Assier Raphaël C, Shanin Andrey V
Department of Mathematics, The University of Manchester, Oxford Road, Manchester M13 9PL, UK.
Department of Physics (Acoustics Division), Moscow State University, Leninskie Gory, 119992 Moscow, Russia.
Proc Math Phys Eng Sci. 2021 Jan;477(2245):20200681. doi: 10.1098/rspa.2020.0681. Epub 2021 Jan 6.
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.
研究了在分支曲面（索末菲曲面）上服从二维亥姆霍兹方程的波场。通过将反射方法应用于具有理想边界条件的直线散射体的衍射问题，这类曲面自然出现。例如，对于半直线或线段衍射的经典规范问题就是这种情况。在本工作中，表明这类波场允许解析延拓到两个复坐标的区域。给出并详细研究了这种延拓的分支集。对于一般的散射问题，表明场的多值解析延拓的所有分支集有一个有限基。每个基函数明确地表示为沿所谓双八轮廓的格林积分。有限基性质在作者先前引入和使用的坐标方程的背景下很重要，如本文针对线段衍射的特殊情况所示。
