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 Zap. Nauchn. Sem. POMI, 2011, Volume 393, Pages 234–258 (Mi znsl4627)

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

A. V. Shanin

M. V. Lomonosov Moscow State University, Moscow, Russia

Abstract: Diffraction of a plane harmonic scalar wave by a cone with ideal boundary condition is studied. A flat cone or a circular cone is chosen as a scatterer. It is known that the diffarcted field contains different components: a spherical wave, geometrically reflected wave, multiply diffracted cylindrical waves (for a flat cone), creepind waves (for a circular cone). The main task 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 representaition uses a Green's function of the problem on a unit sphere with a cut. This Green's function can be presented in the form of 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.

Key words and phrases: diffraction by cone, diffraction series, uniform asymptotics.

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English version:
Journal of Mathematical Sciences (New York), 2012, 185:4, 644–657

Bibliographic databases:

Document Type: Article
UDC: 534.26

Citation: A. V. Shanin, “Asymptotics of waves diffracted by a cone and diffraction series on a sphere”, Mathematical problems in the theory of wave propagation. Part 41, Zap. Nauchn. Sem. POMI, 393, POMI, St. Petersburg, 2011, 234–258; J. Math. Sci. (N. Y.), 185:4 (2012), 644–657

Citation in format AMSBIB
\Bibitem{Sha11} \by A.~V.~Shanin \paper Asymptotics of waves diffracted by a~cone and diffraction series on a~sphere \inbook Mathematical problems in the theory of wave propagation. Part~41 \serial Zap. Nauchn. Sem. POMI \yr 2011 \vol 393 \pages 234--258 \publ POMI \publaddr St.~Petersburg \mathnet{http://mi.mathnet.ru/znsl4627} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2870216} \transl \jour J. Math. Sci. (N. Y.) \yr 2012 \vol 185 \issue 4 \pages 644--657 \crossref{https://doi.org/10.1007/s10958-012-0949-2} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84866534066} 

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This publication is cited in the following articles:
1. I. V. Andronov, D. Bouche, “Diffraction by a narrow circular cone as by a strongly elongated body”, J. Math. Sci. (N. Y.), 185:4 (2012), 517–522
2. Assier R.C., Peake N., “Precise Description of the Different Far Fields Encountered in the Problem of Diffraction of Acoustic Waves by a Quarter-Plane”, IMA J. Appl. Math., 77:5, SI (2012), 605–625
3. Lyalinov M.A., “Scattering of Acoustic Waves by a Sector”, Wave Motion, 50:4 (2013), 739–762
4. Lyalinov M.A., “Electromagnetic Scattering By a Plane Angular Sector: i. Diffraction Coefficients of the Spherical Wave From the Vertex”, Wave Motion, 55 (2015), 10–34
5. Korolkov A.I. Shanin A.V., “Diffraction By a Thin Cone in the Parabolic Approximation. Method of the Boundary Integral Equation”, 2017 International Conference on Electromagnetics in Advanced Applications (Iceaa), IEEE, 2017, 696–699
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