Reflection of a high-amplitude solitary wave at a vertical wall

(Received 29 February 1996 and in revised form 25 October 1996) The collision of a solitary wave, travelling over a horizontal bed, with a vertical wall is investigated using a boundary-integral method to compute the potential fluid flow described by the Euler equations. We concentrate on reporting new results for that part of the motion when the wave is near the wall. The wall residence time, i.e. the time the wave crest remains attached to the wall, is introduced. It is shown that the wall residence time provides an unambiguous characterization of the phase shift incurred during reflection for waves of both small and large amplitude. Numerically computed attachment and detachment times and amplitudes are compared with asymptotic formulae developed using the perturbation results of Su & Mirie (1980). Other features of the flow, including the maximum run-up and the instantaneous wall force, are also presented. The numerically determined residence times are in good agreement with measurements taken from a cine film of solitary wave reflection experiments conducted by Maxworthy (1976). In this paper we consider the reflection at a vertical wall of a solitary wave, using a boundary-integral numerical code, a perturbation method, and re-analysis of cine film taken during the study by Maxworthy (1976). Most attention is given to that part of the motion during which the point of greatest free-surface elevation (the crest) lies close to the wall. The problem of solitary wave reflection has received attention in studies of the interaction between solitary waves, of which the head-on collision of two equal waves is a special case equivalent to that studied here. When weakly nonlinear solitary waves overtake or collide with one another there may be a spatial phase shift but no loss of energy from either wave once sucient time has passed for the two waves to separate; this is the feature by which a soliton is defined (Zabusky & Kruskal 1965). Recent studies have shown that large-amplitude solitary water waves do not behave like solitons. A long time after the collision between two equal waves there is a loss to secondary waves, and a reduction in wave speed. The reduced wave speed necessarily produces a spatial phase shift that increases without bound as t U¢ . It is useful to briefly review certain aspects of the phenomena we wish to study, some of which bear on the interpretation of results to be presented later. In what follows, f(x, t) is the free-surface elevation about the quiescent fluid level and e fl a}h is the dimensionless solitary wave amplitude, a being the amplitude of the incident wave travelling on a fluid of constant still-water depth h. Byatt-Smith (1971) investigated the interaction between two weakly nonlinear solitary waves travelling in opposite