The propagation of long waves over an undulating bed

Abstract

The propagation of long surface water waves over an undulating bed is governed by a Mathieu equation. Instabilities of solutions of this equation indicate resonant interactions associated with the strong reflection of the surface waves by the bedforms. The strongest interaction occurs when the bed wavenumber (l) is approximately twice the surface wavenumber (k). This interaction is known in solid-state physics as Bragg reflection. In this paper the interaction of incident surface waves with a patch of ripples on an otherwise flat bed is considered. For the general nonresonant case (l≠2k), a perturbation solution of the Mathieu equation is obtained. This solution indicates that the reflection coefficient R, defined as the ratio of the magnitudes of the reflected and incident waves, is both oscillatory in the ratio of the overall length of the ripple patch to the surface wavelength and also resonant in the vicinity of l=2k. The condition of exact resonance (l=2k) is studied separately and the solution obtained predicts that R tends uniformly to unity as the number of ripples (having l=2k) is increased. The present approach is simpler and more readily applicable than the use of previous more complicated methods based on potential theory. It also provides some insight into the higher-order resonant interactions between waves on a free surface and an undulating bed. The results are particularly important in oceanography, in connection with the reflection of incident surface waves by sandbars off beaches.