Mathematics of Wave Propagation

Julian L. Davis is the author of Wave Propagation in Solids and Fluids and Wave Propagation in Electromagnetic Media, in addition to two texts on the dynamics of continuous media. Following many years as a research scientist at the Army Armament Research Laboratory and the Army's Ballistic Research Laboratory, he works as a consultant in the engineering sciences. He has also served as an aerodynamicist in the commercial aircraft industry and taught applied mathematics at the Stevens Institute of Technology.

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 PART I ONE-DIMENSIONAL 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 PART II THE WAVE EQUATION IN TWO AND THREE DIMENSIONS 125 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 PART I INVISCID 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 PART II VISCOUS FLUIDS 183 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