A higher-order on-surface radiation condition derived from an analytic representation of a Dirichlet-to-Neumann map

On-surface radiation conditions are useful for obtaining approximate solutions to scattering problems involving compact obstacles. An analytic representation of the Dirichlet-to-Neumann map for a circle is derived and used to construct a higher-order on-surface radiation. condition for a generally convex perfectly conducting body in two dimensions. This approach is based on a Hankel function in which a tangential operator appears in the index. In the high-frequency limit, this analytic representation approaches the square root of a differential operator which commonly arises in the application of parabolic equation techniques to propagation problems. Treating the scattered field propagation angle relative to the surface normal and the-surface curvature as independent parameters, the representation is fit to a rational function to provide an accurate and efficient on-surface radiation condition that is tested for various examples.