Diffraction by a Right-Angled No-Contrast Penetrable Wedge: Analytical Continuation of Spectral Functions

TL;DR

This paper investigates the diffraction of waves by a right-angled penetrable wedge using a two-complex-variable Wiener-Hopf method, revealing the spectral functions' analytic continuation and singularities in complex space.

Contribution

It introduces the analytic continuation of spectral functions onto a two-complex-dimensional manifold and demonstrates the additive crossing concept for the penetrable wedge diffraction problem.

Findings

Spectral functions can be analytically continued in two complex variables.

Singularities of spectral functions are characterized in ^2.

The additive crossing concept is established for this diffraction problem.

Abstract

We study the problem of diffraction by a right-angled no-contrast penetrable wedge by means of a two-complex-variable Wiener-Hopf approach. Specifically, the analyticity properties of the unknown (spectral) functions of the two-complex-variable Wiener-Hopf equation are studied. We show that these spectral functions can be analytically continued onto a two-complex dimensional manifold, and unveil their singularities in $\mathbb{C}^2$. To do so, integral representation formulae for the spectral functions are given and thoroughly used. It is shown that the novel concept of additive crossing holds for the penetrable wedge diffraction problem and that we can reformulate the physical diffraction problem as a functional problem using this concept.