Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 
 

 

A geometric-optical series and a WKB paradox


Author: Samuel H. Gray
Journal: Quart. Appl. Math. 40 (1982), 73-81
MSC: Primary 34E05; Secondary 58G15, 78A45
DOI: https://doi.org/10.1090/qam/652051
MathSciNet review: 652051
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Abstract | References | Similar Articles | Additional Information

Abstract: We discuss a solution of the one-dimensional reduced wave equation with non-constant velocity. We show that, for sufficiently small total velocity variations, this solution is exact. Furthermore, it lends itself to (high-frequency) asymptotic analysis and to elementary numerical analysis in a natural way. For reflected waves, we show that asymptotically small reflection implies numerically small reflection, thus resolving a paradox of classical WKB theory.


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  • [1] S. H. Gray, A second-order procedure for one-dimensional velocity inversion, SIAM J. Appl. Math. 39, 456-462 (1980) MR 593682
  • [2] S. H. Gray and F. Hagin, Toward precise solution of one-dimensional velocity inverse problems, SIAM J. Appl. Math, (to appear) MR 650229
  • [3] R. E. Meyer, Gradual reflection of short waves, SIAM J. Appl. Math. 29, 481-492 (1975) MR 0398292
  • [4] R. E. Meyer, Exponential asymptotics, SIAM Review 22, 213-224 (1980) MR 564565
  • [5] J. J. Mahony, The reflection of short waves in a variable medium, Quart. Appl. Math. 25, 313-316 (1967)
  • [6] A. Erdelyi, Asymptotic expansions, Dover, New York, 1956 MR 0078494
  • [7] H. Bremmer, The W.K.B. approximation as the first term of a qeometric-optical series, Comm. Pure Appl. Math. 4, 105-115 (1951) MR 0044696
  • [8] M. Foster, Transmission effects in a continuous one-dimensional seismic model, Geophysics J. Royal Astro. Soc. 42, 519-527 (1975)
  • [9] N. Bleistein and R. Handelsman, Asymptotic expansions of integrals, Holt, Rinehart and Winston, New York (1975)
  • [10] C. R. Chester and J. B. Keller, Asymptotic solution of systems of linear ordinary differential equations with discontinuous coefficients, J. Math. Mech. 10, 557-567 (1961) MR 0125266
  • [11] S. H. Gray and N. Bleistein, One-dimensional velocity inversion for acoustic waves; numerical results, J. Acoust. Soc. Am. 67, 1141-1144 (1980)

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DOI: https://doi.org/10.1090/qam/652051
Article copyright: © Copyright 1982 American Mathematical Society

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