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
Introduction | p. 1 |
Linear Waves | p. 5 |
Basic Ideas | p. 7 |
Exercises | p. 15 |
Waves on a Stretched String | p. 17 |
Derivation of the Governing Equation | p. 17 |
Standing Waves on Strings of Finite Length | p. 20 |
D'Alembert's Solution for Strings of Infinite Length | p. 26 |
Reflection and Transmission of Waves by Discontinuities in Density | p. 29 |
A Single Discontinuity | p. 29 |
Two Discontinuities: Impedance Matching | p. 31 |
Exercises | p. 33 |
Sound Waves | p. 36 |
Derivation of the Governing Equation | p. 36 |
Plane Waves | p. 40 |
Acoustic Energy Transmission | p. 42 |
Plane Waves In Tubes | p. 45 |
Acoustic Waveguides | p. 50 |
Reflection of a Plane Acoustic Wave by a Rigid Wall | p. 50 |
A Planar Waveguide | p. 51 |
A Circular Waveguide | p. 53 |
Acoustic Sources | p. 57 |
The Acoustic Source | p. 58 |
Energy Radiated by Sources and Plane Waves | p. 62 |
Radiation from Sources in a Plane Wall | p. 64 |
Exercises | p. 70 |
Linear Water Waves | p. 74 |
Derivation of the Governing Equations | p. 74 |
Linear Gravity Waves | p. 78 |
Progressive Gravity Waves | p. 78 |
Standing Gravity Waves | p. 85 |
The Wavemaker | p. 87 |
The Extraction of Energy from Water Waves | p. 91 |
The Effect of Surface Tension: Capillary--Gravity Waves | p. 94 |
Edge Waves | p. 97 |
Ship Waves | p. 99 |
The Solution of Initial Value Problems | p. 104 |
Shallow Water Waves: Linear Theory | p. 109 |
The Reflection of Sea Swell by a Step | p. 112 |
Wave Amplification at a Gently Sloping Beach | p. 114 |
Wave Refraction | p. 117 |
The Kinematics of Slowly Varying Waves | p. 118 |
Wave Refraction at a Gently Sloping Beach | p. 121 |
The Effect of Viscosity | p. 123 |
Exercises | p. 124 |
Waves in Elastic Solids | p. 130 |
Derivation of the Governing Equation | p. 130 |
Waves in an Infinite Elastic Body | p. 132 |
One-Dimensional Dilatation Waves | p. 133 |
One-Dimensional Rotational Waves | p. 134 |
Plane Waves with General Orientation | p. 134 |
Two-Dimensional Waves in Semi-infinite Elastic Bodies | p. 135 |
Normally Loaded Surface | p. 135 |
Stress-Free Surface | p. 137 |
Waves in Finite Elastic Bodies | p. 143 |
Flexural Waves in Plates | p. 144 |
Waves in Elastic Rods | p. 148 |
Torsional Waves | p. 150 |
Longitudinal Waves | p. 155 |
The Excitation and Propagation of Elastic Wavefronts | p. 156 |
Wavefronts Caused by an Internal Line Force in an Unbounded Elastic Body | p. 157 |
Wavefronts Caused by a Point Force on the Free Surface of a Semi-infinite Elastic Body | p. 161 |
Exercises | p. 168 |
Electromagnetic Waves | p. 173 |
Electric and Magnetic Forces and Fields | p. 173 |
Electrostatics: Gauss's Law | p. 177 |
Magnetostatics: Ampere's Law and the Displacement Current | p. 179 |
Electromagnetic Induction: Farady's Law | p. 180 |
Plane Electromagnetic Waves | p. 182 |
Conductors and Insulators | p. 186 |
Reflection and Transmission at Interfaces | p. 189 |
Boundary Conditions at Interfaces | p. 189 |
Reflection by a Perfect Conductor | p. 191 |
Reflection and Refraction by Insulators | p. 194 |
Waveguides | p. 199 |
Metal Waveguides | p. 199 |
Weakly Guiding Optical Fibres | p. 202 |
Radiation | p. 208 |
Scalar and Vector Potentials | p. 208 |
The Electric Dipole | p. 210 |
The Far Field of a Localised Current Distribution | p. 212 |
The Centre Fed Linear Antenna | p. 213 |
Exercises | p. 216 |
Nonlinear Waves | p. 219 |
The Formation and Propagation of Shock Waves | p. 221 |
Traffic Waves | p. 221 |
Derivation of the Governing Equation | p. 221 |
Small Amplitude Disturbances of a Uniform State | p. 224 |
The Nonlinear Initial Value Problem | p. 226 |
The Speed of the Shock | p. 236 |
Compressible Gas Dynamics | p. 239 |
Some Essential Thermodynamics | p. 239 |
Equations of Motion | p. 243 |
Construction of the Characteristic Curves | p. 245 |
The Rankine--Hugoniot Relations | p. 249 |
Detonations | p. 256 |
Exercises | p. 266 |
Nonlinear Water Waves | p. 269 |
Nonlinear Shallow Water Waves | p. 269 |
The Dam Break Problem | p. 270 |
A Shallow Water Bore | p. 275 |
The Effect of Nonlinearity on Deep Water Gravity Waves: Stokes' Expansion | p. 280 |
The Korteweg-de Vries Equation for Shallow Water Waves: the Interaction of Nonlinear Steepening and Linear Dispersion | p. 285 |
Derivation of the Korteweg-de Vries Equation | p. 287 |
Travelling Wave Solutions of the KdV Equation | p. 290 |
Nonlinear Capillary Waves | p. 298 |
Exercises | p. 306 |
Chemical and Electrochemical Waves | p. 308 |
The Law of Mass Action | p. 310 |
Molecular Diffusion | p. 314 |
Reaction-Diffusion Systems | p. 315 |
Autocatalytic Chemical Waves with Unequal Diffusion Coefficients* | p. 326 |
Existence of Travelling Wave Solutions | p. 327 |
Asymptotic Solution for [delta] [[ 1 | p. 330 |
The Transmission of Nerve Impulses: the Fitzhugh-Nagumo Equations | p. 334 |
The Fitzhugh-Nagumo Model | p. 339 |
The Existence of a Threshold | p. 342 |
Travelling Waves | p. 343 |
Exercises | p. 349 |
Advanced Topics | p. 355 |
Burgers' Equation: Competition between Wave Steepening and Wave Spreading | p. 357 |
Burgers' Equation for Traffic Flow | p. 357 |
The Effect of Dissipation on Weak Shock Waves in an Ideal Gas | p. 362 |
Simple Solutions of Burgers' Equation | p. 369 |
Travelling Waves | p. 369 |
Asymptotic Solutions for v [[ 1 | p. 370 |
Exercises | p. 375 |
Diffraction and Scattering | p. 378 |
Diffraction of Acoustic Waves by a Semi-infinite Barrier | p. 379 |
Preliminary Estimates of the Potential | p. 380 |
Pre-transform Considerations | p. 383 |
The Fourier Transform Solution | p. 385 |
The Diffraction of Waves by an Aperture | p. 391 |
Scalar Diffraction: Acoustic Waves | p. 391 |
Vector Diffraction: Electromagnetic Waves | p. 394 |
Scattering of Linear, Deep Water Waves by a Surface Piercing Cylinder | p. 399 |
Exercises | p. 403 |
Solitons and the Inverse Scattering Transform | p. 405 |
The Korteweg-de Vries Equation | p. 406 |
The Scattering Problem | p. 406 |
The Inverse Scattering Problem | p. 410 |
Scattering Data for KdV Potentials | p. 416 |
Examples: Solutions of the KdV Equation | p. 418 |
The Nonlinear Schrodinger Equation | p. 424 |
Derivation of the Nonlinear Schrodinger Equation for Plane Electromagnetic Waves | p. 424 |
Solitary Wave Solutions of the Nonlinear Schrodinger Equation | p. 431 |
The Inverse Scattering Transform for the Nonlinear Schrodinger Equation | p. 435 |
Exercises | p. 448 |
Useful Mathematical Formulas and Physical Data | p. 451 |
Cartesian Coordinates | p. 451 |
Cylindrical Polar Coordinates | p. 451 |
Spherical Polar Coordinates | p. 452 |
Some Vector Calculus Identities and Useful Results for Smooth Vector Fields | p. 453 |
Physical constants | p. 454 |
Bibliography | p. 455 |
Index | p. 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