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A Modern Introduction to the Mathematical Theory of Water Waves
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Preface page xi 1 Mathematical preliminaries 1 1.1 The governing equations of fluid mechanics 2 1.1.1 The equation of mass conservation 3 1.1.2 The equation of motion: Euler's equation 5 1.1.3 Vorticity, streamlines and irrotational flow 9 1.2 The boundary conditions for water waves 13 1.2.1 The kinematic condition 14 1.2.2 The dynamic condition 15 1.2.3 The bottom condition 18 1.2.4 An integrated mass conservation condition 19 1.2.5 An energy equation and its integral 20 1.3 Nondimensionalisation and scaling 24 1.3.1 Nondimensionalisation 24 1.3.2 Scaling of the variables 28 1.3.3 Approximate equations 29 1.4 The elements of wave propagation and asymptotic expansions 31 1.4.1 Elementary ideas in the theory of wave propagation 31 1.4.2 Asymptotic expansions 35 Further reading 46 Exercises 47 2 Some classical problems in water-wave theory 61 I Linear problems 62 2.1 Wave propagation for arbitrary depth and wavelength 62 2.1.1 Particle paths 67 vn viii Contents 2.1.2 Group velocity and the propagation of energy 69 2.1.3 Concentric waves on deep water 75 2.2 Wave propagation over variable depth 80 2.2.1 Linearised gravity waves of any wave number moving over a constant slope 85 2.2.2 Edge waves over a constant slope 90 2.3 Ray theory for a slowly varying environment 93 2.3.1 Steady, oblique plane waves over variable depth 100 2.3.2 Ray theory in cylindrical geometry 105 2.3.3 Steady plane waves on a current 108 2.4 The ship-wave pattern 117 2.4.1 Kelvin's theory 120 2.4.2 Ray theory 134 II Nonlinear problems 138 2.5 The Stokes wave 139 2.6 Nonlinear long waves 146 2.6.1 The method of characteristics 148 2.6.2 The hodograph transformation 153 2.7 Hydraulic jump and bore 156 2.8 Nonlinear waves on a sloping beach 162 2.9 The solitary wave 165 2.9.1 The sech2 solitary wave 171 2.9.2 Integral relations for the solitary wave 176 Further reading 181 Exercises 182 3 Weakly nonlinear dispersive waves 200 3.1 Introduction 200 3.2 The Korteweg-de Vries family of equations 204 3.2.1 Korteweg-de Vries (KdV) equation 204 3.2.2 Two-dimensional Korteweg-de Vries (2D KdV) equation 209 3.2.3 Concentric Korteweg-de Vries (cKdV) equation 211 3.2.4 Nearly concentric Korteweg-de Vries (ncKdV) equation 214 3.2.5 Boussinesq equation 216 3.2.6 Transformations between these equations 219 3.2.7 Matching to the near-field 221 Contents ix 3.3 Completely integrable equations: some results from soli ton theory 223 3.3.1 Solution of the Korteweg-de Vries equation 225 3.3.2 Soliton theory for other equations 233 3.3.3 Hirota's bilinear method 234 3.3.4 Conservation laws 243 3.4 Waves in a nonuniform environment 255 3.4.1 Waves over a shear flow 255 3.4.2 The Burns condition 261 3.4.3 Ring waves over a shear flow 263 3.4.4 The Korteweg-de Vries equation for variable depth 268 3.4.5 Oblique interaction of waves 277 Further reading 284 Exercises 285 4 Slow modulation of dispersive waves 297 4.1 The evolution of wave packets 298 4.1.1 Nonlinear Schrodinger (NLS) equation 298 4.1.2 Davey-Stewartson (DS) equations 305 4.1.3 Matching between the NLS and KdV equations 308 4.2 NLS and DS equations: some results from soliton theory 312 4.2.1 Solution of the Nonlinear Schrodinger equation 312 4.2.2 Bilinear method for the NLS equation 318 4.2.3 Bilinear form of the DS equations for long waves 323 4.2.4 Conservation laws for the NLS and DS equations 325 4.3 Applications of the NLS and DS equations 331 4.3.1 Stability of the Stokes wave 332 4.3.2 Modulation of waves over a shear flow 337 4.3.3 Modulation of waves over variable depth 341 Further reading 345 Exercises 345 5 Epilogue 356 5.1 The governing equations with viscosity 357 5.2 Applications to the propagation of gravity waves 359 5.2.1 Small amplitude harmonic waves 360 5.2.2 Attenuation of the solitary wave 365 5.2.3 Undular bore - model I 374 5.2.4 Undular bore - model II 378 Contents Further reading 386 Exercises 387 Appendices 393 A The equations for a viscous fluid 393 B The boundary conditions for a viscous fluid 397 C Historical notes 399 D Answers and hints 405 Bibliography 429 Subject index 437 |
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