An analytical solution of the mild-slope equation for waves around a circular island on a paraboloidal shoal

Abstract

In this paper, we develop an analytical technique in terms of series expansions to solve the mild-slope equation on an axi-symmetric topography. This technique is applied to study the combined refraction and diffraction of plane monochromatic waves by a circular cylindrical island mounted on a paraboloidal shoal. By using the direct solution for the wave dispersion equation by Hunt [J. Waterw., Port, Coast., Ocean Div. Proc ASCE 4 (1979) 457], the mild-slope equation becomes explicit and it is then solved in terms of combined Fourier series and Taylor series. It is found that, to calculate the wave elevation along a perimeter with a specific radius, more terms in the Taylor series and angular modes in the Fourier series are needed for shorter waves. On the other hand, for the same incident wave, the outer the solutions are sought, more angular modes are needed to obtain the converged result of Fourier series. The comparison with the analytical solution based on the linear shallow-water equation by Homma [Geophys. Mag. 21 (1950) 199] is made for long wave incidence and excellent agreements are obtained. For long waves and waves in intermediate water depth, comparisons are made with other numerical results of the mild-slope equation and an equally good quality of agreement is achieved.