An analytical investigation of oscillations within a circular harbor over a Conical Island

Abstract

Based on the linearized long-wave equation, an analytical solution for wave oscillations within a circular harbor of constant depth over a conical island is obtained. The analysis divides the study domain into three regions: region I (within the harbor), region II (over the island) and region III (from the island toe to the deep sea). In region I, the water depth is uniform, and the method of separation of variables is used to find the free-surface elevation solution. In Region II, the water depth linearly varies along the radial direction of the island, and the method of separation of variables is adopted again to reveal the free-surface elevation solution expressed in Bessel and Hankel functions. Waves in region III consist of the incident waves and the scattered waves from the island, which are respectively expressed with a Fourier-cosine series and a sum series of Hankel function. The final solution is obtained by matching the free-surface elevation and its normal derivatives at the interfaces between the three regions. To confirm the validity of the analytical solution, via adjusting topographical parameters, it was further utilized to compare with the existing analytical solutions for wave-induced oscillations in a circular harbor of constant depth and the scattering of long waves around a cylindrical island mounted on a conical shoal. Finally, as the topographical features of Maug Islands resembles the geometry studied in this article, the coupled oscillation characteristics for Maug Islands are examined. The existence of the island does aggravate oscillations within the harbor, and the features are addressed in details.