The series solution to the modified mild-slope equation for wave scattering by Homma islands

• This is the first exact analytic solution to MMSE for wave scattering by 3D bathymetries.
• The smoothness imposed to the water depth h(r)h(r) is the second order only.
• A depth-averaged mass-conserving condition at r=ar=a is derived to replace a wrong one.
• Our solution coincides with the solution (Cheng, 2011) [31] for a severe case with T=60s very well.
• This technique can model scattering for very short waves of T=25s with kh=25.7818kh=25.7818.

Based on a newly derived explicit form of the modified mild-slope equation (MMSE), a series solution is constructed for wave scattering by Homma island and the convergence of the solution is analyzed. It is found that an intuitive and commonly adopted depth-averaged zero-flux condition on the cylinder surface is wrong and an alternative Robin-type condition is derived. To validate the present solution, comparisons are conducted against various known solutions to the long-wave equation, Helmholtz equation, the mild-slope equation (MSE) and time-dependent MSE. It is shown that the present approach can model wave scattering from very long waves to very short waves. Comparison between the present MMSE solution and its degenerated MSE version is also conducted. Significant contribution of the mass-conserving matching conditions in improving solution accuracy revealed by Porter and Staziker is confirmed for this particular island, which mainly occurs for intermediate waves. By using the present solution, wave amplification is investigated for original Homma island and a modified Homma island. Finally, contour lines of wave amplification in the vicinity around a modified Homma island are presented.