Journal of Mathematical Sciences

, Volume 185, Issue 4, pp 644–657 | Cite as

Asymptotics of waves diffracted by a cone and diffraction series on a sphere

  • A. V. Shanin

Diffraction of a plane harmonic scalar wave by a cone with an ideal boundary condition is studied. A flat cone or a circular cone is chosen as a scatterer. It is known that the diffracted field contains different components: a spherical wave, geometrically reflected wave, multiply diffracted cylindrical waves (for a flat cone), creeping waves (for a circular cone). The main tack of the paper is to find a uniform asymptotics of all wave components. This problem is solved by using an integral representation proposed in the works by V. M. Babich and V. P. Smyshlyaev. This representation uses a Green’s function of the problem on a unit sphere with a cut. This Green’s function can be represented in the form of a diffraction series. It is shown that different terms of the series correspond to different wave components of the conical diffraction problem. A simple formula connecting the leading terms of the diffraction series for the spherical Green’s function with the leading terms of different wave components of the conical problem is derived. Some important particular cases are studied. Bibliography: 5 titles.


Integral Representation Unit Sphere Simple Formula Spherical Wave Wave Component 
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  1. 1.
    V. M. Babich, D. B. Dement’ev, B. A. Samokish, and V. P. Smyshlyaev, “On evaluation of the diffraction coefficient for arbitrary “nonsingular” directions of a smooth convex cone,” SIAM J. Appl. Math., 60, No. 2, 536–573 (2000).MathSciNetMATHCrossRefGoogle Scholar
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    V. A. Borovikov, Diffraction by Polygons and by Polyhedra [in Russian], Nauka, Moscow (1966).Google Scholar
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    V. M. Babich, “On PC ansatz,” J. Math. Sci., 132, No. 1, 2–10 (2006).MathSciNetCrossRefGoogle Scholar
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    A. Popov, A. Ladyzhensky, and S. Khoziosky, “Uniform asymptotics of the wave diffracted by a cone of arbitrary cross-section,” Russ. J. Math. Phys., 16, No. 2, 296–299 (2009).MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    I. V. Andronov, “Diffraction by some strongly elongated bodies of rotation” (submitted to Acoustical Physics).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2012

Authors and Affiliations

  1. 1.Moscow State UniversityMoscowRussia

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