Diffraction by a quarter-plane. Links between the functional equation, additive crossing and Lamé functions

Raphaël C. Assier, Andrey V. Shanin

Published 2020-04-18Version 1

In our previous work (Assier & Shanin, QJMAM, 2019), we gave a new spectral formulation in two complex variables associated with the problem of diffraction by a quarter-plane. In particular, we showed that the unknown spectral function satisfies a condition of additive crossing about its branch set. In this paper, we study a very similar class of spectral problem, and show how the additive crossing can be exploited in order to express its solution in terms of Lam\'e functions. The solution obtained can be thought of as a tailored vertex Green's function whose behaviour in the near-field is directly related to the eigenvalues of the Laplace-Beltrami operator. This is important since the correct near-field behaviour at the tip of the quarter-plane had so far never been obtained via a Wiener-Hopf approach.