Surface wave scattering by submerged cylinders of arbitrary cross-section

The reflection of surface waves normally incident upon an infinite uniform horizontal cylinder of arbitrary cross-section totally immersed beneath the free surface of a fluid of either finite or infinite depth is considered under the assumptions of linearized theory. The problem is formulated in terms of a first-kind integral equation for an unknown function related to the tangential velocity of the fluid around the cylinder. The kernel of the integral operator is symmetric and only weakly singular, a property that is particularly advantageous when considering thin submerged plates. Numerical results are obtained by approximating the solution using Galerkin's method, and it is shown how only a few terms in the expansion are needed to gain high accuracy in the reflection and transmission coefficients. Similar accuracy is observed for thin plates, where special test functions are introduced to correctly model singular behaviour at the ends of the plate. Results for both infinite and finite depth scattering are compared with existing results and new results are also presented showing zeros of transmission for families of submerged obstacles.