Journal of Applied Mathematics
Volume 2012 (2012), Article ID 742705, 13 pages
Research Article

An Extended Analytic Solution of Combined Refraction and Diffraction of Long Waves Propagating over Circular Island

1Department of Hydraulic and Ocean Engineering, National Cheng Kung University, Tainan 701, Taiwan
2Research Center of Ocean Energy and Strategy, National Taiwan Ocean University, Keelung 202, Taiwan
3Department of Marine Environmental Engineering, National Kaohsiung Marine University, Kaohsiung 811, Taiwan

Received 18 June 2012; Revised 5 September 2012; Accepted 24 September 2012

Academic Editor: Michael Meylan

Copyright © 2012 Yu-Shu Kuo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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 ( ) 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.