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Wave Motion : Cambridge Texts in Applied Mathematics - J. Billingham

Wave Motion

Cambridge Texts in Applied Mathematics


Published: 29th January 2001
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Waves are a ubiquitous and important feature of the physical world, and throughout history it has been a major challenge to understand them. They can propagate on the surfaces of solids and of fluids; chemical waves control the beating of your heart; traffic jams move in waves down lanes crowded with vehicles. This introduction to the mathematics of wave phenomena is aimed at advanced undergraduate courses on waves for mathematicians, physicists or engineers. Some more advanced material on both linear and nonlinear waves is also included, thus making the book suitable for beginning graduate courses. The authors assume some familiarity with partial differential equations, integral transforms and asymptotic expansions as well as an acquaintance with fluid mechanics, elasticity and electromagnetism. The context and physics that underlie the mathematics is clearly explained at the beginning of each chapter. Worked examples and exercises are supplied throughout, with solutions available to teachers.

'... an excellent advanced introduction to the mathematical theory of wave motion. it is ideally suited to advanced undergraduate students and beginning postgraduate students ... one attractive feature of the book is the abundance of worked examples and exercises (with solutions available to teachers) ... this is a wonderful book whose reading I would recommend to any scientist interested in learning the mathematical theory of wave motion.' European Journal of Mechanics '... the great strength of the book ... lies in the clarity of exposition of the mathematical solution of the wave equations and of the physical interpretation of these solutions ... this is an excellent book, which is thoroughly recommended ... it ought to become the standard textbook for anyone taking an undergraduate course in mathematical wave theory.' Journal of Fluid Mechanics '... a clearly written book which covers a surprisingly wide variety of topics ... an excellent introduction to this fascinating area of applied mathematics.' A. Jeffrey, Zentralblatt fur Mathematik 'The rich material is presented in a quite digestible way with clear explanation of physical principles and properties and a formal apparatus which does not overwhelm everything else.' H. Mathsam, Monatshefte fur Mathematik '... Billingham and King ... offer an attractive, thorough discussion of wave phenomena.' J. H. Ellison, Choice '... written very clearly ... will be valuable for students of mathematics who wish to apply their mathematics to physics and other fields.' EMS 'Sections on non-linear wave motion and advanced topics extend the usefulness of this excellent text to the first year of graduate study.' Aslib Book Guide '... a text of great clarity ... I warmly recommend this book as a useful source of reference material for applied mathematicians, physicists and engineers alike.' A. Jueld, Contemporary Physics 'Wave Motion has the potential to become the mathematical text for advanced undergraduate courses on the analytical aspects of waves.' Christopher Howls ' ... very accessible for a reader with some background in applied mathematics. The style and the exercises after each chapter make it perfectly fit as lecture notes for a course in applied mathematics.' Bulletin of the Belgian Mathematical Society 'I'm glad I bought a copy of this book as soon as it was published, because it has informed me, and will go on informing my lecture classes, for years to come.' Mark J. Cooker, University of East Anglia, The Mathematical Gazette

Introductionp. 1
Linear Wavesp. 5
Basic Ideasp. 7
Exercisesp. 15
Waves on a Stretched Stringp. 17
Derivation of the Governing Equationp. 17
Standing Waves on Strings of Finite Lengthp. 20
D'Alembert's Solution for Strings of Infinite Lengthp. 26
Reflection and Transmission of Waves by Discontinuities in Densityp. 29
A Single Discontinuityp. 29
Two Discontinuities: Impedance Matchingp. 31
Exercisesp. 33
Sound Wavesp. 36
Derivation of the Governing Equationp. 36
Plane Wavesp. 40
Acoustic Energy Transmissionp. 42
Plane Waves In Tubesp. 45
Acoustic Waveguidesp. 50
Reflection of a Plane Acoustic Wave by a Rigid Wallp. 50
A Planar Waveguidep. 51
A Circular Waveguidep. 53
Acoustic Sourcesp. 57
The Acoustic Sourcep. 58
Energy Radiated by Sources and Plane Wavesp. 62
Radiation from Sources in a Plane Wallp. 64
Exercisesp. 70
Linear Water Wavesp. 74
Derivation of the Governing Equationsp. 74
Linear Gravity Wavesp. 78
Progressive Gravity Wavesp. 78
Standing Gravity Wavesp. 85
The Wavemakerp. 87
The Extraction of Energy from Water Wavesp. 91
The Effect of Surface Tension: Capillary--Gravity Wavesp. 94
Edge Wavesp. 97
Ship Wavesp. 99
The Solution of Initial Value Problemsp. 104
Shallow Water Waves: Linear Theoryp. 109
The Reflection of Sea Swell by a Stepp. 112
Wave Amplification at a Gently Sloping Beachp. 114
Wave Refractionp. 117
The Kinematics of Slowly Varying Wavesp. 118
Wave Refraction at a Gently Sloping Beachp. 121
The Effect of Viscosityp. 123
Exercisesp. 124
Waves in Elastic Solidsp. 130
Derivation of the Governing Equationp. 130
Waves in an Infinite Elastic Bodyp. 132
One-Dimensional Dilatation Wavesp. 133
One-Dimensional Rotational Wavesp. 134
Plane Waves with General Orientationp. 134
Two-Dimensional Waves in Semi-infinite Elastic Bodiesp. 135
Normally Loaded Surfacep. 135
Stress-Free Surfacep. 137
Waves in Finite Elastic Bodiesp. 143
Flexural Waves in Platesp. 144
Waves in Elastic Rodsp. 148
Torsional Wavesp. 150
Longitudinal Wavesp. 155
The Excitation and Propagation of Elastic Wavefrontsp. 156
Wavefronts Caused by an Internal Line Force in an Unbounded Elastic Bodyp. 157
Wavefronts Caused by a Point Force on the Free Surface of a Semi-infinite Elastic Bodyp. 161
Exercisesp. 168
Electromagnetic Wavesp. 173
Electric and Magnetic Forces and Fieldsp. 173
Electrostatics: Gauss's Lawp. 177
Magnetostatics: Ampere's Law and the Displacement Currentp. 179
Electromagnetic Induction: Farady's Lawp. 180
Plane Electromagnetic Wavesp. 182
Conductors and Insulatorsp. 186
Reflection and Transmission at Interfacesp. 189
Boundary Conditions at Interfacesp. 189
Reflection by a Perfect Conductorp. 191
Reflection and Refraction by Insulatorsp. 194
Waveguidesp. 199
Metal Waveguidesp. 199
Weakly Guiding Optical Fibresp. 202
Radiationp. 208
Scalar and Vector Potentialsp. 208
The Electric Dipolep. 210
The Far Field of a Localised Current Distributionp. 212
The Centre Fed Linear Antennap. 213
Exercisesp. 216
Nonlinear Wavesp. 219
The Formation and Propagation of Shock Wavesp. 221
Traffic Wavesp. 221
Derivation of the Governing Equationp. 221
Small Amplitude Disturbances of a Uniform Statep. 224
The Nonlinear Initial Value Problemp. 226
The Speed of the Shockp. 236
Compressible Gas Dynamicsp. 239
Some Essential Thermodynamicsp. 239
Equations of Motionp. 243
Construction of the Characteristic Curvesp. 245
The Rankine--Hugoniot Relationsp. 249
Detonationsp. 256
Exercisesp. 266
Nonlinear Water Wavesp. 269
Nonlinear Shallow Water Wavesp. 269
The Dam Break Problemp. 270
A Shallow Water Borep. 275
The Effect of Nonlinearity on Deep Water Gravity Waves: Stokes' Expansionp. 280
The Korteweg-de Vries Equation for Shallow Water Waves: the Interaction of Nonlinear Steepening and Linear Dispersionp. 285
Derivation of the Korteweg-de Vries Equationp. 287
Travelling Wave Solutions of the KdV Equationp. 290
Nonlinear Capillary Wavesp. 298
Exercisesp. 306
Chemical and Electrochemical Wavesp. 308
The Law of Mass Actionp. 310
Molecular Diffusionp. 314
Reaction-Diffusion Systemsp. 315
Autocatalytic Chemical Waves with Unequal Diffusion Coefficients*p. 326
Existence of Travelling Wave Solutionsp. 327
Asymptotic Solution for [delta] [[ 1p. 330
The Transmission of Nerve Impulses: the Fitzhugh-Nagumo Equationsp. 334
The Fitzhugh-Nagumo Modelp. 339
The Existence of a Thresholdp. 342
Travelling Wavesp. 343
Exercisesp. 349
Advanced Topicsp. 355
Burgers' Equation: Competition between Wave Steepening and Wave Spreadingp. 357
Burgers' Equation for Traffic Flowp. 357
The Effect of Dissipation on Weak Shock Waves in an Ideal Gasp. 362
Simple Solutions of Burgers' Equationp. 369
Travelling Wavesp. 369
Asymptotic Solutions for v [[ 1p. 370
Exercisesp. 375
Diffraction and Scatteringp. 378
Diffraction of Acoustic Waves by a Semi-infinite Barrierp. 379
Preliminary Estimates of the Potentialp. 380
Pre-transform Considerationsp. 383
The Fourier Transform Solutionp. 385
The Diffraction of Waves by an Aperturep. 391
Scalar Diffraction: Acoustic Wavesp. 391
Vector Diffraction: Electromagnetic Wavesp. 394
Scattering of Linear, Deep Water Waves by a Surface Piercing Cylinderp. 399
Exercisesp. 403
Solitons and the Inverse Scattering Transformp. 405
The Korteweg-de Vries Equationp. 406
The Scattering Problemp. 406
The Inverse Scattering Problemp. 410
Scattering Data for KdV Potentialsp. 416
Examples: Solutions of the KdV Equationp. 418
The Nonlinear Schrodinger Equationp. 424
Derivation of the Nonlinear Schrodinger Equation for Plane Electromagnetic Wavesp. 424
Solitary Wave Solutions of the Nonlinear Schrodinger Equationp. 431
The Inverse Scattering Transform for the Nonlinear Schrodinger Equationp. 435
Exercisesp. 448
Useful Mathematical Formulas and Physical Datap. 451
Cartesian Coordinatesp. 451
Cylindrical Polar Coordinatesp. 451
Spherical Polar Coordinatesp. 452
Some Vector Calculus Identities and Useful Results for Smooth Vector Fieldsp. 453
Physical constantsp. 454
Bibliographyp. 455
Indexp. 459
Table of Contents provided by Syndetics. All Rights Reserved.

ISBN: 9780521632577
ISBN-10: 0521632579
Series: Cambridge Texts in Applied Mathematics : Book 24
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 476
Published: 29th January 2001
Publisher: Cambridge University Press
Country of Publication: GB
Dimensions (cm): 22.86 x 15.24  x 2.54
Weight (kg): 0.91