Water waves over a muddy seabed

Abstract

A generalized viscoelastic model is used to describe the rheological properties of mud and is fitted to the available experimental data, so that its constitutive coefficients are just material properties independent of the frequency of the external forcing. We integrate this model into a perturbation analysis to solve the interaction between a thin layer of viscoelastic mud and sinusoidal waves propagating on top of a water layer of intermediate depth. In contrast with the previous studies the analysis is done for decaying water waves and a rheological model with frequency independent coefficients. The leading order motion and the mean second order motion inside the mud layer is determined analytically together with the first two orders motion in water. The analysis is done in a fixed Eulerian frame and it is shown that both a mean horizontal displacement and a Eulerian mean horizontal velocity exists inside the mud layer at the second order. The effect of elasticity and viscosity on the damping of water waves and on the mean motion of the mud is studied. It is shown that a light mud with a high proportion of elasticity will significantly modify the leading order movement through damping. The results are applied to solve analytically the problem of the evolution of the narrow-banded waves propagating on top of a semi-infinite mud layer. It is shown that the presence of the mud layer gives rise to a negative mean current in water layer and to free waves generated at the edge of the mud layer and propagating at the dimensional velocity ....