Abstract
Wave fields obeying the two-dimensional 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 used by the authors previously, as illustrated in this article for the particular case of diffraction by a segment.
Highlights
In this paper we study two-dimensional diffraction problems for the Helmholtz equation belonging to a special class: namely those that can be reformulated as problems of propagation on branched surfaces with finitely many sheets
We have provided an explicit method to analytically continue two-dimensional wave fields emanating from a broad range of diffraction problems and described the singular sets of their analytical continuation
Even though the analytical continuation may have potentially infinitely many branches, each branch can be expressed as a linear combination of finitely many basis functions
Summary
In this paper we study two-dimensional diffraction problems for the Helmholtz equation belonging to a special class: namely those that can be reformulated as problems of propagation on branched surfaces with finitely many sheets. We prove here that these branches have a finite basis, such that any branch can be expressed as a linear combination of a finite number of basis functions with integer coefficients This property of the analytical continuation is an important property of the initial (real) diffraction problem. The ideas behind this work were inspired in part by our recent investigations of applications of multi-variable complex analysis to diffraction problems [7,8,9,10,11], in part by our work on coordinate equations [1,2,3,4,5] and in part by the work of Sternin and his co-authors [12,13,14], in which the analytical continuation of wave fields is considered.
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