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Thin-layer solutions of the Helmholtz equation

Part of: 30th Anniversary Special Issue Optics, electromagnetic theory - General Optics, electromagnetic theory - Basic methods

Published online by Cambridge University Press:  06 October 2020

J. R. OCKENDON Open the ORCID record for J. R. OCKENDON [Opens in a new window]  and
R. H. TEW
J. R. OCKENDON
Affiliation:
Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK email: ock@maths.ox.ac.uk
R. H. TEW
Affiliation:
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK email: richard.tew@nottingham.ac.uk
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Abstract

This paper gives a brief overview of some configurations in which high-frequency wave propagation modelled by Helmholtz equation gives rise to solutions that vary rapidly across thin layers. The configurations are grouped according to their mathematical structure and tractability and one of them concerns a famous open problem of mathematical physics.

Keywords

Helmholtz equationhigh-frequency asymptoticsthin layersinflection points

MSC classification

Primary: 78M35: Asymptotic analysis 78A05: Geometric optics 78A45: Diffraction, scattering

Information

Type
Papers
Information
European Journal of Applied Mathematics , Volume 32 , Special Issue 5: 30th Anniversary Special Issue , October 2021 , pp. 769 - 783
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

Babich, V. M. & Buldyrev, V. S. (1991) Short Wavelength Diffraction Theory, Springer, Berlin.CrossRefGoogle Scholar
Babich, V. M. & Kirpichnikova, N. Y. (1979) The Boundary Layer Method in Diffraction Problems, Springer-Verlag, Berlin, Heidelberg, New York.CrossRefGoogle Scholar
Babich, V. M. & Smyshlyaev, V s for the Schrödinger equation in the case of a potential linear in time and coordinate I: Asymptotics in the shadow zone. J. Sov. Math. 32, 103–111.CrossRefGoogle Scholar
Babich, V. M. & Smyshlyaev, V. P. (1987) The Scattering problem for the Schrödinger equation with a potential linear in time and space II: correctness, smoothness, behaviour of the solution at infinity. J. Sov. Math. 38, 1562–1576.CrossRefGoogle Scholar
Buchal, R. & Keller, J. B. (1960) Boundary layer problems in diffraction theory. Comm. Pure Appl. Math. 13, 85–144.CrossRefGoogle Scholar
Hewett, D. P. (2015) Shadow boundary effects in hybrid numerical-asymptotic methods for high-frequency scattering. Euro. J. Appl. Math. 26, 773–793.CrossRefGoogle Scholar
Hewett, D. P. (2015) Tangent ray diffraction and the Pekeris caret function. Wave Motion 57, 257–267.CrossRefGoogle Scholar
Hewett, D. P., Ockendon, J. R. & Smyshlyaev, V. P. (2019) Contour integral solution of the parabolic wave equation. Wave Motion 84, 90–109.CrossRefGoogle Scholar
Jones, D. S. (1986) Acoustic and Electromagnetic Waves, Oxford University Press. of the parabolic equation. J. Phys. USSR 10, 13–24.Google Scholar
Kazakov, A. Y. (2003) Special Function related to the concave-convex boundary problems of the diffraction theory. J. Phys. A Math. Gen. 36, 4127–4141.CrossRefGoogle Scholar
Kazakov, A. Ya. (2019) “Separation of variables” in the model problems of the diffraction theory. A formal scheme. J. Math. Sci 243, 715–725.CrossRefGoogle Scholar
Kravtsov, Yu. A. (1964) A modification of the geometrical optics method. Radiofizik 7, 1049–1056 (in Russian).Google Scholar
Leontovich, M. S. & Fock, V. A. (1946) Solution of the problem of propagation of electromagnetic waves along the earth’s surface by the method of the parabolic equation. J. Phys. USSR 10, 13–24.Google Scholar
Ludwig, D. (1965) Wave propagation near a smooth caustic. Bull. Am. Math. Soc. 73, 776–779.CrossRefGoogle Scholar
Nussenszveig, H. M. (1965) High-frequency scattering by an impenetrable sphere. Ann. Phys. 34, 23–95.CrossRefGoogle Scholar
Ockendon, J. R. & Tew, R. H. (2012) Thin-layer solutions of the Helmholtz and related equations. SIAM Rev. 54(1), 3–51.CrossRefGoogle Scholar
Olver, P. J. (1991) Applications of Lie Groups to Partial Differential Equations, Springer, New York.Google Scholar
Popov, M. M. (1979) The problem of whispering gallery waves in a neighbourhood of a simple zero of the effective curvature of the boundary. J. Sov. Math. 11, 791–797.CrossRefGoogle Scholar
Popov, M. M. (1982) Correctness of the problem of Whispering Gallery Waves in a neighbourhood of points of zero curvature of the boundary. J. Sov. Math 19, 1487–1493.CrossRefGoogle Scholar
Popov, M. M. (1986) Whispering Gallery Waves in a neighbourhood of an Inflection Point at the Boundary. Asymptotics of the wave field as t $$\to \infty $$ . J. Sov. Math. 32, 215–219.Google Scholar
Popov, M. M. & Krasavin, V. G. (1986) Direction diagram of radiation in the problem of an inflection Point of the boundary. J. Sov. Math. 32, 215–219.CrossRefGoogle Scholar
Popov, M. M. & Pshenchik, I. (1979) Numerical solution of the problem on whispering gallery waves in a neighborhood of a simple zero of the effective curvature of the boundary. J. Sov. Math. (now J. Math. Sci.) 11, 797–804.Google Scholar
Smyshlyaev, V. P. (1991) Concentration of the solution near a limit ray in the neighbourhood of an Inflection point of the boundary. J. Sov. Math. 55, 1757–1760.CrossRefGoogle Scholar
Tew, R. H. (2019) Friedlander-Keller ray expansions and scalar wave reflection at canonically perturbed boundaries. Euro. J. Appl. Math. 30, 1–22.CrossRefGoogle Scholar
Tew, R. H., Chapman, S. J., King, J. R., Ockendon, J. R., Smith, B. J. & Zafarullah, I. (2000) Scalar wave diffraction by tangent rays. Wave Motion 32, 363–380.CrossRefGoogle Scholar