Theory and Applications of Ocean Surface Waves

The following sections are included:

• Dedication
• Preface to Part 1 of 2017 Edition
• Preface to the Expanded Edition
• Preface to the First Edition
• Acknowledgments
• Contents

Many types of waves involving different physical factors exist in the ocean. As in the elementary problem of a spring-mass system, all waves must be associated with some kind of restoring force. It is therefore convenient to make a crude classification of ocean waves according to the restoring force, as shown in Table 1.1.

Disturbances generated by excitations of finite duration such as submarine earthquakes, landslides, explosions, and so on, produce transient waves. Because of dispersion, the propagation of transient water waves is considerably more complex than many other kinds of waves in nature. To facilitate a clear understanding of the physical consequences of dispersion, we will study in this chapter simple models of source mechanism and ocean depth for which detailed analyses are possible. In Sections 2.1 and 2.2 we will study the so-called Cauchy-Poisson problem of waves due to an impulsive source of some kind, with particular emphasis given to the behavior away from the source. Many practical issues concerning earthquake-induced waves in nature are omitted for simplicity. In Sections 2.3 and 2.4 the role of dispersion in slowly modulated wave groups will be investigated. The mathematical technique of multiple-scale expansions is introduced to arrive very directly at the asymptotic result for large time and to prepare the ground for later inclusion of nonlinearity.

When a train of plane monochromatic waves enters a zone of slowly varying depth, the wavenumber can be expected to change with depth in accordance with Eq. (1.4.8), Chapter One, resulting in a gradual change in the phase velocity. In general, the spacings between equal phase lines and the amplitude of crests and troughs will vary from place to place. Similar changes can also occur for waves riding on a current whose intensity varies in horizontal directions. These phenomena, which are related mainly to the variation in phase velocity, are of course well known in optics and acoustics and are called refraction. In this chapter, we develop an approximate ray theory (or geometrical optics theory) for the effects of varying depth (Sections 3.1–3.4) and of varying current (Sections 3.6 and 3.7) on the propagation of infinitesimal waves. The evolution equations will be deduced by the so-called WKB method which is a special version of the multiple-scales method. While these equations are normally solved numerically for practical problems, we shall extract from them physical insights through a variety of analytical examples. A brief discussion is also made, for varying depth only, on the local remedy needed when the ray approximation fails. In a numerical treatment of natural topography, this failure can be overcome more straightforwardly by incorporating diffraction in the so-called mild-slope equation, which we derive in Section 3.5. Other more mathematical aspects not treated here can be found in the excellent surveys of Meyer (1979a) for varying depth and Peregrine (1976) for varying current.

When a propagating wave enters a region where the fractional change of depth within a wavelength is significant, the phenomenon of scattering occurs in which reflection becomes appreciable. The simple ray theory which ignores reflection is no longer adequate. Before discussing the scattering of dispersive waves, let us examine similar problems for long waves in shallow water where dispersion is not important. For mathematical simplicity we shall deal largely with discontinuous depth variation. An interesting aspect of varying depth is wave trapping, that is, the phenomenon whereby waves are effectively confined in a part of the ocean. This topic has been discussed for short waves in Chapter Three. Long-wave trapping problems associated with sloping beaches, continental shelves, and oceanic ridges will be discussed here by way of several simple models such as a rectangular sill or shelf, and a uniformly sloping beach, and so on. In addition, several general aspects of the scattering matrix will be studied. Since only a few continuous variations can be solved analytically with a good deal of algebra, approximate or numerical methods are needed and will be discussed at the end of this chapter.

The following sections are included:

• Introduction
• Formulation for Harbor Oscillation Problems
• Natural Modes in a Closed Basin of Simple Form and Constant Depth
  • A Rectangular Basin
  • A Circular Basin
• Concept of Radiation Damping — A Model Example
• Diffraction Through a Narrow Gap
• Scattering by a Long and Narrow Canal or a Bay
  • General Solution
  • An Open Narrow Bay
• A Rectangular Harbor with a Narrow Entrance
  • Solution by Matched Asymptotic Expansions
  • Resonant Spectrum and Response for Non-Helmholtz Modes
  • The Helmholtz Mode
  • Numerical Results and Experiments
  • Effects of Finite Entry Channel
• The Effect of Protruding Breakwater
  • Representation of Solution
  • Reduction to an Integral Equation
  • Approximate Solution by Variational Method
  • Numerical Results
• A Harbor with Coupled Basins
• A Numerical Method for Harbors of Complex Geometry
• Harbor Response to Transient Incident Wave
• Appendix 5.A The Source Function for a Rectangular Basin
• Appendix 5.B Summation of the $\tilde{G}$ Series
• Appendix 5.C Proof of a Variational Principle
• Appendix 5.D Evaluation of an Integral

In the idealized theory of wave scattering in an inviscid fluid, it is usually assumed that the fluid flow is always tangential to the solid boundary of a wall or a structure. In reality, however, an adverse pressure gradient and viscosity can decelerate the fluid near a sharp convex corner, forcing the flow to separate and to form eddies of high vorticity hence causing significant energy loss. This natural tendency is the basis of perforated panels on room walls to absorb sound energy. Jarlan (1965) introduced this idea to coastal engineering and patented the design of a caisson breakwater which has a perforated front wall but a solid back wall. Dissipation is enhanced by the water jets falling through the holes when the surface elevations on both sides differ. Breakwaters of similar types have been constructed at Baie Comeau Harbor and Chandler Harbor in Quebec, Canada, and at Roscoff Harbor in France (Richey and Sollitt, 1969). A more recent and dramatic example is the North Sea Ekofisk oil storage tank which is surrounded by a circular perforated breakwater with a diameter of approximately 92 m in water of 70 m (Gerwick and Hognestad, 1973). A breakwater consisting of a row of circular piles of 2 m diameter spaced with 0.5 cm gaps has been in use in Osaka Harbor, Japan (Hayashi, Kano and Shirai, 1966)

A proper account for bathymetric variations is an important task in constructing wave-forcasting models for coastal waters. For slowly varying depth, the ray (or WKB) and mild-slope approximations are useful tools to deal with refraction and diffraction in many practical situations. For abrupt depth changes involving one or a few obstacles, effective numerical schemes have been well developed to give the details of wave scattering near the obstacles. In cases where the bottom irregularities extend over many wavelengths, i.e., tens or hundreds of kilometers, however, one must deal with multiple scattering. In nature both the waves and the depth variations can be highly irregular. A comprehensive wave model should account for the scattering of random waves by randomly irregular bathymetry, in addtion to viscous dissipation. Early works on this topic are due to Hasselmann (1966) and Long (1973)

The following sections are included:

• Introduction
• Linearized Equations of Motion for a Constrained Floating Body
  • The Kinematic Condition
  • Conservation of Linear Momentum
  • Conservation of Angular Momentum
  • Summary of Dynamic Equations for a Floating Body in Matrix Form
• Simple Harmonic Motion
  • Decomposition into Diffraction and Radiation Problems
  • Exciting and Restoring Forces; Added Mass and Radiation Damping for a Body of Arbitrary Shape
• Formal Representations of Velocity Potential when h = Constant
  • Away from the Body
  • The Entire Fluid Domain
• Scattering by a Vertical Cylinder with Circular Cross Section
• General Identities for the Diffraction and Radiation of Simple Harmonic Waves
  • Relations between Two Radiation Problems and their Consequences
  • Relations between Two Diffraction Problems
  • One Diffraction Problem and One Radiation Problem
• Numerical Solution by Hybrid Element Method
  • The Variational Formulation
  • The Approximate Solution
  • A Numerical Example
• Remarks on the Numerical Methods by Integral Equations
  • The Integral Equations
  • Irregular Frequencies
• Wave Power Extraction
  • Introduction
  • Optimum Efficiency of Three-Dimensional Absorbers
  • A Two-Dimensional Beam–Sea Absorber — Salter’s Cam (Duck)
  • Circular Buoy Converter
  • Oscillating Water Column (OWC)
• Trapped Modes Near a Mobile Storm Barrier
  • The Two-Gate Mode in an Infinitely Long Barrier
  • Multi-Gate Modes in a Barrier of Finite Length
• Drift Forces
• Principles of Calculating the Transient Motion of a Floating Body
  • Radiated Waves Caused by Impulsive Motion of a Floating Body
  • Relation to the Frequency Response
  • Exciting Force Caused by Scattering of Transient Incident Waves
  • Linearized Equations of Transient Motion of a Floating Body
• Appendix 8.A Derivation of Green’s Function
• Appendix 8.B Radiation Problem for a Heaving Buoy
• Appendix 8.C Radiation Problem for OWC

The following sections are included:

• Introduction
• Linearized Equations of Viscous Flows and the Laminar Boundary Layer
• Damping Rate and the Process of Energy Transfer
  • The Entire Fluid
  • Meniscus Boundary Layer
  • Wall Boundary Layer
  • Interior Core
  • The Damping Rate
• Damping Rate by a Perturbation Analysis
• Details for Standing Waves in a Circular Basin
• The Effect of Air on the Damping of Deep Water Waves
• The Turbulent Boundary Layer Near a Rough Bottom
  • The Boundary-Layer Structure
  • The Friction Coefficient
  • Bottom Friction on the Damping of Standing Shallow-Water Waves in a Basin
• Waves Through a Model Marine Forest
  • A Simple Model of Interstitial Turbulence
  • Governing Equations for the Interstitial Flow
  • Sinusoidal Waves
  • Transient Waves
  • Further Development
• Appendix 9.A An Equipartition Theorem

The following sections are included:

• Introduction
• Mass Transport Near the Sea Bottom — General Theory
• Bottom Mass Transport Under a Long Crested Wave
• Bottom Mass Transport Near a Small Structure
• Remarks on Induced Streaming Outside the Stokes Boundary Layer
• Creeping Flow Theory of Mass Transport in a Channel of Finite Depth
• Further References

The following sections are included:

• Introduction
• Depth and Time-Averaged Equations for the Mean Motion
  • Averaged Equation of Mass Conservation
  • Averaged Equations of Momentum Conservation
  • Some Preliminary Simplifications
  • Summary of Approximate Averaged Equations
• Radiation Stresses in the Shoaling Zone — Small-Amplitude Waves on Constant or Nearly Constant Depth
• Long Waves Forced by Radiation Stress of Short Waves
  • Set-Down or Bound Long Wave
  • Parasitic Long Seiches in a Wave Flume
• Empirical Knowledge of Breaking Waves
  • Breaking of Standing Waves on a Slope
  • Types of Breakers on Mild Beaches
  • Maximum Wave Height
• The Structure of a Uniform Longshore Current on a Plane Beach
  • Shoaling Zone: x > x_b
  • Surf Zone: x < x_b
• Other Empirical Hypotheses or Improvements
  • Bottom Friction
  • Lateral Turbulent Diffusion $S_{xy}^{″}$
• Currents Behind an Offshore Breakwater
  • The Wave Field
  • The Mean Motion
• Currents Around a Conical Island
  • The Wave Field
  • The Mean Motion
• Related Works on Nearshore Currents

The following sections are included:

• Derivation and Classification of Approximate Equations
• Nondispersive Waves in Water of Constant Depth
  • Analogy to Gas Dynamics
  • Method of Characteristics for One-Dimensional Problems
  • Simple Waves and Constant States
  • Expansion and Compression Waves — Tendency of Breaking
• Nonbreaking Waves on a Slope
  • Standing Waves of Finite Amplitude
  • Matching with Deep Water
  • Transient Responses to Initial Inputs
• Subharmonic Resonance of Edge Waves
  • Perfect Tuning
  • Effects of Detuning
• Dispersive Long Waves of Permanent Form and the Korteweg–De Vries (KdV) Equation
  • Solitary Waves
  • Cnoidal Waves
  • The Korteweg–de Vries (KdV) Equation
• Nonlinear Dispersive Standing Waves on a Horizontal Bottom
• Evolution of an Initial Pulse
• Fission of Solitons by Decreasing Depth
• Viscous Damping of Solitary Waves
• Remarks on Modeling Large-Scale Tsunamis
• Evolution of Periodic Waves Over Constant Depth–Harmonic Generation
  • The Initial Development of Near-Resonant Interaction in Water of Constant Depth
  • Governing Equations for Coupled Harmonics
  • Exact Solution of the Two-Harmonics Problem
• Nonlinear Resonance in a Narrow Bay
• Solitons Ahead of a Ship Advancing in a River
• Localization of Solitons Over a Randomly Rough Seabed
  • Asymptotic Equation for Uni-Directional Waves
  • Gaussian Correlation Function
  • Computed Results of Soliton Evolution Over a Long Rough Seabed
• Appendix 12.A Evaluation of Certain Integrals in Section 12.4
• Appendix 12.B Reduction of an Integral in Section 12.9
• Appendix 12.C The Square of a Fourier Series
• Appendix 12.D Details of Random Forcing
• Appendix 12.E Details of β

The following sections are included:

• Introduction
• Evolution Equations for Slowly Modulated Weakly Nonlinear Waves Over Horizontal Seabed
  • Intermediate Depth
  • Deep Water Limit
• Uniform Stokes’ Waves
• Side-Band Instability of Stokes’ Waves
• Permanent Envelopes in Deep Water: Nonlinear Solutions of the Evolution Equation.
• Transient Evolution of One-Dimensional Wave Envelope on Deep Water
  • Evolution of a Single Pulse
  • Evolution of the Front of a Uniform Wavetrain
  • Periodic Modulation of a Uniform Wavetrain — Evolution Beyond the Initial Stage of Instability
  • Rogue Waves
• Infragravity Waves Over Slowly Varying Depth
  • Equation for Long Waves Forced by One Train of Short Waves
  • Short-Wave Envelope
  • Mean Sea-Level
  • Free and Bound Infragravity Waves
• Infragravity Waves Over Periodic Bars
• Remarks on Third-Order Effects of Short Waves Over Slowing Varying Depth
• Diffraction of Steady Stokes’ Waves by a Thin Wedge or a Slightly Slanted Breakwater
• Soliton Envelopes in the Wake of a Ship
• Second-Order Diffraction by a Vertical Cylinder
  • First-Order Solution
  • The Second-Order Problem
  • Second-Order Forcing
  • Second-Order Boundary-Value Problems
  • Response to $\mathcal{I}$
  • Response to $\mathcal{S}$
  • Sample Numerical Results
• Appendix 13.A Asymptotic Behavior ξ₀ in the Far-Field
• Appendix 13.B Weak Radiation Condition

Wind-generated waves on the open ocean surface are usually too complex to be described by one regular slowly varying wavetrain, and require more elaborate spectral representations to acount for sea states with a broad band of wavelengths and directions. For infinitesimal waves, there are spectral theories to describe the random sea surface based on linear superposition of many simple wavetrains. The amplitude-spectrum (which contains the amplitudes and phases), or sometimes only the energy spectrum, can be used to convey information about the complicated geometry and kinematics of the ocean surface. Statistical stationarity is often assumed. A detailed description of such classical treatments can be found in Ochi (1998)…

The following sections are included:

• Introduction
• General Initial Boundary-Value Problem
• High-Order Spectral (HOS) Method
  • Mathematical Formulation
  • Numerical Implementation
  • Error Considerations
  • Relation to Frequency-Domain Perturbation Results
• Applications of HOS to Nonlinear Wave-Wave, Wave-Current, and Wave-Bottom Interactions
  • Stokes Waves
  • Wave Steepening
  • Modulation of a Stokes Wave Train Due to Type I Instabilities
  • Evolution of a Wave Packet
  • Nonlinear Three-Dimensional Waves Due to a Moving Surface Disturbance
  • Nonlinear Wave Interaction with Ambient Current
  • Generalized Bragg Scattering of Surface Waves by Bottom Ripples
• HOS Method for Nonlinear Wave Interaction with Submerged Bodies
  • Mathematical Formulation
  • Numerical Implementation
  • Application to Nonlinear Wave Diffraction by A Submerged Circular Cylinder
• High-Order Spectral Element (HOSE) Method
  • Mathematical Formulation
  • Numerical Implementation
  • Application of HOSE to the Study of Stability of Standing Waves in a Circular Tank
• Nonlinear Wave-Field Evolution by Large-Scale HOS Computations
  • Direct Wave-Field Simulation Using HOS
  • Spectral Evolution and Nonlinear Wave Statistics
  • Probability of Rogue Wave Occurrence
• Mixed Euler-Lagrangian Method
  • Cauchy’s Integral Formulation
  • Green’s Integral Formulation
  • Numerical Implementation
  • Application to Two-and Three-Dimensional Breaking Waves
  • Application to Steep Crescent Waves
  • Application to Free-Surface Flow Over an Impulsively Started Point Sink
  • Application to Plunging Wave Impact on a Vertical Wall
  • Application to Nonlinear Wave Interaction with Floating Bodies
• PFFT-Accelerated Boundary Element Methods
  • Mathematical Formulation
  • Determination of Interpolation Function $H\left(\overrightarrow{\xi }\right)$
  • Numerical Implementation
  • Accuracy and Efficiency of the PFFT Algorithm
  • Comparison of PFFT-QBEM with PFFT-CPM
  • PFFT-BEMs for Fully-Nonlinear Initial Boundary-Value Problems

The following sections are included:

• References
• Subject Index