An Analytic Solution for Wave Scattering by a Circular Cylinder Mounted on a Conical Shoal

Abstract

In this paper, the combined refraction-diffraction of plane monochromatic waves by a circular cylinder mounted on a conical shoal in an otherwise open sea of constant depth is solved analytically based on the mild-slope equation. At first, Hunt's approximate direct solution for the wave dispersion equation [Hunt, 1979] is used to transform the mild-slope equation with implicit coefficients into an equation with all coefficients being explicitly expressed. This equation is then solved analytically in terms of combined Fourier series and Taylor series. Comparisons are made between the present method and the analytical solutions based on the linear shallow-water equation [Zhu and Zhang, 1996] and the Helmholtz equation [MacCamy and Fuchs, 1954] for long waves and short waves, respectively, and excellent agreements are obtained. For waves in intermediate water depth, comparisons are made with numerical results based on the mild-slope equation and an equally good quality of agreement is achieved. By varying the shoal size, it is found that the wave amplification normally enhances with the increase of shoal, due to the increased contribution from wave refraction. For relatively short wave, the trapped waves by refraction in front of the island interact with the reflected waves and form the partial-standing waves there.