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.

中文翻译：

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二维波场的解析延续

研究了在分支表面（Sommerfeld 表面）上服从二维亥姆霍兹方程的波场。由于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，这样的表面自然出现。例如，对于半线或线段衍射的经典典型问题就是这种情况。在目前的工作中，表明这种波场允许解析延拓进入两个复坐标域。给出并详细研究了这种延续的分支集。对于一般散射问题，表明该域的多值解析连续的所有分支的集合具有有限基。每个基函数都明确表示为沿​​着所谓的双八等高线的格林积分。