Theory and Applications of Ocean Surface Waves (Advanced Series on Ocean Engineering) (Advanced Series on Ocean Engineering)

Preface to the Expanded Edition......Page 8
Preface to the First Edition......Page 10
Acknowledgments......Page 14
Contents......Page 16
1 Introduction......Page 30
1.1.1 Governing Equations......Page 31
1.1.2 Boundary Conditions for an Inviscid Irrotational Flow......Page 33
1.2 Linearized Approximation for Small-Amplitude Waves......Page 35
1.3 Elementary Notions of a Propagating Wave......Page 38
1.4 Progressive Water Waves on Constant Depth......Page 40
1.5.1 A Kinematic View......Page 44
1.5.2 A Dynamic View: Energy Flux......Page 45
2.1 Two-Dimensional Transient Problems......Page 50
2.1.1 Transient Disturbance Due to an Initial Displacement on the Free Surface......Page 53
2.1.2 Energy Propagation, Group Velocity......Page 59
2.1.3 Leading Waves Due to a Transient Disturbance......Page 60
2.1.4 Tsunami Due to Tilting of the Bottom......Page 62
2.2 Three-Dimensional Transient Response to Bottom Disturbances......Page 67
2.2.1 Two-Dimensional Tsunami Due to Impulsive Bottom Displacement......Page 71
2.2.2 Leading Waves of a Two-Dimensional Tsunami......Page 76
2.3 The Propagation of a Dispersive Wave Packet......Page 79
2.4.1 Evolution Equation for the Wave Envelope......Page 83
2.4.2 Evolution of the Front of a Wavetrain......Page 87
3 Refraction by Slowly Varying Depth or Current......Page 92
3.1 Geometrical Optics Approximation for Progressive Waves Over a Gradually Varying Bottom......Page 93
3.2 Ray Theory for Sinusoidal Waves, Fermat’s Principle......Page 97
3.3.1 Geometry of Rays......Page 100
3.3.2 Amplitude Variation......Page 105
3.3.3 The Neighborhood of a Straight Caustic......Page 106
3.4.1 Geometry of Rays......Page 109
3.4.2 Amplitude Variation......Page 117
3.5 An Approximate Equation Combining Diffraction and Refraction on a Slowly Varying Bottom — The Mild-Slope Equation......Page 123
3.6 Geometrical Optics Approximation for Refraction by Slowly Varying Current and Depth......Page 128
3.7.1 Uniform Current on Constant Depth......Page 137
3.7.2 Oblique Incidence on a Shear Current Over Constant Depth......Page 139
3.7.3 Colinear Waves and Current......Page 145
4.1.1 Governing Equations......Page 150
4.1.2 Quasi-One-Dimensional Waves in a Long Channel of Slowly Varying Cross Section......Page 153
4.1.3 Further Remarks on the Radiation Condition......Page 154
4.2.1 The Solution......Page 157
4.2.2 Justification of the Matching Conditions at the Junction......Page 162
4.2.3 The Near Field for a Rectangular Step......Page 166
4.3 Straight Depth Discontinuity — Oblique Incidence......Page 170
4.4 Scattering by a Shelf or Trough of Finite Width......Page 173
4.5 Transmission and Reflection by a Slowly Varying Depth......Page 178
4.6 Trapped Waves on a Stepped Ridge......Page 184
4.7.1 A Qualitative Discussion of Trapped Waves......Page 190
4.7.2 The Scattering Matrix [S( )]......Page 192
4.7.3 Trapped Modes as Imaginary Poles of [S( )]......Page 194
4.7.4 Properties of [S( )] for Real......Page 196
4.8 Edge Waves on a Constant Slope......Page 199
4.9.1 General Aspects......Page 201
4.9.2 Scattering of Plane Incident Waves by a Circular Sill......Page 204
4.10 Head-Sea Incidence on a Slender Topography — The Parabolic Approximation......Page 208
4.11.1 Introduction......Page 213
4.11.2 The Variational Principle......Page 215
4.11.3 Finite-Element Approximation......Page 218
Appendix 4.A Partial Wave Expansion of the Plane Wave......Page 226
5.1 Introduction......Page 228
5.2 Formulation for Harbor Oscillation Problems......Page 230
5.3.1 A Rectangular Basin......Page 232
5.3.2 A Circular Basin......Page 234
5.4 Concept of Radiation Damping — A Model Example......Page 235
5.5 Diffraction Through a Narrow Gap......Page 239
5.6.1 General Solution......Page 245
5.6.2 An Open Narrow Bay......Page 249
5.7 A Rectangular Harbor with a Narrow Entrance......Page 253
5.7.1 Solution by Matched Asymptotic Expansions......Page 255
5.7.2 Resonant Spectrum and Response for Non-Helmholtz Modes......Page 259
5.7.3 The Helmholtz Mode......Page 262
5.7.4 Numerical Results and Experiments......Page 263
5.7.5 Effects of Finite Entry Channel......Page 266
5.8 The Effect of Protruding Breakwater......Page 267
5.8.1 Representation of Solution......Page 268
5.8.2 Reduction to an Integral Equation......Page 270
5.8.3 Approximate Solution by Variational Method......Page 272
5.8.4 Numerical Results......Page 274
5.9 A Harbor with Coupled Basins......Page 279
5.10 A Numerical Method for Harbors of Complex Geometry......Page 282
5.11 Harbor Response to Transient Incident Wave......Page 287
Appendix 5.A The Source Function for a Rectangular Basin......Page 297
Appendix 5.B Summation of the G Series......Page 298
Appendix 5.C Proof of a Variational Principle......Page 300
Appendix 5.D Evaluation of an Integral......Page 301
6 Effects of Head Loss at a Constriction on the Scattering of Long Waves: Hydraulic Theory......Page 304
6.1.1 The Field Equations......Page 305
6.1.2 The Matching Conditions and the Near Field......Page 307
6.1.3 The Coefficients f and L......Page 309
6.1.4 Equivalent Linearization......Page 312
6.1.5 Approximate and Exact Solutions......Page 313
6.2 Effect of Entrance Loss on Harbor Oscillations......Page 320
6.2.1 The Boundary-Value Problem......Page 321
6.2.2 Local and Mean Square Response in the Harbor......Page 323
6.2.3 Approximations for Narrow Entrance......Page 325
6.2.4 Small Radiation and Friction Damping......Page 326
6.2.5 Large Friction Damping......Page 328
6.2.6 Numerical Results for General W......Page 329
Appendix 6.A Approximations of an Integral for ka << 1......Page 331
7 Multiple Scattering by Seabed Irregularities......Page 334
7.1 Field Evidence of Periodic Longshore Bars......Page 335
7.2 Evolution Equations for Bragg-Scattering......Page 337
7.3 Normal Incidence......Page 342
7.3.2 Supercritical Detuning:  >  0......Page 343
7.4 Randomly Rough Seabed–Envelope Equation......Page 350
7.5 Change of Wave Amplitude by Disorder......Page 356
7.6 Change of Wavenumber by Disorder......Page 358
Appendix 7.A Explicit Evaluation of the Coefficient......Page 360
8.1 Introduction......Page 364
8.2.1 The Kinematic Condition......Page 367
8.2.2 Conservation of Linear Momentum......Page 370
8.2.3 Conservation of Angular Momentum......Page 373
Buoyancy Torque......Page 376
Constraining Torque......Page 378
The Two-Dimensional Limit......Page 379
8.3.1 Decomposition into Diffraction and Radiation Problems......Page 383
8.3.2 Exciting and Restoring Forces; Added Mass and Radiation Damping for a Body of Arbitrary Shape......Page 385
Two Dimensions......Page 387
Three-Dimensional Eigensolutions in Cylindrical Polar Coordinates......Page 390
8.4.2 The Entire Fluid Domain......Page 391
8.5 Scattering by a Vertical Cylinder with Circular Cross Section......Page 395
8.6 General Identities for the Diffraction and Radiation of Simple Harmonic Waves......Page 402
8.6.1 Relations between Two Radiation Problems and their Consequences......Page 403
8.6.2 Relations between Two Diffraction Problems......Page 405
Two Dimensions......Page 406
Three Dimensions......Page 407
Haskind–Hanaoka Theorem......Page 410
Bessho–Newman Relations......Page 411
8.7 Numerical Solution by Hybrid Element Method......Page 414
8.7.1 The Variational Formulation......Page 415
8.7.2 The Approximate Solution......Page 417
8.7.3 A Theoretical Property of the Hybrid Element Method......Page 420
8.7.4 A Numerical Example......Page 421
8.8.1 The Integral Equations......Page 425
8.8.2 Irregular Frequencies......Page 427
8.9.1 Introduction......Page 430
Beam–Sea Absorbers......Page 431
Omnidirectional Absorbers......Page 432
8.9.2 A Two-Dimensional Beam–Sea Absorber — Salter’s Cam (Duck)......Page 433
8.9.3 Optimum Efficiency of Three-Dimensional Absorbers......Page 439
Omnidirectional Absorbers......Page 442
A Head–Sea Absorber: Hagen–Cockerell Raft......Page 446
8.10 Trapped Modes Near a Mobile Storm Barrier......Page 452
8.10.1 The Two-Gate Mode in an Infinitely Long Barrier......Page 454
8.10.2 Multi-Gate Modes in a Barrier of Finite Length......Page 458
8.11 Drift Forces......Page 462
8.12.1 Radiated Waves Caused by Impulsive Motion of a Floating Body......Page 469
8.12.2 Relation to the Frequency Response......Page 472
8.12.3 Exciting Force Caused by Scattering of Transient Incident Waves......Page 474
8.12.4 Linearized Equations of Transient Motion of a Floating Body......Page 476
Appendix 8.A Derivation of Green’s Function......Page 477
9.2 Linearized Equations of Viscous Flows and the Laminar Boundary Layer......Page 482
9.3 Damping Rate and the Process of Energy Transfer......Page 486
9.3.1 The Entire Fluid......Page 490
9.3.3 Wall Boundary Layer......Page 491
9.3.4 Interior Core......Page 492
9.3.5 The Damping Rate......Page 493
9.4 Damping Rate by a Perturbation Analysis......Page 494
9.5 Details for Standing Waves in a Circular Basin......Page 500
9.6 The Effect of Air on the Damping of Deep Water Waves......Page 505
9.7.1 The Boundary-Layer Structure......Page 510
9.7.2 The Friction Coefficient......Page 513
9.7.3 Bottom Friction on the Damping of Standing Shallow-Water Waves in a Basin......Page 514
Appendix 9.A An Equipartition Theorem......Page 517
10.1 Introduction......Page 566
10.2 Mass Transport Near the Sea Bottom — General Theory......Page 567
10.3 Bottom Mass Transport Under a Long Crested Wave......Page 575
10.4 Bottom Mass Transport Near a Small Structure......Page 584
10.5 Remarks on Induced Streaming Outside the Stokes Boundary Layer......Page 589
10.6 Creeping Flow Theory of Mass Transport in a Channel of Finite Depth......Page 593
10.7 Further References......Page 602
11.1 Introduction......Page 604
11.2 Depth and Time-Averaged Equations for the Mean Motion......Page 606
11.2.1 Averaged Equation of Mass Conservation......Page 607
11.2.2 Averaged Equations of Momentum Conservation......Page 608
Viscous Stress Terms......Page 612
Dynamic Pressure pdh/dxj......Page 613
The Excess Momentum Flux Tensor — The Radiation Stresses for Small Fluctuations......Page 614
11.2.4 Summary of Approximate Averaged Equations......Page 617
11.3 Radiation Stresses in the Shoaling Zone — Small-Amplitude Waves on Constant or Nearly Constant Depth......Page 618
11.4 Long Waves Forced by Radiation Stress of Short Waves......Page 622
11.4.1 Set-Down or Bound Long Wave......Page 623
11.4.2 Parasitic Long Seiches in a Wave Flume......Page 624
11.5.1 Breaking of Standing Waves on a Slope......Page 627
11.5.2 Types of Breakers on Mild Beaches......Page 629
11.5.3 Maximum Wave Height......Page 630
11.6.1 Shoaling Zone: x > xb......Page 632
11.6.2 Surf Zone: x < xb......Page 635
The Friction Factor in the Joint Presence of Waves and Currents......Page 640
Large Angle of Incidence and Strong Current......Page 642
11.7.2 Lateral Turbulent Diffusion S´´xy......Page 645
11.8 Currents Behind an Offshore Breakwater......Page 647
11.8.1 The Wave Field......Page 649
11.8.2 The Mean Motion......Page 654
11.9 Currents Around a Conical Island......Page 659
11.9.2 The Mean Motion......Page 660
11.10 Related Works on Nearshore Currents......Page 665
12.1 Derivation and Classification of Approximate Equations......Page 668
Airy’s Theory for Very Long Waves: ρ  0, E = O(1)......Page 673
Boussinesq Theory: O(E) = O(u2) < 1......Page 674
12.2.1 Analogy to Gas Dynamics......Page 677
12.2.2 Method of Characteristics for One-Dimensional Problems......Page 678
12.2.3 Simple Waves and Constant States......Page 682
12.2.4 Expansion and Compression Waves — Tendency of Breaking......Page 683
12.3 Nonbreaking Waves on a Slope......Page 687
12.3.1 Standing Waves of Finite Amplitude......Page 690
12.3.2 Matching with Deep Water......Page 694
12.3.3 Transient Responses to Initial Inputs......Page 697
12.4 Subharmonic Resonance of Edge Waves......Page 699
12.4.1 Perfect Tuning......Page 700
12.4.2 Effects of Detuning......Page 707
12.5 Dispersive Long Waves of Permanent Form and the Korteweg–De Vries (KdV) Equation......Page 712
12.5.1 Solitary Waves......Page 714
12.5.2 Cnoidal Waves......Page 715
12.5.3 The Korteweg–de Vries (KdV) Equation......Page 722
12.6 Nonlinear Dispersive Standing Waves on a Horizontal Bottom......Page 723
12.7 Evolution of an Initial Pulse......Page 727
12.8 Fission of Solitons by Decreasing Depth......Page 733
12.9 Viscous Damping of Solitary Waves......Page 738
12.10 Remarks on Modeling Large-Scale Tsunamis......Page 745
12.11 Evolution of Periodic Waves Over Constant Depth–Harmonic Generation......Page 751
12.11.1 The Initial Development of Near-Resonant Interaction in Water of Constant Depth......Page 754
12.11.2 Governing Equations for Coupled Harmonics......Page 758
12.11.3 Exact Solution of the Two-Harmonics Problem......Page 760
12.12 Nonlinear Resonance in a Narrow Bay......Page 767
12.13 Solitons Ahead of a Ship Advancing in a River......Page 775
12.14.1 Asymptotic Equation for Uni-Directional Waves......Page 787
12.14.2 Gaussian Correlation Function......Page 792
12.14.3 Computed Results of Soliton Evolution Over a Long Rough Seabed......Page 794
Appendix 12.A Evaluation of Certain Integrals in Section 12.4......Page 795
Appendix 12.B Reduction of an Integral in Section 12.9......Page 796
Appendix 12.C The Square of a Fourier Series......Page 797
Appendix 12.D Details of Random Forcing......Page 799
Appendix 12.E Details of......Page 800
13.1 Introduction......Page 802
13.2.1 Intermediate Depth......Page 804
13.2.2 Deep Water Limit......Page 813
13.3 Uniform Stokes’ Waves......Page 815
13.4 Side-Band Instability of Stokes’ Waves......Page 817
13.5 Permanent Envelopes in Deep Water: Nonlinear Solutions of the Evolution Equation......Page 827
13.6 Transient Evolution of One-Dimensional Wave Envelope on Deep Water......Page 831
13.6.1 Evolution of a Single Pulse......Page 836
13.6.2 Evolution of the Front of a Uniform Wavetrain......Page 841
13.6.3 Periodic Modulation of a Uniform Wavetrain — Evolution Beyond the Initial Stage of Instability......Page 843
13.7 Infragravity Waves Over Slowly Varying Depth......Page 846
13.7.1 Equation for Long Waves Forced by One Train of Short Waves......Page 847
13.7.2 Short-Wave Envelope......Page 851
13.7.3 Mean Sea-Level......Page 854
13.7.4 Free and Bound Infragravity Waves......Page 855
13.8 Infragravity Waves Over Periodic Bars......Page 861
13.9 Remarks on Third-Order Effects of Short Waves Over Slowing Varying Depth......Page 865
13.10 Diffraction of Steady Stokes’ Waves by a Thin Wedge or a Slightly Slanted Breakwater......Page 866
13.11 Soliton Envelopes in the Wake of a Ship......Page 873
13.12 Second-Order Diffraction by a Vertical Cylinder......Page 882
13.12.1 First-Order Solution......Page 883
13.12.2 The Second-Order Problem......Page 884
13.12.3 Second-Order Forcing......Page 885
13.12.5 Response to I......Page 887
13.12.6 Response to S......Page 889
13.12.7 Sample Numerical Results......Page 892
Appendix 13.A Asymptotic Behavior E0 in the Far-Field......Page 893
Appendix 13.B Weak Radiation Condition......Page 894
14 Broad-Banded Nonlinear Surface Waves in the Open Sea......Page 896
14.1 Background......Page 897
14.2 Fourier Formulation......Page 899
14.3 Multiple Time Scales......Page 903
14.4 Conditions for Quartet Resonance......Page 908
14.5 Simple Solutions......Page 911
14.6 Interaction of Two Waves......Page 913
14.7 Interaction of Four Waves (Quartet Interaction)......Page 914
14.7.1 Reduction to One Unknown......Page 915
14.7.2 Solution for Periodic Envelopes......Page 918
14.8 The Cubic Schrodinger Equation......Page 921
14.9 Benjamin–Feir Instability of Stokes Waves......Page 923
14.10 Kinetic Equation of Hasselmann......Page 929
14.11 Extensions......Page 933
Appendix 14.A.1 Fourier Transforms of the Free Surface Conditions......Page 935
Appendix 14.A.2 Surface Properties for Waves of Small Steepness......Page 936
Appendix 14.A.3 Inverting (14.2.9) by Iteration......Page 937
Appendix 14.B Kernels......Page 939
15.1 Introduction......Page 942
15.2 General Initial Boundary-Value Problem......Page 944
15.3 High-Order Spectral (HOS) Method......Page 945
15.3.1 Mathematical Formulation......Page 947
High-order expansions......Page 948
Spectral approach for the boundary-value problem solution......Page 949
Basis function in deep water......Page 950
Basis function in the presence of a varying bottom......Page 951
15.3.2 Numerical Implementation......Page 953
Errors due to truncation in the number of modes N and order M......Page 954
Amplification of round-off and truncation errors......Page 955
Errors due to numerical time integration......Page 956
15.3.4 Relation to Frequency-Domain Perturbation Results......Page 957
15.4.1 Stokes Waves......Page 959
(b) Convergence of the numerical time integration......Page 960
15.4.2 Wave Steepening......Page 962
15.4.3 Modulation of a Stokes Wave Train Due to Type I Instabilities......Page 965
15.4.4 Evolution of a Wave Packet......Page 969
15.4.5 Nonlinear Three-Dimensional Waves Due to a Moving Surface Disturbance......Page 973
15.4.6 Nonlinear Wave Interaction with Ambient Current......Page 980
15.4.7 Generalized Bragg Scattering of Surface Waves by Bottom Ripples......Page 987
Class I Bragg resonance......Page 988
Class II Bragg resonance......Page 993
Class III Bragg resonance......Page 995
15.5.1 Mathematical Formulation......Page 1000
15.5.2 Numerical Implementation......Page 1002
15.5.3 Application to Nonlinear Wave Diffraction by a Submerged Circular Cylinder......Page 1004
Diffracted waves......Page 1005
Oscillating forces......Page 1008
Mean forces......Page 1010
15.6 High-Order Spectral Element (HOSE) Method......Page 1012
15.6.1 Mathematical Formulation......Page 1013
Spectral-element approach for the boundary-value solution......Page 1014
Choice of spectral basis functions......Page 1015
Evaluation of high z-derivatives of  (m)......Page 1016
Determination of the base flow for a nonlinear standing wave......Page 1017
Convergence test......Page 1018
Transition Matrix Method......Page 1020
Instability of Standing Waves in a Circular Tank......Page 1022
(a) Axisymmetric standing waves......Page 1023
(b) Non-axisymmetric waves......Page 1025
15.7 Mixed Euler-Lagrangian Method......Page 1029
15.7.1 Cauchy’s Integral Formulation......Page 1031
15.7.2 Green’s Integral Formulation......Page 1033
Constant panel method (CPM)......Page 1035
Quadratic boundary element method (QBEM)......Page 1037
15.7.3 Numerical Implementation......Page 1040
Far-field closure......Page 1041
Sawtooth instability......Page 1042
Double-node technique for intersections of two surfaces......Page 1043
Intersection updating......Page 1044
15.7.4 Application to Two- and Three-Dimensional Breaking Waves......Page 1045
Two-dimensional overturning waves — direct comparison to experiments......Page 1047
Three-dimensional overturning waves......Page 1055
(a) Basic features......Page 1056
(b) Effect of three-dimensionality......Page 1058
15.7.5 Application to Steep Crescent Waves......Page 1064
Development of L2 crescent waves......Page 1066
Development of L3 crescent waves......Page 1071
15.7.6 Application to Free-Surface Flow Over an Impulsively Started Point Sink......Page 1072
Sub-critical regime: Fr < Fr1 = 0.1924......Page 1075
Sub-critical regime: Fr < Fr2 = 0.1930......Page 1076
Trans-critical regime: 0.1924 = Fr1 < Fr < Fr2 = 0.1930......Page 1079
15.7.7 Application to Plunging Wave Impact on a Vertical Wall......Page 1080
Simulation scheme......Page 1083
Direct comparison with experiments......Page 1085
15.7.8 Application to Nonlinear Wave Interaction with Floating Bodies......Page 1090
First- and second-harmonic forces......Page 1093
Third-harmonic force......Page 1094
References......Page 1100
Subject Index......Page 1130