Surface-wave propagation over sinusoidally varying topography

TLDR

Surface waves in finite‑depth water can be scattered by undulating bottom topography; this study focuses on long‑crested waves incident on a patch of regular bottom ripples. The study investigates how much incident wave energy is reflected by the ripple patch, characterizes the near‑field wave pattern, and explores the implications for sediment transport on erodible beds. Using linear perturbation theory, the authors derive an oscillatory reflection coefficient that depends on the ratio of patch length to surface wavelength and on the surface‑to‑bed wavelength ratio, and validate the predictions against extensive laboratory wave‑tank data. They find a Bragg resonance that strongly reflects incident energy, observe that the standing wave upstream evolves into a progressive transmitted wave downstream, and report that laboratory measurements confirm the theoretical predictions.

Abstract

Surface waves travelling in water of finite depth may be scattered by a region of undulating bottom topography. The present study is concerned with the idealized two-dimensional situation in which long-crested surface waves are incident upon a patch of long-crested regular bottom ripples. The principal question examined concerns the amount of incident wave energy that is reflected by the ripple patch. Linear perturbation theory is used to show that the reflection coefficient is both oscillatory in the quotient of the length of the patch and the surface wavelength, and also strongly dependent upon the quotient of the surface and bed wavelengths. In particular, there is a Bragg resonance between the surface waves and the ripples, which is associated with the reflection of incident wave energy. A secondary question concerns the nature of the wave field in the immediate vicinity of the ripple patch. In resonant cases, it is shown how the partially standing wave on the upwave side of the ripple patch gives way, in an almost linear manner over the patch itself, to a progressive transmitted wave on the downwave side. The theoretical predictions are compared with an extensive set of laboratory observations made in a wave tank. Comparisons relating both to the reflection coefficient, and also to the wave field over the ripple patch, are shown to give consistently good agreement. Finally, the implications of the results for sediment transport on an erodible bed are examined.