Homotopy Perturbation Method applied to the Berkhoff equation for wave propagation on uneven bottoms

The paper is focussed on the application of the Homotopy Perturbation Method (HPM) to the wave propagation on uneven bottoms. The corresponding Berkhoff equation is solved on sloping bottoms with different types of boundary conditions: Dirichlet, Neuman or Robin. The matrix topology of the solution appears to be very effective in obtaining the water level inside the domain, as well as the reflection and the transmission coefficients. Different solutions are studied in the case of successive multiple slopes, with reference to Booij's experiments. The first part of the study is in the shallow water regime, where an analytical solution is known, suitable to validate the results obtained by the homotopy method. The second part is more general and necessitates to use an approximate solution of the wave dispersion relation: Hunt’s continuous function is chosen. The very good results obtained allow us to discretize any bottom bathymetry by a series of linear slopes. These experiments are completed with considerations on the great interest provided by the power of the semi-analytical solution HPM, which can be applied on smaller subdomains, avoiding restricted criteria in space, in order to give the solution on the infinite domain. This depends on the type of boundary conditions and the number of the homotopy orders.