The parabolic approximation method

  • Fred D. Tappert
Part of the Lecture Notes in Physics book series (LNP, volume 70)


This article has dealt with various aspects of parabolic approximation methods in underwater acoustics, mostly for propagation of sinusoidal signals. Extensions of these methods to time-dependent problems are also available: pulse propagation, moving sources and receivers, frequency shifting effects due to rapid temporal variations of oceanic conditions, and so forth. However, an adequate description of these extensions would require another long section and it was felt that the principles involved in making parabolic approximations have been sufficiently illustrated. Parabolic equation methods in underwater acoustics were developed only in the last few years, and as more and more use is made of these methods we may expect that many of the important modelling problems in ocean acoustics may be solved.


Parabolic Equation Sound Speed Acoustic Field Acoustic Model Parabolic Approximation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [1]
    M. Leontovich and V. Fock, Zh. Eks. Teor. Fix. 16 (1946), 557–573 [J. Phys. USSR 10 (1946), 13–24; also Ref. [2], Chap. 11.]Google Scholar
  2. [2]
    V. A. Fock, Electromagnetic Diffraction and Propagation Problems, Pergamon Press, 1965 [Chaps. 11, 13, 14].Google Scholar
  3. [3]
    G. D. Malyuzhinets, Usp. Fiz. Nauk 69 (1959), 321–334 [Sov. Phys. Usp. 2 (1959), 749–758].Google Scholar
  4. [4]
    L. A. Vainshtein, Zh. Techn. Fiz. 34 (1964), 193 [Sov. Phys. Techn. Phys. 9 (1964), 157].Google Scholar
  5. [5]
    L. A. Vainshtein, Open Resonators and Waveguides, Golem Press, Boulder, Colorado, 1969.Google Scholar
  6. [6]
    P. L. Kelley, Phys. Rev. Lett. 15 (1965), 1005.Google Scholar
  7. [7]
    V. I. Talanov, Zh. E. T. Fiz. Pis. Red. 2 (1965), 218 [JETP lett. 2 (1965), 138].Google Scholar
  8. [8]
    A. Hasegawa and F. D. Tappert, Appl. Phys. Lett. 23 (1973), 142–144; 23 (1973), 171–172.Google Scholar
  9. [9]
    O. Svelto, in Progress in Optics, Vol. 12 (ed. E. Wolf), North-Holland Pub., 1974 [Chap. I, pp. 1–51].Google Scholar
  10. [10]
    J. H. Marburger, Prog. Quant. Elect. 4 (1975), 35–110.Google Scholar
  11. [11]
    V. I. Karpman, Nonlinear Waves in Dispersive Media, Pergamon Press, 1975.Google Scholar
  12. [11a]
    V. E. Zakharov, Zh. Eksp. Teor. Fiz. 62 (1972), 1745–1759 [Sov. Phys. JETP 35 (1972), 908–914].Google Scholar
  13. [12]
    G. Morales and Y. C. Lee, Phys. Rev. Lett. 33 (1974), 1016–1019.Google Scholar
  14. [13]
    H. H. Chen and C. S. Liu, Phys. Rev. Lett. 37 (1976), 693–697Google Scholar
  15. [14]
    F. D. Tappert and C. N. Judice, Phys. Rev. Lett. 29 (1972), 1308–1311.Google Scholar
  16. [15]
    V. I. Tatarskii, Zh. Eksp. Teor. Fiz. 56 (1969), 2106 [Sov. Phys. JETP 29 (1969), 1133].Google Scholar
  17. [16]
    L. A. Chernov, Sov. Phys. Acoust. 15 (1970), 511.Google Scholar
  18. [17]
    V. I. Klyatskin and V. I. Tatarskii, Zh. Eksp. Teor. Fiz. 58 (1970), 624–634 [Sov. Phys. JETP 31 (1970), 335–339].Google Scholar
  19. [18]
    Yu. N. Barabanenkov, Yu. A. Kravtsov, S. N. Rytov, and V. I. Tatarskii, Sov. Phys. Uspekhi 13 (1971), 551–580.Google Scholar
  20. [19]
    P. L. Chow, J. Math. Phys. 13 (1972), 1224–1236.Google Scholar
  21. [20]
    G. C. Papanicolaou, D. McLaughlin, and R. Burridge, J. Math. Phys. 14 (1973), 84–89.Google Scholar
  22. [21]
    I. M. Besieris and F. D. Tappert, J. Math. Phys. 14 (1973), 1829–1836.Google Scholar
  23. [22]
    R. H. Hardin and F. D. Tappert, “Analysis, simulation, and models of ionospheric scintillation”, Joint Radar Propagation Study, Bell Lab Report, March 1974, 76pp.Google Scholar
  24. [23]
    J. F. Claerbout, Geophys. 35 (1970), 407–418.Google Scholar
  25. [24]
    J. F. Claerbout, “Numerical holography”, in Acoustical Holography, Vol. 3, ed. A. F. Metherell, Plenum Press, N. Y., 1971, pp. 273–283.Google Scholar
  26. [24a]
    J. F. Claerbout, Fundamentals of Geophysical Data Processing, McGraw-Hill, N N.Y., 1976.Google Scholar
  27. [25]
    H. Bremmer, Comm. Pure Appl. Math. 4 (1951), 105–115.Google Scholar
  28. [26]
    H. Bremmer, Radio Sci. 8 (1973), 511–534.Google Scholar
  29. [27]
    P.v.d. Woude and H. Bremmer, Radio Sci. 10 (1975), 23–28.Google Scholar
  30. [28]
    F. W. Sluijter, J. Opt. Soc. Am. 60 (1970), 8.Google Scholar
  31. [29]
    J. Corones, J. Math. Anal. Appl. 50 (1975), 361–372.Google Scholar
  32. [30]
    P. G. Bergman, J. Acoust. Soc. Am. 17 (1946), 329.Google Scholar
  33. [31]
    L. M. Brekhovskikh, Waves in Layered Media, Academic Press, N. Y., 1970.Google Scholar
  34. [32]
    I. Tolstoy and C. S. Clay, Ocean Acoustics, McGraw-Hill, N. Y., 1966.Google Scholar
  35. [33]
    R. J. Urick, Principles of Underwater Sound, McGraw-Hill, N. Y., 1975.Google Scholar
  36. [34]
    P. R. Tatro and C. W. Spofford, Proc. IEEE Ocean Environment Conf. (1973), pp. 206–216.Google Scholar
  37. [35]
    F. D. Tappert and R. H. Hardin, in “A synopsis of the AESD workshop on acoustic modeling by non-ray tracing techniques”, C. W. Spofford, AESD TN-73-05, Arlington, Va. (1973).Google Scholar
  38. [36]
    R. H. Hardin and F. D. Tappert, SIAM Rev. (Chronicle) 15 (1973), 423; F. D. Tappert, SIAM Rev. (Chronicle) 16 (1974), 140.Google Scholar
  39. [37]
    F. D. Tappert, J. Acoust. Soc. Am. 55 (1974), 534.Google Scholar
  40. [38]
    F. D. Tappert and R. H. Hardin, Proc. Eighth Inter. Congress on Acoustics (London, 1974), Vol. 2, p. 452.Google Scholar
  41. [39]
    G. W. Benthien, D. F. Gordon, and L. E. McCleary, J. Acoust. Soc. Am. 55 (1974), 545.Google Scholar
  42. [40]
    S. M. Flatté and F. D. Tappert, J. Acoust. Soc. Am. 58 (1975), 1151–1159.Google Scholar
  43. [41]
    C. Garrett and W. H. Munk, Geophys. Fluid Dyn. 2 (1972), 225–264.Google Scholar
  44. [42]
    W. H. Munk, J. Acoust. Soc. Amer. 55 (1974), 220–226.Google Scholar
  45. [43]
    S. T. McDaniel, J. Acoust. Soc. Amer. 57 (1975), 307–311.Google Scholar
  46. [44]
    S. T. McDaniel, J. Acoust. Soc. Amer. 58 (1975), 1178–1185.Google Scholar
  47. [45]
    A. O. Williams, Jr., J. Acoust. Soc. Amer. 58 (1975), 1320–1321.Google Scholar
  48. [46]
    R. M. Fitzgerald, J. Acoust. Soc. Amer. 57 (1975), 839–842.Google Scholar
  49. [47]
    E. A. Polyanskii, Sov. Phys. Acoust. 20 (1974), 90.Google Scholar
  50. [48]
    J. A. DeSanto, “Connection between the solutions of the Helmholtz and parabolic equations for sound propagation”, Proc. SACLANTCEN Conf. (1975).Google Scholar
  51. [49]
    H. K. Brock, “The AESD parabolic equation model”, AESD TN-75-07, ONR, Arlington, Va. (1975).Google Scholar
  52. [50]
    H. K. Brock, R. N. Buchal, and C. W. Spofford, “Modifying the sound speed profile to improve the accuracy of the parabolic equation technique”, AESD Tech. Memo (1975).Google Scholar
  53. [51]
    D. R. Palmer, J. Acoust. Soc. Am. 60 (1976), 343–354.Google Scholar
  54. [52]
    J. S. Hanna, J. Acoust. Soc. Am. 60 (1976), 1024–1031.Google Scholar
  55. [53]
    R. N. Buchal and F. D. Tappert, “A variable range step in the split-step Fourier algorithm”, AESD Tech. Memo (1975).Google Scholar
  56. [54]
    F. Jensen and H. Krol, “The use of the parabolic equation method in sound propagation modelling”, SACLANTCEN Memo SM-72 (1975).Google Scholar
  57. [55]
    P. M. Volk, “Solutions of acoustic wave propagation in the ocean by the parabolic approximation”, Thesis, Univ. of Hawaii, Dept. of Oceanography (1975).Google Scholar
  58. [56]
    D. J. Thomson, “Parabolic equation acoustic model”, Defense Research Establishment Pacific Report (1975).Google Scholar
  59. [57]
    G. A. Kreigsmann and E. N. Larsen, “On the parabolic approximation to the reduced wave equation”, SIAM J. Appl. Math. (to appear).Google Scholar
  60. [58]
    H. F. Harmuth, J. Math. Phys. (MIT) 36 (1957), 269–278.Google Scholar
  61. [59]
    A. V. Popov, Akust. Zh. 15 (1969), 265–274 [Sov. Phys. Acoust. 15 (1969), 226–233].Google Scholar
  62. [60]
    A. L. Dyshko, Zh. vychisl. Mat. mat. Fiz. 8 (1968), 238–242 [USSR Comput. Math. Phys. 8 (1968), 340–346].Google Scholar
  63. [61]
    A. Goldberg, H. M. Schey, and J. L. Schwartz, Amer. J. Phys. 35 (1967), 177.Google Scholar
  64. [62]
    R. D. Richtmyer and K. W. Morton, Difference Methods for Initial-Value Problems, Second Ed., Interscience Pub., N. Y., 1967.Google Scholar
  65. [63]
    T. Talpay, Bell Laboratories Report, 1972 (unpublished).Google Scholar
  66. [64]
    R. L. Holford and C. W. Spofford, Bell Laboratories Report, Long Range Acoustic Propagation Project, April 1973 (unpublished).Google Scholar
  67. [65]
    R. W. Hamming, Numerical Methods for Scientists and Engineers, 2nd Ed., McGraw-Hill, N.Y., 1973.Google Scholar

Copyright information

© Springer-Verlag 1977

Authors and Affiliations

  • Fred D. Tappert
    • 1
  1. 1.Courant Institute of Mathematical SciencesNew York UniversityNew York

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