On long-wave propagation over a fluid-mud seabed

Using the Boussinesq approximation, a set of depth-integrated wave equations for long-wave propagation over a mud bed is derived. The wave motions above the mud bed are assumed to be irrotational and the mud bed is modelled as a highly viscous fluid. The pressure and velocity are required to be continuous across the water-mud interface. The resulting governing equations are differential-integral equations in terms of the depth-integrated horizontal velocity and the free-surface displacement. The effects of the mud bed appear in the continuity equation in the form of a time integral of weighted divergence of the depth-averaged velocity. Damping rates for periodic waves and solitary waves are calculated. For the solitary wave case, the velocity profiles in the water column and the mud bed at different phases are discussed. The effects of the viscous boundary layer above the mud-water interface are also examined.