High frequency waves near cusp caustics
Authors:
E. Kalligiannaki, Th. Katsaounis and G. N. Makrakis
Journal:
Quart. Appl. Math. 61 (2003), 111-129
MSC:
Primary 78A05; Secondary 35Q60, 76Q05, 78A40
DOI:
https://doi.org/10.1090/qam/1955226
MathSciNet review:
MR1955226
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Abstract: It is well known that the usual harmonic ansatz of geometrical optics fails near caustics. However, uniform expansions exist which are valid near and on the caustics, and reduce asymptotically to the usual geometric field far enough from them. In this paper, we apply the Kravtsov-Ludwig technique for computing high-frequency fields near cusp caustics. We compare these fields, with those predicted by the geometrical optics, for a couple of model problems: first, the cusp generated by the evolution of a parabolic initial front in a homogeneous medium, a problem which arises in the high-frequency treatment of cylindrical aberrations, and second, the cusp formed by refraction of the rays emitted from a point source in a stratified medium with a weak interface. It turns out that inside and near the cusp, the geometrical optics solution is significantly different than the Kravtsov-Ludwig solution, but far enough from the caustic that the two solutions are, in fact, in very good agreement.
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G. Avila and J. Keller, The high-frequency asymptotic field of a point source in an inhomogeneous medium, Comm. Pure Appl. Math. XVI (1963), 363–381.
A. A. Asatryan and Yu. A. Kravtsov, Longitudinal caustic scale and boundaries of applicability of uniform Airy-asymptotics, Wave Motion 19 (1994), 1–10.
V. I. Arnold, A. N. Varchenko and S. M. Husein-Zade, Singularities of Differentiable Maps, Vol. 1, Birkhäuser Verlag, Basel, 1985.
V. B. Babich, The short wave asymptotic form of the solution for the problem of a point source in an inhomogeneous medium, USSR J. Comp. Math. and Math. Phys. 5(5) (1965), 949–951.
N. Bleistein and R. Handelsman, Asymptotic Expansions of Integrals, Dover Publications Inc., New York, 1986.
R. N. Buchal and J. B. Keller, Boundary layer problems in diffraction theory, Comm. Pure Appl. Math. 13 (1960), 85–144.
V. A. Borovikov, Uniform stationary phase methods, The Institute of Electrical Engineers, London, 1994.
L. M. Brekhovskikh, Waves in Layered Media, Academic Press, New York, 1980.
M. Brown and F. Tappert, Catastrophe theory, caustics and traveltime diagrams in seismology, Geophys. J. R. Astr. Soc. 88 (1987), 217–229.
V. Cěrvenỳ, I. A. Molotkov, and I. Pšenčik, Ray Method in Seismology, Univerzita Karlova, Praha, 1977.
C. H. Chapman and R. Drummond, Body-wave seismograms in inhomogeneous media using Maslov asymptotic theory, Bull. Seism. Soc. Amer. 72(6) (1982), S277–S317.
C. H. Chapman, Ray theory and its extensions: WKBJ and Maslov seismograms, J. Geophys. 58 (1985), 27–43.
D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering, Springer-Verlag, New York, 1992.
J. N. L. Connor and D. Farrelly, Molecular collisions and cusp catastrophes: Three methods for the calculation of Pearcey’s integral and its derivatives, Chem. Phys. Let. 81(2) (1981), 306–310.
J. N. L. Connor and D. Farrelly, Theory of cusped rainbows in elastic scattering: Uniform semiclassical calculations using Pearcey’s integral, J. Chem. Phys. 75(6) (1981), 2831–2846.
J. J. Duistermaat, Oscillatory integrals, Lagrangian immersions and unfolding of singularities, Comm. Pure Appl. Math XXVII (1974), 207–281.
L. Gárding, Singularities in Linear Wave Propagation, Lecture Notes in Math. 1241, Springer-Verlag, Berlin, 1987.
V. Guillemin and S. Sternberg, Geometric Asymptotics, Amer. Math. Soc., Providence, R.I., 1977.
V. Guillemin and D. Schaeffer, Remarks on a paper of D. Ludwig, Bull. Amer. Math. Soc. 79(2) (1973), 382–385.
X. Huang and G. West, Effects of weighting functions on Maslov uniform seismograms: A robust weighting method, Bull. Seism. Soc. Amer. 87(1) (1997), 164–173.
L. Hormander, The Analysis of Linear Partial Differential Operators I, Springer-Verlag, New York, 1983.
S. Izumiya, G. T. Kossioris, and G. N. Makrakis, Multivalued solutions to the eikonal equation in stratified media, (submitted).
E. Kalligiannaki, Computation of high frequency fields near cusp caustics, Diploma Thesis, Dept. Math., Univ. Crete (Feb. 1999).
D. Kaminski, Asymptotic expansion of the Pearcey integral near the caustic, SIAM J. Math. Anal. 20(4) (1989), 987–1005.
T. Katsaounis, G. T. Kossioris and G. N. Makrakis, Computation of high frequency fields near caustics, Math. Models Methods in Appl. Sci. 11 (2001), 199–228.
Yu. A. Kravtsov and Yu. I. Orlov, Caustics, Catastrophes and Wave Fields, Springer Series on Wave Phenomena, Springer-Verlag, Berlin, 1993.
Yu. A. Kravtsov and Yu. I. Orlov, Geometrical Optics of Inhomogeneous Media, Springer Series on Wave Phenomena 6, Springer-Verlag, Berlin, 1990.
Yu. A. Kravtsov, Two new asymptotic methods in the theory of wave propagation in inhomogeneous media (review), Sov. Phys. Acoust. 14(1) (1968), 1–17.
V. V. Kucherenko, Quasiclassical asymptotics of a point-source function for the stationary Schrödinger equation, Theoret. Math. Phys. (English Translation) 1(3) (1969), 294–310.
R. M. Lewis, Asymptotic theory of wave propagation, Arch. Rat. Mech. Anal. 20 (1965), 191–250.
N. N. Lebedev, Special Functions and Their Applications, Dover Publications, Inc., New York, 1972.
D. Ludwig, Uniform asymptotic expansions at a caustic, Comm. Pure Appl. Math. XIX (1966), 215–250.
I. Tolstoy and C. S. Clay, Ocean Acoustics. Theory and Experiment in Underwater Sound, American Institute of Physics, New York, 1966.
B. R. Vainberg, Quasiclassical approximation in stationary scattering problems, Func. Anal. Appl. 11 (1977), 247–257.
B. R. Vainberg, Asymptotic Methods in Equations of Mathematical Physics, Gordon and Breach, New York, 1989.
R. Weder, Spectral and Scattering Theory for Wave Propagation in Perturbed Stratified Media, Springer-Verlag, New York, 1991.
R. Wong, Asymptotic Approximations of Integrals, Academic Press, Inc., New York, 1989.
R. W. Ziolkowski and G. A. Deschamps, Asymptotic evaluation of high-frequency fields near a caustic: An introduction to Maslov’s method, Radio Sci. 19(4) (1984), 1001–1025.
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