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Uniform progressing wave solutions of the wave equation

Published online by Cambridge University Press:  24 October 2008

Erich Zauderer
Erich Zauderer
Affiliation:
Polytechnic Institute of Brooklyn, Brooklyn, New York 11201
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Extract

The solution of problems involving the propagation of discontinuities and other singularities for hyperbolic partial differential equations by means of progressing wave expansions is discussed in the book by Courant(l). He also refers to the work of Hadamard, Friedlander, Ludwig and others on this subject. More recently, Ludwig (2), Lewis(3) and others have considered 'uniform' progressing wave expansions for various problems. These expansions are valid in regions where the standard expansions are not suitable and they can be re-expanded in the standard form outside these regions. Examples of such regions are given by envelopes of bicharacteristic curves or, equivalently, caustics and by shadow boundaries such as occur in diffraction problems. In each of these regions, which we term 'transition regions' different types of uniform expansions are required.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 70 , Issue 3 , November 1971 , pp. 455 - 465
Copyright
Copyright © Cambridge Philosophical Society 1971

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References

REFERENCES

(1)Courant, R. and Hilbert, D.Methods of mathematical physics, vol. ii (New York, Inter-Science, 1962).Google Scholar
(2)Ludwig, D.Uniform asymptotic expansions at a caustic. Comm. Pure Appl. Math. 19 (1966), 215–250.CrossRefGoogle Scholar
(3)Ahluwalia, D. S. and Lewis, R. M.Diffraction of progressing waves by a screen. J. Inst. Math. Appl. 5 (1969), 113–139.CrossRefGoogle Scholar
(4)Zauderer, E.Uniform asymptotic solutions of the reduced wave equation. J. Math. Anal. Appl. 30 (1970), 157–171.CrossRefGoogle Scholar
(5)Wasow, W.Asymptotic Expansions for Ordinary Differential Equations (John Wiley, New York, 1965).Google Scholar
(6)Zauderer, E. Uniform progressing wave expansions. J. Math. Anal. Appl. (To appear.)Google Scholar
(7)Brekhovskikh, L. M.Waves in layered media (Academic Press, New York, 1960).Google Scholar
(8)Felsen, L. B. Lateral waves. Electromagnetic wave theory, part I, ed. Brown, J., pp. 11–44 (Pergamon Press, New York, 1967).CrossRefGoogle Scholar
(9)Hadamard, J.Lectures on Conchy's problem in linear partial differential equations (Dover Publications, New York, 1952).Google Scholar