Scattering of gravity-capillary waves on a bottom step

Abstract

Under the linear approximation, we study the scattering of gravity-capillary waves on a bottom step. The boundary conditions can be satisfied by considering a countable number of evanescent modes that are localized near the bottom step. Such modes have certain specific features compared to similar modes for pure gravity waves. In particular, there exists an extra mode. When a numerical solution is calculated, this extra mode allows us to match in the fluid region and impose the additional condition at the surface, which arises for capillary gravity waves (and not for pure gravity waves). The approximate formulas are suggested for the transformation coefficients. By means of numerical calculations, we find the reflection and transmission coefficients of traveling waves and compare the results with the predictions of the approximate formulas. The approximate formulas are shown to agree well with the numerical data. Both numerical and approximate results agree with the energy flux conservation. We also study the wave transformation on the rectangular underwater bump and trench, which are straightforward extensions of our step calculations. We calculate the transformation coefficients numerically and illustrate the transformation by time-dependent results, which are also shown as animations in the supplementary material.