The deformation of steep surface waves on water - I. A numerical method of computation

Abstract

Plunging breakers are beyond the reach of all known analytical approximations. Previous numerical computations have succeeded only in integrating the equations of motion up to the instant when the surface becomes vertical. In this paper we present a new method for following the time-history of space-periodic irrotational surface waves. The only independent variables are the coordinates and velocity potential of marked particles at the free surface. At each time-step an integral equation is solved for the new normal component of velocity. The method is faster and more accurate than previous methods based on a two dimensional grid. It has also the advantage that the marked particles become concentrated near regions of sharp curvature. Viscosity and surface tension are both neglected. The method is tested on a free, steady wave of finite amplitude, and is found to give excellent agreement with independent calculations based on Stokes's series. It is then applied to unsteady waves, produced by initially applying an asymmetric distribution of pressure to a symmetric, progressive wave. The freely running wave then steepens and overturns. It is demonstrated that the surface remains rounded till well after the over-turning takes place.