Water waves over a compressible muddy seabed

An asymptotic theory is developed to investigate the interaction of surface water waves and a compressible muddy seabed in water of intermediate depth. The water column is treated as inviscid outside a thin bottom boundary layer, the thickness of which is assumed to be of the same order of magnitude as the wave amplitude. The seabed is modelled as an isotropic and homogeneous poro-viscoelastic layer with an incompressible solid skeleton and a compressible pore fluid. The thickness of the seabed is assumed to be comparable to the wave amplitude. Using a perturbation approach, the leading-order analytical solutions for the wave and mud flow are derived, while the evolution of the wave envelope, the mean Eulerian velocity and the mass transport velocity beneath progressive waves are obtained at the second order. Based on the present solution, the wave motion and the induced mass transport are analysed and compared with previous solutions that ignore the compressibility of the seabed. The results demonstrate that the compressibility of the seabed plays a critical role in modulating the flows within both the seabed and the overlying bottom boundary layer. Neglecting compressibility may lead to an underestimation of the interface vertical displacement in highly elastic beds and an overestimation in viscous-dominated cases. Consequently, the Reynolds stress distributions in these regions deviate significantly from predictions based on incompressible seabed theory. This inaccuracy further propagates to the prediction of second-order steady currents and mass transport velocities.