Interaction of Water Waves with Permeable Barrier using Galerkin Approximation

Abstract

In this paper we analyze the interaction of water waves with a permeable barrier which is slightly perturbed from its vertical position within the framework of linearised water wave theory. The barrier is placed in water of finite depth. Two different kinds of barriers are examined, namely, (I) a partially immersed barrier and (II) a submerged bottom standing barrier. The governing boundary value problem involving the velocity potential function is split into two boundary value problems involving the zeroth order as well as the first order velocity potential functions by using a simplified perturbation technique. The zeroth order reflection and transmission coefficients which are due to a vertical permeable barrier are evaluated by solving a Fredholm integral equation of second kind numerically by using a one term Galerkin approximation. Green’s theorem is applied to evaluate the first order reflection and transmission coefficients. The first order transmission coefficient vanishes irrespective of the shape of the barrier. The numerical values for the first order reflection coefficient are determined by choosing some appropriate shape functions. The numerical results for the zeroth order reflection coefficient which stand for the case of a vertical barrier are validated against the known results for both the permeable and impermeable barriers. The first order reflection curves are also compared by making the porosity constant to be zero with those available in the literature for an impermeable nearly vertical barrier.