Slow Interactions of Gravity Waves and a Corrugated Sea Bed

Abstract

The effect of a corrugated sea bed on the linear theory of gravity water waves is considered. By straining the time variable, a perturbation solution is found in ε (the ratio of corrugation amplitude to mean water depth), through first order, for a wave system that is arbitrarily oriented with respect to the corrugations. That solution breaks down when the wave number k normal to the corrugation is a half-integer multiple of the wave number 2ω of the corrugations, i.e., when k = (ω, 2ω, .... Of these singularities, the first (k = ω) appears at the first order. To obtain a uniformly valid zeroth-order solution we include a zeroth-order reflected wave system, and obtain an alternation between incident and reflected waves on a time scale of order 0(ε−1). As representative of the other singular wave numbers, we consider k = 3ω, which singularity appears at the third order, and obtain a uniformly valid solution through second order (for the shallow water limit). Nonlinear effects are considered to the extent of noting that the zeroth-order linear and nonlinear results are identical, even for the first singular wave number k = ω.