D. H. Peregrine

Long waves on a beach

Equations of motion are derived for long waves in water of varying depth. The equations are for small amplitude waves, but do include non-linear terms. They correspond to the Boussinesq equations for water of constant depth. Solutions have been calculated numerically for a solitary wave on a beach of uniform slope. These solutions include a reflected wave, which is also derived analytically by using the linearized long-wave equations.

Calculations of the development of an undular bore

If a long wave of elevation travels in shallow water it steepens and forms a bore. The bore is undular if the change in surface elevation of the wave is less than 0·28 of the original depth of water. This paper describes the growth of an undular bore from a long wave which forms a gentle transition between a uniform flow and still water. A physical account of its development is followed by the results of numerical calculations. These use finite-difference approximations to the partial differential equations of motion. The equations of motion are of the same order of approximation as is necessary to derive the solitary wave. The results are in general agreement with the available experimental measurements.

Interaction of Water Waves and Currents

Publisher Summary This chapter discusses the varied physical circumstances in which interactions among water waves and currents occur. Different mathematical approaches, relevant observations, and experiments that are applicable to all or some of these physical circumstances are described. The emphasis is on waves and their interaction with preexisting currents rather than on wave-generated currents. Common simplifying assumption is that the waves are of sufficiently small amplitude for the free-surface boundary conditions to be linearized and evaluated at, or close to, the mean free surface. Most progress can be made in this subject with such a constraint, but wherever possible, finite-amplitude effects are discussed. Unlike some other common forms of wave motion, water waves involve water motion varying with direction perpendicular to the space in which they propagate. The chapter concludes on the interaction of waves generated by a ship with the flow around it.

Surf and run-up on a beach: a uniform bore

A numerical solution is obtained to describe the behaviour of a uniform bore over a sloping beach and the subsequent run-up and back-wash. The results exhibit features which have only previously been described in a qualitative manner. These include the formation’ of a landward-facing bore in the back-wash. A comprehensive set of results are presented for a typical initial subcritical bore height ratio.

Steep, steady surface waves on water of finite depth with constant vorticity

2-D steady surface waves on a shearing flow are computed for the special case where the flow has uniform vorticity, i.e. in the absence of waves the velocity varies linearly with height. A boundary-integral method is used in the computation which is similar to that of Simmen & Saffman (1985) who describe such waves on deep water. Particular attention is given to the effects of finite depth with descriptions of waves of limiting steepness, waves with eddies beneath their crests and extremely high waves on high-speed flows. Many qualitative features of these waves are relevant to steep waves propagating in shallow water, or on a strong wind-induced drift current. An important practical point in the interpretation of wave measurements of wind driven waves is that mean kinetic energy and potential energy densities are unequal even for infinitesimal waves. This may mean that wave energy spectra deduced from surface-elevation measurements in the conventional way may sometimes be misleading.

The dynamics of strong turbulence at free surfaces. Part 1. Description

A free surface may be deformed by fluid motions; such deformation may lead to surface roughness, breakup, or disintegration. This paper describes the wide range of free-surface deformations that occur when there is turbulence at the surface, and focuses on turbulence in the denser, liquid, medium. This turbulence may be generated at the surface as in breaking water waves, or may reach the surface from other sources such as bed boundary layers or submerged jets. The discussion is structured by consideration of the stabilizing influences of gravity and surface tension against the disrupting effect of the turbulent kinetic energy. This leads to a two-parameter description of the surface behaviour which gives a framework for further experimental and theoretical studies. Much of the discussion is necessarily heuristic, and is often limited by a lack of appropriate experimental observations. It is intended that such experiments be stimulated, to test the value or otherwise of our two-parameter description.

Unsteady water wave modulations: fully nonlinear solutions and comparison with the nonlinear Schrödinger equation

The time evolution of a uniform wave train with a small modulation which grows is computed with a fully nonlinear irrotational flow solver. Many numerical runs have been performed varying the initial steepness of the wave train and the number of waves in the imposed modulation. It is observed that the energy becomes focussed into a short group of steep waves which either contains a wave which becomes too steep and therefore breaks or otherwise having reached a maximum modulation then recedes until an almost regular wave train is recovered. This latter case typically occurs over a few hundred time periods. We have also carried out some much longer computations, over several thousands of time periods in which several steep wave events occur. Several features of these modulations are consistent with analytic solutions for modulations ! ;

Swash overtopping a truncated plane beach

Swash on a plane beach is modelled by using a solution of the shallow-water equations due to Shen & Meyer (1963). The equations are used in a form appropriate for a plane at a finite angle to the horizontal. The beach is cut-off at a level below that of the maximum run-up, and the water is taken to fall freely over the end of the beach. An explicit solution is found which permits evaluation of the overtopping flow and total volume for one swash event.

Pressure-impulse theory for liquid impact problems

A mathematical model is presented for the high pressures and sudden velocity changes which may occur in the impact between a region of incompressible liquid and either a solid surface or a second liquid region. The theory rests upon the well-known idea of pressure impulse, for the sudden initiation of fluid motion in incompressible fluids. We consider the impulsive pressure field which occurs when a moving fluid region collides with a fixed target, such as when an ocean wave strikes a sea wall. The boundary conditions are given for modelling liquid-solid and liquid-liquid impact problems. For a given fluid domain, and a given velocity field just before impact, the theory gives information on the peak pressure distribution, and the velocity after impact. Solutions for problems in simple domains are presented, which give insight into the peak pressures exerted by a wave breaking against a sea wall, and a wave impacting in a confined space. An example of liquid-liquid impact is also examined. Results of particular interest include a relative insensitivity to the shape of the incident wave, and an increased pressure impulse when impact occurs in a confined space. The theory predicts that energy is lost from the bulk fluid motion and we suggest that this energy can be transferred to a thin jet of liquid which is projected away from the impact region.

Computations of overturning waves

The numerical method of Longuet-Higgins & Cokelet (1976), for waves on deep water, is extended to account for a horizontal bottom contour, and used to investigate breaking waves in water of finite depth. It is demonstrated that a variety of overturning motions may be generated, ranging from the projection of a small-scale jet at the wave crest (of the type that might initiate a spilling breaker) to large-scale plunging breakers involving a significant portion of the wave. Although there seems to be a continuous transition between these wave types, a remarkable similarity is noticed in the overturning regions of many of the waves. Three high-resolution computations are also discussed. The results are presented in the form of interrelated space-, velocity- and acceleration-plane plots which enable the time evolution of individual fluid particles to be followed. These computations should be found useful for the testing of analytical theories, and may also be applied, for example, to studies of slamming forces on shipping and coastal structures.