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Mathematics of Wave Propagation

Julian L. Davis
Princeton University Press
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Preface xiii
CHAPTER ONE Physics of Propagating Waves 3
Introduction 3
Discrete Wave-Propagating Systems 3
Approximation of Stress Wave Propagation in a Bar by a Finite System of Mass-Spring Models 4
Limiting Form of a Continuous Bar 5
Wave Equation for a Bar 5
Transverse Oscillations of a String 9
Speed of a Transverse Wave in a Siting 10
Traveling Waves in General 11
Sound Wave Propagation in a Tube 16
Superposition Principle 19
Sinusoidal Waves 19
Interference Phenomena 21
Reflection of Light Waves 25
Reflection of Waves in a String 27
Sound Waves 29
Doppler Effect 33
Dispersion and Group Velocity 36
Problems 37

CHAPTER TWO Partial Differential Equations of Wave Propagation 41
Introduction 41
Types of Partial Differential Equations 41
Geometric Nature of the PDEs of Wave Phenomena 42
Directional Derivatives 42
Cauchy Initial Value Problem 44
Parametric Representation 49
Wave Equation Equivalent to Two First-Order PDEs 51
Characteristic Equations for First-Order PDEs 55
General Treatment of Linear PDEs by Characteristic Theory 57
Another Method of Characteristics for Second-Order PDEs 61
Geometric Interpretation of Quasilinear PDEs 63
Integral Surfaces 65
Nonlinear Case 67
Canonical Form of a Second-Order PDE 70
Riemann's Method of Integration 73
Problems 82

CHAPTER THREE The Wave Equation 85

Factorization of the Wave Equation and Characteristic Curves 85
Vibrating String as a Combined IV and B V Problem 90
D'Alembert's Solution to the IV Problem 97
Domain of Dependence and Range of Influence 101
Cauchy IV Problem Revisited 102
Solution of Wave Propagation Problems by Laplace Transforms 105
Laplace Transforms 108
Applications to the Wave Equation 111
Nonhomogeneous Wave Equation 116
Wave Propagation through Media with Different Velocities 120
Electrical Transmission Line 122

Two-Dimensional Wave Equation 125
Reduced Wave Equation in Two Dimensions 126
The Eigenvalues Must Be Negative 127
Rectangular Membrane 127
Circular Membrane 131
Three-Dimensional Wave Equation 135
Problems 140

CHAPTER FOUR Wave Propagation in Fluids 145

Lagrangian Representation of One-Dimensional Compressible Gas Flow 146
Eulerian Representation of a One-Dimensional Gas 149
Solution by the Method of Characteristics: One-Dimensional Compressible Gas 151
Two-Dimensional Steady Flow 157
Bernoulli's Law 159
Method of Characteristics Applied to Two-Dimensional Steady Flow 161
Supersonic Velocity Potential 163
Hodograph Transformation 163
Shock Wave Phenomena 169

Elementary Discussion of Viscosity 183
Conservation Laws 185
Boundary Conditions and Boundary Layer 190
Energy Dissipation in a Viscous Fluid 191
Wave Propagation in a Viscous Fluid 193
Oscillating Body of Arbitrary Shape 196
Similarity Considerations and Dimensionless Parameters; Reynolds' Law 197
Poiseuille Flow 199
Stokes' Flow 201
Oseen Approximation 208
Problems 210

CHAPTER FIVE Stress Waves in Elastic Solids 213
Introduction 213
Fundamentals of Elasticity 214
Equations of Motion for the Stress 223
Navier Equations of Motion for the Displacement 224
Propagation of Plane Elastic Waves 227
General Decomposition of Elastic Waves 228
Characteristic Surfaces for Planar Waves 229
Time-Harmonic Solutions and Reduced Wave Equations 230
Spherically Symmetric Waves 232
Longitudinal Waves in a Bar 234
Curvilinear Orthogonal Coordinates 237
The Navier Equations in Cylindrical Coordinates 239
Radially Symmetric Waves 240
Waves Propagated Over the Surface of an Elastic Body 243
Problems 247

CHAPTER SIX Stress Waves in Viscoelastic Solids 250
Introduction 250
Internal Friction 251
Discrete Viscoelastic Models 252
Continuous Marwell Model 260
Continuous Voigt Model 263
Three-Dimensional VE Constitutive Equations 264
Equations of Motion for a VE Material 265
One-Dimensional Wave Propagation in VE Media 266
Radially Symmetric Waves for a VE Bar 270
Electromechanical Analogy 271
Problems 280

CHAPTER SEVEN Wave Propagation in Thermoelastic Media 282
Introduction 282
Duhamel-Neumann Law 282
Equations of Motion 285
Plane Harmonic Waves 287
Three-Dimensional Thermal Waves; Generalized Navier Equation 293

CHAPTER EIGHT Water Waves 297
Introduction 297
Irrotational, Incompressible, Inviscid Flow; Velocity Potential and Equipotential Surfaces 297
Euler's Equations 299
Two-Dimensional Fluid Flow 300
Complec Variable Treatment 302
Vortex Motion 309
Small-Amplitude Gravity Waves 311
Water Waves in a Straight Canal 311
Kinematics of the Free Surface 316
Vertical Acceleration 317
Standing Waves 319
Two-Dimensional Waves of Finite Depth 321
Boundary Conditions 322
Formulation of a Typical Surface Wave Problem 324
Example of Instability 325
Approximation Aeories 327
Tidal Waves 337
Problems 342

CHAPTER NINE Variational Methods in Wave Propagation 344
Introduction; Fermat's Principle 344
Calculus of Variations; Euler's Equation 345
Configuration Space 349
Kinetic and Potential Energies 350
Hamilton's Variational Principle 350
Principle of Virtual Work 352
Transformation to Generalized Coordinates 354
Rayleigh's Dissipation Function 357
Hamilton's Equations of Motion 359
Cyclic Coordinates 362
Hamilton-Jacobi Theory 364
Extension of W to 2n Degrees of Freedom 370
H-J Theory and Wave Propagation 372
Quantum Mechanics 376
An Analogy between Geometric Optics and Classical Mechanics 377
Asymptotic Theory of Wave Propagation 380
Appendix: The Principle of Least Action 384
Problems 387

Bibliography 389
Index 391