Bragg scattering of gravity waves by a sea bed of varying depth in the presence of uniform current covered by a floating membrane

Abstract

We present an asymptotic approach to solving the water wave scattering by undulating bottom topography in the presence of uniform currents where a flexible membrane covers the upper surface. The interest in this problem lies in the development of asymptotic solutions using the Fourier transform under the action of uniform currents. The method allows the physical processes involved in the sea-bed topography and membrane-covered surface and wave interactions to be studied. In particular, we identify the existence of Bragg resonance between gravity waves and the bottom ripples, which are associated with the reflection of incident wave energy. We consider the impacts of uniform current, and we highlight the central role of the asymptotic expansion method in the evolution of the response of current. For depth Froude number in the range of 0.4–0.7, the effects of bottom topography on Bragg resonance dominate. The current shifts the frequency of the most reflected wave components, and wave action conservation results in amplified reflected wave energies for the following currents. The theory developed in the frequency domain is illustrated in the time domain using discrete Fourier transform with the Joint North Sea Wave Observation Project (JONSWAP) spectrum [Hasselmann et al., J. Phys. Oceanogr. 10, 1264 (1980)] to analyze wave propagation through the whole system.