Embedding formulae in diffraction theory

Embedding formulae are remarkable as they allow one to decompose scattering problems apparently dependent upon several angular variables (angles of incidence and observation) into those dependent upon fewer angular variables. In terms of facilitating rapid computations across considerable parameter regimes, this is a considerable advantage. Our aim is to derive embedding formulae for scattering and diffraction problems in acoustics, electromagnetism and elasticity. Here we construct a general approach to formulating and using embedding formulae. We do this using complementary approaches: overly singular states and a physical interpretation in terms of sources. The crucial point we identify is the form of the auxiliary state used in the reciprocal theorem; this is unphysically singular at the edge and is reminiscent of weight-function methods used in fracture mechanics. Illustrative implementations of our approach are given using Wiener-Hopf techniques for semi-infinite model problems in both elasticity and acoustics. We also demonstrate our approach using a numerical example from acoustics and we make connections with high-frequency asymptotic methods.