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Computational Ocean Acoustics pp 457–529Cite as

Parabolic Equations

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Part of the book series: Modern Acoustics and Signal Processing ((MASP))

Abstract

The pioneering work on parabolic wave equations goes back to the mid-1940s when Leontovich and Fock [1] applied a PE method to the problem of radio wave propagation in the atmosphere. Since then, parabolic equations have been used in several branches of physics, including the fields of optics, plasma physics, seismics, and underwater acoustics. It is the application of PE methods to wave-propagation problems in the ocean that is the subject of this chapter.

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References

  1. M.A. Leontovich, V.A. Fock, Zh. Eksp. Teor. Fiz. 16, 557–573 (1946) [Engl. transl.: J. Phys. USSR 10, 13–24 (1946)]

    Google Scholar 

  2. R.H. Hardin, F.D. Tappert, Applications of the split-step Fourier method to the numerical solution of nonlinear and variable coefficient wave equations. SIAM Rev. 15, 423 (1973)

    Google Scholar 

  3. F.D. Tappert, The parabolic approximation method. in Wave Propagation in Underwater Acoustics, ed. by J.B. Keller, J.S. Papadakis (Springer, New York, 1977), pp. 224–287

    Google Scholar 

  4. J.A. Davis, D. White, R.C. Cavanagh, NORDA parabolic equation workshop. Rep. TN-143. Naval Ocean Research and Development Activity, Stennis Space Center, MS, 1982

    Google Scholar 

  5. D. Lee, S.T. McDaniel, Ocean Acoustic Propagation by Finite Difference Methods (Pergamon, New York, 1988)

    MATH  Google Scholar 

  6. A.P. Rosenberg, A new rough surface parabolic equation program for computing low-frequency acoustic forward scattering from the ocean surface. J. Acoust. Soc. Am. 105, 144–153 (1999)

    ADS  Google Scholar 

  7. J.F. Claerbout, Fundamentals of Geophysical Data Processing (Blackwell, Oxford, 1985), pp. 194–207

    Google Scholar 

  8. R.R. Greene, The rational approximation to the acoustic wave equation with bottom interaction. J. Acoust. Soc. Am. 76, 1764–1773 (1984)

    ADS  Google Scholar 

  9. D.F. St. Mary, Analysis of an implicit finite difference scheme for very wide angle underwater acoustic propagation. in Proceedings of the 11th IMACS World Congress on System Simulation and Scientific Computing (Oslo, Norway, 1985), pp. 153–156

    Google Scholar 

  10. G.H. Knightly, D. Lee, D.F. St. Mary, A higher-order parabolic wave equation. J. Acoust. Soc. Am. 82, 580–587 (1987)

    Google Scholar 

  11. A. Bamberger, B. Engquist, L. Halpern, P. Joly, Higher order parabolic wave equation approximations in heterogeneous media. SIAM J. Appl. Math. 48, 129–154 (1988)

    MathSciNet  MATH  Google Scholar 

  12. M.D. Collins, Applications and time-domain solution of higher-order parabolic equations in underwater acoustics. J. Acoust. Soc. Am. 86, 1097–1102 (1989)

    ADS  Google Scholar 

  13. L. Halpern, L.N. Trefethen, Wide-angle one-way wave equations. J. Acoust. Soc. Am. 84, 1397–1404 (1988)

    ADS  MathSciNet  Google Scholar 

  14. M.D. Feit, J.A. Fleck Jr., Light propagation in graded-index fibers. Appl. Opt. 17, 3990–3998 (1978)

    ADS  Google Scholar 

  15. D.J. Thomson, N.R. Chapman, A wide-angle split-step algorithm for the parabolic equation. J. Acoust. Soc. Am. 74, 1848–1854 (1983)

    ADS  MATH  Google Scholar 

  16. D.H. Berman, E.B. Wright, R.N. Baer, An optimal PE-type wave equation. J. Acoust. Soc. Am. 86, 228–233 (1989)

    ADS  MathSciNet  Google Scholar 

  17. L. Nghiem-Phu, F.D. Tappert, Modeling of reciprocity in the time domain using the parabolic equation method. J. Acoust. Soc. Am. 78, 164–171 (1985)

    ADS  MATH  Google Scholar 

  18. S.T. McDaniel, Propagation of normal mode in the parabolic approximation. J. Acoust. Soc. Am. 57, 307–311 (1975)

    ADS  Google Scholar 

  19. R.J. Hill, Wider-angle parabolic wave equation. J. Acoust. Soc. Am. 79, 1406–1409 (1986)

    ADS  MathSciNet  Google Scholar 

  20. M.D. Collins, A higher-order parabolic equation for wave propagation in an ocean overlying an elastic bottom. J. Acoust. Soc. Am. 86, 1459–1464 (1989)

    ADS  Google Scholar 

  21. M.D. Collins, Higher-order parabolic approximations for accurate and stable elastic parabolic equations with application to interface wave propagation. J. Acoust. Soc. Am. 89, 1050–1057 (1991)

    ADS  Google Scholar 

  22. B.T.R. Wetton, G.H. Brooke, One-way wave equations for seismoacoustic propagation in elastic waveguides. J. Acoust. Soc. Am. 87, 624–632 (1990)

    ADS  MathSciNet  Google Scholar 

  23. H. Kolsky, Stress Waves in Solids (Dover, New York, 1963), p. 11

    MATH  Google Scholar 

  24. R.R. Greene, A high-angle one-way wave equation for seismic wave propagation along rough and sloping interfaces. J. Acoust. Soc. Am. 77, 1991–1998 (1985)

    ADS  MATH  Google Scholar 

  25. J. Miklowitz, The Theory of Elastic Waveguides (North-Holland, Amsterdam, The Netherlands, 1984), p. 198

    Google Scholar 

  26. M.D. Collins, W.L. Siegmann, A complete energy conserving correction for the elastic parabolic equation. J. Acoust. Soc. Am. 104, 687–692 (1999)

    ADS  Google Scholar 

  27. W. Jerzak, W.L. Siegmann, M.D. Collins, Modeling Rayleigh and Stoneley waves and other interface and boundary effects with the parabolic equation. J. Acoust. Soc. Am. 117, 3497–3503 (2005)

    ADS  Google Scholar 

  28. D.A. Outing, W.L. Siegmann, M.D. Collins, E.K. Westwood, Generalization of the rotated parabolic equation to variable slopes. J. Acoust. Soc. Am. 120, 3534–3538 (2006)

    ADS  Google Scholar 

  29. S.A. Chin-Bing, D.B. King, J.A. Davis, R.B. Evans, PE Workshop II: Proceedings of the Second Parabolic Equation Workshop (Naval Research Laboratory, Washington, DC, 1993)

    Google Scholar 

  30. C.T. Tindle, H. Hobaek, T.G. Muir, Normal mode filtering for downslope propagation in a shallow water wedge. J. Acoust. Soc. Am. 81, 287–294 (1987)

    ADS  Google Scholar 

  31. M.D. Collins, A self starter for the parabolic equation method. J. Acoust. Soc. Am. 92, 2069–2074 (1992)

    ADS  Google Scholar 

  32. M.D. Collins, A stabilized self starter. J. Acoust. Soc. Am. 106, 1724–1726 (1999)

    ADS  Google Scholar 

  33. H.K. Brock, The AESD parabolic equation model. Rep. TN-12. Naval Ocean Research and Development Activity, Stennis Space Center, MS, 1978

    Google Scholar 

  34. D.J. Thomson, Wide-angle parabolic equation solutions to two range-dependent benchmark problems. J. Acoust. Soc. Am. 87, 1514–1520 (1990)

    ADS  Google Scholar 

  35. M.D. Collins, A split-step Padé solution for the parabolic equation method. J. Acoust. Soc. Am. 93, 1736–1742 (1993)

    ADS  Google Scholar 

  36. P.G. Bergman, The wave equation in a medium with a variable index of refraction. J. Acoust. Soc. Am. 17, 329–333 (1946)

    ADS  Google Scholar 

  37. D. Huang, Finite element solution to the parabolic wave equation. J. Acoust. Soc. Am. 84, 1405–1413 (1988)

    ADS  Google Scholar 

  38. G. Botseas, D. Lee, K.E. Gilbert, IFD: Wide angle capability. Rep. TR-6905. Naval Underwater Systems Center, New London, CT, 1983

    Google Scholar 

  39. S.T. McDaniel, D. Lee, A finite-difference treatment of interface conditions for the parabolic wave equation: The horizontal interface. J. Acoust. Soc. Am. 71, 855–858 (1982)

    ADS  MathSciNet  MATH  Google Scholar 

  40. F.B. Jensen, C.M. Ferla, Numerical solutions of range-dependent benchmark problems in ocean acoustics. J. Acoust. Soc. Am. 87, 1499–1510 (1990)

    ADS  Google Scholar 

  41. M.B. Porter, F.B. Jensen, C.M. Ferla, The problem of energy conservation in one-way models. J. Acoust. Soc. Am. 89, 1058–1067 (1991)

    ADS  Google Scholar 

  42. M.D. Collins, E.K. Westwood, A higher-order energy-conserving parabolic equation for range-dependent ocean depth, sound speed, and density. J. Acoust. Soc. Am. 89, 1068–1075 (1991)

    ADS  Google Scholar 

  43. D. Lee, S.T. McDaniel, A finite-difference treatment of interface conditions for the parabolic wave equation: The irregular interface. J. Acoust. Soc. Am. 73, 1441–1447 (1983)

    ADS  MATH  Google Scholar 

  44. R.N. Baer, Propagation through a three-dimensional eddy including effects on an array. J. Acoust. Soc. Am. 69, 70–75 (1981)

    ADS  Google Scholar 

  45. J.S. Perkins, R.N. Baer, An approximation to the three-dimensional parabolic-equation method for acoustic propagation. J. Acoust. Soc. Am. 72, 515–522 (1982)

    ADS  Google Scholar 

  46. W.L. Siegmann, G.A. Kriegsmann, D. Lee, A wide-angle three-dimensional parabolic wave equation. J. Acoust. Soc. Am. 78, 659–664 (1985)

    ADS  MathSciNet  Google Scholar 

  47. A. Tolstoy, 3-D propagation issues and models. J. Comput. Acoust. 4, 243–271 (1996)

    MATH  Google Scholar 

  48. F. Sturm, Numerical study of broadband sound pulse propagation in three-dimensional oceanic waveguides. J. Acoust. Soc. Am. 117, 1058–1079 (2005)

    ADS  Google Scholar 

  49. D. Lee, G. Botseas, W.L. Siegmann, Examination of three-dimensional effects using a propagation model with azimuth-coupling capability (FOR3D). J. Acoust. Soc. Am. 91, 3192–3202 (1992)

    ADS  Google Scholar 

  50. C.F. Chen, Y.-T. Lin, D. Lee, A three-dimensional azimuthal wide-angle model. J. Comput. Acoust. 7, 269–288 (1999)

    Google Scholar 

  51. M.D. Collins, S.A. Chin-Bing, A three-dimensional parabolic equation model that includes the effects of rough boundaries. J. Acoust. Soc. Am. 87, 1104–1109 (1990)

    ADS  MathSciNet  Google Scholar 

  52. F.B. Jensen, H. Schmidt, Subcritical penetration of narrow Gaussian beams into sediments. J. Acoust. Soc. Am. 82, 574–579 (1987)

    ADS  Google Scholar 

  53. F.B. Jensen, W.A. Kuperman, Sound propagation in a wedge-shaped ocean with a penetrable bottom. J. Acoust. Soc. Am. 67, 1564–1566 (1980)

    ADS  Google Scholar 

  54. F.B. Jensen, C.T. Tindle, Numerical modeling results for mode propagation in a wedge. J. Acoust. Soc. Am. 82, 211–216 (1987)

    ADS  Google Scholar 

  55. F.B. Jensen, P.L. Nielsen, M. Zampolli, M.D. Collins, W.L. Siegmann, Benchmark scenarios for range-dependent seismo-acoustic models. in Theoretical and Computational Acoustics 2007, ed. by M.I. Taroudakis, P. Papadakis (University of Crete, Heraklion, Greece, 2008), see addendum

    Google Scholar 

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Author information

Authors and Affiliations

  1. NATO Undersea Research Centre, Viale San Bartolomeo 400, La Spezia, 19126, Italy

    Finn B. Jensen

  2. Marine Physical Lab., Scripps Institution of Oceanography, Mail Code 0205 Gilman Dr. 9500, La Jolla, CA, 92093-0238, USA

    William A. Kuperman

  3. Heat, Light, and Sound Research, Inc., 3366 North Torrey Pines Court, Suite 310, La Jolla, CA, 92037, USA

    Michael B. Porter

  4. Massachusetts Institute of Technology (MIT), Massachusetts Avenue 77, Cambridge, MA, 02139, USA

    Henrik Schmidt

Authors
  1. Finn B. Jensen
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  2. William A. Kuperman
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  3. Michael B. Porter
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  4. Henrik Schmidt
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Correspondence to Finn B. Jensen .

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Jensen, F.B., Kuperman, W.A., Porter, M.B., Schmidt, H. (2011). Parabolic Equations. In: Computational Ocean Acoustics. Modern Acoustics and Signal Processing. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-8678-8_6

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  • DOI: https://doi.org/10.1007/978-1-4419-8678-8_6

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  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4419-8677-1

  • Online ISBN: 978-1-4419-8678-8

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