Wave Propagation in Solids and Fluids

395 Pages·1988·8.884 MB·English

by  Julian L. Davis (auth.)

Wave Propagation in Solids and Fluids
Forthcoming from Springer-Verlag:
Wave Propagation in Electromagnetic Media
by Julian L. Davis
Julian L. Davis
Wave Propagation in
Solids and Fluids
With 58 Illustrations
Sp ringer-Verlag
New York Berlin Heidelberg
London Paris Tokyo
Julian L. Davis
22 Libby Avenue
Pompton Plains, NJ 07444
U.S.A.
Library of Congress Cataloging-in-Publication Data
Davis, Julian L.
Wave propagation in solids and fluids.
Bibliography: p.
1. Solids-Mathematics. 2. Fluids-Mathematics.
3. Waves. 4. Differential equations, Partial.
5. Hamilton-Jacobi equations. 1. Title.
QC176.8.W3D38 1988 531'.1133 88-4618
© 1988 by Springer-Verlag New York Inc.
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9 8 765 432 1
ISBN-I3 978-1-4612-8390-4 e-ISBN-I3 978-1-4612-3886-7
DOl 10.1007/978-1-4612-3886-7
Preface
The purpose of this volume is to present a clear and systematic account of
the mathematical methods of wave phenomena in solids, gases, and water
that will be readily accessible to physicists and engineers. The emphasis is
on developing the necessary mathematical techniques, and on showing how
these mathematical concepts can be effective in unifying the physics of wave
propagation in a variety of physical settings: sound and shock waves in
gases, water waves, and stress waves in solids. Nonlinear effects and
asymptotic phenomena will be discussed.
Wave propagation in continuous media (solid, liquid, or gas) has as its
foundation the three basic conservation laws of physics: conservation of
mass, momentum, and energy, which will be described in various sections
of the book in their proper physical setting. These conservation laws are
expressed either in the Lagrangian or the Eulerian representation depending
on whether the boundaries are relatively fixed or moving. In any case, these
laws of physics allow us to derive the "field equations" which are expressed
as systems of partial differential equations. For wave propagation
phenomena these equations are said to be "hyperbolic" and, in general,
nonlinear in the sense of being "quasi linear" . We therefore attempt to
determine the properties of a system of "quasi linear hyperbolic" partial
differential equations which will allow us to calculate the displacement,
velocity fields, etc. The technique of investigating these equations is given
by the method of characteristics, which will be described in various portions
of the book where appropriate. This method is essential in investigating
large amplitude waves which are by nature nonlinear.
One of the unique features of this book is the treatment of the Hamilton
Jacobi theory in the setting of the variational calculus. It will be shown that
this theory is a natural vehicle for relating classical mechanics and geometric
optics; it guides us in developing the Schrodinger equation of quantum
mechanics which will be taken up more fully in the second volume. The
asymptotic expansion approach of J. Keller and his associates shows us
that geometric optics is the infinite frequency approximation of an
asymptotic expansion in powers of the wave number. The succeeding terms
vi Preface
yield diffraction information. An asymptotic approach which handles non
linear phenomena is also used in the chapter on water waves, following the
methods of K.O. Friedrichs, J.J. Stoker, and their associates.
We envision a two-volume set. This first volume is concerned with
wave propagation phenomena in nonconducting media. By a nonconducting
medium we mean a continuum (solid, liquid, gas) that is not electrically or
magnetically conducting. The second volume will be concerned with wave
propagation in electromagnetically conducting media, and will cover such
topics as wave propagation in electromagneti'C media, plasmas, magnetohy
drodynamics, and quantum mechanics including relativistic effects. Each
volume is relatively self-contained and independent of the other.
The books are designed to be interdisciplinary in nature, in the spirit
best expressed by the following quote from the Preface to Methods of
Mathematical Physics, Volume 1, 1953, by Richard Courant:
Since the seventeenth century, physical intuition has served as a vital
source for mathematical problems and methods. Recent trends and fashions
have, however, weakened the connection between mathematics and physics
(and the engineering sciences); mathematicians turning away from the roots
of mathematics in intuition, have concentrated on refinement and emphasized
the postulational side of mathematics, and at times have overlooked the unity
of their science with physics (and the engineering sciences). In many cases,
physicists (and engineers) have ceased to appreciate the attitudes of
mathematicians. This rift is unquestionably a serious threat to science as a
whole; the broad stream of scientific development may split into smaller and
smaller rivulets and dry out. It seems therefore important to direct our efforts
toward reuniting divergent trends by clarifying the common features and
interconnections of many distinct and diverse scientific facts. Only thus can
the student attain some mastery of the material and the basis be prepared for
further organic development of research.
These remarks of Professor Courant are as timely today as they were
thirty-four years ago.
Julian L. Davis
January 15, 1988
Contents
Preface v
CHAPTER 1
Oscillatory Phenomena
Introduction
1.1. Harmonic Motion
1.2. Forced Oscillations 7
1.3. Combination of Wave Forms 11
1.4. Oscillations in Two Dimensions 14
1.5. Coupled Oscillations 16
1.6. Lagrange's Equations of Motion 19
1.7. Formulation of the Problem of Small Oscillations for
Conservative Systems 22
1.8. The Eigenvalue Equation 25
1.9. Similarity Transformation and Normal Coordinates 28
CHAPTER 2
The Physics of Wave Propagation 34
Introduction 34
2.1. The Conservation Laws of Physics 37
2.2. The Nature of Wave Propagation 38
2.3. Discretization 40
2.4. Sinusoidal Wave Propagation 41
2.5. Derivation of the Wave Equation 43
2.6. The Superposition Principle, Interference Phenomena 45
2.7. Concluding Remarks 49
CHAPTER 3
Partial Differential Equations of Wave Propagation 50
Introduction 50
3.1. Wave Equation as an Equivalent First-Order System 52
3.2. Method of Characteristics for a Single First-Order Quasilinear
Partial Differential Equation 54
viii Contents
3.3. Second-Order Quasilinear Partial Differential Equation 61
3.4. Method of Characteristics for Second-Order Partial
Differential Equations 63
3.5. Propagation of Discontinuities 68
3.6. Canonical Form for Second-Order Partial Differential Equations
with Constant Coefficients 70
3.7. Conservation Laws, Weak Solutions 73
3.8. Divergence Theorem, Adjoint Operator, Green's Identity,
Riemann's Method 76
CHAPTER 4
Transverse Vibrations of Strings 84
Intro{iu~tion 84
4.1. Solution of the Wave Equation, Characteristic Coordinates 84
4.2. D' Alembert's Solution 86
4.3. Nonhomogeneous Wave Equation 93
4.4. Mixed Initial Value and Boundary Value Problem, Finite String 96
4.5. Finite or Lagrange Model for Vibrating String 101
CHAPTER 5
Water Waves 108
Introduction 108
5.1. Conservation Laws 109
5.2. Potential Flow 116
5.3. Two-Dimensional Flow, Complex Variables 118
5.4. The Drag Force Past a Body in Potential Flow 122
5.5. Energy Flux 127
5.6. Small Amplitude Gravity Waves 129
5.7. Boundary Conditions 134
5.8. Formulation of a Typical Surface Wave Problem 135
5.9. Simple Harmonic Oscillations in Water at Constant Depth 137
5.10. The Solitary Wave 143
5.11. Approximation Theories 148
CHAPTER 6
Sound Waves 159
Introduction 159
6.1. Linearization of the Conservation Laws 159
6.2. Plane Waves 162
6.3. Energy and Momentum 163
6.4. Reflection and Refraction of Sound Waves 169
6.5. Sound Wave Propagation in a Moving Medium 176
6.6. Spherical Sound Waves 180
6.7. Cylindrical Sound Waves 183
6.8. General Solution of the Wave Equation 186
6.9. Huyghen's Principle 188
Contents ix
CHAPTER 7
Fluid Dynamics 192
Introduction 192
Part I. Inviscid Fluids 194
Introduction 194
7.1. One-Dimensional Compressible Inviscid Flow 194
7.2. Two-Dimensional Steady Flow 209
7.3. Shock Wave Phenomena 220
Part II. Viscous Fluids 237
Introduction 237
7.4. Viscosity, Elementary Considerations 237
7.5. Conservation Laws for a Viscous Fluid 239
7.6. Flow in a Pipe, Poiseuille Flow 250
7.7. Dimensional Considerations 252
7.8. Stokes's Flow 255
7.9. Oscillatory Motion 262
7.10. Potential Flow 268
CHAPTER 8
Wave Propagation in Elastic Media 274
Introduction 274
Historical Introduction to Wave Propagation 275
8.1. Fundamental Concepts of Elasticity 280
8.2. Equations of Motion for the Stress Components 296
8.3. Equations of Motion for the Displacement, Navier Equations 297
8.4. Propagation of a Plane Elastic Wave 299
8.5. Spherically Symmetric Waves 303
8.6. Reflection of Plane Waves at a Free Surface 304
8.7. Surface Waves, Rayleigh Waves 309
CHAPTER 9
Variational Methods in Wave Phenomena 312
Introduction 312
9.1. Principle of Least Time 313
9.2. One-Dimensional Treatment, Euler's Equation 314
9.3. Euler's Equations for the Two-Dimensional Case 316
9.4. Generalization to Functionals with More Than One
Dependent Variable 319
9.5. Hamilton's Variational Principle 322
9.6. Lagrange's Equations of Motion 324
9.7. Principle of Virtual Work 324
9.8. Transformation to Generalized Coordinates 328
9.9. Rayleigh's Dissipation Function 332
9.10. Hamilton's Equations of Motion 334
9.11. Cyclic Coordinates 338
x Contents
9.12. Lagrange's Equations of Motion for a Continuum 340
9.13. Hamilton's Equations of Motion for a Continuum 345
9.14. Hamilton-Jacobi Theory 351
9.15. Characteristic Theory in Relation to Hamilton-Jacobi Theory 358
9.16. Principle of Least Action 364
9.17. Hamilton-Jacobi Theory and Wave Propagation 368
9.18. Application to Quantum Mechanics 373
9.19. Asymptotic Phenomena 375
Bibliography 381
Index 383