A Modern Introduction to the Mathematical Theory of Water Waves (Cambridge Texts in Applied Mathematics, Series Number 19)

The theory of water waves has been a source of intriguing and often difficult mathematical problems for at least 150 years. Virtually every classical mathematical technique appears somewhere within its confines. Beginning with the introduction of the appropriate equations of fluid mechanics, the opening chapters of this text consider the classical problems in linear and non-linear water-wave theory. This sets the ground for a study of more modern aspects, problems that give rise to soliton-type equations. The book closes with an introduction to the effects of viscosity. All the mathematical developments are presented in the most straightforward manner, with worked examples and simple cases carefully explained. Exercises, further reading, and historical notes on some of the important characters in the field round off the book and help to make this an ideal text for a beginning graduate course on water waves. Cover......Page 1 About......Page 2 Cambridge Texts in Applied Mathematics......Page 5 A Modern Introduction to the Mathematical Theory of Water Waves......Page 6 052159832X......Page 7 Contents......Page 10 Preface......Page 14 1 Mathematical preliminaries......Page 18 1.1 The governing equations of fluid mechanics......Page 19 1.1.1 The equation of mass conservation......Page 20 1.1.2 The equation of motion: Euler's equation......Page 22 1.1.3 Vorticity, streamlines and irrotational flow......Page 26 1.2 The boundary conditions for water waves......Page 30 1.2.1 The kinematic condition......Page 31 1.2.2 The dynamic condition......Page 32 1.2.3 The bottom condition......Page 35 1.2.4 An integrated mass conservation condition......Page 36 1.2.5 An energy equation and its integral......Page 37 1.3 Nondimensionalisation and scaling......Page 41 1.3.2 Scaling of the variables......Page 45 1.3.3 Approximate equations......Page 46 1.4 The elements of wave propagation and asymptotic expansions......Page 48 1.4.2 Asymptotic expansions......Page 52 Further reading......Page 63 Exercises......Page 64 2 Some classical problems in water-wave theory......Page 78 I Linear problems......Page 79 2.1 Wave propagation for arbitrary depth and wavelength......Page 0 2.1.1 Particle paths......Page 84 2.1.2 Group velocity and the propagation of energy......Page 86 2.1.3 Concentric waves on deep water......Page 92 2.2 Wave propagation over variable depth......Page 97 2.2.1 Linearised gravity waves of any wave number moving over a constant slope......Page 102 2.2.2 Edge waves over a constant slope......Page 107 2.3 Ray theory for a slowly varying environment......Page 110 2.3.1 Steady, oblique plane waves over variable depth......Page 117 2.3.2 Ray theory in cylindrical geometry......Page 122 2.3.3 Steady plane waves on a current......Page 125 2.4 The ship-wave pattern......Page 134 2.4.1 Kelvin's theory......Page 137 2.4.2 Ray theory......Page 151 II Nonlinear problems......Page 155 2.5 The Stokes wave......Page 156 2.6 Nonlinear long waves......Page 163 2.6.1 The method of characteristics......Page 165 2.6.2 The hodograph transformation......Page 170 2.7 Hydraulic jump and bore......Page 173 2.8 Nonlinear waves on a sloping beach......Page 179 2.9 The solitary wave......Page 182 2.9.1 The sech^2 solitary wave......Page 188 2.9.2 Integral relations for the solitary wave......Page 193 Further reading......Page 198 Exercises......Page 199 3.1 Introduction......Page 217 3.2 The Korteweg-de Vries family of equations......Page 221 3.2.2 Two-dimensional Korteweg-de Vries (2D KdV) equation......Page 226 3.2.3 Concentric Korteweg-de Vries (cKdV) equation......Page 228 3.2.4 Nearly concentric Korteweg-de Vries (ncKdV) equation......Page 231 3.2.5 Boussinesq equation......Page 233 3.2.6 Transformations between these equations......Page 236 3.2.7 Matching to the near-field......Page 238 3.3 Completely integrable equations: some results from soliton theory......Page 240 3.3.1 Solution of the Korteweg-de Vries equation......Page 242 3.3.2 Soliton theory for other equations......Page 250 3.3.3 Hirota's bilinear method......Page 251 3.3.4 Conservation laws......Page 260 3.4 Waves in a nonuniform environment......Page 272 3.4.2 The Burns condition......Page 278 3.4.3 Ring waves over a shear flow......Page 280 3.4.4 The Korteweg-de Vries equation for variable depth......Page 285 3.4.5 Oblique interaction of waves......Page 294 Further reading......Page 301 Exercises......Page 302 4 Slow modulation of dispersive waves......Page 314 4.1 The evolution of wave packets......Page 315 4.1.2 Davey-Stewartson (DS) equations......Page 322 4.1.3 Matching between the NLS and KdV equations......Page 325 4.2 NLS and DS equations: some results from soliton theory......Page 329 4.2.2 Bilinear method for the NLS equation......Page 335 4.2.3 Bilinear form of the DS equations for long waves......Page 340 4.2.4 Conservation laws for the NLS and DS equations......Page 342 4.3 Applications of the NLS and DS equations......Page 348 4.3.1 Stability of the Stokes wave......Page 349 4.3.2 Modulation of waves over a shear flow......Page 354 4.3.3 Modulation of waves over variable depth......Page 358 Further reading......Page 362 5 Epilogue......Page 373 5.1 The governing equations with viscosity......Page 374 5.2 Applications to the propagation of gravity waves......Page 376 5.2.1 Small amplitude harmonic waves......Page 377 5.2.2 Attenuation of the solitary wave......Page 382 5.2.3 Undular bore - model I......Page 391 5.2.4 Undular bore - model II......Page 395 Further reading......Page 403 Exercises......Page 404 A The equations for a viscous fluid......Page 410 B The boundary conditions for a viscous fluid......Page 414 C Historical notes......Page 416 D Answers and hints......Page 422 Bibliography......Page 446 Subject index......Page 454

For over a hundred years, the theory of water waves has been a source of intriguing and often difficult mathematical problems. Virtually every classical mathematical technique appears somewhere within its confines. Beginning with the introduction of the appropriate equations of fluid mechanics, the opening chapters of this text consider the classical problems in linear and nonlinear water-wave theory. This sets the stage for a study of more modern aspects, problems that give rise to soliton-type equations. The book closes with an introduction to the effects of viscosity. All the mathematical developments are presented in the most straightforward manner, with worked examples and simple cases carefully explained. Exercises, further reading, and historical notes on some of the important characters in the field round off the book and make this an ideal text for a beginning graduate course on water waves.

The theory of water waves has been a source of intriguing mathematical problems for at least 150 years. This text considers the classical problems in linear and non-linear water-wave theory, as well as more modern aspects - problems that give rise to soliton-type equations. Lastly it examines the effects of viscosity. Beginning with the introduction of the appropriate equations of fluid mechanics, Johnson considers classical problems in linear and non-linear water-wave theory, goes on to look at soliton type equations and closes with an introduction to viscosity