Mathematics of Wave Propagation

By: Julian L. Davis

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Hardcover | 17 July 2000

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416 Pages

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25.4 x 19.6 x 3.18

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Earthquakes, a plucked string, ocean waves crashing on the beach, the sound waves that allow us to recognize known voices. Waves are everywhere, and the propagation and classical properties of these apparently disparate phenomena can be described by the same mathematical methods: variational calculus, characteristics theory, and caustics. Taking a medium-by-medium approach, Julian Davis explains the mathematics needed to understand wave propagation in inviscid and viscous fluids, elastic solids, viscoelastic solids, and thermoelastic media, including hyperbolic partial differential equations and characteristics theory, which makes possible geometric solutions to nonlinear wave problems. The result is a clear and unified treatment of wave propagation that makes a diverse body of mathematics accessible to engineers, physicists, and applied mathematicians engaged in research on elasticity, aerodynamics, and fluid mechanics.

This book will particularly appeal to those working across specializations and those who seek the truly interdisciplinary understanding necessary to fully grasp waves and their behavior. By proceeding from concrete phenomena (e.g., the Doppler effect, the motion of sinusoidal waves, energy dissipation in viscous fluids, thermal stress) rather than abstract mathematical principles, Davis also creates a one-stop reference that will be prized by students of continuum mechanics and by mathematicians needing information on the physics of waves.

Industry Reviews

"An excellent book that covers seemingly diverse wave phenomena in a unified, coherent manner. Students and practicing engineers and physicists will find this book a useful addition to their collections."--Applied Mathematics Reviews

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Preface xiii
Physics of Propagating Waves	p. 3
Introduction	p. 3
Discrete Wave-Propagating Systems	p. 3
Approximation of Stress Wave Propagation in a Bar by a Finite System of Mass-Spring Models	p. 4
Limiting Form of a Continuous Bar	p. 5
Wave Equation for a Bar	p. 5
Transverse Oscillations of a String	p. 9
Speed of a Transverse Wave in a Siting	p. 10
Traveling Waves in General	p. 11
Sound Wave Propagation in a Tube	p. 16
Superposition Principle	p. 19
Sinusoidal Waves	p. 19
Interference Phenomena	p. 21
Reflection of Light Waves	p. 25
Reflection of Waves in a String	p. 27
Sound Waves	p. 29
Doppler Effect	p. 33
Dispersion and Group Velocity	p. 36
Problems	p. 37
Partial Differential Equations of Wave Propagation	p. 41
Introduction	p. 41
Types of Partial Differential Equations	p. 41
Geometric Nature of the PDEs of Wave Phenomena	p. 42
Directional Derivatives	p. 42
Cauchy Initial Value Problem	p. 44
Parametric Representation	p. 49
Wave Equation Equivalent to Two First-Order PDEs	p. 51
Characteristic Equations for First-Order PDEs	p. 55
General Treatment of Linear PDEs by Characteristic Theory	p. 57
Another Method of Characteristics for Second-Order PDEs	p. 61
Geometric Interpretation of Quasilinear PDEs	p. 63
Integral Surfaces	p. 65
Nonlinear Case	p. 67
Canonical Form of a Second-Order PDE	p. 70
Riemann's Method of Integration	p. 73
Problems	p. 82
The Wave Equation	p. 85
One-Dimensional Wave Equation	p. 85
Factorization of the Wave Equation and Characteristic Curves	p. 85
Vibrating String as a Combined IV and B V Problem	p. 90
D'Alembert's Solution to the IV Problem	p. 97
Domain of Dependence and Range of Influence	p. 101
Cauchy IV Problem Revisited	p. 102
Solution of Wave Propagation Problems by Laplace Transforms	p. 105
Laplace Transforms	p. 108
Applications to the Wave Equation	p. 111
Nonhomogeneous Wave Equation	p. 116
Wave Propagation through Media with Different Velocities	p. 120
Electrical Transmission Line	p. 122
The Wave Equation in Two and Three Dimensions	p. 125
Two-Dimensional Wave Equation	p. 125
Reduced Wave Equation in Two Dimensions	p. 126
The Eigenvalues Must Be Negative	p. 127
Rectangular Membrane	p. 127
Circular Membrane	p. 131
Three-Dimensional Wave Equation	p. 135
Problems	p. 140
Wave Propagation in Fluids	p. 145
Inviscid Fluids	p. 145
Lagrangian Representation of One-Dimensional Compressible Gas Flow	p. 146
Eulerian Representation of a One-Dimensional Gas	p. 149
Solution by the Method of Characteristics: One-Dimensional Compressible Gas	p. 151
Two-Dimensional Steady Flow	p. 157
Bernoulli's Law	p. 159
Method of Characteristics Applied to Two-Dimensional Steady Flow	p. 161
Supersonic Velocity Potential	p. 163
Hodograph Transformation	p. 163
Shock Wave Phenomena	p. 169
Viscous Fluids	p. 183
Elementary Discussion of Viscosity	p. 183
Conservation Laws	p. 185
Boundary Conditions and Boundary Layer	p. 190
Energy Dissipation in a Viscous Fluid	p. 191
Wave Propagation in a Viscous Fluid	p. 193
Oscillating Body of Arbitrary Shape	p. 196
Similarity Considerations and Dimensionless Parameters; Reynolds' Law	p. 197
Poiseuille Flow	p. 199
Stokes' Flow	p. 201
Oseen Approximation	p. 208
Problems	p. 210
Stress Waves in Elastic Solids	p. 213
Introduction	p. 213
Fundamentals of Elasticity	p. 214
Equations of Motion for the Stress	p. 223
Navier Equations of Motion for the Displacement	p. 224
Propagation of Plane Elastic Waves	p. 227
General Decomposition of Elastic Waves	p. 228
Characteristic Surfaces for Planar Waves	p. 229
Time-Harmonic Solutions and Reduced Wave Equations	p. 230
Spherically Symmetric Waves	p. 232
Longitudinal Waves in a Bar	p. 234
Curvilinear Orthogonal Coordinates	p. 237
The Navier Equations in Cylindrical Coordinates	p. 239
Radially Symmetric Waves	p. 240
Waves Propagated Over the Surface of an Elastic Body	p. 243
Problems	p. 247
Stress Waves in Viscoelastic Solids	p. 250
Introduction	p. 250
Internal Friction
Table of Contents provided by Publisher. All Rights Reserved.

Mathematics of Wave Propagation

At a Glance

Hardcover

Industry Reviews

Shipping

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Defective items

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