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Mathematics of Wave Propagation - Julian L. Davis

Mathematics of Wave Propagation

Hardcover Published: 7th May 2000
ISBN: 9780691026435
Number Of Pages: 416

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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.

"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

Preface xiii
Physics of Propagating Wavesp. 3
Introductionp. 3
Discrete Wave-Propagating Systemsp. 3
Approximation of Stress Wave Propagation in a Bar by a Finite System of Mass-Spring Modelsp. 4
Limiting Form of a Continuous Barp. 5
Wave Equation for a Barp. 5
Transverse Oscillations of a Stringp. 9
Speed of a Transverse Wave in a Sitingp. 10
Traveling Waves in Generalp. 11
Sound Wave Propagation in a Tubep. 16
Superposition Principlep. 19
Sinusoidal Wavesp. 19
Interference Phenomenap. 21
Reflection of Light Wavesp. 25
Reflection of Waves in a Stringp. 27
Sound Wavesp. 29
Doppler Effectp. 33
Dispersion and Group Velocityp. 36
Problemsp. 37
Partial Differential Equations of Wave Propagationp. 41
Introductionp. 41
Types of Partial Differential Equationsp. 41
Geometric Nature of the PDEs of Wave Phenomenap. 42
Directional Derivativesp. 42
Cauchy Initial Value Problemp. 44
Parametric Representationp. 49
Wave Equation Equivalent to Two First-Order PDEsp. 51
Characteristic Equations for First-Order PDEsp. 55
General Treatment of Linear PDEs by Characteristic Theoryp. 57
Another Method of Characteristics for Second-Order PDEsp. 61
Geometric Interpretation of Quasilinear PDEsp. 63
Integral Surfacesp. 65
Nonlinear Casep. 67
Canonical Form of a Second-Order PDEp. 70
Riemann's Method of Integrationp. 73
Problemsp. 82
The Wave Equationp. 85
One-Dimensional Wave Equationp. 85
Factorization of the Wave Equation and Characteristic Curvesp. 85
Vibrating String as a Combined IV and B V Problemp. 90
D'Alembert's Solution to the IV Problemp. 97
Domain of Dependence and Range of Influencep. 101
Cauchy IV Problem Revisitedp. 102
Solution of Wave Propagation Problems by Laplace Transformsp. 105
Laplace Transformsp. 108
Applications to the Wave Equationp. 111
Nonhomogeneous Wave Equationp. 116
Wave Propagation through Media with Different Velocitiesp. 120
Electrical Transmission Linep. 122
The Wave Equation in Two and Three Dimensionsp. 125
Two-Dimensional Wave Equationp. 125
Reduced Wave Equation in Two Dimensionsp. 126
The Eigenvalues Must Be Negativep. 127
Rectangular Membranep. 127
Circular Membranep. 131
Three-Dimensional Wave Equationp. 135
Problemsp. 140
Wave Propagation in Fluidsp. 145
Inviscid Fluidsp. 145
Lagrangian Representation of One-Dimensional Compressible Gas Flowp. 146
Eulerian Representation of a One-Dimensional Gasp. 149
Solution by the Method of Characteristics: One-Dimensional Compressible Gasp. 151
Two-Dimensional Steady Flowp. 157
Bernoulli's Lawp. 159
Method of Characteristics Applied to Two-Dimensional Steady Flowp. 161
Supersonic Velocity Potentialp. 163
Hodograph Transformationp. 163
Shock Wave Phenomenap. 169
Viscous Fluidsp. 183
Elementary Discussion of Viscosityp. 183
Conservation Lawsp. 185
Boundary Conditions and Boundary Layerp. 190
Energy Dissipation in a Viscous Fluidp. 191
Wave Propagation in a Viscous Fluidp. 193
Oscillating Body of Arbitrary Shapep. 196
Similarity Considerations and Dimensionless Parameters; Reynolds' Lawp. 197
Poiseuille Flowp. 199
Stokes' Flowp. 201
Oseen Approximationp. 208
Problemsp. 210
Stress Waves in Elastic Solidsp. 213
Introductionp. 213
Fundamentals of Elasticityp. 214
Equations of Motion for the Stressp. 223
Navier Equations of Motion for the Displacementp. 224
Propagation of Plane Elastic Wavesp. 227
General Decomposition of Elastic Wavesp. 228
Characteristic Surfaces for Planar Wavesp. 229
Time-Harmonic Solutions and Reduced Wave Equationsp. 230
Spherically Symmetric Wavesp. 232
Longitudinal Waves in a Barp. 234
Curvilinear Orthogonal Coordinatesp. 237
The Navier Equations in Cylindrical Coordinatesp. 239
Radially Symmetric Wavesp. 240
Waves Propagated Over the Surface of an Elastic Bodyp. 243
Problemsp. 247
Stress Waves in Viscoelastic Solidsp. 250
Introductionp. 250
Internal Friction
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9780691026435
ISBN-10: 0691026432
Audience: Tertiary; University or College
Format: Hardcover
Language: English
Number Of Pages: 416
Published: 7th May 2000
Country of Publication: US
Dimensions (cm): 24.28 x 16.38  x 3.12
Weight (kg): 0.73