Nonlinear diffusive surface waves in porous media

Abstract

A fully nonlinear, diffusive, and weakly dispersive wave equation is derived for describing gravity surface wave propagation in a shallow porous medium. Darcy's flow is assumed in a homogeneous and isotropic porous medium. In deriving the general equation, the depth of the porous medium is assumed to be small in comparison with the horizontal length scale, i.e. O(μ²) =O(h₀/L)²[Lt ]1. The order of magnitude of accuracy of the general equation is O(μ⁴). Simplified governing equations are also obtained for the situation where the magnitude of the free-surface fluctuations is also small, O(ε)=O(a/h₀)[Lt ]1, and is of the same order of magnitude as O(μ²). The resulting equation is of O(μ⁴, ε²) and is equivalent to the Boussinesq equations for water waves. Because of the dissipative nature of the porous medium flow, the damping rate of the surface wave is of the same order magnitude as the wavenumber. The tide-induced groundwater fluctuations are investigated by using the newly derived equation. Perturbation solutions as well as numerical solutions are obtained. These solutions compare very well with experimental data. The interactions between a solitary wave and a rectangular porous breakwater are then examined by solving the Boussinesq equations and the newly derived equations together. Numerical solutions for transmitted waves for different porous breakwaters are obtained and compared with experimental data. Excellent agreement is observed.