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Friedlander–Keller ray expansions and scalar wave reflection at canonically perturbed boundaries

Published online by Cambridge University Press:  04 January 2018

R.H. TEW
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 concerns the reflection of high-frequency, monochromatic linear waves of wavenumber k(≫ 1) from smooth boundaries which are O(k−1/2) perturbations away from either a specified near-planar boundary or else from a given smooth, two-dimensional curve of general O(1) curvature. For each class of perturbed boundary, we will consider separately plane and cylindrical wave incidence, with general amplitude profiles of each type of incident field. This interfacial perturbation scaling is canonical in the sense that a ray approach requires a modification to the standard WKBJ ‘ray ansatz’ which, in turn, leads to a leading-order amplitude (or ‘transport’) equation which includes an extra term absent in a standard application of the geometrical theory of diffraction. This extra term is unique to this scaling, and the afore-mentioned modification that is required is an application of a generalised type of ray expansion first posed by F. G. Friedlander and J. B. Keller (1955 Commun. Pure Appl. Math.6, 387–394).

Keywords

Ray theoryWKBJ-method
Type
Papers
Information
European Journal of Applied Mathematics , Volume 30 , Issue 1 , February 2019 , pp. 1 - 22
Copyright
Copyright © Cambridge University Press 2018 

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References

[1] Friedlander, F. G. & Keller, J. B. (1955) Asymptotic expansions of solutions of (∇2 + k 2)u = 0. Commun. Pure Appl. Math. 6, 387–394.Google Scholar
[2] Keller, J. B. & Lewis, R. M. (1995) Asymptotic methods for partial differential equations: The reduced wave equation and Maxwell's equations. In: Keller, J. B., McLaughlin, D. W. & Papanicolaou, G. C. (editors), Surveys in Applied Mathematics, Vol. 1, Plenum Press, New York, pp. 1–82Google Scholar
[3] Engineer, J. C., King, J. R. & Tew, R. H. (1998) Diffraction by slender bodies. Euro J. Appl. Math. 9, 129–158.Google Scholar
[4] Keller, J. B. (1962) Geometrical theory of diffraction. J. Opt. Soc. Am. A 52, 116–130.Google Scholar
[5] Levy, B. R. & Keller, J. B. (1959) Diffraction by a smooth object. Commun. Pure Appl. Math. 12, 159–209.Google Scholar
[6] Keller, J. B. (1957) Diffraction by an aperture. J. Appl. Phys. 28, 426–444.Google Scholar
[7] Keller, J. B., Lewis, R. M. & Seckler, B. D. (1956) Asymptotic solution of some diffraction problems. Commun. Pure Appl. Math. 9 (9), 207–265.Google Scholar
[8] Bleistein, N. (1984) Mathematical Methods for Wave Phenomena, Academic Press, Orlando.Google Scholar
[9] Zauderer, E. (1989) Partial Differential Equations of Applied Mathematics, 2nd ed., Wiley Interscience Publications, New York, USA.Google Scholar
[10] Leontovich, M. & Fock, V. (1945) Solution of the problem of propagation of electromagnetic waves along the Earth's surface by the method of the parabolic equation. J. Phys. 10, 13–24.Google Scholar
[11] Keller, J. B. (1956) Diffraction by a convex cylinder. Inst. Radio Eng.–Trans. Antennas and Propagation AP-4, 312–321.Google Scholar
[12] 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.Google Scholar
[13] Ockendon, J. R. & Tew, R. H. (2012) Thin-layer solutions of the Helmholtz and related equations. SIAM Rev. 54, 3–51.Google Scholar
[14] Chapman, S. J., Lawry, J. M. H., Ockendon, J. R. & Tew, R. H. (1999) On the theory of complex rays. SIAM Rev. 41, 417–509.Google Scholar
[15] Buchal, R. N. & Keller, J. B. (1960) Boundary layer problems in diffraction theory. Comm. Pure Appl. Math. 13, 85–114.Google Scholar
[16] Ludwig, D. (1966) Uniform asymptotic expansions at a caustic. Comm. Pure Appl. Math. 19, 215–250.Google Scholar