Note on the reflexion of water waves at a nearly vertical cliff in the presence of surface tension

Abstract

In this note the two-dimensional problem of incoming surface waves against an approximately vertical wall or cliff in water of infinite depth is examined, using velocity potential formulation and linearized boundary-value problem theory for time-harmonic motion. The cliff has arbitrary profile but for simplicity is taken to be vertical at the free surface. The approximate first-order solution is determined subject to a dynamical edge condition apposite to the presence of surface tension, and contains partially reflected outgoing waves. The solution is obtained by perturbation theory in a form involving known unperturbed and first-order correction potentials that is applicable also to water of finite constant depth. The motivation for the note is to point out a correction in principle to the results of a recent investigation for a specific profile, in which reflexion is ignored and another error made in obtaining a first-order solution by a method that is restricted to infinite depth.