Dispersive shallow water waves over a porous sea bed

Abstract

Nonlinear shallow water equations of Airy and Boussinesq types are derived to predict wave propagation over a porous sea bed. The flow in the porous sea bed follows Darcy's law. The shallow water theories have important advantages over the integral equation theory. For a 2-D linear wave propagation problem an integral equation method can be used. However, extension to a nonlinear 3-D problem increases the complexity and the numerical difficulty of the integral equation method significantly. The 3-D problem can more easily be handled by the shallow water equations. Unlike the integral equation the shallow water equations can in addition be extended in order to consider boundary layer effects. The shallow water equations are tested for 2-D linear periodic waves against an integral equation method which solves the problem without any approximations for depth variations. The linear, dispersive long wave and the integral equation method show good agreement even for wavelengths less than ten times the water depth. Bragg resonance due to interaction between reflected and transmitted waves and porous sinusoidal ripples are also considered.