A new solution for waves incident to a circular island on an axi-symmetric shoal

Based on the mild-slope type equation, a general solution is presented in this paper for waves incident to a circular cylindrical island mounted on an axi-symmetric shoal. The solution is presumed to be composed of the Fourier cosine expansion on account of the symmetric wave scattering field. The solution process then involves the use of the Frobenius method while the variable coefficients of the governing equation are approximated by the polynomial functions. A solution is achieved in the form of combined Fourier cosine expansion and power series. Comparisons with numerical methods, as well as with the analytical solution of Liu et al. (2004), imply the accuracy of the present results based on Berkhoff (1972) mild-slope equation. By adopting the extended mild-slope equation of Chandrasekera and Cheung (1997), the present method is then used to investigate the development of wave run-up with an increasing circular shoal area for a designed island topography.

► A quasi-analytical solution to the mild-slope type equation is presented.
► Scattering by a circular cylindrical island on an axi-symmetric shoal is studied.
► Our method is applicable to a wide range of bed shape and mild-slope type equation.
► Run-up behaviour in connection with varying shoal area is investigated.