A new family of embedding formulae for diffraction by wedges and polygons

Abstract

Embedding formulae represent the solution to certain linear plane-wave scattering problems in terms of solutions to other auxiliary scattering problems. Here the method of [R.V. Craster, A.V. Shanin, E.M. Doubravsky, Embedding formulae in diffraction theory, Proc. R. Soc. Lond. A 459 (2003) 2475–2496] and, in particular, [R.V. Craster, A.V. Shanin, Embedding formulae for diffraction by rational wedge and angular geometries, Proc. R. Soc. Lond. A 461 (2005) 2227–2242], whose auxiliary problems are forced by multipoles located at the corners of the scatterer, is adapted and extended to determine formulae relating solutions corresponding to different incident plane wave angles. As in [R.V. Craster, A.V. Shanin, Embedding formulae for diffraction by rational wedge and angular geometries, Proc. R. Soc. Lond. A 461 (2005) 2227–2242], wedge geometries of rational angle and polygons with rational internal angle are considered and appropriate embedding formulae derived. The method is also shown to extend to the more general class of polygon whose internal angles are all multiples of the same rational angle.