On gravity-wave scattering by non-secular changes in depth

Abstract

The reflection of a straight-crested gravity wave by a non-secular perturbation h₁(x) in depth relative to an otherwise flat bottom of depth h₀ is calculated through an expansion in ε∝h₁/h₀. Explicit results are developed up to second order for the sinusoidal patch h₁=−bsin(mπx/l), 0<x<l, and reduced for Bragg resonance. Trapped modes are absent at first order but appear at second order and contribute O(ε²)/O(ε³) to the maximum (Bragg-resonant) reflection coefficient for odd/even m. A third-order approximation that includes the dominant contributions of the third-order components of the resonant peak of the reflection coefficient for large m, but neglects the trapped modes, predicts resonant peaks that agree with the values measured by Davies & Heathershaw (1984).