1 Preliminaries 1
1.1 Water Wave Theories in Historical Perspective 1
1.1.1 The Mild-Slope Equations 2
1.1.2 The Boussinesq-Type Equations 3
1.2 The Governing Equations 4
1.3 Lagrangian Formulation 5
1.4 Hamiltonian Formulation 5
References 6
2 Weakly Nonlinear Water Waves Propagating over Uneven Bottoms 9
2.1 Modified Third-Order Evolution Equations of Liu and Dingemans 9
2.2 Fourth-Order Evolution Equations and Stability Analysis 15
2.3 Third-Order Evolution Equations for Wave-Current interactions 26
References 35
3 Resonant Interactions Between Weakly Nonlinear Stokes Waves and Ambient Currents and Uneven Bottoms 37
3.1 Introduction 37
3.2 Governing Equations and WKBJ Perturbation Expansion 38
3.3 Subharmonic Resonance 40
3.4 Dynamical System 46
References 51
4 The Mild-Slope Equations 53
4.1 Introduction 53
4.2 Three-Dimensional Currents over Mildly Varying Topography 54
4.3 Two-Dimensional Currents over Rapidly Varying Topography 58
4.4 Three-Dimensional Currents over Rapidly Varying Topography 65
4.5 Two-Dimensional Currents over Generally Varying Topography 70
4.6 A Hierarchy for Two-Dimensional Currents over Generally Varying Topography 73
References 77
5 Linear Gravity Waves over Rigid,Porous Bottoms 79
5.1 Introduction 79
5.2 A Rapidly Varying Bottom 80
5.3 Generally Varying Bottom 85
References 93
6 Nonlinear Unified Equations over an Uneven Bottom 95
6.1 Introduction 95
6.2 Nonlinear Unified Equations 95
6.3 Explicit Special Cases 97
6.3.1 Generalized Nonlinear Shallow-Water Equations of Airy 97
6.3.2 Generalized Mild-Slope Equation 98
6.3.3 Stokes Wave Theory 98
6.3.4 Higher-Order Boussinesq-Type Equations 99
References 102
7 Generalized Mean-Flow Theory 103
7.1 Introduction 103
7.2 Governing Equations and Boundary Conditions 104
7.3 Averaged Equations of Motion 105
7.4 Generalized Wave Action Conservation Equation and Its Wave Actions 109
References 110
8 Hamiltonian Description of Stratified Wave-Current Interactions 113
8.1 Introduction 113
8.2 Two-Layer Wave-Current Interactions 114
8.3 n-Layer Pure Waves 119
8.4 n-Layer Wave-Current Interactions over Uneven Bottoms 122
References 126
9 Surface Capillary-Gravity Short-Crested Waves with a Current in Water of Finite Depth 127
9.1 Introduction 128
9.2 An Incomplete Match and Its Solution 128
9.3 Linear Capillary-Gravity Short-Crested Waves 132
9.3.1 System Formulation 132
9.3.2 Analytical Solutions and Kinematic and Dynamical Variables 134
9.3.3 Special Cases 136
9.4 Second-Order Capillary-Gravity Short-Crested Waves 138
9.5 Third-Order Gravity Short-Crested Waves 146
9.5.1 The System Equations and the Perturbation Method 146
9.5.2 Third-Order Solution 149
9.5.3 Special Cases 161
9.5.4 Short-Crested Wave Quantities 165
9.5.5 Short-Crested Wave Forces on Vertical Walls 171
9.6 Third-Order Pure Capillary-Gravity Short-Crested Waves 178
9.6.1 Formulation 178
9.6.2 Solution 180
9.6.3 Kinematical and Dynamical Variables 197
References 208
Appendices 211
A γ，μand v in(2.1.4) 211
B ζ（3,1），？p(3.1),A(3,2)，ηj，τj，μj，λj and vj in Chapter 2 212
C λ1 and λ2 in(2.3.44) 218
D μj in(3.3.22) 219
E I23,I33,I35,I36 in Chapter 5 220
F Coefficients in(9.4.33)and(9.4.34) 223
G Coefficients in(9.5.136)-(9.5.138) 225
H Coefficients in(9.5.139)and(9.5.140) 227
Subject Index 231