Analytical continuation of two-dimensional wave fields

TL;DR

This paper demonstrates that wave fields satisfying the 2D Helmholtz equation on branched surfaces can be analytically continued into complex domains, with a finite basis of branches explicitly constructed via Green's integrals, advancing diffraction analysis methods.

Contribution

It introduces a method to analytically continue 2D Helmholtz wave fields on Sommerfeld surfaces and constructs a finite basis of branches explicitly using Green's integrals.

Findings

Wave fields admit an explicit multi-valued analytic continuation.

The set of all branches has a finite basis.

Each basis function is expressed as a Green's integral along double-eight contours.

Abstract

Wave fields obeying the 2D Helmholtz equation on branched surfaces (Sommerfeld surfaces) are studied. Such surfaces appear naturally as a result of applying the reflection method to diffraction problems with straight scatterers bearing ideal boundary conditions. This is for example the case for the classical canonical problems of diffraction by a half-line or a segment. In the present work, it is shown that such wave fields admit an analytical continuation into the domain of two complex coordinates. The branch sets of such continuation are given and studied in detail. For a generic scattering problem, it is shown that the set of all branches of the multi-valued analytical continuation of the field has a finite basis. Each basis function is expressed explicitly as a Green's integral along so-called double-eight contours. The finite basis property is important in the context of coordinate equations, introduced and utilised by the authors previously, as illustrated in this article for the particular case of diffraction by a segment.