Boussinesq equations for wave transformation on porous beds

Abstract

A set of time-dependent vertically-integrated equations is derived to model the horizontally two-dimensional transformation of waves on a porous bed. The basic equations, called the Boussinesq equations for porous beds, contain the leading orders of nonlinearity and dispersivity. A general resistance equation has been used for the porous medium. The applicability bounds of the basic equations, limited by weak dispersivity and underestimated porous damping rates in deeper waters, have been extended by adding dispersion terms to the momentum equations and calibrating the resulting dispersion relation with a linear theory for porous beds. A numerical method based on finite differences is employed to solve the equations for two dimensions. The extended equations are verified for damped wave propagation on a horizontal bed, wave transformation on uniform porous slopes and combined refraction, diffraction, shoaling and damping around a submerged porous breakwater with an opening.