A Series Solution for Wave Scattering by a Circular Island on a Shoal Based on the Mild-Slope Equation

Abstract

ABSTRACTA series solution based on the mild-slope equation is produced in this study of wave scattering produced by a circular cylindrical island mounted on an axi-symmetrical shoal. The solution is presumed to be a Fourier cosine expansion with variable coefficients in the radial direction on account of the symmetric scattering field, which translates the original 2-D boundary-value problem to a 1-D one in which an ordinary differential equation is in effect treated. Approximations to the coefficients of the governing equation with the Taylor expansions enable the use of the Frobenius method, and consequently the solution is obtained in a combined Fourier and power series. For verification, the present method is mainly compared with Zhu and Zhang's [1] analytical solution of the linearised shallow water equation for a conical shoal, and with a different analytical solution of the mild-slope equation developed by Liuet al.[2] for a paraboloidal shoal. Fine agreements are achieved. The present method is then used to investigate the variation pattern of the wave run-up when the shoal profile varies from conical to paraboloidal, and some interesting phenomena are observed.