Surface wave damping by a Robin boundary condition at a permeable seabed

We investigate the damping of inviscid surface gravity waves when they propagate over a permeable seabed. Traditionally, this problem is solved by considering the wave motion in the upper water layer interacting with the Darcy flow in the porous bottom layer through matching of the solutions at the permeable interface. The novel approach in this study is that we describe the interaction between the upper fluid layer and the permeable bottom layer by the application of a Robin boundary condition. By varying the magnitude of the Robin parameter $R$, we can model the bottom structure of the fluid layer from rigid to completely permeable. In the case when the bottom is a porous medium where Darcy’s law applies, or composed by densely packed vertical Hele-Shaw cells, $R$ is small, and can by determined by comparison with analytical results for a two-layer structure. In this case the wave damping is small. For larger values of $R$, the damping increases, and for very large $R$, the wave is almost critically damped. For the nonlinear transport in spatially damped shallow-water waves, increasing bottom permeability (larger $R$), reduces the magnitude of the horizontal Stokes drift velocity, while the drift profile tends to become parabolic with height. The vertical Stokes drift in the limit of large permeability is linear with height, with a magnitude that is larger than the horizontal drift. It is suggested that the implementation of a permeable bottom bed in some cases could prevent shorelines from damaging erosion by incoming surface waves.