, returning to the algebraic representation (4.34): of S3(u) and using the completeness relation (4.15): and Eq. (4.26) to eliminate the vectors 14(u), 15(u), and 16(u), we find that 3

S3(u)= 2i

C In(u) o in(u). n=

(7.11)

1

With this result we can infer from Eq. (7.10), via the vector identity (2.30), that [I,(u), 12(u), I3(u)] = 0 at u = u s . (7.12) The vectors I,(us), 12(uS), and l3(uS) are therefore linearly dependent and we are thereby assured of the existence of amplitude coefficients yl, y 2 , y 3 , not all zero, for which the solution (7.2) [or (7.3)] complies with the boundary condition (7.8). It has now been proved that the existence of a subsonic speed us at which S3(u) is singular is a necessary and sufficient condition for an anisotropic elastic half-space to transmit a free surface wave. The form of the secular equation presented by Eqs. (7.10) and (7.11) is due to Stroh (1962, Section 8). Previously Synge (1956a, Section 5) had developed the theory of harmonic free surface waves from first principles and arrived at the secular equation in the guise (7.12). He then argued that this is a complex equation entailing two independent restrictions on the speed of propagation. The fact that (7.12) is equivalent to a real equation for u s , namely (7.10), was expressly pointed out by Stroh (1962, Section 8). Unfortunately the fundamental significance of Stroh’s work on elastic surface waves was overlooked until the early 197Os, and much of the research in this area published in the meantime was confused by attempts to reconcile Synge’s view of the exclusive character of free surface waves on crystalline substrates with increasingly strong evidence from numerical studies of the widespread existence of such solutions. The amplification of Stroh’s ideas by Malen and Lothe (1971) and by Barnett et al. (1973a,b) has brought to light the important fact that not only is S3(u) singular at a free surface wave speed, but its rank falls to one. This property emerges from Eqs. (7.9) by the following reasoning. Let aC(O, us) denote the real part and a-(O, us) the imaginary part of the polarization vector a(0, us). Then, since S,(us) is a real 3-tensor, Eq. (7.9), yields the relations Sl(us)a+(O,U S ) = -a-(O,

US),

Sl(us)a-(O,us) = a+@, us), (7.13)

352

P . Chadwick and G . D. Smith

making apparent the fact that a+(O, us) and a-(O, us) cannot be real scalar multiples of one another (Barnett et al., 1973b, Section 4). This establishes the existence of a plane of polarization of the free surface wave, spanned by the linearly independent vectors a+(O, us) and a-(O, us). It is now seen from Eq. (7.9)2that the real and imaginary parts of a(0, us), and hence any vector lying in the plane of polarization, belong to the null space of S3(us).The rank of S3(us) is thus either 1 or 0. However, the latter possibility would require S3(us) to vanish and Eqs. (4.30)* and (4.36): would then give det S(us)= det Sl(us)det ST(us) = det S:(us)

= det(-I) = - 1.

But Eq. (4.27)* shows the product of the eigenvalues of S(us) to be i3(-i)3 = 1. Hence rank S3(us) = 1

(7.14)

and the plane of polarization may be identified with the null space of S3(us) (Barnett et al., 1973b, Section 2). Since this result implies and is implied by (7.10), holding at u = us, (7.14) is also a necessary and sufficient condition for the existence of a free surface wave. A further noteworthy consequence of Eqs. (7.13) is that the polarization vector a(0, us) can be constructed from either its real or its imaginary part: from (7.13)', for instance, a(0, us) = {I - iSl(us))a+(O,us).

(7.15)

In conjunction with the prescribed function U(k), a(0, us) determines the path traced out in the plane of polarization by a representative particle in the surface of the body. For a harmonic free surface wave, resulting from the choice U ( k )= 6(k - OU, '),

(7.16)

where o is the angular frequency, the particle paths are ellipses.

C. THEDISLOCATION-SURFACE WAVEANALOGY The connection between uniformly moving line dislocations and surface waves in anisotropic elastic materials, already evident from their common roots in the general solution (6.7), is reinforced by the realization that the 3-tensor S 3 ( u ) entering the prelogarithmic factor A(u) in the Lagrangian function (6.37) of a translating dislocation also fixes the speed of travel of a free surface wave. It has been pointed out in Section VII,B that any real 3-vector b situated in the plane of polarization of a free surface wave belongs to the null space of S3(us), and we now observe from Eq. (6.37) that if b is regarded as the Burgers vector of a line dislocation moving with speed u s ,

Surface Waves in Anisotropic Elastic Materials

353

then the Lagrangian function L(u)vanishes at u = u s . Put in slightly different terms, if a free surface wave exists in an anisotropic elastic half-space, then for any Burgers vector lying in the plane of polarization of the wave the energy associated with a line dislocation moving in the same direction as the wave with the same speed and in the same material is equally shared between kinetic energy and elastic strain energy. This striking extension of the dislocation-surface wave analogy is due to Malen and Lothe (1971). Alongside the above-mentioned similarities we should also take note of a significant difference between the solutions representing a moving line dislocation and a free surface wave. Whereas in the former the speed is at our disposal (within the subsonic range) and the six constants c , are then completely specified by the source condition (5.14), in the latter the speed is effectively decided by the boundary condition (7.8) and the three amplitude coefficients yu are determined only to within an arbitrary complex multiplier (Lothe, 1972, Section 5). In parallel with the theory of vibrations, the distinction here is between forced and free motion. At bottom, however, the importance of the dislocation-surface wave analogy resides in the possibility of basing upon the solution for a line dislocation proofs of the positive definiteness of -S,(O) and S;(v). An alternative approach in which, for similar purposes, the forced surface wave discussed in Section VII,A replaces the line dislocation has been worked out by Ingebrigtsen and Tonning (1969) and reformulated, with numerous extensions and clarifications, by Lothe and Barnett (1976a). This theory has an attractive cohesiveness, but we regard the line of thought followed in the preceding text as offering the best available combination of physical immediacy and mathematical rigour. In particular, the detailed study of the uniformly moving line dislocation, undertaken in Section VI, introduces concepts indispensible to a full appreciation of the issues affecting the existence of free surface waves. Before resuming this development in Section VIII we outline, in Section VII,D, some interesting facets of the alternative theory.

IMPEDANCETENSOR D. THESURFACE We consider here, as in Section VII,A, the forced motion of an anisotropic elastic half-space driven by a traveling surface load. First we return to Eq. (7.7),, rewritten as L(u) = iz(v)a,(u),

(7.17)

where Z(v) = S; ' ( u )

+ is; ' ( v ) S , ( u ) .

(7.18) Since Z(v)expresses a linear relationship .between the vectors aJu) and I,(v),

354

P . Chadwick and G. D. Smith

generating, respectively (through Eqs. (7.5), (7.6), and (6.4)2), the surface displacement u(xl - ut, 0 ) and the surface traction T T ( x l - ut, O)e,, it is referred to as the surface impedance tensor (Lothe and Barnett, 1976a, Section 111; see also Ingebrigtsen and Tonning, 1969, Section 111). We recall that, in the subsonic range 0 Iu < ir, S2(v)is symmetric and positive definite and, from Eq. (4.37)*, that S ; '(u)S,(u) is skew-symmetric. Hence

ZT(U) = Z(u),

(7.19)

proving that the surface impedance is a Hermitian tensor. Equations (7.4) and (7.17) jointly supply the relation I ( k x 2 , u ) = iZ(u)a(kx, , 0).

(7.20)

Suppose now that the motion takes on the form of a free surface wave when u = u s , the polarization being no longer prescribed. Then the boundary condition (7.8) applies arid Eq. (7.20) yields

Z(us)a(O, us) = 0.

(7.21)

det Z(us) = 0

(7.22)

We deduce that and that the polarization vector a(0, us) belongs to the null space of Z(us) (cf. Ingebrigtsen and Tonning, 1969, Section V). Because of the Hermitian symmetry of Z(u), the alternative form (7.22) of the secular equation is, like (7.10), a real equation. However, since the surface impedance tensor is complex we cannot conclude from Eq. (7.21) that rank Z(us) = 1. The existence and uniqueness of free surface waves can be studied through the behavior in the subsonic range [0, ij) of the surface impedance tensor Z ( v ) rather than the real tensor S3(u), provided that Z ( u ) can be shown to have properties analogous to the positive definiteness of - S 3 ( 0 ) and S3(u).It may be established in the following manner that Z(0) is positive definite (Ingebrigtsen and Tonning, 1969, Section IV,B; Lothe and Barnett, 1976a, Section V). The real form of Eq. (7.3) appropriate to a harmonic surface wave of angular frequency o is (u(xl - vt, x 2 ) , t ( x , - vt, x,))

= $&a(kx,,

u), I(kx2, 0))

+ $e-n(ir(kx,

, v ) , T(kx, , u ) ) ,

(7.23)

where R

=

i k ( x , - vr)

and

o = kv.

(7.24)

For a motion induced by steadily moving loads the right-hand side of Eq. (5.7) is known to represent the negative of the Lagrangian function. On

Surface Waves in Anisotropic Elastic Materials

355

entering the solution (7.23) into the expression I+Zn(ku)- 1

-4(2n)-'ku

(tp, 1 up, 2 f

- c p , 2 u p , 1) dt,

(7.25)

we therefore obtain the density of the Lagrangian averaged over a period of the motion. The result is ik(dldxd(a(kx2

?

4, Z(u)a(kx,

9

v)),

(7.26)

use being made of Eq. (7.20). Due to the exponents in Eq. (7.4) having negative real parts, the inner product in (7.26) tends to zero as x 2 -,00. Hence, integrating with respect to x 2 , we find that the total averaged Lagrangian of the material within a semi-infinite cylinder of unit crosssectional area extending from the boundary x2 = 0 is ==

L(u) = -$k(a(O, u), Z(u)a(O, u)).

(7.27)

When the surface load is stationary the kinetic energy is zero and - E(0) is the total averaged strain energy of the cylindrical sub-body. Subject to the strong convexity condition (3.3) this is a positive quantity regardless of the polarization vector a(0, u ) which, as pointed out in Section VII,A, can be given any preassigned value. The surface impedance tensor is thus positive definite at u = 0. Finally we verify that, in the harmonic surface wave motion described by Eq. (7.23), the mean energy flux is everywhere parallel to the boundary x2 = 0. The time average of the energy flux in the direction normal to the boundary is derived from Eq. (6.25) in the form

J; = - ( 2 ~ ) '-k d

jff+2n(ku'-'

tp, 1 u p , 1

dt.

(7.28)

On substituting from the solution (7.23) into this expression and appealing once more to Eq. (7.20) we duly confirm that T2 = 0 (cf. Lothe and Barnett, 1976a, Section V; see also Farnell, 1970, Section VII). From the beginnings sketched in this section the alternative theory of elastic surface waves can be brought to quite an advanced state of development. The paper by Lothe and Barnett (1976a) gives further details. VIII. The Uniqueness and Related Properties of Free Surface Waves

The discussion given in Section VII,B shows that the questions of the existence and uniqueness of free elastic surface waves devolve upon the number of real roots possessed by the secular equation (7.10) in the range

P . Chadwick and G . D. Smith

356

0 < u < 6 of subsonic speeds. We now take up these basic problems, dealing in the present section with uniqueness and allied topics and in Section IX with the more difficult and elaborate problem of existence. The analysis that follows is grounded on work by Barnett et al. (1973a,b) and hinges on Eq. (7.14). This result is shown in Section VII1,A to lead directly to a proof of uniqueness. We then evolve, in Sections VII1,B and C, convenient methods of calculating the speed and the polarization vector of a free surface wave. A. UNIQUENESS

1 . The General Case Let J,(u) be the real eigenvalues of the real symmetric 3-tensor S,(u) and let si(u) be an orthonormal set of associated real eigenvectors. Differentiation with respect to u of the eigen-relation (8.1)

s 3 ( ~ k ( u )= Ji(ubi(u), followed*byan obvious manipulation, leads to

J:(u)

= (si(v),S;(u)si(u)).

(8.2) Since S,(O) is negative definite and S3(u) positive definite for 0 < u < ij, we deduce from Eq. (8.2) that the eigenvalues L,(u) increase monotonically with u in the subsonic range from negative values at u = 0. If there is a free surface wave speed us in the interval (0, ;), Eq. (7.14) holds and two of the eigenvalues A,(u) must vanish together at u = u s . It then follows that the secular equation (7.10) can have no root other than us in the subsonic range. For if there were two roots, at least one of the eigenvalues Li(u) would have to vanish twice in (0,t?) which is contrary to the monotonicity property just established (cf. Barnett et al., 1973b, Appendix). It should be noted that this uniqueness proof holds without regard to the values of the li(u)’s at u = 0. It rests, therefore, entirely upon the positive definiteness of S3(u)which has been shown in Section VI,D to involve no restriction on the linear elasticities of the transmitting material. 2. Hexagonal Symmetry Returning to the illustrative example introduced in Section VI,E, we observe that the expression (6.57) for S,(u) is in spectral form, enabling us to read off the eigenvalues as &(u)

=

~ I / : F ( u ’ / ‘v( u:)), ~ ; J 2 ( u ) = p‘V:F(u’/V:)r; J3(u) = - pV:r 3 ( u ).

‘(u),

(8.3)

357

Surface Waves in Anisotropic Elastic Materials

By virtue of the properties

F ( 0 ) = - 2 ( 1 - cc), F( 1) = 1, ~ ‘ ( x=) 1 + 2x-’( 1 - x ) - ”’(1 - a x ) - ’”{( 1 - ax)l” - ( 1

- x)1/2}2,

(8.4)

where 0 < cc = (VZ/Vl)* < 1, the function F(x) defined by Eq. (6.59) increases monotonically from negative to positive values as x increases from 0 to 1. Thus F ( x ) has exactly one zero, say x s , in the interval (0,1). When one of the cases A(i) and A(ii), specified by the conditions (6.49)and (6.50),applies, the simultaneous zero xs”’V2 of Al(u) and A2(u) falls within the subsonic range which, from (6.52),is 0 < u < V, . A unique free surface wave therefore exists. This conclusion extends to case B (thereby embracing the isotropic elastic half-space) if xs < (V, /V,)’( Il), the critical speed now being V,. If xs 2 ( V 3 / V 2 ) 2however, , A,(IJ) and &(u) vanish together outside the subsonic range.+ The significance of this behavior is discussed later in Section IX,E. B. CALCULATION OF THE SPEED OF PROPAGATION Equation (7.14) is equivalent to the statement that there exists a nonzero real 3-vector s such that S,(u,) = s 0 s.

(8.5)

Alternatively, in relation to the orthonormal basis e, we can characterize (7.14) as saying that every 2 x 2 submatrix of the matrix of components of S 3 ( u ) is singular at u = u s . These facts provide the basis for a simplified method ofcomputing the wave speed us, first devised by Barnett et al. (1973b, Section 2 and Appendix). Let

denote any nonzero principal submatrix of the matrix of components of S,(u). Then, in view of Eq. (8.5), one eigenvalue of M(u,) is zero and the other eigenvalue is positive, being the sum of squares of two components of s. The smaller eigenvalue p ( u ) of M(u), given by P(U) = i { M l h ) + M,,(~)l- [*{Mll(U)- M22(412 + M:z(~)11iz9(8.7)

’

If xs = (V3/V2)’,all three eigenvalues of S3(u) vanish at u = x ~ ” V 2= V, . However, this value of u marks the transonic state, so our earlier conclusion that S,(u) # 0 in the subsonic range is not contravened.

358

P . Chadwick and G. D. Smith

consequently vanishes at u = us. M'(u) is a principal submatrix of the matrix of components of S3(u). It is therefore positive definite for 0 < u < ij, which means, by the argument produced at the beginning of Section VIII,A, that the eigenvalues of M(u) are steadily increasing functions of u over the subsonic range. The determination of the free surface wave speed us thus reduces to the computationally straightforward exercise of locating the one zero in the interval (0, i) of the monotone function &I). The construction of p(u) from Eq. (8.7) calls for the evaluation of only three of the six independent components of S3(u) and appropriate integral representations are provided by Eq. (6.30).

C. CALCULATION OF THE POLARIZATION VECTOR 1. The General Case

The vector s in Eq. (8.5) is orthogonal to the null space of S3(us) and is therefore aligned with the normal to the plane of polarization. For some pair of unit vectors e,, e , , taken from the basis e, S3(us)ek is a nonzero scalar multiple of s and {S,(us)ek)A e, is a nonzero real 3-vector orthogonal to s and hence in the plane of polarization. Since the amplitude coefficients in the solution (7.3) and (7.4) are arbitrary to within a complex scalar multiplier, no loss of generality is involved in identifying the latter vector with the real part of the polarization vector a+@, us) = (S,(vs)e,) A e, .

(8.8)

On combining this expression with Eq. (7.15) we then arrive at the formula 409 us) = {I - i s , ( ~ s ) l [ { S , ( ~ s )Ae kellf

(8.9)

for the polarization vector itself (cf. Barnett et al., 1973a; 1973b, Section 4). A useful alternative form of Eq. (8.9) isf

(8.10) In order to compute the polarization vector from Eq. (8.10) we need six components of S,(us) and three components of S3(us) relative to e. The relevant integral representations are furnished in this instance by Eqs. (4.32); and (6.30). When a(0, us) is known the path of a representative We make use here of the identity A(n A b) = [A'e!, a, b]e, , where A is an arbitrary 3-tensor and a, b are arbitrary 3-vectors.

Surface Waves in Anisotropic Elastic Materials

359

surface particle can be determined.+When interest is confined to the orientation of the plane of polarization, however, it suffices to compute S3(vs)ek,a task requiring only three components of S3(vs).

2. Hexagonal Symmetry Resuming our discussion of this particular case, we now evaluate the polarization vector by substituting the explicit forms (6.55) and (6.57) of S , ( v ) and S,(v) into Eq. (8.10). It is assumed here that one of the cases A(i) and A(ii) applies so that the existence of a unique free surface wave is assured. The choice k = 3, 1 = 2 is satisfactory, inasmuch as it leads to a nonzero result, and simplifications can be effected with the help of the relation F(x,) = 0. After discarding an unessential factor we find that a(O, us) = (1 - xS)'/'el - i( 1 - fxs)ez ,

(8.1 1)

demonstrating that the plane of polarization coincides with the reference plane R. For a harmonic free surface wave the elliptical paths of the surface particles are described in the retrograde sense, the major and minor axes being aligned with the x 2 and x 1 directions, respectively. The quotient of the major to the minor axis is +( 1 - x s ) $( 1 - xS)'/'. These findings apply, of course, to Rayleigh waves in an isotropic elastic half-space when, in Eq. (6.59), (V, /V1)' is equated to p/(A 2 p ) [ = $( 1 - 2 v ) / ( 1 - v ) ] .

+

+

IX. The Existence o f Free Surface Waves In this penultimate section we complete our account of basic aspects of the theory of free surface waves in anisotropic elastic materials by considering the fundamental question of the existence of such waves. The substantial progress which has been made toward the resolution of this problem is largely due to the penetrating work of Barnett and Lothe (1974a; see also Lothe and Barnett, 1976a, Section VI) which has inevitably dictated the form of the present treatment. The method of investigating the existence of free surface waves followed by Barnett and Lothe is based on an integral r e p resentation of the real quadratic form associated with the symmetric 3-tensor S,(v). The main technical problem is to determine the behavior of the integral in the transonic limit v T 6, and we deal with this matter in Section IX,B after preparing the ground in Section IX,A. Our investigation of the central integral differs somewhat from the original analysis of Barnett and Lothe, and we adopt a mode of presentation which avoids as far as

' A neat account of this calculation, as applied to a harmonic wave, has been given by Toupin and Rivlin (1961, Section 8).

360

P . Chadwick and G. D . Smith

possible the need to examine seriatim the six possible types of transonic states. The initial conclusion which emerges is that a unique free surface wave exists provided that no limiting body wave satisfies the condition of zero traction at the boundary of the transmitting half-space. The exceptional transonic states arising when this proviso is not fulfilled are studied in Section IX,C, and a summary of the final conclusions is provided in Section IX,D. The remaining sections are of subsidiary interest. In Section IX,E we round off our discussion of the special case of hexagonal symmetry initiated in Section VI,E and continued in Sections VIII,A,2 and C,2. Finally, in Section IX,F, we give a precise statement and proof of an asymptotic lemma used in Section 1X.B. A. FORMULATION OF THE EXISTENCE PROBLEM 1. Behavior of the Prelogarithvnic Factor A(v) It has been shown in Section VI1,B that a necessary and sufficient condition for the existence of a free surface wave in an anisotropic elastic halfspace is the simultaneous vanishing, at some subsonic speed, of two eigenvalues of the real symmetric 3-tensor S3(us). We also know, from Section VIII,A,l, thaf there can be at most one motion of this type. The existence problem for free surface waves can thus be approached by investigating the behavior of S,(u) in the subsonic range (0, ij). We use for this purpose the real quadratic form associated with S,(u), that is, K- times the prelogarithmic factor A(u) appearing in the expression (6.37)’ for the Lagrangian per unit length of a translating line dislocation. The positive definiteness of -S,(O) and S3(u)implies that, for any choice of the Burgers vector b, A(u) increases monotonically over the interval of interest from a negative value at u = 0. If for some Burgers vector, A(u) takes positive values in the subsonic range, at least one eigenvalue of S,(u) must have a zero between 0 and ij. Equation (7.10) then holds for some subsonic speed u = us and, as proved in Sections V11,B and VIII,A,l, there is a unique free surface wave. Existence is guaranteed, as well as uniqueness, if afortiori we can find a Burgers vector for which A(u) -+ co as u 7 ij. We now look into this possibility.

’

2. Integrul Representation of A(u) Equations (6.37)z and (6.30) together supply the following integral representation of the prelogarithmic factor A(u):

Surface Waves in Anisotropic Elastic Materials

36 1

The eigenvalues of the modified acoustical tensor Q(cp,v ) are given in (6.12) and associated unit eigenvectors are qi(cp). From these ingredients we can assemble the inverse Q-'(cp, u ) in the spectral form

valid when 0 i u < i,. The result of combining Eqs. (9.1) and (9.2) is

use being made of the algebraic identities (2.12) and (2.10).

B. BEHAVIOR OF I\(u) IN

THE

TRANSONIC LIMIT

1. Preliminary Analysis From the definition (6.5), of the real 3-tensor R(cp, u ) we see that the first inner product, in Eq. (9.3), like the terms an the second line, is linear in uz. It is clear, therefore, that if the prelogarithmic factor A(v) is unbounded in the transonic limit, the cause lies in the first factors of the integrands. The Bth integral in Eq. (9.3), where B is 1,2, or 3, corresponds to the same branch of the slowness section as the body wave speed cB(cp),and on looking back to Eq. (6.34), and Fig. 2 we perceive that any divergent behavior as u t i, must arise from the contributions to the integral of neighborhoods of the critical angles belonging to the relevant branch. For each of th'e possible types of transonic states the ranges of the subscripts A and B are determined by the number of critical angles' and the number of branches that are tangent to one another on the associated radius vectors. The number of pairs ( A , B) for which singularities of the integrands in Eq. (9.3) occur is equal to L, the number of limiting body waves. The pairs themselves can be gathered from the descriptions of the six types of transonic states given in Section VI,C,3 or read off directly from the diagrams in Fig. 2: for ease of reference we list them in Table 1.

' When there is only one critical angle it is denoted here by in Section VI.

The subscript 1 was dropped

362

P . Chadwick and G . D. Smith TABLE 1 RANGE OF THE SUM IN EQ. (9.4) FOR THE Six POSSIBLE TRANSONIC STATES ~

Type

~~~~

Values of (A, E )

L

Nonoverlapping subintervals of [-&, 4x1 centered on the critical angles belonging to the branch of the slowness section representing ~ ~ ( c may p) be defined as follows. Let zAB be the smallest absolute difference between distinct members of the set of real numbers consisting of @ A and any angles between -% and & at which c)B((p)is discontinuous.+ Then the intervals [iA - $CAB, @ ,, +TAB] have at most an end point in common and we can pick out neighborhoods of the critical angles + A by dividing the range of integration of the Bth integral at cp = f$zAB. The foregoing considerations lead to the rearranged form

@A

fh,

+

of the integral representation (9.3).The summation here extends over the t pairs (A, B) enumerated in Table 1 and the remainder term A*(u), which need not be written out explicitly, approaches a finite limit A*(;) as u 7 b.

2. Application of the Asymptotic Lemma An appropriate tool for estimating the leading term on the right-hand side of Eq. (9.4) at values of u approaching the limiting speed 6 is the asymptotic lemma proved in Section IX,F. The application of this result is facilitated by the introduction of some auxiliary notation. Recalling that the inner product in Eq. (9.4) is linear in u2 we set 2

p-'lb, R(cp, uhB(V))2

seC2

cp =

fBK((P)(c2 - U2)K K=O

(9-5)

' The outer profile of the slowness section may have kinks. Cases in point are illustrated in Figs. 2b, 2d, and 2f.

Surface Waves in Anisotropic Elastic Materials

363

fBo(V) = P-’(b, R(P, fi)q~(CP))’SeC’ CO 2 0.

(9.6)

and note that

Also we define gB(q)

=

ci(cp)sec2 cp.

(9.7)

It then follows from Eqs. (6.14) and (6.151, suitably generalized, that gB(@A)

=?’ ;

= O,

&(@A)

d(@A)

= 2cGAB

(9.8)

3

where

Expressed in terms of the new functions, Eq. (9.4) reads 2

A(u) =

1B ) C (i,’ - u ’ ) ~

(A,

K=O

(9.10) It can easily be verified, from Eqs. (9.5) and (9.7)-(9.9), that, for all possible pairs ( A , B), the functions fB&) (K = 0, 1, 2) and g B ( ( p ) satisfy the analytical requirements (ab(d) stated in Section IX,F, x, c, 2, a, and b being replaced in turn by cp, @ A , 6’ - u’, @ A - $A,,, and @ A + $ T A B - The asymptotic formula (9.24) can therefore be used in Eq. (9.10), giving A(U)

7‘C6-1’2(fi’ - U z ) - ” z

Gii”fBo(@A)

4-

A*(fi)

as

U

c.

(A, 8)

(9.11) Bearing in mind the inequalities in (9.6) and (9.9), we deduce from (9.11) that A(u) + co as u t i, if fBO(@A) > 0 for at least one of the pairs (A, B) entered in the appropriate row of Table 1. This condition is met if W @ A , G) qB( @ A)

#0

for Some

(A, B),

(9.12)

since it then suffices to equate the Burgers vector b, which is freely disposable, to the nonzero real 3-vector on the left. Part (ii) of the asymptotic lemma implies that A(u) remains bounded in the transonic limit if and only iffBO(@A)= 0 for all the relevant pairs (A, B) in Table 1. In view of the arbitrariness of b it follows from Eq. (9.6)that this is the case if and only if

R(@, , i ) ) q B ( @ A )

=0

for all

(A, B).

(9.13)

364

P . Chadwick and G . D. Smith

Before considering the significance of the criteria (9.12) and (9.13) we interpolate a remark about the tensor S,(u). Equation (4.32);, in conjunction with the spectral relation (9.2) and Eq. (2.12)2, provides the integral representation n/2

3

11

S,(U) = - ( n p ) - 1

{ci((P) - v 2 cos2 q}-'

p = l "n12

x

qp(d

(9.14)

0 {R(% ~ ~ , ( 4 4

d(P9

which, when rearranged in accordance with the procedure described above, becomes

c1

@A + r A d 2

Sib) = -(Wv

(A. B)

qB((P)

{gB((P)

- g B ( @ A ) + ijz

- uz})-

@A-rAB/2

@ iR((P, uhB((P)} sec2 (P dq + s:(0).

(9.15)

Invoking once more part (ii) of the asymptotic lemma, we infer from Eq. (9.15) that (9.13) is a necessary and sufficient condition for each of the components of S,(u) relative to the real orthonormal basis e to remain bounded as Y t 6. In short, when h(u),and hence S3(v), is bounded in the transonic limit, so is S,(u) (Barnett and Lothe, 1974a, Section 4).' 3. Interpretation of the Conditions (9.12)and (9.13) The analysis of limiting body waves, initially developed in Sections VI,B,3 and VI,C,3 in the context of the moving line dislocation, extends in its entirety to the theory of surface waves. We return now to the discussion given in Section VI,B,3 of the transonic state of Type 1, the simplest and most common of the possibilities that can occur. Equation (6.23) shows that if R(@, ij)ql(@)is zero, the stress vector associated with the limiting body wave vanishes. There is consequently no contribution to the stress components T2i,as we see from Eq. (6.4), , and the condition of zero traction is satisfied at the boundary x 2 = 0. When converted to the present notation, R(& ij)q,(@) becomes R(@A,6)PB(GA),where A = B = 1, corresponding to the single entry in Table 1 for a transonic state of Type 1. The generalization to all possible transonic states of the deduction made from Eq. (6.23) is at once apparent; the vanishing of R(GA, ij)qB(GA) means that the limiting body wave characterized by the pair (A, B) meets at x 2 = 0 the condition of zero traction appropriate to a free surface wave (cf. Barnett and Lothe, 1974a, Sections 4 and 5). We say that the transonic state is exceptional when all the limiting body

' Note, however, that this statement does not extend to S2(u).

Surface Waves in Anisotropic Elastic Materials

365

waves conform to the traction-free boundary condition at x2 = 0 and normal when there is at least one limiting body wave which violates this condition. Equations (9.12) and (9.13) are then recognized as necessary and sufficient conditions for the transonic state to be normal and exceptional respectively. With these interpretations we can now conclude that: a. Subject to the material satisfying the strong convexity condition (3.3), an anisotropic elastic half-space admits a unique free surface wave if the transonic state is normal (Lothe and Barnett, 1976a, Section VII). b. The prelogarithmic factor A(u) is bounded as v T iI if and only if the transonic state is exceptional.

To the extent that a body wave which leaves the surface x2 = 0 tractionfree is likely to occur only when the wave and surface normals are symmetrically orientated in relation to the crystallographic axes of the material, conclusion (a) indicates that the existence of a unique free surface wave is usually to be expected. Statement (b) does not settle the question of whether or not an exceptional transonic state is compatible with the existence of a unique free surface wave. Further investigation is needed and this is pursued in Section IX,C. First, however, we make a closer examination of the condition (9.13). It was proved in Section VI,B,3 that when the transonic state is of Type 1 the limiting body wave has zero energy flux in the x2 direction. In transonic states of the other types this property applies to each limiting body wave and Eq. (6.26), generalizes to (%(@A),

R(@A

9

bkB(@A)) =

B,

(9.16)

(cf. Barnett and Lothe, 1974a, Section 3). Let C and D denote the integers forming with B a permutation of { 1, 2, 3). Then qA = { q B ( @ A ) , qC(@”), ( I D ( @ ” ) } is a real orthonormal basis of C 3 . When the 3-vector R(@A,iI)QB(@”) is referred to q A , Eq. (9.13) yields three scalar conditions, one of which is automatically satisfied on account of (9.16). In each of the other conditions two distinct, and hence orthogonal, members of qA appear, so R(+A, b) can be replaced by R(@”), the difference between these tensors, as shown by Eq. (6.5),, being spherical. The condition (9.13) is therefore equivalent to the simpler requirement

(cf. Barnett and Lothe, 1974a, Section 3). When compared with (2.20)2,Eqs. (9.17) are seen to imply the vanishing of two of the off-diagonal components of R(@J relative to qA . For a transonic state of Type 3, A = 1, reflecting the existence ofjust one critical angle,

366

P . Chadwick and G. D.Smith

but B has the maximal range (1, 2, 3) since belongs to all three branches of the slowness section (see Fig. 2c). In this case Eqs. (9.17) require all the offdiagonal components of R(@,) relative to q1 to be zero, and Eq. (9.16), with (6.5),, states that all the diagonal components are equal to p i 2 sin @ 1 cos C1. We have proved, however, in Section III,A, that R(cp) can never be spherical. Thus there is no exceptional transonic state of Type 3 and, granted the strong convexity property, a unique free surface wave exists unconditionally (Barnett and Lothe, 1974a, Section 2). C. EXCEPTIONAL TRANSONIC STATES

When the transonic state is exceptional the stress vector associated with each limiting body wave is zero and the 3-tensors S1@) and S3(ij) are bounded. On letting o tend to ij in Eq. (7.7), and using the generalized form = aA qB(@A),

a = 1, 2, f . .

9

(9.18)

of Eq. (6.22), we thus find that s3(ijhB(@A1

=

(A, B,

(9.19)

(Barnett and Lothe, 1974a, Section 4). 1. Types 2, 4, 5, and 6 For transonic states of these types, L = 2 or 3. Assuming for the present that the L vectors qB(@A) are linearly independent, we deduce from Eq. (9.19) that either two or all three of the eigenvalues of S3(u)are zero at the limiting speed ij. The simultaneous vanishing of two eigenvalues of S3(u)within the subsonic range is then precluded and no free surface wave is possible. When the transonic state is of Type 2, 4, 5, or 6, normality is thus a necessary as well as a sufficient condition for the existence of a unique free surface wave. As regards the linear independence of the real unit 3-vectors qB(@A),we first observe that there is nothing to prove when the transonic state is of Type 2 since ql(@l) and q2(G1) are orthogonal. For a state of Type 4 the relevant vectors are ql(@l) and q1(e2). The vanishing of the corresponding stress vectors means that (ql(@l), 0) and (ql(@,), 0) are eigenvectors of N(0, c), the associated eigenvalues, tan dl and tan @,, being distinct. The eigenvectors, and hence q1(G1) and q1(@,), are therefore linearly independent (Barnett and Lothe, 1974a,Section 5). Similar reasoning establishes the linear independence ofq,(@,),ql(@,), and q1(@3)in a transonic state ofType 6 and the fact that, in a state of Type 5, ql(@,l) does not belong to the subspace spanned by the orthogonal vectors (Il(@,) and q2(4j2). The linear independence of ql(Gl), ql(@,), and q2(@,) then follows.

Surface Waves in Anisotropic Elastic Materials

367

2. Type 1 When L = 1 Eq. (9.19) tells us that S,(u) has at least one zero eigenvalue at u = ij. If the number of zero eigenvalues is two or three, the argument used in the last subsection rules out the possibility of there being a free surface wave. When a single eigenvalue of S , ( u ) vanishes at u = i; the variations of the three eigenvalues A,(u) (known, from Section VIII,A, to be monotone functions) over the interval 0 Iu 5 ij have the two possible forms illustrated in

(a)

(b)

FIG. 3. Possible variations with o of the eigenvalues &(u) of S,(u) when the transonic state is exceptional and of Type 1.

Fig. 3 (cf. Barnett and Lothe, 1974a, Fig. 2). A unique free surface wave exists when Fig. 3a is applicable and the two nonzero eigenvalues of S,(i) are patently positive. There can be no free surface wave when the second possibility pertains and Fig. 3b shows that the nonzero eigenvalues of S,(h) are, in this case, negative. As noted by Barnett and Lothe (1974a, Section 6), the situation can thus be conveniently summarized in terms of the trace of S3(C): when the transonic state is exceptional and of Type 1 a unique free surface wave exists if tr S,(C) > 0; otherwise no such wave exists. Examples of the behavior represented in Figs. 3a and 3b are encountered in the numerical results reported by Farnell (1970, Section V). For propagation along the [100] direction on the (001) plane of the cubic crystal nickel, for instance, a surface wave coexists with a faster body wave satisfying the traction-free boundary condition, in full qualitative accord with Fig. 3a. On the other hand, for propagation in nickel along the [I101 direction on the same plane, a body wave meeting the condition of zero traction at x2 = 0 is accompanied by a faster wave which, apart from the fact that it travels with a supersonic speed, has all the properties of an authentic surface wave (see

368

P . Chadwick and G . D. Smith

Farnell's Fig. 4).This combination is consistent with Fig. 3b when the two eigenvalues of S,(u) which are negative at u = 6 become zero together at some supersonic speed us. The two patterns of behavior just described recur in other orientations of cubic crystals and in noncubic materials (see, for example, Farnell's Figs. 13, 26, and 28). But the configurations in which a body wave obeys the traction-free boundary condition invariably enjoy a high degree of symmetry. The Occurrence of a supersonic surface wave is made possible by the vanishing, due to the prevailing symmetry, of the amplitude coefficient y1 associated with the real eigenvalue pl(O, us) in the solution (7.2). At neighboring directions of propagation the symmetry is lost and all three of the 7:s will normally be nonzero. The real eigenvalue pl(O, us) then enters the solution, giving the corresponding partial wave a direction of propagation inclined at an angle tan-' p,(O, us) to the plane x2 = 0. As a result the displacement-stress field no longer decays toward zero as xt + a~and there is a nonzero component of energy flux in the x2 direction. Because of these properties the name pseudo-, or leaky, surface waue is given to the overall disturbance. Such a wave is inherently a supersonic phenomenon.

D. SUMMARY OF CONCLUSIONS At present no general statement can be made about the existence of a free surface wave in an anisotropic elastic half-space unless the material satisfies the strong convexity condition (3.3). When this requirement is fulfilled, as of course it is by all known elastic materials, the governing factor is the nature of the transonic state. This is determined by the slowness section of the material which is, in its turn, specified by the linear elastic constants and the orientations of the surface and wave normals (these latter directions fixing the reference plane R and the base vector el). The six possible types of transonic states are characterized in Section VI,C,3, and all except Type 3 may be either normal or exceptional; these terms are defined in Section IX,B,3. A transonic state of Type 3 is necessarily normal. A complete solution of the existence problem is provided by the following conclusions, established in Sections IX,B and C. (a) If the transonic state is normal, a unique free surface wave exists. (b) If the transonic state is exceptional and not of Type 1, no free surface wave exists. (c). If the transonic state is exceptional and of Type 1, a unique free surface wave exists if tr S,(6) =- 0. Otherwise there is no free surface wave.

Surface Waves in Anisotropic Elastic Materials

369

Each of the conclusions (a) to (c) is stated or implied in the work of Barnett and Lothe (1974a, Sections 5 and 6).

SYMMETRY E. HEXAGONAL The properties of free surface waves in a hexagonal elastic half-space, situated with the zonal axis of the material normal to the reference plane R, have been worked out in stages in Sections VIII,A,2 and VIII,C,2. We now conclude our study of this special case by verifying that the earlier results are consistent with the general findings summarized above. It is presumed here that the transverse body wave speeds V, and V3 are unequal, so that the transonic state is of Type 1. In cases. A(i) and A(ii) the limiting body wave travels in the x1 direction with speed V, and a short calculation, using Eqs. (7.2), (6.16), (6.22), (6.23), and (6.47), , shows that the associated displacement-stress field is given by m

(u(xl - V, t), t(xl - V, t))

=

U ( k )exp{ik(x, - V, t)} dk(e,, p V i el).

-a 0

(9.20) Since t , l(xl - V, t) # 0, the traction on the plane x2 = 0 is nonzero, proving that the transonic state is normal. It follows from part (a) of the general theorem stated in Section IX,D that a unique free surface wave exists, and this is the conclusion reached by direct calculation in Section VIII,A,2. In case B the speed with which the limiting body wave propagates in the x1 direction is V, and in place of Eq. (9.20) we obtain

U ( k ) exp{ik(x, - V,t)} dk(e, , 0 ) .

(u(x, - V,t), t(xl - V 3 t ) )= a JOm

(9.21) The vanishing of the stress vector indicates that the transonic state is exceptional. We therefore proceed to calculate tr S3(6). The result, derived from Eqs. (6.57) and (6.58), is tr S,(V,) = p V f ( V l ( V : - V:)-’”

+ V,(V$ - V:)-”2)F(V:/V2),

(9.22)

and we recall that the function F(x), defined by Eq. (6.59), is monotone increasing in the interval [0, 11with a zero at xs . By appeal to part (c) of the general theorem, a unique free surface wave hence exists if ( V3/V# > xs and there is no free surface wave if (V, /V,)* 5 xs .Again, full agreement with our earlier results is secured.

370

P . Chudwick and G . D. Smith

As noted in Section VIII,A,2, when xs 2 (V, /V2)2the eigenvalues of S,(u) which are negative at v = V, vanish together outside the subsonic range, at u = x,”’V2. We thus have an example of the behavior described in Section IX,C,2; the body wave represented by Eq. (9.21) is accompanied by a disturbance traveling at a supersonic speed but otherwise displaying all the properties of a genuine free surface wave. The numerical results for hexagonal materials given by Farnell (1970; see, for example, the data for zinc in Fig. 28) lead us to expect that, for directions of propagation in a zonal plane, the supersonic free surface wave is contiguous to a branch of pseudosurface waves while the body wave marks the degeneration, in the highly symmetric configuration obtaining here, of a branch of legitimate free surface waves.+ An existence-uniqueness theorem for free surface waves applicable to certain symmetric placements of semi-infinite elastic bodies with orthorhombic (or higher)symmetry has been proved, by ad hoc methods, by Chadwick (1976). In interpreting this result in the light of the present work it has to be borne in mind that Chadwick’s proof does not distinguish between genuine and supersonic surface modes. When this distinction is ignored in the example under consideration there is, in all cases, precisely one free surface wave, in agreement with Chadwick’s theorem. F. AN ASYMPTOTIC LEMMA We prove here the following result, used in Section IX,B. Let rZ be a positive real parameter and let f o ( x ) ,f l ( x ) , . ..,f N ( x )and g ( x ) be real-valued functions with the following properties. (a)f o ( x ) ,f i ( x ) , .. .,fN(x) and g ( x ) are continuous in the interval a 5 x I b. (b) g “ ( x ) is continuous in the interval a < x < b. ( c ) g ( x ) has a minimum at x = c, where a < c < 6, and g’(c) = 0, g”(c) > 0. (d) The infimum of g(x) in any closed subinterval of [a, b ] not containing x = c exceeds g(c). Define (9.23)

(ii) iff,(c) = 0 a n d f g ( x ) is continuous in (a, b), I(A) is bounded as 1 1 0. T h e proof is patterned on Laplace’s method for definite integrals containing a large exponential (see Copson, 1965, Chapter 5).

’

Note Added in Proof Lothe and Alshits (1977) have made detailed calculations confirming these expectations.

371

Surface Waves in Anisotropic Elastic Materials

(i) First, observing condition (c), we select a positive real number E less than g”(c). Since fK(x) ( K = 0, 1, . .., N ) and g”(x) are continuous for a < x < 6, there then exists a positive real 6 < min(c - a, b - c ) such that

+

fK(c) -

0 < g”(c) - E I g ” ( x ) I g”(c)+ &

i

for

-

II

+

(9.25) (9.26)

Conditions (b) and (c) allow us to apply Taylor’s theorem to g ( x ) in the form x Ic + 6, + f ( x - c)’g”({) for c - 6 I where L‘ - 6 < 4 < c + 6. Hence, from (9.26) and (9.27),

g(x) = g(c)

+A(x

-

c)’

I g(x)

-

for c - 6 I x I c

g ( c ) I @(x - c)’

(9.27)

+ 6, (9.28)

where A

and

= g”(c) - E

B

= g”(c)

+E

(9.29)

are positive constants. From the inequalities (9.25) and (9.28) we now have

(9.30) or, evaluating the first and third integrals,

(9.31) In the remainder of the interval [a, b], g ( x ) - g(c)

+ A > G - g ( c ) + 1,

(9.32)

where G=

inf

(9.33)

g(x) > dc),

1a.c-dl u [ c + 6 , b l

by condition (d). Hence F

(9.34)

P. Chadwick and G . D. Smith

372

where

(9.35) The results (9.29), (9.31), and (9.34) combine to give

-

+ &}‘I2

(g”(c)

< -

{g”(c)- E ) ” Z

I”2F

I”?F + G - g(c) + A ’

7c

and on proceeding to the limit I

(9.36)

10 we reach the inequalities

(9.37) But

E

can be made as small as we please. Hence

(9.38) and the asymptotic formula (9.24) follows directly. (ii) Iffo@) = 0 andf;(x) is continuous in (a, b) we replacef,(x) byfG(x) in the inequalities (9.25) and obtain, by a second application of Taylor’s theorem, f o ( x )= (x - c)fb(c)

+ t(x - c)y&)

for c - 6 Ix Ic + 6, (9.39)

where c - 6 < r] < c + 6. The bounds in the inequalities replacing (9.30)can be integrated and are found to approach 268-’{f$(c) - E } and 2 6 A - ’ { f ; ( c ) + E } as I 1 0. The bounds for the remainder of the interval [a, b] corresponding to (9.34)also tend to finite limits as I 1 0 and, as before, the contributions to I ( I ) of the functionsf,(x), . . .,fN(x)vanish in this limit. The stated conclusion is thus proved.

Surface Waves in Anisotropic Elastic Materials

373

X. Supplementary Topics

From a strictly theoretical standpoint a complete account of the existence problem for free elastic surface waves requires an understanding of the incidence of nonexistence when the strong convexity condition (3.3) is relaxed. As long as interest is focused on infinitesimal deformations from a natural configuration of an elastic half-space this is a somewhat academic question, but the case is different when consideration turns to the propagation of surface waves in a highly deformable body which has previously received a finite static strain.? The elasticity tensor entering the equation of motion is then interpretable as the instantaneous rate of change of the nominal stress with respect to the transposed deformation gradient as the material passes through the prestrained state, and the strong convexity of this tensor is arguably too severe a restriction on elastic response to be generally acceptable (Wang and Truesdell, 1973,Chapter 111, Sections 8 and 9). Furthermore, unless the boundary of the semi-infinite body is tractionfree in the finitely deformed equilibrium configuration, the constant fourthorder tensors appearing in the equation of motion and the boundary condition are not the same. Free surface waves may therefore fail to exist in certain circumstances and particular cases have been exhibited by Flavin (1963, Section 4) and Willson (1972a, Section 5 ; 1972b, Section 6). General results are still lacking, however, in this area. A different elaboration of the theory of elastic surface waves arises when the transmitting material is assumed to have piezoelectric properties, a generalization of some importance in view of the widespread use of piezoelectric crystals in physics and technology. The enlargement of the system of field and constitutive equations occasioned by the cross-couplingof the deformation and the electric field is accompanied by an increase in the number of possible boundary conditions. Each of these factors complicates the analysis of free surface waves, but it has been found possible to develop an eightdimensional formalism in parallel with the treatment given herein by adjoining the electric potential to the mechanical displacement, and the normal component of the electrical displacement to the traction on the boundary (Kraut, 1969; Barnett and Lothe, 1975b). In recent work Lothe and Barnett

' The analysis of elastic surface waves within the framework of the theory of small amplitude motions superimposed on a finite elastic deformation has been presented by Eringen and Suhubi (1974, Section 4.4) and a number of concrete examples have been worked out by Willson (1972a,b, 1973a,b, 1974a.b). Related studies of the stability of a deformed elastic halfspace are described in papers by Usmani and Beatty (1974) and Beatty and Usmani (1975). All this work refers to elastic materials which are initially isotropic.

374

P . Chadwick and G. D. Smith

(1976b) have investigated the existence problem in this more complicated setting and made substantial progress toward its solution. The third topic which may appropriately be mentioned here is the propagation of interfacial waves in a bicrystalline medium formed by welding together two semi-infinite elastic bodies composed of different anisotropic materials or the same material in distinct orientations. Barnett and Lothe (1974b)have briefly examined the implications for this problem of the solution of the equations of linearized elastodynamics representing a line dislocation moving steadily in a crystalline interface. They have shown in this way that the secular equation determining the speed of plane interfacial waves can be expressed as a single real relation, a conclusion also reached, by a different argument, by Chadwick and Currie (1974). It follows that the directions in which waves may propagate are not discrete, but more incisive methods will clearly be needed to elucidate the actual conditions under which interfacial waves can exist. ACKNOWLEDGMENTS We wish to thank Drs. P. K. Currie and M.J. P. Musgrave for stimulating our interest in this subject and the Science Research Council of the U.K. for the award of Studentships to one of the authors (GDS). REFERENCES

ASARO,R. J., HIRTH, J. P., BARNETT, D. M., and LOTHE, J. (1973). A further synthesis of sextic and integral theories for dislocations and line forces in anisotropic media. Phys. Status Solidi B 60, 261-271. BARNETT,D. M., and LOTHE, J. (1973). Synthesis of the sextic and the integral formalism for dislocations, Greens functions, and surface waves in anisotropic elastic solids. Phys. Noru. 7, 13-19. BARNETT,D. M., and LOTHE,J. (1974a). Consideration of the existence of surface wave (Rayleigh wave) solutions in anisotropic elastic crystals. J. Phys. F. 4, 671-686. BARNETT, D. M., and LOTHE, J. (1974b). An image force theorem for dislocations in anisotropic bicrystals. J. Phys. F. 4, 1618-1635. BARNETT,D. M., and LOTHE, J. (1975a). Line force loadings on anisotropic half-spaces and wedges. Phys. Norv. 8, 13-22. BARNETT, D. M., and LOTHE, J. (1975b). Dislocations and line charges in anisotropic piezoelectric insulators. Phys. Status M i d i B 67, 105-111. BARNETT, D. M., and SWANGER, L.A. (1971). The elastic energy of a straight dislocation in an infinite anisotropic elastic medium. Phys. Status Solidi B 48, 419-428. BARNETT, D. M., NISHIOKA, K., and LOTHE,J. (1973a). Surface waves in anisotropic media. A formulation for the purpose of rapid numerical calculations. Phys. Status Solidi B 55, K115-KI17. BARNEW,D. M.,LOTHE,J., NISHIOKA,K., and ASARO,R. J. (1973b). Elastic surface waves in anisotropic crystals: A simplified method for calculating Rayleigh velocities using dislocation theory. J. Phys. F. 3, 1083-1096.

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BEATTY, M. F., and USMANI, S. A. (1975). On the indentation of a highly elastic half-space. Q. J . Mech. Appl. Math. 28, 47-62. BELTZ,R. J., DAVIS,T. L., and MALBN,K. (1968). Some unifying relations for moving dislocations. Phys. Status Solidi 26, 621-637. BRAEKHUS, J., and LOTHE,J. (1971). Dislocations at and near planar interfaces. Phys. Status Solidi B 43, 651-657. CHADWICK, P. (1976). The existence of pure surface modes in elastic materials with orthorhombic symmetry. J. Sound Vib. 47, 39-52. CHADWICK, P., and CURRIE, P. K. (1974). Stoneley waves at an interface between elastic crystals. Q. J. Mech. Appl. Math. 27, 497-503. COPSON,E. T. (1965). “Asymptotic Expansions.” Cambridge Univ. Press, London and New York. CURRIE,P. K. (1974). Rayleigh waves on elastic crystals. Q. J. Mech. Appl. Marh. 27.489-496. DE KLERK, J. (1972). Elastic surface waves. Phys. Today 25 ( N o . ll), 32-39. DUFF,G. F. D. (1960). The Cauchy problem for elastic waves in an anisotropic medium. Philos. Trans. R. SOC.London, Ser. A 252, 249-273. ERINGEN,A. C., and SUHUBI,E. S. (1974). “Elastodynamics,” Vol. 1. Academic Press. New York. ESHELBY, J. D. (1956). The continuum theory of lattice defects. Solid State Phys. 3. 79-144. ESHELBY, J. D., READ,W. T., and SHOCKLEY, W.(1953). Anisotropic elasticity with applications to dislocation theory. Acta Metall. 1, 251-259. FARNELL, G. W. (1970). Properties of elastic surface waves. Phys. Acoust. 6, 109-166. FLAVIN, J. N. (1963). Surface waves in pre-stressed Mooney material. Q. J . Mech. Appl. Math. 16, 44-449. HALMOS, P. R. (1958). “ Finite-Dimensional Vector Spaces,” 2nd ed. Van Nostrand-Reinhold, Princeton, New Jersey. HIRTH,J. P. (1976). Anisotropic elastic theory of line defects. Met. Sci. 10, 222-224. K. A., and TONNING, A. (1969). Elastic surface waves in crystals. Phys. Rev. 184, INGEBRIGTSEN, 942-95 1. KRAUT,E. A. (1969). New mathematical formulation for piezoelectric wave propagation. Phys. Rev. 188, 145C1455. LARDNER, R. W. (1974). “Mathematical Theory of Dislocations and Fracture.” Univ. of Toronto Press, Toronto. LOTHE,J. (1972). Dislocations in anisotropic media. In “Computational Solid State PhySics (F. Herman, N. W. Dalton, and T. Koehler, eds.), pp. 425-440. Plenum, New York. LOTHE,J., and ALSHITS,V. I. (1977).The criterion for the existence of quasi-bulk surface waves. Sou. Phys. (Crystall.). (To be published.) LOTHE,J., and BARNETT, D. M. (1976a). On the existence of surface-wave solutions for anisotropic half-spaces with free surface. J. Appl. Phys. 47, 428-433. D. M. (1976b). Integral formalism for surface waves in piezoelectric LOTHE, J., and BARNETT, crystals. Existence considerations. J. Appl. Phys. 47, 1799-1807. MALBN,K. (1970). Stability and some characteristics of uniformly moving dislocations. Narl. Spec. Publ. 317, 23-33. Bur. Stand. (US.), MALBN,K. (1971). A unified six-dimensional treatment of elastic Green’s functions and dislocations. Phys. Status Solidi B 44,661-672. MALBN,K., and LOTHE,J. (1970). Explicit expressions for dislocation derivatives. Phys. Starus Solidi 39, 287-296. MALBN,K., and LOTHE,J. (1971). Comment on dislocations moving with the Rayleigh wave velocity. Phys. Status Solidi B 43, K139-K142. MITRA,T. (1963). Continuous distributions of moving dislocations. Philos. Mag. [8] 8,843-857. ”

3 76

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MURA,T. (1975). A note on the strain field of a dislocation line in anisotropic media. Phys. Status Solidi B 70, Kl-K6. MUSCRAVE,M. J. P. (1970). “Crystal Acoustics.” Holden-Day, San Francisco, California. NISHIOKA, K., and LOTHE,J. (197%). Isotropic limiting behaviour of the six-dimensional formalism of anisotropic dislocation theory and anisotropic Green’s function theory. 1. Sum rules and their applications. Phys. Status Solidi B 51,645-656. K., and LOTHE, J. (1972b). Isotropic limiting behaviour of the six-dimensional formaNISHIOKA, lism of anisotropic dislocation theory and anisotropic Green’s function theory. 11. Perturbation theory on the isotropic N-matrix. Phys. Status Solidi B 52, 45-54. PEASF. M. C., 111. (1965). “Methods of Matrix Algebra.” Academic Press, New York. SCHOUTEN, J. A. (1954). “Tensor Analysis for Physicists,” 2nd ed., pp. 178-179. Oxford Univ. Press (Clarendon), London and New York. SMITH,H. I. (1970). The physics and technology of surface elastic waves. Int. J. Nondestr. Test. 2, 31-59. STEEDS, J. W. (1973). “Anisotropic Elasticity Theory of Dislocations,’’ p. 26. Oxford Univ. Press (Clarendon), London and New York. STROH,A. N. (1958). Dislocations and cracks in anisotropic elasticity. Philos. Mag. [7] 3, 625-646. STROH,A. N. (1962). Steady state problems in anisotropic elasticity. J . Math. Phys. 41, 77-103. SYNGE,J. L. (1956a). Elastic waves in anisotropic media. J. Marh. Phys. 35, 323-334. SYNGE,J. L. (1956b). Flux of energy for elastic waves in anisotropic media. Proc. R. Ir. Acad., Sect. A 58, 13-21. SYNGE,J. L. (1957). “The Hypercircle in Mathematical Physics,” pp. 411-413. Cambridge Univ. Press, London and New York. TEUTONICO, L. J. (1962). Uniformly moving dislocations of arbitrary orientation in anisotropic media. Phys. Rev. 127,413-418. TOUPIN,R.*A., and RIVLIN,R. S. (1961). Electro-magneto-optical effects. Arch. Ration. Mech. Anal. 7, 434-448. USMANI,S . A,, and BEAITY,M. F. (1974).On the surface instability of a highly elastic half-space. J . Elasticity 4, 249-263. VIKTOROV, I. A. (1967). “Rayleigh and Lamb Waves.” Plenum, New York. C. (1973). “Introduction to Rational Elasticity.” Noordhoff Int., WANG,C.-C., and TRUESDELL, Leyden. WHITE,R. M. (1970). Surface elastic waves. Proc. IEEE 58, 1238-1276. WILLSON,A. J. (1972a). Wave-propagation in uniaxially-stressed elastic media. Pure Appl. Geophys. 93, 5-18. WILLSON,A. J. (1972b). Wave-propagation in biaxially-stressed elastic media. Pure Appl. Geophys. 95,48-58. WILLSON, A. J. (1973a). Surface and plate waves in biaxially-stressed elastic media. Pure Appl. Geophys. 102, 182-192. WILLSON, A. J. (1973b). Surface waves in restricted Hadamard materials. Pure Appl. Geophys. 110, 1967-1976. WILLSON,A. J . (1974a). Surface waves in uniaxially stressed Mooney material. Pure Appl. Geophys. 112, 352-364. WILLSON,A. J. (1974b). The anomalous surface wave in uniaxially-stressed elastic material. Pure Appl. Geophys. 112,665-674.

Author Index Numbers in italics refer to the pages on which the complete references are listed.

A

Bolotin, V. V., 246, 248, 299 Bolton, W. E., 133, 186 Born, M., 235, 241 Borodai, I. K., 169, 186 Boussinesq, J., 22, 85 Bouvard, M., 63.85 Bradley, E. F., 18, 49, 63, 64, 79, 84, 85 Braekhus, J., 315, 375 Breaux, D. K., 65.88 Bridgman, P. W., 4, 85 Browning, K. A., 27, 85 Brush, L. M., Jr., 51, 53, 85, 89 Businger, J. A., 3, 18, 49, 63, 64, 79, 84, 85

Abramowitz, M., 95, 101, 108, 185 Adachi, H., 166, 185 Adkins, J. E., 212, 220, 221, 241, 242 Alshits, V. I., 370, 375 Antman, S. S., 240, 241 Arnol'd, V. I., 258, 275, 298 Arons, A. B., 28, 89 Arya, S. P. S.,3, 48, 63, 64, 83, 84, 84, 85 Asaro, R. J., 329, 336, 351, 352, 356, 357, 358, 3 74 Ashley, H., 96, 186 Asmis, K. G., 247, 301

C

B Baba, E., 162, 164, 168, 186 Baines, W. D., 53, 84 Ball, J. M., 228, 241 Barnett, D. M., 304, 305, 311, 314, 315, 318, 319, 320, 324, 329, 331, 332, 336, 331, 338, 340, 347, 348, 350, 351, 352, 353, 354, 355, 356, 357, 358, 359, 364, 365, 366, 367, 369, 373, 374,374, 37s Batra, R. C., 227, 241 Beatty, M. F., 223, 241, 373, 375, 376 Bechov, R., 35, 85 Bell, J. F., 221, 229, 241 Belt?., R. J., 343, 375 Benard, H., 21, 28, 34, 85 Bernussov, J., 283, 298 Birkhoff, G., 4, 85 Birkhoff, G. D., 258, 298 Black, H. F., 246, 298 Blanchard, D., 28.89 Bogoliubov, N. N., 247, 248, 299 Bogy, D. B.,203, 241 Bohn, M.P., 246,299

Carbone, E. R., 240, 241 Carlson, D. E., 194, 241 Cauchy, A.-L., 191, 229, 234, 235, 241 Cesari, L., 246, 299 Chadwick, P., 305, 370, 374, 375 Chapman, R. B., 164, 166, 174, 183, 186 Cheng, D. K., 270, 299 Cheng, W. G., 286,300 Cheng, W. H., 275, 278, 279, 282, 283, 286, 290, 291, 292, 294, 296, 297, 298, 300 Chhatpar, C. K., 247, 299 Chu, T. Y., 34, 40, 50, 66, 85, 86 Coddington, E. A,, 280, 299 Coleman, B. D., 225, 241 Copson, E. T.. 370, 375 Corcos, G. M., 27, 88 Corrsin, S., 66, 85 Cosart, W. A,, 20. 88 Cosserat, E., 196, 241 Cosserat, F., 196, 241 Cote, 0. R., 84, 90 Crapper. P. F., 30, 51, 53, 54, 59, 85 Croft, J. F., 34, 36, 85 377

Author Index

378 Cromwell, T., 52, 85 Csanady, G. T., 31, 33, 85 Cummins, W. E., 146, 147, 160, 186 Currie, P. K., 350, 374, 375

Frank, W., 121, 176, 186 Friedmann, P., 246, 299 Fiirth, R.,235, 241

G

D Dafermos, C. M., 191, 241 Dagan, G., 168, 186 Daly, B. J., 73, 85 Daoud, N., 92, 164, 166, 186 Davis, T. L., 343, 375 Day, A., 212,229, 244 Deardorff, J. W., 24, 34, 35, 36, 73, 85 de Klerk, J., 305, 375 Dodu, J., 51, 61, 88 Donaldson, C. DuP., 73. 85 Dropkin, D., 34, 86, 89 Duff, G. F. D., 339, 375 Dugundji, J., 247, 299 Dumas, H., 63,85 Dyer, A. J., 49, 65, 85, 86

E Edelen, D. G. M., 196, 241 Ekman, V. W., 32, 86 Ellison, T. H., 63, 65, 67, 68, 77, 80, 86 Elmaraghy, R.,246, 299 Ericksen, J. L., 190, 191, 211, 212, 218, 220, 225,226,228,236,237,239, 241, 242 Eringen, A. C., 373, 375 Eshelby, J. D., 199, 242, 315, 325, 326, 327, 330,375 Evan-Iwanowski, R. M., 246, 247, 299,301

F Faltinsen, 0. M., 128, 136, 137, 140, 167, 169, 172, 181, 186, 188 Farnell, G. W., 305, 355, 367, 370, 375 Federov, K. N., 50,89 Flavin, J. N., 373, 375 Fletcher, D. C., 195, 242 Fomin, S. V., 195, 242 Fosdick, R. L., 211, 242 Fox, D. C., 72, 86

Garon, A. M., 28, 34, 37, 40, 42, 86 Gazda, I. W., 34, 37, 89 Gelfand, I. M., 195, 242 Gent, A. N., 240, 242 Ghobarah, A. A., 246, 247, 299 Gille, J., 21, 50, 66, 86 Globe, S., 34, 86 Goldstein, R. J., 28, 34, 37, 40, 42, 50, 66, 85, 86 Green, A. E., 220, 221, 242 Gumowski, I., 275, 299 Gurtin, M. E., 191, 227, 242

H Hagedorn, P., 247, 248, 299 Haight, E. C., 247, 299 Hale, J. K., 246, 299 Halmos, P. R.,307, 375 Hamel, G., 199, 242 Hanaoka, T., 126, 186 Harlow, F. H., 73,85 Havelock, T. H., 92, 186 Hayashi, C., 246, 299 Hazel, P., 27, 86 Herring, J. R.,72, 86 Herrmann, G., 246, 299 Hicks, D. B., 65, 86 Hill, R., 191, 192, 224, 228, 237, 242 Hinze, J. O., 13, 71, 86 Hirata, M. H., 164, 186 Hirth, J. P., 305, 329, 374, 375 Holton, J. R.,30, 86 Hook, D. E., 223, 241 Houben, H., 246,299 Howard, L. N., 40, 86 Hsu, C. S., 246, 247, 248, 250, 252, 253, 255, 256, 257, 273, 274, 275, 278, 279, 282, 283, 285, 286, 290, 291, 292, 293, 294, 296, 297, 298, 299, 300, 301 Hua, L. K., 230, 242 Humphreys, H. W., 45, 47,88

Author Index I

Iwatsubo, T., 246, 300 Izakson, A. A,, 16, 55, 86 Izumi. Y . , 18, 49, 63, 64, 79, 84, 85

J

Jacobsen, J. P., 65, 86 Jeffreys, H., 34, 86 Jones, W. P., 73, 86 Joosen, W. P. A., 113, 186 Jury, E. I., 261, 300

K Kajitani, H., 169, 186 Kalnins, A,, 246, 300 Kantha, L. H., 55, 86 Kao, B.-C., 220, 242 Kato, H., 51, 55, 56, 61, 86 Kauderer, H., 247, 300 Kazanski, A. B., 33,86 Kehagioglou, T., 247, 299 Kelly, A., 235, 243 Khajeh-Nouri, B., 66, 73, 87 Khaskind, M. D., 92, 126, 186 King, W. W., 247, 299 Kirchoff, G., 197, 242 Kitaigorodskii, S . A,, 66, 86 Knops, R. J., 191, 222, 242 Knowles, J. K., 195, 199, 203, 229, 242, 244 Kolmogoroff, A. N., 77,86 Korvin-Kroukovsky, B. V., 182, 186 Koschmieder, F. L., 34, 86 Kraichnan, R. H., 35, 42, 72, 86 Kraus, E. B., 51, 87 Kraut, E. A,, 373, 375 Krawietz, A,, 228, 242 Krishnamurti, R., 34, 87 Kullenberg, G., 27, 63, 65, 69, 70, 87

379 L

Landahl, M., 96, 186 Langenecker, B., 239, 243 Lardner, R. W . , 330, 331, 332, 375 Laslett, L. J., 275, 300 Laufer, J., 16, 87 Launder, B. E., 73, 84, 86, 87 Laurence, F. J., 18, 88 Levinson, N., 280, 299 Linden, P. F., 30, 51, 53, 54, 59, 63, 85, 87 Lindley, P. B., 240, 242 Liu, H., 283, 298 Long, R. R., 6, 7, 23, 25, 26, 27, 36, 47, 51, 57, 60,61, 65, 68, 70, 72, 87 Lothe, J., 304, 305, 311, 314, 315, 317, 318, 319, 320, 323, 324, 329, 332, 333, 336, 337, 338, 340, 343, 347, 348, 350, 351, 352, 353, 354, 355, 356, 357, 358, 359, 364, 365, 366, 367, 369, 370, 373, 374, 374, 375 Love, A. E. H., 230, 243 Lovelock, D., 195, 243 Lumley, J. L., 2, 3, 6, 18, 66, 73, 87, 89

M Macmillan, N. H., 235, 243 McTernan, A. J., 246, 298 Malen, K., 304, 311, 315, 332, 333, 340, 343, 348, 351, 353, 375 Malkus, W. V. R., 34, 36, 43, 47, 50, 87 Markus, L., 258, 300 Marris, A. E., 212, 243 Maruo, H., 113, 128, 130, 142, 146, 162, 187 May, R. M., 293,300 Mellor, G. L., 73, 75, 79, 82, 87 Mettler, E., 246, 300 Michal, A. D., 207, 243 Michell, J. H., 92, 187 Michelsen, F. C., 128, 186 Miles, J. W., 27, 87 Miller, K. S., 266, 300 Millikan, C. B., 16, 55, 87 Milverton, S . W., 34, 88 Mira, C., 275, 283, 298, 299

Author Index

380

Mitropolsky, Y. A., 247, 248, 299 Monin, A. S., 7, 9, 22, 23, 31, 33, 48, 55, 86, 87 Montgomery, R. B., 69, 88 Moon, H., 222, 243, 244 Moore, M. J., 26, 27, 51, 57, 60, 65, 87 Morgan, D. L., 18, 88 Morrey, C. B., Jr., 228, 243 Mote, C. D., Jr., 246, 300 Mukhopadhyay, V., 247,299 Munk, M., 95, 187 Mura, T., 329, 376 Murdoch, I., 227, 242 Musgrave, M. J. P., 305, 339,376

Pease, M. C., 111, 315, 322, 323, 376 Peters, A. S., 92, 187 Phillips, 0. M.,23, 26, 51, 55, 56, 61, 86, 88 Pipkin, A. C., 220, 229, 242, 243 Plate, E. J., 18, 33, 48, 63, 64, 84, 84, 88 Poincare, H., 230, 243, 258, 300 Pollard, R. T., 56, 88 Porter, W. R., 121, 176, 187 Prandtl, L., 17, 35, 88 Priestley, C. H. B., 18, 35, 48, 88 Proudman, J., 26, 65, 88 Pruitt, W. O., 18, 88 Pun, L., 283, 300

N

R

Naot, D., 73, 88 Netsvetayev, Y. A,, 169, 186 Newman, J. N., 92, 104, 113, 126, 128, 130, 131, 137, 166, 171, 175, 181,187, 188 Ng, K. H., 73,88 Niiler, P. P., 57, 88 Nikuradse, J., 19, 88 Nishikawa, Y.,246, 300 Nishioka, K., 315, 317, 323, 336, 351, 352, 356, 357, 358,374,376 Noether, E., 194, 195, 196, 243 Noll, W., 191, 212, 220, 221, 228, 244

0 Ogilvie, T. F., 94, 126, 137, 160, 161, 165, 168, 172, 174, 175, 177, 178, 180, 182, 187, 188 Ogino, S., 246, 300 Orszag, S. A., 72, 86, 88

P Page, F., 65, 88 Paidoussis, M.P., 246, 300 Panov, A. M., 266,270,300 Pao, Y. H., 72, 88 Parry, G. P., 237, 240, 243 Patterson, G. S., Jr., 72, 88 Payne, L. E., 222, 242

Rayleigh, Lord, 34, 88 Read, W. T., 315, 325, 326, 327, 375 Reece, G. J., 13, 87 Reed, A. M., 167, 187 Rehm, R. C., 69,89 Reiner, I., 231, 242 Reynolds, O., 8, 88 Reynolds, W. C., 9, 73, 88 Rhines, P. B., 56, 88 Rhodes, J. E., Jr., 246, 301 Rispin, P. P., 94, 187 Rivlin, R. S., 196, 222, 240, 243, 359, 376 Robertson, J. M.,9, 55, 64, 88 Rodi, W., 73, 87 Rogallo, R. S., 104, I87 Rosenbaum, H., 73,85 Rossby, H. T., 30, 37, 69, 88 Rotta, J. C., 75, 88 Rouse, H., 45, 47, 51, 61, 88 Rund, H., 195,243

S Sage, B. H., 65,88 St.-Venant, A.-J.-C. B. de, 220, 243 Salvesen, N., 136, 169, 188 Sanchez, V. M.,220, 229, 243 Sanford, W. F., 247, 299 Sasaki, N., 142, 187 Sawyers, K., 222,243

38 1

Author Index Schlinger, W. G., 65, 88 Schmid, E., 239, 243 Schmidt, G., 246, 301 Schmidt, R. J., 34, 88 Schouten, J. A,, 340, 376 Schwarz, H. R., 247, 301 Schwarz, W. H., 20,88 Scotti, R. S., 27, 88 Seitz, F., 230, 231, 243 Sewell, M. J., 227, 243, 244 Shavit, A., 73, 88 Shen, Y. T., 94, 188 Shield, R. T., 194, 199, 212, 220, 241, 244 Shockley, W., 315, 325, 326, 327,375 Shtokalo, I. Z., 246, 301 Silverthorn, L. J., 246, 299 Silveston. P. L., 34, 89 Smale, S., 258, 301 Smith, H. I., 305, 376 Sijding, H., 169, 188 Somerscales, E. F. C., 34, 37,89 Somerset, J. H., 247, 301 Spalding, D. B., 73, 88 Spencer, A. J. M., 220, 244 Spiegel, E. A,, 23, 89 Stakgold, I., 191, 230, 244 Starzhinskii, V. M., 246, 301 Steeds, J. W., 330, 376 Stegun, I., 95, 101, 108, 185 Stem, M., 29, 89 Sternberg, E., 195, 199, 203, 229,242, 244 Stoker, J. J., 92, 106, 151, 174, 187, 188, 254,301 Stommel, H., 28, 50, 89 Stroh, A. N., 304, 310, 315, 325, 328, 330, 332, 333, 338, 339, 345, 347, 348, 351, 376 Sugiyama, Y., 246, 300 Suhubi, E. S., 373, 375 Sullivan, R. D., 73, 85 Sundaram, T. R., 66, 69,89 Sundararajan, C., 246,300 Swanger, L. A,, 329, 331,374 Synge, J. L., 332, 340, 351, 376

T Tabarrok, B., 246, 299 Takagi, M., 169,188 Takekuma, K., 168, 186

Takens, F., 258,301 Taneda, S., 168, 188 Tani, J., 246, 301 Tasai, F., 121, 169, 176, 188 Tennekes, H., 2, 3, 6, 15, 18, 19, 89, 90 Teutonico, L. J., 339, 376 Thomas, D. B., 34,89 Thompson, J. L., 204, 244 Thompson, R., 56,88 Thompson, S. M., 59, 63, 89 Thorpe, S. A., 27, 28, 59.89 Threlfall, D. C., 28, 34, 37, 40,42, 89 Timman, R., 171, 181,188 Tondl, A,, 247,301 Tonning, A., 310, 315, 348, 353, 354, 375 Toupin, R. A,, 204, 244, 359, 376 Townsend, A. A,, 21, 34, 36, 37,66,67,89 Troesch, A. W., 133, 141, 182, 188 Troger, H., 247, 301 Truesdell, C., 191, 196, 197, 212, 220, 221, 222, 223, 226,228, 243, 244, 373, 376 Tso, W. K., 246, 247,299, 301 Tuck, E. O., 113, 136, 137, 146, 147, 149, 162, 164, 167, 169, 172, 174, 177, 178, 180, 182, 187, 188 Tulin, M. P., 168, 186 Turner, J. S., 21, 23, 27, 50, 51, 52, 53, 54, 59, 61, 63, 65, 67, 68, 86, 87, 89

U Ursell, F., 121, 124, 125, 130, 133, 134, 136, 137, 139, 141, 145, 163, 167, 186,188 Usmani, S. A,, 373, 375, 376

V Van Dyke, M., 95, 188 Varley, E., 212, 229, 244 Veronis, G., 23, 89 Viktorov, I. A., 305, 376 von Karman, T., 17,89 Vossers, G., 128, 130, 146, 188

W Wan& C.-C., 196, 197, 220, 228, 244, 373, 376 Ward, G. N., 93, 188

382

Author Index

Watkins, C. D., 27, 85 Wehausen, J. V . , 106, 107, 149, 151, 163, 177, 188 Weidenhammer, F., 247, 301 White, R. M., 305, 376 Wiley, R. L., 27, 89 Wilkes, E. W., 191, 222, 242 Willems, P. Y., 246, 300 Willis, G. F., 24, 34, 35, 36, 85 Willson, A. J., 373, 376 Wolanski, E., 51, 53, 54, 89 Wolfshtein, M., 73, 88 Woods, J. D., 27, 50, 65, 89 Wu, J., 58, 61, 90 Wu, T. Y., 94, 104, 187, 188 Wyngaard, J. C., 18, 49, 63, 64,66, 79, 84, 85,90

Y Yaglom, A. M., 7, 9, 22, 23, 31, 35, 48, 55, 85, 87 Yakubovich, V. A., 246, 301 Yamada, T., 68, 82, 90 Yee, H. C., 266, 273, 275, 278, 279, 282, 283, 286, 290, 291, 292, 293, 294, 296, 297, 298, 300,301 Yih, C. S., 22, 45, 47, 88, 90 Young, L. C., 196, 244

L

Zeman, O., 15, 19,90 Zilitinkevich, S. S., 33, 90

Subject Index

Boundary layer theory, matching in, 102 Boussinesq approximation, 22 Boussinesq equations, 23 Bow-flow problem, for slender ship in steady forward motion, 156-166 Buoyancy, in turbulence experiments without shear, 52 Buoyancy difference, across mixed layer, 62 Buoyancy flux, due to wake collapse, 7C72

A

Aerodynamics slender-body theory in, 95-105, 145 slender-ship problem and, 93 3-D potential theory and, 100-101 3-D problem in, 134, 163 2-D problems in, 163-164 1-D solutions in, 99, 103 d’Alembert’s principle, 208 Anisotropic elastic materials, surface waves in, 303-374 see also Surface waves Artificial line distribution, in slendership theory, 153 Asymptotic analysis and parametrically excited hanging string in fluid, 252-257 for weakly nonlinear systems, 247-257 Asymptotic stability global regions of, 276-283 for one-one higher-order systems, 280 for systems not one-one, 28C283 Atmosphere Ekman layer in, 31-33 mean Reynolds stress model of surface layer in, 73-80 surface layer in, 31, 73-80 Available potential energy, in turbulent flow, 25

L

Cascade of energy, in turbulent flow, 13 Cauchy-Green tensor, in elastostatics, 199 Cauchy-Poisson problem, 158 Cauchy relations, in elastostatics, 199, 234-235 Convection double diffusive, 28-29 similarity theory of, 35-37 thermal, see Thermal convection Convection motion, 3 Convection theory, in laboratory observations, 35-37 Correlation coefficient, in turbulent flow, 1 1 Couette flow, turbulent plane, 9-10

D Density variations governing equations for fluids with, 22-24 turbulence and, 21-33 Difference equations analysis of nonlinear parametric excitation problems by, 257-266 discrete time formulation and, 258-259 Diffraction problems, for slender ship in unsteady motion, 125-145

B Backward images, of periodic points in nonlinear parasitic excitation problems, 265-266 Blaha effect, 239 Boundary layer, viscous, see Viscous boundary layer

383

Subject Index

384

Diffusion molecular, 29-30 turbulent, 29-30 Dimensional analysis generalized, 6 Pi theorem in, 4 turbulence and, 4-6 Discrete time formulation, 258-259 Discrete trajectory, 259 Dislocation-surface wave analogy, 352-353 Displaced mass ratio, defined, 256 Dissipation function E, 12 Drag coefficient, in turbulent flow, 16-20

E Eddies earth rotation and, 2 horizontal dimensions of, 2 Eddy diffusivity, in stably stratified fluids, 66-70 Eddy viscosity vs. eddy velocity, 66-67 in stably stratified fluids, 66-70 in turbulent flow, 19-20 Effective damping coefficient, 256 Eigenvalue problem, for surface waves in anisotropic elastic materials, 3 14-325 Ekman layer, properties of, 31-33 Elasticity tensors in reference plane, 3 12-3 15 for surface waves in anisotropic elastic materials, 311-314 Elasticity theory, 190 Elastic materials, anisotropic, see Anisotropic elastic materials Elastic surface waves see also Free surface waves; Surface waves basic analysis of, 347-355 free surface waves and, 350-352 general solution in, 348-350 Elastodynamics, d‘Alembert’s principle and, 208 Elastostatic equations, 197-200 Elastostat ics arena in, 220-224 averages in semi-inverse methods for, 2 13-220 basic equations of, 189-190, 192-200

Blaha effect in, 239 Cauchy relations in, 199, 230, 233-235 crystal lattices in, 229-232 energy method in, 225-236 Euler-Lagrange operators in, 192-197 208-2 10 experiment and mechanistic theory in, 220-24 1 homogeneous isotropic materials in, 210-213 inaccessibility in, 224-229 infinitesimal disturbances in, 227 instabilities in, 237-241 kinematics in, 200-207, 232-234 lattice configurations in, 232-234 molecular theory in, 234-237 null Lagrangian in, 196-197 perturbation, 221 plane, see Plane elastostatics point group in, 231 reduced equations in, 207-210 St.-Venant’s principle in, 222-223 semi-invariant function in, 196 semi-inverse methods in, 200-220 space group in, 231 special topics in, 189-241 Elastostatic solution, modification of, 333-335 Euler-Lagrange operators, in electrostatics, 192-196

F Flux Richardson number, 26, 63 see also Richardson number Forced oscillations horizontal body oscillations and, 116 low-frequency, 111-118 nonuniformity of near-field solution in, 117 for slender ships in unsteady motion, 111-125 3-D Laplace equation for, 114 very high frequency, 123-125 Free surface waves, 3 S 3 5 3 asymptotic lemma in, 370-372 and behavior of h ( u ) in transonic limit, 361-366 exceptional transonic states in, 366-368

Subject Index existence of, 359-372 formulation of existence problem in, 3-361 hexagonal symmetry in, 369-370 polarization vector in, 358-359 speed of propagation in, 357-358 uniqueness and related properties of, 355-359 Friction velocity, in turbulent flow, 11 Froude number, 150, 156-157, 166-167 inertial forces and, 157 G Geophysical systems, turbulence in, 1-84 see also Turbulence Gradient Richardson number, 26,6465 see also Richardson number

H Heave/pitch forced-oscillation problem, 176182 Heaviside step function, 151, 154, 159 High-frequency oscillations, for slender ships in unsteady motion, 118-120 Hinged bar asymptotic stability in, 293-296 bifurcation and, 293 damped but elastically unrestrained, 288-296 elastically restrained, 296298 periodic solutions for, 288-293 subjected to periodic impact load, 286-298 Homogeneous fluids, 7-20 Reynolds stresses in, 7-9 logarithmic layer and drag coefficient for, 16-20 modified pressure P in, 10 Homogeneous isotropic materials, in elastostatics, 210-213 I Incompressible fluid, continuity equation for, 7 Incremental potential energy, in turbulent flow, 25

385

Instability regions, responses to, 248-252 Internal wave field, defined, 3

K Kelvin-Helmholtz instability, 27 Kolmogorov scales, 14

L Laminar motion, 3 change to turbulent motion, 8 Lattice kinematics, in elastostatics, 232-234 “Law of the wall,” in turbulent flow, 16 Local isotropy, in turbulent flow, 14 Logarithmic friction law, 18 Logarithmic layer, in turbulent flow, 16-20 Log law, for turbulent flow, 17-18 Long-wave theory, for slender ships in unsteady motion, 117, 130-131

M Marginal stability, defined, 34 Matched asymptotic expansions, 102 Matching, in boundary layer theory, 102 Modified pressure P, in homogeneous fluids, 10 Molecular boundary layers thick conductive layer in, 38 in thermal convection, 37-40 thick viscous layer in, 38-40 Molecular theory, in electrostatics, 234-237 Moving line singularity breakdown of solution in, 339-342 limiting speed in, 335-339

N Naval architecture, “strip theory” in, 118-120 Navier-Stokes equations, 8, 16, 72 Near field, 2-D Laplace equation for, 120-121 Near-field behavior, for slender ships in unsteady motion, 117, 120

386

Subject Index

Nonlinear parametric excitation problems, 245 -298 analysis of by difference equations, 257-266 asymptotic analysis for weakly nonlinear systems, 247-251 backward images of periodic points in, 265-266 bifurcation and birth of new periodic solutions in, 261-265 global regions of asymptotic stability in, 276-283 gradient vectors in, 263 hinged bar subject to periodic impact load in, 286-298 impulsive parametric excitation for, 283-298 linear system stability and response in, 284-286 new periodic solutions in, 265 periodic solutions and local stability in, 259-261 P-K point in, 260 P-K solutions in, 261-265 second order difference systems in, 266-276 steady state response in, 255-257 Nonlinear second order systems, 270-276 bifurcation and geometric visualization in, 270-273 Nonlinear systems impulsively and parametrically excited, 283-284 parametric excitation in, see Nonlinear parametric excitation problems with single degree of freedom, 247-252 steady state response in, 255-257 Numerical simulations, of turbulent flows, 72-73 Nusselt number-Rayleigh number relation, in thermal convection, W 4 3

0 Oceans, quasihorizontal motion in, 2 see also Surface waves Order-of-magnitude symbol, 6-7

P Parametrically excited hanging string in field, 252-257 Parametric excitation impulse, 283-298 nonlinear, see Nonlinear parametric excitation problems Partial differential equation, with periodic coefficient, 252-255 Pipes and channels, turbulent flows in, 9-16 Pi theorem, in dimensionless analysis, 4-6 P-K point in nonlinear parametric excitation problems, 260 in second order difference systems, 275 P-K solutions in nonlinear parametric excitation problems, 260-265 for second-order systems, 279 Plane elastostatics displacement gradient and stress fields in, 328-329 stationary line dislocation energy in, 331-332 stationary line singularity in, 327-332 for surface waves in anisotropic materials, 325-332 total strain energy in, 329-331 Plumes, in thermal convection, 44 Point group, in elastostatics, 231 Polarization vector, in free surface waves, 358-359 Potential energy, available and incremental, 25 Potential theory, 2-D and 3-D, 100-101 Prandtl number, 36

R Random wave field, defined, 3 Rayleigh number marginal stability and, 34 Nusselt number and, 40-43 range of, 36 temperature and, 66 Reynolds number in convection, 3 in dimensional analysis, 5

Subject Index laminar-to-turbulent change and, 8 maximum, 72 plumes and, 44 “running,” 157 turbulence and, 3 Reynolds stresses in atmosphere surface layer, 73-80 in homogeneous fields, 7-9 Richardson number buoyancy difference and, 62-63 critical flux, 80-83 flux, see Flux Richardson number gradient, see Gradient Richardson number in Moore and Long experiment, 65-69 in shear experiments, 57 turbulence and, 26-28 Rotation, neglect of, in turbulence, 30-3 1 S Saddle point in second order difference systems, 268 separatrices of, 273-276 Second order difference systems asymptotic stability for, 276-279 nonlinear, 270-276 in nonlinear parametric excitation problems, 266-276 P-K points in, 275 P-K solutions in, 279 saddle point of first kind in, 268 saddle point of second kind in, 268 saddle point separatrices in, 273-276 singular points and periodic solutions in, 266-268 stable spiral point in, 269 unstable node of first kind in, 268 unstable spiral point in, 268-269 Semi-inverse methods averages in, 213-220 homogeneous isotropic materials in, 210-213 raison d’bre for, 213 Separatrices defined, 273 of negative orders, 275-276 Shear thermal convection with, 48-50 turbulence experiments with, 54-58 turbulence experiments without, 51-54

387

Ship hydrodynamics see also Slender ship in steady forward motion, etc foundation of, 92 potential theory in, 94 singular perturbation problems in, 91-185 “slenderness” concept in, 96, 129 thin-ship theory in, 92-93 Short waves, for slender ships in unsteady motion, 131-136 Similarity theory new derivation of, 46-47 for unstable case, 79-80 Singular-perturbation problems, in ship hydrodynamics, 91-185 Slender body forward motion of, 104 in steady longitudinal motion, 97 in transverse direction, 98 zero thickness in, 103-104 Slender body theory in aerodynamics, 95-105 basic concept of, 96-97 most obvious problem in, 145 as singular perturbation problem, 100 “slenderness” concept in, 96, 129 Slender ship in incident wave, 127 in steady forward motion, see Slender ship in steady forward motion Slender ship in steady forward motion, 145-169 bow-flow problem in, 156166 low-speed problem in, 166-169 as most obvious problem, 145 slender-ship theory and, 147-150 Slender ship in unsteady forward motion, 169-185 boundary condition for, 171 coordinate systems for, 170 heave/pitch forced oscillation problem for, 176-182 high-speed slender-ship theory and, 182-1 85 3-D Laplace equation for, 175 Slender ship in unsteady motion diffraction problems for, 125-145 extremely short waves and, 144-145 forced oscillations in, 111-125 high-frequency oscillations and, 118-123

388

Subject Index

Slender ship in unsteady motion (Cont’d) long waves and, 130-131 near field vs. far field in, 121-123 problem formulation in, 105-106 radiation patterns in, 107-111 short waves and, 131-144 ‘strip theory” and, 118-120 3-D Laplace solution for, 134 2 - D cross-sectional flows for, 136 2 - D problem in, 120-123 very high frequency oscillations for, 123-125 at zero speed, 105-145 Slender ship problem, defined, 93 Slender ship theory artificial line distribution in, 153-154 basic assumptions of, 146 failure of, 151-156 high-speed, 18 1- 185 ordinary, 147-150 thin-ship theory and, 155-156, 160 3-D Laplace equation for, 174-175 3-D source in, 163 2-D Helmholtz equation for, 168, 175 2-D Laplace equation Tor, 162-163, 174 2-D problems in, 163 wedge-bow problem for, 159-161 Space group, elastostatics in, 231 Stably stratified fluids buoyancy flux due to wave collapse in, 70-72 eddy viscosity and eddy diffusivity in, 66-70 energy arguments for, 60-63 laboratory experiments vs. observations in atmosphere and ocean, 63-66 turbulence in, 50-72 turbulence experiments without shear in, 5 1-54 Stationary line singularity, in plane elastostatics, 327-332 Stratified fluid, erosion of, 53 “Strip theory,” for slender ships in unsteady motion, 118-120 Surface impedance tensor, in elastic surface waves, 353 Surface layer, rotation in, 31 Surface waves acoustical tensors in, 311-312 algebraic notations in, 307-310

in anisotropic elastic materials, 303-374 breakdown of solution in, 339-342 completeness and spectral relations in, 317 degeneracy of eigenvalue problem in, 321-325 dislocation-surface wave analogy in, 352-353 eigenvalue problem in, 314-325 eigenvalues and eigenvectors of N(4) in, 315-317 elastic, see Elastic surface waves elastic constants for, 311-312 elasticity tensors in, 310-314 free, 304, 350-352 fundamental eigenvalue problem for, 3 14-325 hexagonal symmetry in, 345-347 inner and tensor products for, 308 integral representations for. 318-321 non-semisimple degeneracy in, 322 plane elastostatics for, 325-332 progressive, 304 representations and decompositions in, 309-3 10 special tensors for, 309 stationary line singularity in, 327-332 supplementary topics on, 373-374 surface impedance tensor for, 353-355 tensor S,(u) and SJu) in, 342-345 uniformly moving line singularity in, 332-347

T Thermal, defined, 33, 44 Thermal convection, 33-50 and buoyant convection from isolated source, 44-47 comparison of theories of, 47-48 molecular bonding layers in, 37-40 new derivation of similarity theory in, 46-47 Nusselt number-Rayleigh number relation in, 40-43 plumes and thermals in, 44 with shear, 48-50 turbulence and, 21, 28 Thin-ship theory, 92-93 see also Slender ship in steady forward motion, etc.

Subject Index Third-order closure schemes derivation of critical flux Richardson number in, 80-83 and mixed layer of finite depth, 83-84 Turbulence see also Turbulent flow available potential energy in, 25-26 density variations and rotation in, 21-33 dimensional analysis and, 4-6 double diffusion convection in, 28-29 Ekman layer in, 31-33 energy arguments in, 6&63 introductory concepts in, 7-20 molecular and turbulent diffusion in, 29-30 order-of-magnitude symbol in, 6-7 Reynolds number and, 3 Richardson number and, 26-28 and rotation in surface layer, 30-31 in stably stratified fluids, 50-72 surface layer and, 30-31 in, 6-7 symbol thermal convection and, 21, 28 Turbulence experiments homogeneous fluid in, 59 with and without shear, 54-60 Turbulence research mixed-layer-of-finite-depth theory in, 83-84 third-order closure schemes in, 72-84 Turbulent flow see also Turbulence cascade of energy in, 13 common characteristics of, 2-3 convection and, 3 correlation coefficient in, 11 eddy viscosity in, 19-20 friction velocity in, 11 incremental potential energy in, 25 Kolmogorov scales in, 14 law of the wall in, 16-17 local isotropy in, 14 logarithmic friction law in, 18

-

A 8 7

c a

D 9 E O

F 1 6 H 1 J

2 3 4 5

389

logarithmic layer and drag coefficient in, 16-20 log law for, 17-18 nature of, 2-3 numerical simulations of, 72 in pipes or channels, 9-16 Turbulent plane Couette flow, 9-10

U Uniformly moving line singularity hexagonal symmetry in, 345-347 limiting speed in, 335-339 modification of elastostatic solution in, 333-335 possible modes of solution breakdown in, 339- 342 properties of tensors S,(u) and S,(c) in, 342-344 for surface waves in anisotropic elastic materials. 332-347

v Van der Waals’ fluid theory, 190-191 Velocity defect, in turbulent flow, 16 Viscous boundary layer, defined, 12 Von Karman constant, 48, 56

W

Wake collapse, buoyancy flux due to, 70-72 Wave fields, nondissipative, 3 Weakly nonlinear systems, asymptotic analysis for, 247-257

Z Zero-Froude-number problem, 167