An extended analytic solution of combined refraction and diffraction of long waves propagating over circular island

An analytic solution of long waves scattering by a cylindrical island mounted on a permeable circular shoal was obtained by solving the linear long wave equation (LWE). The solution is in terms of the Bessel function expressed by complex variables. The present solution is suitable for arbitrary bottom configurations described by a power function with two independent parameters. For the case of the paraboloidal shoal, there exists a singular point (α = 2) which can be removed using Frobenius series, where α is a real constant. The present solution is reduced to Yu and Zhang's (2003) solution for impermeable circular shoal. The numerical results show some special features of the combined effect of wave refraction and diffraction caused by a porous circular island. The effect of key parameters of the island dimension, the shoal slope, and permeability on wave scattering was discussed based on the analytic solution.