Wave scattering by circular arc shaped plates

This work investigates the reflection and transmission properties of a circular arc plate which is submerged in deep water. The purpose is to compare the reflective properties of a circular arc plate with those for a submerged, circular cylinder in order to assess the suitability of using circular arc plates when constructing a water wave lens. Linear theory is assumed and two separate techniques are used to determine the wave field. The first involves expanding the potential as a series of multipole potentials outside a circular region and a series of nonsingular solutions of Laplace's equation within the region and matching the expansions on the boundary. The second technique is based on a variational procedure and is used to derive an explicit, approximate expression for the reflection coefficient, under the assumption that the plate is short compared with the other length scales in the problem. Results are presented which compare the approximate solution with the full numerical method for a variety of plates. Finally, the full numerical calculations of the reflection and transmission coefficients for a plate are compared with those for a submerged, circular cylinder.