Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-24T01:12:31.007Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic solutions of the Helmholtz equation: Generalised Friedlander–Keller ray expansions of fractional order

Part of: Asymptotic theory Optics, electromagnetic theory - General

Published online by Cambridge University Press:  18 September 2018

R. H. TEW
R. H. TEW*
Affiliation:
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Applications of a WKBJ-type ‘ray ansatz’ to obtain asymptotic solutions of the Helmholtz equation in the high-frequency limit are now standard and underpin the construction of ‘geometrical optics’ ray diagrams in many electromagnetic, acoustic and elastic reflection, transmission and other scattering problems. These applications were subsequently extended by Keller to include other types of rays – called ‘diffracted’ rays – to provide an accessible and impressively accurate theory which is relevant in wide-ranging sets of circumstances. Friedlander and Keller then introduced a modified ray ansatz to extend yet further the scope of ray theory and its applicability to certain other classes of diffraction problems (tangential ray incidence upon an obstructing boundary, for instance) and did so by the inclusion of an extra term proportional to a power of the wave number within the exponent of the initial ansatz. Our purpose here is to generalise this further still by the inclusion of several such terms, ordered in a natural sequence in terms of strategically chosen fractional powers of the large wave number, and to derive a systematic sequence of boundary value problems for the coefficient phase functions that arise within this generalised exponent, as well as one for the leading-order amplitude occurring as a pre-exponential factor. One particular choice of fractional power is considered in detail, and waves with specified radially symmetric or planar wavefronts are then analysed, along with a boundary value problem typifying two-dimensional radiation whereby arbitrary phase and amplitude variations are specified on a prescribed boundary curve. This theory is then applied to the scattering of plane and cylindrical waves at curved boundaries with small-scale perturbations to their underlying profile.

Keywords

Asymptotic methodsray theoryWKBJ method

MSC classification

Primary: 34E10: Perturbations, asymptotics
Secondary: 34E15: Singular perturbations, general theory 34E20: Singular perturbations, turning point theory, WKB methods 78A45: Diffraction, scattering
Type
Papers
Information
European Journal of Applied Mathematics , Volume 31 , Issue 1 , February 2020 , pp. 1 - 25
Copyright
© Cambridge University Press 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abboud, T. & Ammari, H. (1996) Diffraction at a curved grating: approximation by an infinite plane grating. J. Math. Anal. Appl. 202, 1076–1100.CrossRefGoogle Scholar
Achenbach, J. D., Gautesen, A. K. & McMaken, H. (1982) Ray Methods for Waves in Elastic Solids, Pitman Publishing, Inc., Marshfield, MA.Google Scholar
Bleistein, N. (1984) Mathematical Methods for Wave Phenomena, Academic Press, Orlando.Google Scholar
Buchal, R. N. & Keller, J. B. (1960) Boundary layer problems in diffraction theory. Commun. Pure Appl. Math. 13, 85–114.CrossRefGoogle Scholar
Friedlander, F. G. & Keller, J. B. (1955) Asymptotic solutions of (∇2 + k 2)u = 0. Commun. Pure Appl. Math. 6, 387–394.Google Scholar
Keller, J. B. (1962) Geometrical theory of diffraction. J. Opt. Soc. Am. A 52, 116–130.CrossRefGoogle ScholarPubMed
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, NY, pp. 1–82.CrossRefGoogle Scholar
Keller, J. B., Lewis, R. M. & Seckler, B. D. (1956) Asymptotic solutions of some diffraction problems. Commun. Pure Appl. Math. 9, 207–265.CrossRefGoogle Scholar
Kline, M. & Kay, I. W. (1965) Electromagnetic Theory and Geometrical Optics, John Wiley and Sons, Interscience, New York.Google Scholar
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
Levy, B. R. & Keller, J. B. (1959) Diffraction by a smooth object. Commun. Pure Appl. Math. 12, 159–209.CrossRefGoogle Scholar
Lewis, R. M., Bleistein, N. & Ludwig, D. (1967) Uniform asymptotic theory of creeping waves. Commun. Pure Appl. Math. 20, 295–328.CrossRefGoogle Scholar
Ludwig, D. (1966) Uniform asymptotic expansions at a caustic. Commun. Pure Appl. Math. 19, 215–250.CrossRefGoogle Scholar
Luneberg, R. K. (1965) Mathematical Theory of Optics, Cambridge University Press, Cambridge, UK.CrossRefGoogle Scholar
Molinet, F., Andronov, I. & Bouche, D. (2005) Asymptotic and Hybrid Methods in Electromagnetics. IEE Electromagnetic Waves Series, Vol. 51, The Institution of Electrical Engineers, London, UK.CrossRefGoogle Scholar
Rulf, B. (1968) Uniform asymptotic theory of diffraction at an interface. Commun. Pure Appl. Math. 21, 67–76.Google Scholar
Tew, R. H. (2018) Friedlander–Keller ray expansions and scalar wave reflection at canonically-perturbed boundaries. Euro. J. Appl. Math. 1–22. doi:10.1017/S0956792517000353.Google 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