Interaction of waves with two-dimensional obstacles: a relation between the radiation and scattering problems

A relation connecting the reflexion and transmission coefficients for scattering of water waves by a fixed body with the far-field radiated waves due to forced motions of the same body is derived. Two alternative derivations are given, including a simple argument based on the analysis of an appropriate linear superposition of the two problems, and a more formal application of Green's theorem to the two potentials. For bodies with horizontal symmetry, the transmission and reflexion coefficients are related to the phase angles of the far-field radiated waves associated with symmetric and antisymmetric forced motions of the body. Some general conclusions follow for arbitrary symmetric bodies, and these are verified in specific cases by comparison with existing solutions. The applicability of these relations to other types of wave problem is noted.