Random waves in water of variable depth

Abstract

The propagation of linear random gravity waves in water of variable depth is considered, to determine the explicit effect of an uneven topography on the free surface, its spectral representation, and probability structure. The method of analysis is based on a perturbation series expansion of governing equations of wave motion in powers of a small parameter which characterizes the slow variation of the still water depth over a dominant wave length. Systematic solutions of the resulting perturbation equations lead us first to the conclusion that the description of the wave field reduces to an orthogonal spectral representation in a slowly varying medium just as in homogeneous media. Subsequently, solutions to first order in depth nonuniformities are utilized to explore how the free surface, its spectrum, and general probability structure are modified spatially, as waves propagate from a homogeneous region into an inhomogeneous regime with variable water depth.