North holland series in applied mathematics and mechanics 22 the theory of elastic waves and waveguides

THE THEORY OF

ELASTIC WAVES
AND WAVEGUIDES

by

JULIUS MTKLOWITZ
Division of Engineering and Applied Science
California Institute of Technology
Pasadena, California

NORTH-HOLLAND PUBLISHING COMPANY
AMSTERDAM · NEW YORK · OXFORD


© North-Holland Publishing Company 1978
All rights reserved. No part of this publication may be reproduced, stored in a retrieval
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North-Holland ISBN: 0 7204 0551 3

First edition 1978
Second printing 1980

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A M S T E R D A M · NEW YORK · O X F O R D
Sole distributors for the U.S.A. and Canada:
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Library of Congress Cataloging in Publication Data
Miklowitz, Julius, 1919The theory of elastic waves and waveguides.
(North-Holland series in applied mathematics and mechanics)
Includes bibliographical references.
1. Elastic waves. 2. Boundary value problems.
I. Title.
QA935.M626 53Γ.33 76-54637
ISBN 0 7204 0551 3

PRINTED IN THE NETHERLANDS


To Gloria, Paul and David


SPEC. P2

\ $ & & ' '

V.SECOND BREAK
NEAR HEAD

Second fracture due to unloading waves from the first.

PROBABLE
* FIRST BREAK
COINCIDENT
WITH KNIFE

EDGli IMPRESSION


PREFACE

The primary objective of this book is to give the reader a basic understanding of waves and their propagation in a linear elastic continuum.
The studies presented here, of elastodynamic theory and its application to
fundamental boundary value problems, should prepare the reader to tackle
many physical problems of modern general interest in engineering and
geophysics, and particular interest in mechanics and seismology.
The book focuses on transient wave propagation reflecting the strong
interest in this topic exhibited in the literature and my research interest
for the past twenty years. Chapters 5-8, and part of 2 are exclusively on
transient waves, bringing to the reader a detailed physical and mathematical
exposition of the fundamental boundary value problems in the subject.
The approach is through the governing partial differential equations with
integral transforms, integral equations and analytic function theory and
applications being the tools. Transient waves in the infinite and semi-infinite
medium and waveguides (rods, plates, etc.) are covered, as well as pulse
diffraction problems. Chapters 3 and 4 with their extensive discussions of time
harmonic waves in a half space, two half spaces in welded contact and
waveguides are of interest per se. They are also important as necessary
background for the later chapters.
The book will also serve as a reference source for workers in the subject
since many important works are involved in the presentation. Many others
are cited, but I make no claim to an extensive literature search since time
precluded that. In this connection my survey covers the literature through
1964 (see reference [4.4] at the end of Chapter 4).
I found my way into this subject long ago and quite accidentally. In experiments with plexiglas tension specimens, preliminary ones in an investigation
of dynamic stress-strain properties, a few of the specimens in these static

tests broke suddenly and in a brittle manner in two places. The frontispiece
p. VI) depicts this phenomenon (also for high-speed tool steel). Simple wave


VIII

PREFACE

analysis showed the second fracture was created through a series of reflections of the unloading wave from the ends of the remaining elastic cantilever,
the source being the first fracture [details in my paper, Journal of Applied
Mechanics, 20 (1953) 122-130]. Needless to say this interesting phenomenon dramatizes in a simple way the severe damage that can be created
by more complicated unloading (and loading) elastic waves, for example in
earthquakes.
The book had its beginnings in a first year graduate course on elastic
waves I initiated at the California Institute of Technology in the late fifties.
In its present form the course (a full academic year, three lectures a week)
draws on a good share of the material presented in this book. Prerequisites
for the course have been introductory courses in the theory of elasticity and
complex variables. Chapter 1 helps in this since it presents a brief introductory treatment of elasticity. Further, later material involving integral
transforms and analytic function theory and applications is quite self-contained.
A one semester or a two quarter course on elastic wave propagation can
be based on chapters 2 to 5 with selected material from the beginning
of Chapters 6 and 7. Each Chapter has exercises, some problems and proofs
primarily designed to involve the reader in the text material.
I would like to thank my colleagues Professors Thomas K. Caughey,
James K. Knowles, Eli Sternberg and Theodore T. Y. Wu for reading
certain of the chapters and making helpful suggestions. Similar acknowledgment is extended to Professor W. Koiter and my former graduate
students Dr. David C. Gakenheimer and Professor Richard A. Scott. It will
also become apparent to the reader that my graduate students have made
a substantial input to the book for which I am grateful. Last but not least

I should like to thank Mrs. Carol Timkovich and Mrs. Joan Sarkissian for
their excellent and patient typing of the manuscript, and Cecilia S. J. Lin
for her outstanding art work appearing in the major share of the figures
herein.
Lastly, let me say I have found working in elastic wave propagation more
than exciting. I hope that my book conveys this to the reader and, in particular, leads other young people into the subject with the same fascination
that I found in it.
Pasadena, California

J. MIKLOWITZ


INTRODUCTION

Purpose of the book
This book is intended to give the reader a basic understanding of waves
and their propagation in a linear elastic continuum. Elastodynamic theory,
and its application to fundamental problems, are developed here. They
underlie the approaches to many physical problems of modern general
interest in engineering and geophysics, and particular interest in mechanics
and seismology. The challenge in most of these problems stems from the
complicated wave reflection, refraction and diffraction processes that occur
at a boundary or interface in the continuum. This complexity evidences
itself in the partial mode conversion of an elastic wave upon reflection from
a traction-free or rigid boundary which converts, for example, compression
into compression and shear. When there is a neighboring parallel boundary
(forming then a waveguide), the so-created waves undergo multiple reflections between the two boundaries. This leads to dispersion, a further
complicating geometric effect, which is characterized by the presence of
a characteristic length (like the thickness of a plate). In the case of timeharmonic waves, dispersion leads to a frequency or phase velocity dependence on wavelength, and is responsible for the change in shape of a pulse
as it travels along a waveguide. As the title of this book indicates, a healthy

share of the material presented will focus on waveguide problems and hence
a detailed study of elastic wave dispersion.
It will become apparent in studying the various topics here that obtaining
solutions to elastodynamic problems depends strongly on having the
appropriate mathematical techniques. It follows that in addition to the
analysis of these solutions, the mathematical techniques per se form a natural
part of our studies. In particular an understanding of these techniques, and
in turn creating still others, lays the ground work for furthering our knowledge in the present subject.


2

INTRODUCTION

The early history of the subject
The study of elastic wave propagation had its origin in the age-old search
for an explanation of the nature of light. In the first half of the nineteenth
century light was thought to be the propagation of a disturbance in an
elastic aether. As pointed out in Love's interesting historical introduction
of the theory of elasticity [Ι^,ρ.?] 1 , the researches of Fresnel (1816) and
Thomas Young (1817) showed that two beams of light, polarized in
planes perpendicular to one another, do not interfere with each other.
Fresnel concluded that this could be explained only by transverse waves,
i.e., waves having displacement in direction normal to the direction of
propagation. Fresnel's conclusion gave the study of elasticity a powerful
push, in particular attracting the great mathematicians Cauchy and Poisson
to the subject.
Fundamental representations of elastic waves
By late in the year 1822 Cauchy (cf. [1.2,p.8]) had discovered most of the
elements of the classical theory of elasticity, including the stress and displacement equations of motion2. In 1828 Poisson presented his important first mémoire [2.1] (published in 1829) on numerous applications

of the general theory to special problems. An addition to this mémoire [2.2]
disclosed that Poisson was the first to recognize that an elastic disturbance
was in general composed of both types of fundamental displacement waves,
the dilatational (longitudinal) and equivoluminal (transverse) waves. His
work showed that every sufficiently regular solution of the displacement
equation of motion can be represented by the sum of two component
displacements, the first being the gradient of a scalar potential function and
the second representing a solenoidal field, where the potential function and
1

Use will be made of bracketed numbers to identify references throughout
the book. The references will be found only at the end of the chapter in
which they occur first. An exception is this Introduction which also draws
on many references appearing in later chapters, e.g., [1.2], the second
reference of Chapter 1.
2
According to Love [1.2, p. 6] Navier (1821) was the first to derive the
general equations of equilibrium and vibration of elastic solids.


INTRODUCTION

3

solenoidal displacement satisfy wave equations having the dilatational and
equivoluminal wave speeds, respectively.
Poisson's general solution does not involve the vector potential appropriate
to the solenoidal displacement component. Such a solution, i.e., one using
both a scalar and vector potential, was apparently first given by Lamé
in 1852 [2.5]. Thus through the efforts of Poisson and Lamé it was shown

that the general elastodynamic displacement field is represented as the sum
of the gradient of a scalar potential and the curl of a vector potential, each
satisfying a wave equation. Since its inception this representation of the
displacement field has been the core of most advances made through the
solution of boundary value problems in linear elastodynamics, the obvious
appeal being the wealth of knowledge that exists concerning solutions
of the wave equation. The question of completeness of Lamé's solution
was raised by Clebsch (1863) [I.l], but his proof was inconclusive. A rigorous
completeness proof was given in 1892 by Somigliana [1.2] and subsequently
by Tedone (1897) [1.3] and Duhem (1898) [1.4]. In 1885 Neumann [1.7]
gave the proof of the uniqueness for the solutions of the three fundamental
boundary-initial value problems for the finite elastic medium.
Important early investigations on the propagation of elastic waves were
those contributed by Poisson (1831), Ostrogradsky (1831) and Stokes (1849)
on the isotropic infinite medium [1.2,p.l8]. Poisson and Ostrogradsky
solved the initial value problem by synthesis of simple harmonic solutions
obtaining the displacement at any point and at any time in terms of the
initial distribution of displacement and velocity. Stokes pointed out that
Poisson's resulting two waves were waves of the dilatation and rotation.
Cauchy (1830) and Green (1839) investigated the propagation of a plane
wave through a crystalline medium, obtaining equations for the velocity
of propagation in terms of the direction of the normal to the wavefront
[1.2, pp. 18, 299]. In general the wave surface (a surface bounding the
disturbed portion of the medium) was shown to have three sheets corresponding to the three values of the wave velocity. In the case of isotropy two
of the sheets are coincident, and all of the sheets are concentric spheres.
The coincident ones correspond to transverse plane waves (in modern
nomenclature SV and SH, vertically and horizontally polarized shear
waves, respectively), and the third the dilatational wave (the P wave).
Exploiting the strain-energy function, Green also showed that for a particular form of this function the wave surface is made up of a sphere representing the dilatational wave and two sheets corresponding to equivoluminal

4

INTRODUCTION

waves. ChristofTel (1877) [1.2, pp. 18, 295-299] discussed the propagation
of a surface of discontinuity through an elastic medium. He showed the
surface moved normally to itself with a velocity that is determined, at any
point, by the direction of the normal to the surface, à la same law that holds
for plane waves propagated in that direction.
Investigation of elastic wave motion due to body forces was first carried
out by Stokes (1849) [2.15], and later by Love (1903) [2.16]. On the basis
of wave equations on the dilatation and rotation, and Poisson's integral
formula (for the solution of the three dimensional initial value problem
of the scalar potential), Stokes was the first to derive the basic singular
solution for the displacements generated by a suddenly applied concentrated
load at a point of the unbounded elastic medium. Love made an independent
exhaustive study, solving the point load problem with the aid of retarded
potentials. He showed that Poisson's integral formula yields correct results
for a quantity only when it is continuous at its wavefront, hence invalidating
Stokes' results for the dilation and rotation with (admissible) singular
wavefronts. Love confirmed Stokes' solution, gave corrected expressions
for the dilatation and rotation when they are singular, and added considerably to the interpretation of the solution. Love's work contained still
another important part. This was his extension of Kirchhoff's well-known
integral representation (1882) [cf. 2.18] for the potential governed by the inhomogeneous wave equation to one for the displacement in elastodynamics.
In recent years this representation has found particular usefulness in wave
diffraction problems.
Half space
In 1887 Rayleigh [3.8] made the very important finding of his now weilknown surface wave. This wave is generated by a pair of plane harmonic
waves, dilatational and equivoluminal (P and SV), in grazing incidence

at the surface of an elastic half space. The resultant wave is not plane since
it decays exponentially into the interior of the half space. It travels parallel
to the surface with a wave speed that is slightly less than that of the equivoluminal body (interior of medium) wave. Rayleigh's wave is a core
disturbance in elastodynamic problems involving a traction free surface.
Lamb (1904) [6.1] was the first to study the propagation of a pulse in an
elastic half space. The paper was a major advance, one of prime importance
in seismology. In it Lamb treated four basic problems, the surface normal


INTRODUCTION

5

line and point load sources, and the buried line and point sources of dilatation. He derived his solutions through Fourier synthesis of the steady
propagation solutions. For the surface source problems Lamb evaluated
the surface displacements (horizontal and vertical) which showed that the
response was composed of a front running dilatational wave, followed by
the equivoluminal and Rayleigh surface waves. Lamb also brought forth
the important fact that in the far field (from the source) the largest disturbance was the Rayleigh surface wave. He noted the nondispersive nature
of this Rayleigh wave, and in the case of the point-load excitation that
it decayed as the inverse of the square root of the radial coordinate, a
property typical of two-dimensional wave propagation. Other later studies
of note on Lamb's problem were those of Nakano (1925, 1930) [1.5, 1.6]
and Lapwood (1949) [6.8] who investigated Lamb's formal solutions
involving internal sources as integrals. Nakano showed the Rayleigh wave
does not appear at places near the source. Lapwood treated the step input
case. Russian work on Lamb's problem began in the early 1930's. The
notable works by Sobolev (1932, 1933), Smirnov and Sobolev (1932),
/Nariskina (1934) and Schermann (1946) focused on the response for the
half space interior. Smirnov and Sobolev gave a fundamentally new method

for attacking the problem and other elastodynamic problems (not involving
a characteristic length) based on similarity solutions. In 1949 Petrashen
generalized this new method, employing Fourier integrals and contour
integration, which enabled him to separate the Rayleigh wave from the terms
in the solution representing the dilatational and equivoluminal waves3.
In 1916 Lamb [1.8] extended his work to the cases of impulsive line and
point sources traveling in a fixed direction with constant velocity.
The ingenious technique of Cagniard for solving transient wave problems
of the half space, and two half spaces in contact, came along in 1939 [3.19].
The technique uses the Laplace transform on time, with spatial variables
as parameters. The Laplace transformed solution to a problem is then an
integral (e. g., Fourier) containing these parameters and the Laplace transform
parameter, which through certain integrand transformations results in the
Laplace integral operator (Carson's integral equation). This is then solved
for the inverse Laplace transform (the solution) by inspection. As we shall
3

Further detail on these Russian works may be found in Goodier [1.7]
and Ewing et al [3.11].


6

INTRODUCTION

see in this book, Cagniard's method is basic to much of the modern work
in transient elastodynamic problems.
Two half spaces in contact
The reflection and refraction of plane harmonic waves from a planar
interface between two-welded half spaces was first studied by Knott in 1899

[1.9]. Walker (1919) [1.10] treated the reflection of such waves from a free
planar boundary of a half space, i.e., the special case of Knott's problem
when one of the half spaces is a vacuum. Jeffreys (1926) [1.11], [1.12],
Muskat and Meres (1940) [3.15] and Gutenberg (1944) [3.4] elaborated
on Knott's work, the latter two works evaluating numerically energy ratios
of reflected and refracted seismic waves for a variety of half space combinations (e.g., fluid-solid). In 1911 Love [3.17] in the course of an investigation
on the effect of a surface layer on the propagation of Rayleigh waves found
another wave of the same type. For short wavelengths compared to the
thickness of the layer Love showed a modified Rayleigh wave existed with
velocity dependent on the properties of both media. Later Stoneley (1924)
[3.18] showed that this generalized Rayleigh wave had a motion which was
greatest near the interface. It is now referred to as the Stoneley wave.
Cagniard's exhaustive study of the problem of impulsive radiation from
a point source in a space composed of the two-welded elastic solid half
spaces was first published in 1939 [3.19]. It was a work of major importance
in seismology.
Waveguides
The study of elastic waveguides had its beginnings in the subject of
vibrations of elastic solid bodies with the simplest one-dimensional approximate theories being developed first. Euler (1744) and Daniel Bernoulli
(1751) derived the governing partial differential equation for the flexural
(lateral) vibration of bars (or rods) by variation of a strain-energy function,
and then determined the normal modes and the frequency equation for all
types of end conditions (combinations of free, clamped and simply supported
ends) [1.2, p. 4]. Navier (1824) derived the basic approximate equation for
extensional (longitudinal) vibrations [1.2, p. 25]. Chladni(1802) investigated
these modes of vibration experimentally, as well as those of extensional
(longitudinal) and torsional vibrations. Earlier Chladni (1787) published


INTRODUCTION

7

his experimental results on nodal figures of vibrating plates which were
a challenge to theoreticians of that era. In 1821 Germain published the
partial differential equation for the flexural vibrations in plates [1.2, p. 5].
In his 1829 mémoire Poisson [2.1] showed that the theory of vibrations
of thin rods was covered by the exact equations of motion of linear elasticity.
Poisson assumed the rod was a circular cylinder of small cross section, and
expanded all quantities in powers of the radial coordinate in the section.
When terms above the fourth power of the radius were neglected, the exact
equations yielded the approximate ones for flexural vibrations, which were
identical with those of Euler mentioned earlier. Similarly, the equation for
extensional (longitudinal) vibrations was that found earlier by Navier.
The analogous equation for torsional vibrations was obtained first by
Poisson in the work being discussed.
The exact theory work of Pochhammer (1876) [4.8] for the general
vibrations of an infinitely long circular cylinder with a traction free lateral
surface was a major advance, one that underlies much of the exact and
approximate theory modern research on steady and transient wave propagation in the elastic rod. Using separation of variables r, 0, z and /, the radial
and circumferential sectional coordinates, axial coordinate and time,
respectively, Pochhammer solved the exact displacement equations of
motion. His displacements were represented by an infinite harmonic (in z
and t) wave train with amplitude being a product of sinusoidal (in 0) and
Bessel (in r) functions. Both the wave train and its amplitude were parametrically dependent on the frequency and wave number. Making use of the
conditions of a traction free cylindrical surface, Pochhammer obtained
the frequency equations (frequency as a function of wave number) for
extensional, torsional and flexural wave trains. (Superposition of two trains
of waves traveling in opposite directions along the cylinder gives the steady
free vibrations of the infinite cylinder).

Pochhammer also derived the first and second approximations to the
lowest branch of the frequency equation for extensional waves, and the first
approximation to the lowest branch of the frequency equation for flexural
waves. Through the years these have been a guide in the construction and
use of approximate wave theories for the rod. Analogous work for the
infinite plate in plane strain with traction free faces was carried out by
Rayleigh [1.13] and Lamb [1.14] in 1889 for time-harmonic straight crested
waves. Since their writing, the Pochhammer and Rayleigh-Lamb frequency
equations with their infinite number of branches (roots) have been the


8

INTRODUCTION

subject of many studies, an almost complete understanding of these spectra
being achieved only recently.
Lamb (1917) [1.15] was first to analyze the lowest symmetric and antisymmetric modes of propagation in the plate pointing out that for high
frequency and short waves they become Rayleigh surface waves. To improve
the elementary theory for extensional vibrations (and waves) in a rod,
Rayleigh (1894) [7.1] obtained a correction to the frequency equation based
on consideration of the radial inertia of the rod element. Love (1927)
[1.2, p. 428] derived the equation of motion and end (boundary) conditions
for this theory. The importance of the theory stems from the fact that its
dispersion relation models exactly the Pochhammer second approximation
to the lowest branch of the frequency equation for extensional waves
mentioned earlier. Similary, Rayleigh in 1894 [7.1, §§ 161, 162] corrected the
Bernoulli-Euler flexural wave theory for the thin rod (the dispersion relation
of which models Pochhammer's first approximation to the lowest branch
of the frequency equation for flexural waves) by considering the rotatory

inertia of the rod element. Subsequently, Timoshenko (1921, 1922) [7.3]
showed it was equally important to take account of shear deformation of
the element. His approximate theory, accounting for both effects, has
played the greater role in modern work on flexural waves in a rod.
Another major advance in waveguides, of prime importance to seismology,
was Love's finding in 1911 [3.17, pp. 160-165] of his now well known Love
waves. As pointed out in Ewing et al. [3.11, §§ 4.5] the first long-period
seismographs, which measured horizontal motion only, exhibited large
transverse components in the main disturbance of an earthquake. Love's
work showed the waves involved were SH waves confined to a superficial
layer of an elastic half space.
Pulse propagation in an elastic waveguide involving dispersion had its
beginnings in wave group analysis. In his interesting early monograph on
the propagation of disturbances in dispersive media, Havelock [4.1] points
out that Hamilton as early as 1839, in his work on the theory of light,
investigated the velocity of propagation of a finite train of waves in a dispersive medium. However, the work in the form of short abstracts was overlooked until the early 1900's. Russell (1844) seems to have been the first
to observe the wave group phenomenon noting that in water, individual
waves moved more quickly than the group as a whole. Stokes (1876) is
usually credited with setting down the first analytical expression for group
velocity, and Rayleigh with subsequent development. Kelvin's group


INTRODUCTION

9

method of approximating integral representations of dispersive waves came
along in (1887) in his work on water waves. This was an important advance,
which is now known as the method of stationary phase. Later, Lamb (1900,
1912) presented enlightening graphical methods in the wave group concept.

We leave to Havelock's monograph discussions of the contributions of
Reynolds (1877), Gibbs (1886), Havelock (1908, 1910), Green (1909),
Sommerfeld (1912, 1914), Brillouin (1914) and others to the theory and
applications of wave group analysis to a variety of physical fields including
elasticity. One will find the references on all of the foregoing contributions
to wave group analysis in Havelock's book [4.1].
Impact
As Love [1.2, pp. 25-26] points out many early studies were concerned
with the phenomena of impact when two bodies collide, interest initially
being in the collision of two rods as a system involving longitudinal waves.
It was studied first by Poisson (1833), and later by Saint-Venant (1867).
The results of these investigations did not agree satisfactorily with experiment, hence it appeared the impact phenomena could not be described by
longitudinal wave theory. In 1882 Voigt suggested that the impacting
rods should be thought of as separated by a transition layer with the geometric shape of the interface having an influence on the impact process.
His correction led to a little better agreement with experiment. Hertz
(1882) was more successful in his treatment of two bodies pressed together. He assumed that the strain produced in each body was a local
statical effect, produced gradually and subsiding gradually, and found the
duration of impact and the size and shape of the parts that come into contact. They compared favorably with experiment. Later, Sears (1908, 1912)
[1.2, p. 440] conducted an extensive investigation of the problem with
experiments on longitudinal impact of metal rods with rounded ends, and
proposed a theory assuming the ends of the rods come into contact according
to Hertz's theory, whereas away from the ends the earlier longitudinal wave
theory of Saint-Venant applies. Sears' theory was confirmed by experiment.
Further experiments on impact were carried out by Hopkinson (1905)
[1.2, p. 117].
Other related problems were treated by longitudinal wave theory. The
longitudinal impact of a large body upon one end of a rod was treated by
Sébert and Hugoniot (1882), Boussinesq (1883) and Saint-Venant (1883)

10

INTRODUCTION

[1.2, p. 26]. In 1930 Donnell [1.16] extended the solution to the case of a
conical rod. Saint-Venant also solved several other problems by vibration theory that involved a body striking the rod transversely. As Love
points out [1.2, p. 26] the problem of a transverse load traveling along a
string (modeling a train crossing a bridge) was first treated by Willis (1849)
who wrote the differential equation for the problem neglecting the inertia of
the wire. Stokes (1849) solved the equation. The importance of the inertia
was brought out later, Phillips (1855) and Saint-Venant (1883) writing more
complete solutions. Further contributions to the impact problem are discussed in the survey by Goldsmith [I. 17].
Wave diffraction
The study of the diffraction of an elastic wave from an obstacle, like
the beginning general studies of elastic wave propagation, had its origin in
the elastic solid (aether) theory of light. Famous early works on the diffraction of light waves were the paper by Stokes (1849) mentioned earlier, which
treated the diffraction of light by an aperture in a screen, and a series of
papers by Rayleigh beginning in 1871 on the diffraction of light by small
particles. These works and the further progress in studies on elastic wave
diffraction are discussed in an interesting history in the book by Pao and
Mow [8.11] on the topic.
Modern work and reading
Interesting in the history of the contributions to this subject is the relatively fallow period lying between the work of the classical elasticians
in the nineteenth century and early twentieth century, and modern work
which has been expanding at an increasing rate since World War II days.
Aside from the fact that the subject offers intrigue and challenge there are
several practical reasons for the modern expansion. One of the strongest,
at least from the mechanics and engineering point of view, has been the
continually growing need for information on the performance of structures
subjected to high rates of loading. In geophysics the expanding research

activity in elastic waves has also had strong underlying practical reasons
such as the need for more accurate information on earthquake phenomena
and improved prospecting techniques. Further, seismologists have been con-


INTRODUCTION

11

cerned with the nuclear detection problem. Developments in the related
fields of acoustics and electromagnetic waves, and in applied mathematics
in general, have also influenced the interest and progress made in the
study of elastic waves. Last but not least, the electronic computer has
been of considerable influence. As in many other fields, it has given numerical
information for otherwise intractable problems.
The vastness of the modern contributions to the subject of waves in
a linear, homogeneous, isotropic elastic medium precludes presenting an
abstracted report on them here. The bulk of these studies were, and continues
to be, concerned with problems involving boundaries and dispersion. The
survey by Miklowitz [4.4] gives a fairly complete coverage of the pertinent
literature to 1965. It shows that extended information exists now on transient wave propagation in the elastic half space. Through integral transforms, the Cagniard-deHoop inversion technique, similarity solutions,
related asymptotics and other analytical and experimental methods, solutions
for most cases of Lamb's problem (surface and buried sources of most types
including traveling loads) have been derived and evaluated. Advances were
also made on the problem of a buried spherical cavity source in the half
space. Concern over underground protective construction created new
interest in wave diffraction and scattering by an obstacle in the half space,
hence in the related infinite medium problems involving cylindrical and
spherical cavities. Some gains were also made in our understanding of
elastic wave propagation in a wedge.

Concerning waveguides, extended information exists now on the frequency equations governing extensional, flexural and torsional waves in
the infinite elastic rod, plate and cylindrical shell. Recent efforts have
established the character of the higher real branches of the frequency
spectrum (real frequency vs. real wave number), and the existence and
character of the imaginary and complex branches of the spectrum (real
frequency vs. imaginary and complex numbers) for these waveguides. This
information, basic to transient excitation, and multi-integral transform
and other methods, have produced integral solutions for various edge excited
semi-infinite waveguides based on both the exact and approximate theories.
Evaluation of the solution through asymptotics and numerical integration
have produced a significant amount of information on the response of these
waveguides.
Important advances have been made in the theory and solution of problems on the diffraction of a plane pulse by a semi-infinite plane boundary


12

INTRODUCTION

(a slit or rigid barrier) in the infinité elastic solid. In two-dimensional diffraction and related crack problems, methods (1) using similarity solutions
with and without solutions of integral equations of the Wiener-Hopftype, and (2) involving the Laplace transform, Wiener-Hopf-type integral
equations and the Cagniard-deHoop inversion technique, have been
productive. Related interesting advances were made in the study of finite
plane cracks and obstacles using other integral equation techniques. In
addition, three-dimensional integral representations for the displacement
and acceleration vectors have been used in particular problems. Further
detail is left to [4.4] which also contains references to earlier surveys.

Contents of present book
Chapter 1 of the present book is entitled Introduction i> linear elastodynamics. Since the theory of elastic waves and waveguides is based on

the classical theory of elasticity, the chapter sets down from the latter
the definition of basic quantities, governing equations, the fundamental
problems and the uniqueness of their solutions. The treatment presents
what is need from the classical theory to attain the objectives of our subject.
Chapter 2, entitled The fundamental waves of elastodynamics and their
representations, is concerned with certain basics of integrating the elastodynamic displacement equations of motion, and analyzing the general
nature of wave solutions so found. The chapter begins with a treatment of
body waves, i.e., interior medium waves, dilatational and equivoluminal,
and the Lamé solution of the displacement equations of motion which is
comprised of both waves. Types of these waves, plane, cylindrical etc.,
their symmetries and time nature are then discussed. A treatment of propagation of surfaces of discontinuity, and related wavefronts, characteristics and rays follows. An important class of problems, those due to
body force disturbances (interior disturbances due to an external source,
e.g., gravity), are then discussed at length, followed by a treatment of the
one-, three- and two-dimensional initial value problems. The chapter concludes with a study of the method of characteristics for one-dimensional
initial value and boundary-initial value problems.
Chapter 3, is entitled Reflection and refraction of time harmonic waves at
an interface. It presents an extensive study of wave reflection, refraction
and generation at a single planar interface for mostly plane waves, harmonic


INTRODUCTION

13

(sinusoidal) in time and (two-dimensional) space. It will be seen that
these relatively simple waves (P, SV and SH) bring out basic information
on reflection and refraction phenomena, and new waves peculiar to the
interface, that are general properties of the more complicated time-dependent waves studied later in the book. The chapter begins with consideration
of wave reflection from the boundary of an elastic half space, i.e., the
interface is one between two half spaces, one elastic and the other a vacuum.

The more general case of two half spaces in welded contact is treated
later in the chapter. In both of these problems, all special cases are treated
in detail, e.g., normal and grazing incidence, reflection and refraction at
critical angles, total reflection, reflection and refraction of wave pairs and
their generation of Rayleigh surface and Stoneley interface waves.
Chapter 4, entitled Time harmonic waves in elastic waveguides, is a natural
extension of Chapter 3. Here we introduce a second planar boundary
parallel to the surface of an elastic half space creating an infinite plate or
layer. Now the P, SV and SH waves, studied in Chapter 3, reflect from
boundary to neighboring boundary, generally (in the Rand SV wave cases)
undergoing mode conversion at each reflection, and progressing along the
length of the plate. The neighboring parallel boundaries are in effect guiding
the waves along the plate. This example of a waveguide, as well as others,
e.g., rod, cylindrical shell and layered elastic solid, have the common feature
of two or more parallel boundaries which introduce one or more characteristic lengths into a problem. These characteristic lengths lead to wave
dispersion which is characterized by a dependence of frequency on wavelength. We study in detail the plate in plane strain and the rod, drawing
on the modern works of Mindlin, Onoe and coworkers and noting the
other waveguides can be treated similarly.
Chapter 5, entitled Integral transforms, related asymptotics and introductory applications, paves the way for solving the fundamental time-dependent
boundary value problems of the subject carried out in Chapters 6, 7 and
8. As the title indicates we set down here the basics of integral transforms
and related asymptotics and the beginnings of their applications in our
subject. Starting with the Fourier integral theorem, the theory and properties
of the (one-sided) Laplace transform, the bilateral (or two-sided) Laplace
transform, the exponential Fourier transforms (of real and complex argument), the Fourier sine and cosine transforms and the Hankel transforms are
developed. Then, after a brief introduction to asymptotic expansions and
their properties, we discuss asymptotic expansions of integrals and, in


14

INTRODUCTION

particular, the Laplace and Fourier integrals of prime interest to our
subject. In this, detailed discussions are given of Laplace's method, the
method of steepest descents, the method of stationary phase and the asymptotics of the Laplace transform, which form the tools for long and
short time (after the wavefront arrival) approximations in wave problems.
Lastly, the problems of spherical and cylindrical cavity sources in the
infinite solid are treated. Contour integrations produce the exact solutions
and asymptotics the short- and long-time approximations.
Chapter 6, entitled Transient waves in an elastic half space treats the
basic boundary value problems for the half space through modern integral
transform methods. The plane-strain (Lamb's) problems for the surface
normal line load source and buried line dilatational source are treated
by the Cagniard-deHoop method. In the first of these problems, wavefront approximations are worked out by two methods, a special method
used with the Cagniard-deHoop technique and the method of steepest
descents. For Lamb's axially symmetric problems for the surface and buried
vertical point loads we follow Pekeris' work and method, related to Cagniard's
method, but independently developed. Lastly, the chapter presents part
of a complete exact solution and its derivation for the problem of the
suddenly applied normal point load that travels on the surface of the half
space. The solution by Gakenheimer and Miklowitz represents the first
application of the Cagniard-deHoop method to a nonaxisymmetric problem.
Chapter 7, entitled Transient waves in elastic waveguides, a natural extension of Chapter 4 with its addition of the time variable, enables us
to discuss more physically realistic waveguide problems. In effect, here
we learn the techniques for integrating over the frequency spectra set down
in Chapter 4. The chapter begins with a discussion of approximate theories
and one-dimensional problems. Derivation of the classical approximate
theories for extensional and flexural waves in a thin rod (Love-RayJeigh,
Bernoulli-Euler and Timoshenko theories) is carried out by Hamilton's

principle, and boundary-initial value problems based on these theories are
solved exactly and approximately using the Laplace transform, contour integration and asymptotics. Problems for the infinite plate in plane strain
follow, being solved by a technique given by Lloyd and Miklowitz involving
a double integral transform (Laplace on time, exponential Fourier on propagation coordinate). Contour integrations in the planes of the two transform
parameters (related to frequency and wave number) lead to a direct corre-


INTRODUCTION

15

spondence between the component parts of the frequency spectrum and the
individual integrals comprising a transient wave solution. This permits a
study to be made of component waves in the solution through numerical
evaluation of the related integrals. Such evaluations, and related approximations obtained with the aid of the method of stationary phase, are
discussed.
Edge load problems for the semi-infinite waveguide form the next topic
of Chapter 7. Such problems with their basic corner difficulties have been
solved only recently by Skalak (1957), Folk et al. (1958) and De Vault and
Curtis (1962) for the rod with mixed edge conditions (a mixture of stress
and displacement components specified), and by Miklowitz (1969) and
Sinclair and Miklowitz (1975) for the plate in plane strain with nonmixed
edge conditions (stress or displacement components specified). The former
class of problems are separable. They are solved directly through a double
integral transform technique. The latter class of problems are nonseparable.
They require in addition to a double transform, integral equations on the
edge unknowns, and a boundedness condition on the solution to solve the
integral equations. Both techniques and their applications to these problems
are examined in detail with long time-far field and short time-near field
approximations being deduced. A discussion of related work for other

waveguide problems, including axially symmetric ones, is also presented
in the chapter.
Chapter 8, entitled Pulse scattering by half-plane, cylindrical and spherical
obstacles, treats elastodynamic scattering problems that are related to the
classical ones in optics, acoustics and electromagnetic waves. The methods
presented, however, are modern ones involving integral transforms, integral
quations and certain other associated techniques. The first half of the
chapter is devoted to cases of diffraction of a plane-elastic pulse by a halfplane scatterer. They are mixed boundary value problems. Specifically, we
reat the diffraction of a plane SH-pulse (a Sommerfeld-type problem)
by a traction-free half plane, followed by the analogous, but more -complicated, case involving the P-pulse. The Laplace transform of the solution
in each case is obtained as the solution of a set of dual integral equations,
reduced to algebraic ones by a Wiener-Hopf technique inherent in a procedure presented by Clemmow (1951) in related electromagnetic wave problems. Inversion is accomplished by the Cagniard-deHoop method, essentially following earlier work by deHoop (1958). Finally the wave systems in
each are discussed.


16

INTRODUCTION

The second part of the chapter is concerned with diffraction of an elastic
pulse from circular cylindrical and spherical scatterers. Such problems have
only recently been attacked, and the treatment here reflects this in the
methods and results presented. The cylindrical cavity scatterer is treated
first by an integral transform method given by Miklowitz (1963, 1966) and
Peck and Miklowitz (1969) incorporating Friedlander's (1954) representation
of the solution (used in related acoustics problems). Friedlander's representation of the solution is a series having terms of wave form corresponding
to propagation in the circumferential direction (about the cavity). This
representation is very accurate for early times (not necessarily the earliest)
at a station. Two problems are treated, the suddenly applied, normal line
load, and the plane wave impingement, on the cavity wall. For long time

in the far field (in Θ) it is shown that Rayleigh waves are predominant
in both problems. For the second problem an exact inversion for the shorter
times and the near field, in the form of integrals over modes of propagation,
show that the lowest modes (those with smallest imaginary wave numbers)
for dilatation, equivoluminal and Rayleigh waves predominate. The technique and results of related earlier work by Baron, Matthews and Parnes
(1961, 1962), using a Fourier series technique for the second problem, are
also discussed and compared with the foregoing method and results of
Miklowitz and Peck. Further related works on approximations are reviewed
for the rigid and elastic cylindrical scatterers. Finally a brief review of work
is presented on wave scattering by the circular cylindrical elastic inclusion
with resultant wave focusing, and wave diffraction from a spherical cavity.
Lastly a section on Supplementary Reading is presented. Its purpose is to
guide the reader to other important works that are (1) natural extensions
of the text material, and (2) on topics dealing with additional effects in the
linear elastic medium not treated in the book, because of limitations on
time and space, e.g., waves in anisotropic media.
Other books on elastic waves and related subjects
Some brief remarks are in order on some other books in this subject
as well as those on related subjects. The book by Kolsky [3.1] published
in 1952 contains an introduction to elastic waves. The book by Ewing et al.
[3.11], published in 1956 and oriented toward seismology, is a comprehensive
treatment of elastic waves with material and extensive references on most
topics. Of note also in seismology are the books by Bullen [I.18] published


INTRODUCTION

17

in 1954, Cagniard, first published in French in 1939, and translated and

revised by Dix and Flinn in 1962 [3.19], and Brekhovskik [1.19] the translated
Russian counterpart of [3.11] published in 1960. More recent comprehensive
treatments of elastic waves are the books by Achenbach [2.14] published
in 1973, Eringen and Suhubi [1.20] and Graff [1.21] published in 1975.
The following books on special topics in elastic waves are of note:
Redwood [3.3] on waveguides, Viktorov [3.12] on Rayleigh and Lamb (plate)
waves, Pao and Mow [8.11] on wave diffraction and Auld [1.22] on the
theory of waves in a piezoelectric-elastic solid and its application to problems
in scattering, waveguides and resonators.
As for related topics the following books will be of interest: On acoustics,
the classic treatise by Lord Rayleigh [7.1, both volumes], Friedlander's
book [2.10] dealing with modern mathematical techniques for solving
problems involving sound pulse reflection and diffraction, and the book
by Morse and Ingard [1.23] on theoretical acoustics. On optics, we reference,
of course, Sommerfeld [3.7], for waves in water, Stoker [1.24], and for
electromagnetic waves, Jones [2.18]. On methods in wave propagation, for
dispersive waves we have already referenced Havelock's monograph [4.1],
The recent book of Brillouin [4.2] on this topic is also of note. Finally,
we reference two books important to mathematical methods in basic wave
phenomena. The first is the Courant-Hilbert volume [2.20] on partial
differential equations (by Courant), in particular its chapters on hyperbolic
equations in two or more independent variables, and its discussions of the
theory of characteristics and rays that are fundamental to wave propagation
phenomena. The second is the recent comprehensive treatment by Whitham
[1.25] of the theory of linear and nonlinear waves, with applications drawn
from acoustics, optics, water waves and gas dynamics.
References
[I.I.] A. Clebsch, Journal für Reine und Angewandte Mathematik 61 (1863), 195.
[I.2.] C. Somigliana, Atti Reale Accad. Line. Roma, Ser 5,1 (1892), 111.
[I.3.] O. Tedone, Mem. Reale Accad, Scienze Torino, Ser 2,47 (1897), 181.

[I.4.] P. Duhem, Mém. Soc. Sei. Bordeaux, Ser. V, 3 (1898), 316.
[I.5.] H. Nakano, Japan Journal of Astronomy and Geophysics 2 (1925), 233-326.
[I.6.] H. Nakano, Geophysics Magazine (Tokyo) 2 (1930), 189-348.
[I.7.] J. N. Goodier, The Mathematical Theory of Elasticity, Surveys in Applied Mathematics I, John Wiley and Sons, Inc., New York (1958), 1-47.
[I.8.] H. Lamb, Philosophical Magazine [6] 13 (1916), 386-399, 539-548.
[I.9.] C. G. Knott, Philosophical Magazine [5] 48 (1899), 64-97.


18

REFERENCES

[1.10.] G. W. Walker, Philosophical Transactions of the Royal Society (London) A 218
(1919), 373-393.
[1.11.] H. Jeffreys, Monthly Notices of the Royal Astronomical Society: Geophysics Supplement 1 (1926), 321-334.
[1.12.] H. Jeffreys, Proceedings of the Cambridge Philosophical Society 22 (1926), 472-481.
[1.13.] Lord Rayleigh, Proceedings of the London Mathematical Society 20 (1888-1889),
225-234.
[1.14.] H. Lamb, Proceedings of the London Mathematical Society 21 (1889-1890), 85.
[1.15.] H. Lamb, Proceedings of the Royal Society of London A 93 (1917), 114-128.
[1.16.] L. H. Donell, Transactions of the American Society of Mechanical Engineers 52
(1930), 153-167.
[1.17.] W. Goldsmith, Impact, The Collision of Solids. In: Applied Mechanics Surveys,
eds. H. Abramson, H. Liebowitz, J. M. Crowley and S. Juhasz, Spartan Books,
Washington D. C. (1966), 785-802.
[1.18.] K. E. Bullen, An Introduction to the Theory of Seismology, 2nd Edition. Cambridge
University Press (1953).
[1.19.] L. M. Brekhovskikh, Waves in Layered Media, Applied Mathematics and Mechanics
6. Academic Press, New York (1960).
[1.20.] A. C. Eringen and E. S. Suhubi, Elastodynamics Volume 2: Linear Theory. Academic

Press, New York (1975).
[1.21.] K. F. Graff, Wave Motion in Elastic Solids. Ohio State University Press, Columbus,
Ohio (1975).
[1.22.] B. A. Auld, Acoustic Fields and Waves in Solids, Volumes I, 2. John Wiley and
Sons, New York (1973).
[1.23.] P. M. Morse and K. U. Ingard, Theoretical Acoustics. McGraw-Hill Book Company, New York (1968).
[1.24.] J. J. Stoker, Water Waves. Interscience Publishers, Inc., New York (1957).
[1.25.] G. B. Whitham, Linear and Nonlinear Waves. John Wiley and Sons, New York
(1974).


CHAPTER 1

INTRODUCTION
TO LINEAR ELASTODYNAMICS

1.1. Introduction; Description of deformation and motion
The theory of elastic waves and waveguides is based on the classical
theory of elasticity. This chapter therefore sets down from the latter the
definition of basic quantities, governing equations, and problems and the
uniqueness of their solutions. The treatment presents just the material that
is needed to attain the objectives of our subject. Recommended references
for the reader are the books by Sokolnikoff [1.1, Chapters 1, 2 and 3],
Love [1.2, Chapters I, II, III and VII], and Nowacki [1.3, Chapter 1]. The
historical introduction to the theory of elasticity in Love's book is very
interesting. In reading it one is impressed by the array of great mathematicians of the 17th, 18th and 19th centuries, who found intrigue in the subject.
In the introductory chapter we mentioned those early contributions pertaining to the history of elastic wave propagation. In this chapter we will
mention some of the other early contributions to classical elasticity theory.
In general however, we will depend on Love's historical introduction to
earmark most of them.

There are two basic ways of describing deformation and motion in continuum mechanics. They are known as the Lagrangian, or material, and
Eulerian, or spatial, descriptions. The Lagrangian description uses the
coordinates of a material point or particle, injts undeformed position, and
time, as independent variables. Assuming, for example, rectangular Cartesian
coordinates xu x2, x3, denoting them collectively as xf ( / = 1 , 2, 3), or the
position vector x, with axes fixed in space, a later deformed position of the
particle would be given by x\ = x\{x, t). In the Eulerian description the
coordinates of the particle in the deformed position, and time, are taken as
independent variables, i.e., χ^χ^χ',
t), the undeformed position of the particle is a function of its deformed position and time. As Sokolnikoff [1.1, §11],