Mass transport in water waves over a thin layer of soft viscoelastic mud

A theory is presented for the mass transport induced by a small-amplitude progressive wave propagating in water over a thin layer of viscoelastic mud modelled as a Voigt medium. Based on a sharp contrast in length scales near the bed, the boundary-layer approximation is applied to the Navier-Stokes equations in Lagrangian form, which are then solved for the first-order oscillatory motions in the mud and the near-bed water layers. On extending the analysis to second order for the mass transport, it is pointed out that it is inappropriate, as was done in previous studies, to apply the complex viscoelastic parameter to a higher-order analysis, and also to suppose that a Voigt body can undergo continuous steady motion. In fact, the time-mean motion of a Voigt body is only transient, and will stop after a time scale given by the ratio of the viscosity to the shear modulus. Once the mud has attained its steady deformation, the mass transport in the overlying water column can be found as if it were a single-layer system. It is found that the near-bed mass transport has non-trivial dependence on the mud depth and elasticity, which control the occurrence of resonance. Even when the resonance is considerably damped by viscosity, the mass transport in water over a viscoelastic layer can be dramatically different, in terms of magnitude and direction, from that over a rigid bed.