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
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
3.3 Completely integrable equations: some results from
soliton 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
Further reading ............................................ 386
Exercises .................................................. 387
Appendices .................................................... 393
A The equations for a viscous fluid .......................... 393
В The boundary conditions for a viscous fluid ................ 397
С Historical notes ........................................... 399
D Answers and hints .......................................... 405
Bibliography .................................................. 429
Subject index ................................................. 437
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