Three discrete problems for Helmholtz equation is studied analytically using Sommerfeld integral approach. They are the problem with point source on a whole plane, the problem of diffraction by a half-plane, and the problem of diffraction by a right-angled wedge. It is showed that total field is represented as an integral from an algebraic function on a manifold. The latter is torus. For the problem with a point source a recursive relation is introduced. For half-plane and wedge problems solutions are obtained in terms of Sommerfeld integral.

中文翻译：

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用于离散衍射问题的 Sommerfeld 型积分

使用 Sommerfeld 积分方法解析地研究了 Helmholtz 方程的三个离散问题。它们是整个平面上的点源问题、半平面衍射问题和直角楔形衍射问题。结果表明，总场表示为流形上代数函数的积分。后者是环面。对于点源问题，引入了递归关系。对于半平面和楔形问题的解决方案是根据 Sommerfeld 积分获得的。