A 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 Liu et 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.

A second-order nonlinear frequency-domain model extending the linear complementary mild-slope equation (CMSE) is presented. The nonlinear model uses the same streamfunction formulation as the CMSE. This allows the vertical profile assumption to accurately satisfy the kinematic bottom boundary condition in the case of nonlinear triad interactions as well as for the linear refraction–diffraction part. The result is a model with higher accuracy of wave–bottom interactions including wave–wave interaction. The model's validity is confirmed by comparison with accurate numerical models, laboratory experiments over submerged obstacles and analytical perturbation solutions for class III Bragg resonance.

This study elucidated the complicated phenomena of wave refraction and diffraction around a circular island due to random incident waves traveling with a current. Various combinations of random incident wave and current conditions were used to investigate the wave height distributions around a circular island numerically and experimentally. Numerical calculations were carried out based on the theory derived by Lin & Hsu [1]. According to the results, it shows that numerical calculations can predict experimental data quantitatively well.