On the excitation of long nonlinear water waves by a moving pressure distribution. Part 2. Three-dimensional effects

Abstract

The three-dimensional wave pattern generated by a moving pressure distribution of finite extent acting on the surface of water of depth h is studied. It is shown that, when the pressure distribution travels at a speed near the linear-long-wave speed, the response is governed by a forced nonlinear Kadomtsev-Petviashvili (KP) equation, which describes a balance between linear dispersive, nonlinear and three-dimensional effects. It is deduced that, in a channel of finite width 2w, three-dimensional effects are negligible if w [Lt ] h2/a, a being a typical wave amplitude; in such a case the governing equation reduces to the forced Korteweg-de Vries equation derived in previous studies. For aw/h2 = O(1), however, three-dimensional effects are important; numerical calculations based on the KP equation indicate that a series of straight-crested solitons are radiated periodically ahead of the source and a three-dimensional wave pattern forms behind. The predicted dependencies on channel width of soliton amplitude and period of soliton formation compare favourably with the experimental results of Ertekin, Webster & Wehausen (1984). In a channel for which aw/h2 [Gt ] 1, three-dimensional, unsteady disturbances appear-ahead of the pressure distribution.