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: 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.
- Samuel H. Gray, A second order procedure for one-dimensional velocity inversion, SIAM J. Appl. Math. 39 (1980), no. 3, 456–462. MR 593682, DOI https://doi.org/10.1137/0139038
- Samuel H. Gray and Frank Hagin, Toward precise solution of one-dimensional velocity inverse problems, SIAM J. Appl. Math. 42 (1982), no. 2, 346–355. MR 650229, DOI https://doi.org/10.1137/0142027
- R. E. Meyer, Gradual reflection of short waves, SIAM J. Appl. Math. 29 (1975), no. 3, 481–492. MR 398292, DOI https://doi.org/10.1137/0129039
- R. E. Meyer, Exponential asymptotics, SIAM Rev. 22 (1980), no. 2, 213–224. MR 564565, DOI https://doi.org/10.1137/1022030
J. J. Mahony, The reflection of short waves in a variable medium, Quart. Appl. Math. 25, 313–316 (1967)
- A. Erdélyi, Asymptotic expansions, Dover Publications, Inc., New York, 1956. MR 0078494
- H. Bremmer, The W.K.B. approximation as the first term of a geometric-optical series, Comm. Pure Appl. Math. 4 (1951), 105–115. MR 44696, DOI https://doi.org/10.1002/cpa.3160040111
M. Foster, Transmission effects in a continuous one-dimensional seismic model, Geophysics J. Royal Astro. Soc. 42, 519–527 (1975)
N. Bleistein and R. Handelsman, Asymptotic expansions of integrals, Holt, Rinehart and Winston, New York (1975)
- Clive R. Chester and Joseph B. Keller, Asymptotic solution of systems of linear ordinary differential equations with discontinuous coefficients, J. Math. Mech. 10 (1961), 557–567. MR 0125266
S. H. Gray and N. Bleistein, One-dimensional velocity inversion for acoustic waves; numerical results, J. Acoust. Soc. Am. 67, 1141–1144 (1980)
S. H. Gray, A second-order procedure for one-dimensional velocity inversion, SIAM J. Appl. Math. 39, 456–462 (1980)
S. H. Gray and F. Hagin, Toward precise solution of one-dimensional velocity inverse problems, SIAM J. Appl. Math, (to appear)
R. E. Meyer, Gradual reflection of short waves, SIAM J. Appl. Math. 29, 481–492 (1975)
R. E. Meyer, Exponential asymptotics, SIAM Review 22, 213–224 (1980)
J. J. Mahony, The reflection of short waves in a variable medium, Quart. Appl. Math. 25, 313–316 (1967)
A. Erdelyi, Asymptotic expansions, Dover, New York, 1956
H. Bremmer, The W.K.B. approximation as the first term of a qeometric-optical series, Comm. Pure Appl. Math. 4, 105–115 (1951)
M. Foster, Transmission effects in a continuous one-dimensional seismic model, Geophysics J. Royal Astro. Soc. 42, 519–527 (1975)
N. Bleistein and R. Handelsman, Asymptotic expansions of integrals, Holt, Rinehart and Winston, New York (1975)
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)
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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Article copyright:
© Copyright 1982
American Mathematical Society
