Damping coefficient by long waves - Viscoelastic mud-current interaction

In this work, we study the damping of long linear waves propagating on a thin mud layer, which obeys a viscoelastic behavior, in the presence of a marine current; two cases are considered: (1) a current traveling in an opposite way to the wave propagation and (2) in the same way to the wave propagation. We obtain an asymptotic solution to the dimensionless governing equations, which allows identifying the competition between the different mechanisms involved to gain insight into the physics by which coastal mud responds to water waves. To determine the shear stresses into the viscoelastic mud, we have used the well-known Maxwell rheological model. At the interface water-mud, we assume the presence of a small Stokes' layer, and this condition allows us to consider the continuity of shear stresses and velocities at the interface. The effects of the mud physical parameters as well as the hydrodynamics impact of the water waves on the damping coefficient, fluid velocities, and shear stresses and on the free surface elevation are analyzed. The results show that the maximum value of the damping coefficient occurs when the mud thickness is of the same order of magnitude as the Stokes' layer and for increments of the current magnitude traveling in an opposite way to the wave propagation. The asymptotic solution is compared with solutions reported in the specialized literature and the results adjust properly.