论文标题
二维波场的分析延续
Analytical continuation of two-dimensional wave fields
论文作者
论文摘要
研究了在分支表面上遵守2D Helmholtz方程（Sommerfeld表面）的波场。由于将反射方法应用于带有理想边界条件的直散射器的衍射问题，因此这种表面自然而然地出现。例如，这是通过半线或细分市场衍射的经典规范问题的情况。在目前的工作中，结果表明，这种波场接收了两个复杂坐标的域的分析延续。此类延续的分支集进行了详细的详细研究。对于通用散射问题，可以表明该场的多价分析延续的所有分支的集合都有有限的基础。每个基础函数沿所谓的双八八次轮廓明确表示为绿色的积分。如前所述，在作者先前引入和使用的坐标方程的上下文中，有限的基础属性很重要，如本文所示，针对细分市场的特定衍射情况。
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.
