Surface wave propagation over small bottom undulations in the presence of a submerged flexible porous barrier

Abstract

The problems of scattering of surface gravity waves over an undulated bottom in the presence of either a porous vertical elastic plate or a tensioned porous membrane are addressed. A simplified perturbation technique is employed to split the governing boundary value problem into a series of problems involving the zeroth and the first-order potential functions. The solution to the problem associated with zeroth-order potential is obtained by reducing it to a Fredholm integral equation with the help of Havelock’s theorems. An approximate solution of the integral equation is obtained by applying the multi-term Galerkin technique. Application of Green’s integral theorem leads to finding the first-order reflection and transmission coefficients in terms of integrals involving the shape function representing the bottom deformation and the solution of the zeroth-order problem. The accuracy of the present results is established by generating some previous results for the case of an elastic plate submerged in water with bottom deformation. From the graphical presentation of the results it is observed that at certain values of frequency related to the ripple wavenumber, the first-order reflection coefficient exhibits resonant nature, whereas the net reflection is affected by the flexibility characteristics as well as the porosity of the structure.