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Sommerfeld-type integrals for discrete diffraction problems |
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| Institution: | 1. South Australian Museum, PO Box 825, Normanville, SA 5204, Australia;2. Department of Biological Evolution, Faculty of Biology, Lomonosov Moscow State University, Leninskie Gory 1(12), Moscow 119234, Russia;3. Geological Institute, Russian Academy of Sciences, Pyzhevskiy pereulok 7, Moscow 119017, Russia;4. Borissiak Palaeontological Institute, Russian Academy of Sciences, Moscow 117647, Russia;5. State Key Laboratory of Palaeobiology and Stratigraphy, Nanjing Institute of Geology and Palaeontology, Chinese Academy of Sciences, Nanjing 210008, China;6. University of Chinese Academy of Sciences, Beijing 100049, China |
| Abstract: | Three problems for a discrete analog of the Helmholtz equation are studied analytically using the plane wave decomposition and the Sommerfeld integral approach. They are: (1) the problem with a point source on an entire plane; (2) the problem of diffraction by a Dirichlet half-line; (3) the problem of diffraction by a Dirichlet right angle. It is shown that the total field can be represented as an integral of an algebraic function over a contour drawn on some manifold. The latter is a torus. As a result, explicit solutions are obtained in terms of recursive relations (for the Green’s function), algebraic functions (for the half-line problem), or elliptic functions (for the right angle problem). |
| Keywords: | Discrete Helmholtz equation Discrete green’s function Canonical diffraction problem Reflection method Elliptic integrals Sommerfeld integral Dispersion equation |
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