Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-25T03:03:31.218Z Has data issue: false hasContentIssue false
  • English
  • Français

The asymptotic analysis of canonical problems in high-frequency scattering theory

I. Stratified media above a plane boundary

Published online by Cambridge University Press:  24 October 2008

W. G. C. Boyd
W. G. C. Boyd
Affiliation:
Department of Mathematics, University of Dundee, Dundee DD1 4HN
Get access Rights & Permissions [Opens in a new window]

Abstract

The asymptotic treatment of high-frequency scalar wave problems has in the past been rather unsatisfactory. Typically, the integral representations which arose were evaluated by stationary phase, or as a series of residues. The justification of these methods was usually heuristic and formal. In this paper, a method is advanced which, it is claimed, may be applied to any one-parameter separation of variables problem. The method assumes an integral representation whose contour of integration is the real axis. It is then only necessary to deform this contour in the neighbour-hood of the real axis to derive rigorous asymptotic expansions of the field in both the illumination and shadow. The method is applied to the particular example of scattering by a plane boundary in a general stratified medium with monotonically increasing refractive index.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 74 , Issue 2 , September 1973 , pp. 289 - 312
Copyright
Copyright © Cambridge Philosophical Society 1973

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

REFERENCES

(1)Bleistein, N. and Handelsman, R. A.A generalization of the method of steepest descent. J. Inst. Math. Appl. 10 (1972), 211–230.CrossRefGoogle Scholar
(2)Boyd, W. G. C.Creeping waves in an inhomogeneous medium. Proc. Roy. Soc. Edinburgh Sect. A 69 (1971), 227–245.Google Scholar
(3)Boyd, W. G. C. On high-frequency scattering in a certain stratified medium. Ph.D. Thesis, University of Dundee (1972).Google Scholar
(4)Boyd, W. G. C.High-frequency scattering in a certain stratified medium. Proc. Roy. Soc. Edinburgh Sect. A. (To be published.)Google Scholar
(5)Boyd, W. G. C.The asymptotic analysis of canonical problems in high-frequency scattering theory. H. The circular and parabolic cylinders. Proc. Cambridge Philos. Soc. 74 (1973), 313–332.CrossRefGoogle Scholar
(6)Copson, E. T.An introduction to the theory of functions of a complex variable (Oxford University Press, 1935).Google Scholar
(7)Erdélyi, A.Asymptotic expansions (Dover; New York, 1956).Google Scholar
(8)Hille, E.Lectures on ordinary differential equations (Addison-Wesley; Reading, Massachusetts, 1969).Google Scholar
(9)Jones, D. S.High-frequency refraction and diffraction in general media. Philos. Trans. Roy. Soc. London Ser. A 255 (1963), 341–387.Google Scholar
(10)Jones, D. S.The theory of electromagnetism (Pergamon; Oxford, 1964).Google Scholar
(11)Leppington, F. G.Creeping waves in the shadow of an elliptic cylinder. J. Inst. Math. Appl. 3 (1967), 388–402.CrossRefGoogle Scholar
(12)Seckler, B. D. and Keller, J. B.Asymptotic theory of diffraction in inhomogeneous media. J. Acoust. Soc. Amer. 31 (1959), 206–216.CrossRefGoogle Scholar
(13)Sleeman, B. D.On diffraction at short wavelengths by a prolate spheroid. J. Inst. Math. Appl. 5 (1969), 432–442.CrossRefGoogle Scholar
(14)Titchmarsh, E. C.The theory of functions (Oxford University Press; 2nd edition, 1939).Google Scholar
(15)Ursell, F.Creeping modes in a shadow. Proc. Cambridge Philos. Soc. 64 (1968), 171–191.CrossRefGoogle Scholar
(16)Wait, J. R.Electromagnetic waves in stratified media (Pergamon; Oxford, 2nd edition, 1970).Google Scholar
(17)Wasow, W.Asymptotic expansions for ordinary differential equations (Interscience; New York, 1965).Google Scholar