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Uniform Caustic Asymptotics Derived with Standard Integrals

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Caustics, Catastrophes and Wave Fields

Part of the book series: Springer Series on Wave Phenomena ((SSWAV,volume 15))

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Abstract

By representing a wave field in terms of standard caustic integrals one obtains only a local asymptotic solution. Augmenting the standard integral by its derivatives with respect to external parameters (Kravtsov-Ludwig technique) renders the asymptotic uniformly valid, that is, applicable both at short and at long distances from the caustic. We introduce the concepts of transverse and longitudinal caustic scales to derive uniformly-valid applicability conditions for the asymptotic expressions.

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References

  1. Yu.A. Kravtsov: Izv. VUZ Radiofiz. 8,659 (1965) [English transl.: Sov. Radiophys. 8,467 (1965)]

    Google Scholar 

  2. Yu.A. Kravtsov: Sov. Phys. - Acoust 14, 1 (1968)

    Google Scholar 

  3. Yu.A. Kravtsov: In Analiticheskie Methody v Teorii Difraktsii Rasprostraneniya Voln (analytic methods in diffraction and wave propagation theory) (Nauchnyi Soviet poAkustike, Akad, Nauk SSSR, Moscow 1970) pp. 258–362

    Google Scholar 

  4. G.B. Airy: Trans. Cambridge Philos. Soc. 6, 379 (1838)

    ADS  Google Scholar 

  5. R.N. Bûchai, J.B. Keller: Commun. Pure Appl. Math. 13, 85 (1960)

    Article  Google Scholar 

  6. Yu.L. Gazaryan: In Voprosy Dinamicheskoi Teorii Rasprostranenia Seismicheskikh Voln (topics in the dynamic theory of seismic waves), Vol. 5 (IZd. LGU, Leningrad 1961) p. 73

    Google Scholar 

  7. V.M. Babich, N.Y. Kirpichnikova: The Boundary-Layer Method in Diffraction Problems, Springer Ser. Electrophys. Vol. 3 (Springer, Berlin, Heidelberg 1979)

    Google Scholar 

  8. Yu.A. Kravtsov: Izv. VUZ Radiofiz. 7, 664 (1964) [English transl.: Radiophys. Quantum Electron. 7, no. 4, 104 (1964)]

    Google Scholar 

  9. C. Chester, B. Friedman, F. Ursell: Proc. Cambridge Philos. Soc. 53, 599 (1957)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  10. D. Ludwig: Commun. Pure Appl. Math. 19, 215 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  11. M.V. Berry: Proc. Philos. Soc. 89, 479 (1966)

    Article  ADS  MATH  Google Scholar 

  12. M.V. Berry: Sci. Prog. Oxf. 57, 43 (1969)

    Google Scholar 

  13. M.V. Berry, K.E. Mount: Rep. Prog. Phys. 35, 313 (1972)

    Article  ADS  Google Scholar 

  14. Yu.A. Kravtsov, Yu.I. Orlov: Sov. Phys. - Usp. 26, 1038 (1983)

    Article  MathSciNet  ADS  Google Scholar 

  15. Yu.A. Kravtsov: Izv. VUZ Radiofiz. 10, 1283 (1967) [English transl.: Radiophys. Quantum Electron. 10, 719 (1967)]

    Google Scholar 

  16. Yu.I. Orlov: Trudy MEI No. 194 (Moscow Power Engineering Institute, Moscow 1974) p. 103

    Google Scholar 

  17. V.M. Babich, V.S. Buldyrev: Short-Wavelength Diffraction Theory, Springer Ser. Wave Phenom., Vol. 4 (Springer, Berlin, Heidelberg 1991)

    Google Scholar 

  18. V.M. Babich, S.A. Egorov: In Vo prosy Dinamicheskoi Teorii Rasprostraneniya Seismic he skikh Voln (topics in the dynamic theory of seismic waves) (Izd. LGU, Leningrad 1983) Vol. 3, p. 4

    Google Scholar 

  19. Yu.A. Kravtsov: Radiophys. Quantum Electron. 7, no. 6, 40 (1964)

    Google Scholar 

  20. Yu.A. Kravtsov, Yu.I. Orlov: Geometrical Optics of Inhomogenous Media, Springer Ser. Wave Phenom., Vol. 6 (Springer, Berlin, Heidelberg 1990)

    Google Scholar 

  21. Yu.A. Kravtsov, L.A. Ostrovskii, N.S. Stepanov: Proc IEEE 62, 1492 (1974)

    Article  ADS  Google Scholar 

  22. V.Ya. Groshev, Yu.A. Kravtsov: Radiophys. Quantum Electron. 11, 1019 (1968)

    Article  ADS  Google Scholar 

  23. J.J. Dustermaat: Commun. Pure Appl. Math. 27, 207 (1974)

    Article  Google Scholar 

  24. M.V. Berry: Adv. Phys. 25, 1 (1976)

    Article  ADS  Google Scholar 

  25. G. Dangelmayr, W. Veit: Ann. Phys. 118, 108 (1979)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  26. A.S. Kryukovskii, D.S. Lukin: Radiotekhnika i Elekhtronika 26, 1121 (1981)

    ADS  Google Scholar 

  27. A.S. Kryukovskii, D.S. Lukin: In Voprosy Difraktsii Elecktromagnitnykh Voln (problems of electromagnetic waves diffraction) (MFTI, Moscow 1982) p. 40

    Google Scholar 

  28. A.S. Kryukovskii, D.S. Lukin: In Difraktsia and Rasprostranenie Elektromagnitnykh Voln (diffraction and propagation of electromagnetic waves) (MFTI, Moscow 1984) p. 39

    Google Scholar 

  29. A.S. Kryukovskii, D.S. Lukin, E.A. Palkin: Ravnomernye Asimptotiki Integralov ot Bystro Ostsilliruyushchikh Funktsii s Vyrozhdennymi Sedlovymy Tochkami (uniform asynptotics of integrals of rapidly oscillating functions with degenerated saddle points). Institute of radio Engineering and Electronics AN SSSR, preprint No. 41 (413) (IRE AN SSSR, Moscow 1984)

    Google Scholar 

  30. A.S. Kryukovskii, D.S. Lukin, E.A. Palkin: Izv. VUZ Radiofiz. 29, 79 (1986)

    ADS  Google Scholar 

  31. A.S. Kryukovskii, D.S. Lukin, E.A. Palkin: Sov. J. Numer. Anal, and Math. Model. 2, 219 (1987)

    MathSciNet  Google Scholar 

  32. A.S. Kryukovskii, D.S. Lukin, E.A. Palkin: Kraevye i Uglovye Katastrophy vZadachakh Difraktsii i Rasprostranenia Voln (the edge and corner catastrophes in problems of wave diffraction and propagation (Kazanskii Aviatsionnyi Institut, Kazan 1988)

    Google Scholar 

  33. A.S. Kryukovskii, D.S. Lukin, E.A. Palkin, D.V. Rastyagaev: In Trudy X Shkoly-Seminara po Difraktsii i Rasprostaneniyu Voln (proceedings of X school-seminar on wave diffraction and propagation) Moscow, Feb. 7–15, 1993 (MFTI, Moscow 1993) p. 36

    Google Scholar 

  34. E.B.Ipatov, A.S. Kryukovskii, D.S. Lukin, D.V. Rastyagaev: Radiotekhnika i Elektronika 39, 538 (1994)

    ADS  Google Scholar 

  35. A.S. Kryukovskii, Radiotechnika i Elektronika 41, 59 (1996)

    Google Scholar 

  36. A.S. Kryukovskii, D.V. Rastyagaev: In Rasprostranenie i Diffraktsia Voln v Neodnorodnykh Sredakh (waves propagation and diffraction in inhomogeneous media) (MFTI, Moscow 1989) p. 56

    Google Scholar 

  37. A.S. Kryukovskii: In Difraktsia i Rasprostranenie Eletromagnitnykh Voln (diffraction and propagation of electromagnetic waves) (MFTI, Moscow 1993) p. 4

    Google Scholar 

  38. A.S. Kryukovskii: In Problemy Difraktsii i Rasprostranenia Voln (problems of waves diffraction and propagation) (MFTI, Moscow 1993, P. 47

    Google Scholar 

  39. I.A. Vergizaev, A.S. Kryukovskii: In Problemy Rasprostranenia i Difraktsii Elektromagnitnykh Voln (problems of propagation and diffraction of electromagnetic waves) (MFTI, Moscow 1995) p. 39

    Google Scholar 

  40. J.B. Keller: A geometrical theory of diffraction, in Calculus of Variations and its Applications, Proc. Symp. Appl. Math., Vol. 8 (McGraw-Hill, New York 1958) pp. 27–52

    Google Scholar 

  41. D. Ludwig: SIAM Rev. 12, 325 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  42. F.W.J. Olver: Philos. Trans. Roy. Soc. London A 247, 328 (1954)

    Article  MathSciNet  ADS  Google Scholar 

  43. B.T. Kormilitsyn: Radio Eng. Electron. Phys. 11, 988 (1966)

    Google Scholar 

  44. Yu.I. Orlov: Sov. Radiosphys. 9, 307, 394 (1966)

    Google Scholar 

  45. Yu.I. Orlov: radiophys. Quantum Electron. 11, 180 (1968)

    Article  ADS  Google Scholar 

  46. Yu.I. Orlov: Radiophys. Quantum Electron. 10, 21 (1967)

    Article  ADS  Google Scholar 

  47. Yu.I. Orlov: Radiophys. Quantum Electron. 20, 1148 (1977)

    Article  ADS  Google Scholar 

  48. Yu.I. Orlov: Radiophys. Quantum Electron. 13, 320 (1970)

    Article  ADS  Google Scholar 

  49. Yu.I. Orlov, S.K. Tropkin: Radiophys. Quantum Electron. 23, 979 (1980)

    Article  ADS  Google Scholar 

  50. V.E. Grikurov: Radiophys. Quantum Electron. 23, 690 (1980)

    Article  MathSciNet  ADS  Google Scholar 

  51. A.S. Kryukovskii, D.S. Lukin, E.A. Palkin: Izv. VUZ Radiofiz. 25, 1375 (1982)

    Google Scholar 

  52. J.M. Martin, S.M. Flatté: Appl. Opt. 27, 2111 (1988)

    Article  ADS  Google Scholar 

  53. S.M.Rytov, Yu.A. Kravtsov; V.I. Tatarskii: Principles of Statistical Radiophysics, Vol. 4: Wave Propagation Through Random Media (Springer, Berlin, Heidelberg 1989)

    Google Scholar 

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Authors and Affiliations

  1. Space Research Institute, Russian Academy of Science, Profsoyyznaya 84/32, 117810, Moscow, Russia

    Yu. A. Kravtsov

  2. Power Engineering Institute, Moscow, Russia

    Yu. I. Orlov

Authors
  1. Yu. A. Kravtsov
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  2. Yu. I. Orlov
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© 1999 Springer-Verlag Berlin Heidelberg

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Kravtsov, Y.A., Orlov, Y.I. (1999). Uniform Caustic Asymptotics Derived with Standard Integrals. In: Caustics, Catastrophes and Wave Fields. Springer Series on Wave Phenomena, vol 15. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-59887-6_5

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  • DOI: https://doi.org/10.1007/978-3-642-59887-6_5

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-64167-1

  • Online ISBN: 978-3-642-59887-6

  • eBook Packages: Springer Book Archive

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