On step approximation of water-wave scattering over steep or undulated slope

Abstract

In this paper, Miless variational formulation (Miles, 1967) is extended to study problems of water-wave scattering by steep slopes. In the procedure for obtaining the solution, the arbitrary bottom profiles are represented by flat shelves separated by abrupt steps. By applying the stationary condition of the variational formulation, a system of linear equations is obtained with unknown coefficients that represent the horizontal velocities. Our variational formulation is extended by considering the evanescent eigenfunctions in the representations of these horizontal velocities. The extended variational formulation is applied to solving water-wave scattering by using Roseaus curved bottom profiles of both mild and steep slopes (Roseau, 1976). The solutions obtained by the extended variational formulation are convergent with Roseaus analytical solution up to four decimal places for the mild and steep cases, while the solutions obtained by the traditional variational formulation (Miles, 1967) and the transfer-matrix method of Devillard et al. (1988) are not accurate enough for the steep case. Furthermore, by using the integral equation method, the improvement of the proposed method is enforced by two types of bottom profiles considered by Porter and Porter (2000). Finally, numerical experiments are performed to compare the present method with the extended mild-slope equation of Porter and Staziker (1995).