Analytical Solution for Long-Wave Scattering by a Circular Island Mounted on a General Shoal

Abstract

In this paper, an analytical solution is given in the form of a Taylor series for long-wave scattering by a cylindrical island mounted on a general shoal, where the water depth in the shoal region can be a real constant plus a power function of the radial distance. In other words, the crest of the shoal is not restricted to be on the still water level. The distribution of singular points becomes complicated; therefore, the solution technique needs to be more sophisticated. Because of the generality of the location of the shoal crest, the analytical solutions find several classical analytical solutions to be its special cases, which include long-wave scattering by a cylindrical island mounted on an idealized paraboloidal shoal, a cylindrical island mounted on an idealized conical shoal, a cylindrical island mounted on an idealized shoal, and a cylindrical island located on a flat bottom. The present analytical solution covers a much wider range of problems, and therefore, is much more useful. Finally, the effect of the shoal size to the wave-scattering pattern is investigated using the present analytical solution.