Reflections of internal gravity waves off bumpy surfaces

Abstract

Waves in a density stratified fluid, termed internal gravity waves, reflected by a sloping plane make an angle with the horizontal equal to the angle between the incident wave and the horizontal. We add a perturbation to the plane small in size when compared to the amplitude of the wave motion. An asymptotic expansion is used and problems are derived up to two orders O(1) and O(ε). The leading order problem is the same as the reflection problem by a flat plane. The problem on O(ε) is a radiation problem that we solve using Fourier transforms.