The Rayleigh hypothesis in the theory of diffraction by a cylindrical obstacle

The Rayleigh hypothesis in the theory of scattering by a cylindrical obstacle of arbitrary cross section is investigated analytically. The hypothesis asserts that outside and on the obstacle the scattered field may be expanded in terms of outward-going wave functions of the circular cylinder. As such, it is analogous to the assumption made by Lord Rayleigh in his treatment of diffraction by a reflection grating. We show that the validity of the Rayleigh hypothesis is governed by the distribution of singularities in the analytic continuation of the exterior scattered field. Conditions are derived under which the Rayleigh hypothesis is rigorously valid. As examples, the elliptic cylinder and the perturbed circular cylinder are considered in detail.