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.

Table of Contents:
1 Oscillatory Phenomena.- 1.1. Harmonic Motion.- 1.2. Forced Oscillations.- 1.3. Combination of Wave Forms.- 1.4. Oscillations in Two Dimensions.- 1.5. Coupled Oscillations.- 1.6. Lagrange’s Equations of Motion.- 1.7. Formulation of the Problem of Small Oscillations for Conservative Systems.- 1.8. The Eigenvalue Equation.- 1.9. Similarity Transformation and Normal Coordinates.- 2 The Physics of Wave Propagation.- 2.1. The Conservation Laws of Physics.- 2.2. The Nature of Wave Propagation.- 2.3. Discretization.- 2.4. Sinusoidal Wave Propagation.- 2.5. Derivation of the Wave Equation.- 2.6. The Superposition Principle, Interference Phenomena.- 2.7. Concluding Remarks.- 3 Partial Differential Equations of Wave Propagation.- 3.1. Wave Equation as an Equivalent First-Order System.- 3.2. Method of Characteristics for a Single First-Order Quasilinear Partial Differential Equation.- 3.3. Second-Order Quasilinear Partial Differential Equation.- 3.4. Method of Characteristics for Second-Order Partial Differential Equations.- 3.5. Propagation of Discontinuities.- 3.6. Canonical Form for Second-Order Partial Differential Equations with Constant Coefficients.- 3.7. Conservation Laws, Weak Solutions.- 3.8. Divergence Theorem, Adjoint Operator, Green’s Identity, Riemann’s Method.- 4 Transverse Vibrations of Strings.- 4.1. Solution of the Wave Equation, Characteristic Coordinates.- 4.2. D’Alembert’s Solution.- 4.3. Nonhomogeneous Wave Equation.- 4.4. Mixed Initial Value and Boundary Value Problem, Finite String.- 4.5. Finite or Lagrange Model for Vibrating String.- 5 Water Waves.- 5.1. Conservation Laws.- 5.2. Potential Flow.- 5.3. Two-Dimensional Flow, Complex Variables.- 5.4. The Drag Force Past a Body in Potential Flow.- 5.5. Energy Flux.- 5.6. Small Amplitude Gravity Waves.- 5.7. Boundary Conditions.- 5.8. Formulation of a Typical Surface Wave Problem.- 5.9. Simple Harmonic Oscillations in Water at Constant Depth.- 5.10. The Solitary Wave.- 5.11. Approximation Theories.- 6 Sound Waves.- 6.1. Linearization of the Conservation Laws.- 6.2. Plane Waves.- 6.3. Energy and Momentum.- 6.4. Reflection and Refraction of Sound Waves.- 6.5. Sound Wave Propagation in a Moving Medium.- 6.6. Spherical Sound Waves.- 6.7. Cylindrical Sound Waves.- 6.8. General Solution of the Wave Equation.- 6.9. Huyghen’s Principle.- 7 Fluid Dynamics.- I. Inviscid Fluids.- 7.1. One-Dimensional Compressible Inviscid Flow.- 7.2. Two-Dimensional Steady Flow.- 7.3. Shock Wave Phenomena.- II. Viscous Fluids.- 7.4. Viscosity, Elementary Considerations.- 7.5. Conservation Laws for a Viscous Fluid.- 7.6. Flow in a Pipe, Poiseuille Flow.- 7.7. Dimensional Considerations.- 7.8. Stokes’s Flow.- 7.9. Oscillatory Motion.- 7.10. Potential Flow.- 8 Wave Propagation in Elastic Media.- Historical Introduction to Wave Propagation.- 8.1. Fundamental Concepts of Elasticity.- 8.2. Equations of Motion for the Stress Components.- 8.3. Equations of Motion for the Displacement, Navier Equations.- 8.4. Propagation of a Plane Elastic Wave.- 8.5. Spherically Symmetric Waves.- 8.6. Reflection of Plane Waves at a Free Surface.- 8.7. Surface Waves, Rayleigh Waves.- 9 Variational Methods in Wave Phenomena.- 9.1. Principle of Least Time.- 9.2. One-Dimensional Treatment, Euler’s Equation.- 9.3. Euler’s Equations for the Two-Dimensional Case.- 9.4. Generalization to Functional with More Than One Dependent Variable.- 9.5. Hamilton’s Variational Principle.- 9.6. Lagrange’s Equations of Motion.- 9.7. Principle of Virtual Work.- 9.8. Transformation to Generalized Coordinates.- 9.9. Rayleigh’s Dissipation Function.- 9.10. Hamilton’s Equations of Motion.- 9.11. Cyclic Coordinates.- 9.12. Lagrange’s Equations of Motion for a Continuum.- 9.13. Hamilton’s Equations of Motion for a Continuum.- 9.14. Hamilton-Jacobi Theory.- 9.15. Characteristic Theory in Relation to Hamilton-Jacobi Theory.- 9.16. Principle of Least Action.- 9.17. Hamilton-Jacobi Theory and Wave Propagation.- 9.18. Application to Quantum Mechanics.-9.19. Asymptotic Phenomena.