Advertisement

Normal Modes

  • Finn B. Jensen
  • William A. Kuperman
  • Michael B. Porter
  • Henrik Schmidt
Chapter
Part of the Modern Acoustics and Signal Processing book series (MASP)

Abstract

Normal-mode methods have been used for many years in underwater acoustics. An early and widely cited reference is due to Pekeris [1], who developed the theory for a simple two-layer model of the ocean. At about the same time Ide et al. [2] had been using normal modes to interpret propagation in the Potomac River and Chesapeake Bay. Progress in the development of normal-mode methods is presented in an excellent summary given by Williams [3]. Numerical techniques now exist which can treat problems with an arbitrary number of fluid and viscoelastic layers.

Keywords

Sound Speed Phase Speed Transmission Loss Scattered Field Richardson Extrapolation 
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.

References

  1. 1.
    C.L. Pekeris, Theory of propagation of explosive sound in shallow water. Geol. Soc. Am. Mem. 27 (1948)Google Scholar
  2. 2.
    J.M. Ide, R.F. Post, W.J. Fry, The propagation of underwater sound at low frequencies as a function of the acoustic properties of the bottom. J. Acoust. Soc. Am. 19, 283 (1947)ADSCrossRefGoogle Scholar
  3. 3.
    A.O. Williams, Normal-mode methods in propagation of underwater sound. in Underwater Acoustics, ed. by R.W.B. Stephens (Wiley-Interscience, New York, 1970), pp. 23–56Google Scholar
  4. 4.
    K. Aki, P.G. Richards, Quantitative Seismology: Theory and Methods (Freeman, New York, 1980)Google Scholar
  5. 5.
    I. Stakgold, Green’s Functions and Boundary Value Problems (Wiley, New York, 1979)MATHGoogle Scholar
  6. 6.
    M.B. Porter, E.L. Reiss, A numerical method for bottom interacting ocean acoustic normal modes. J. Acoust. Soc. Am. 77, 1760–1767 (1985)ADSCrossRefGoogle Scholar
  7. 7.
    W.M. Ewing, W.S. Jardetzky, F. Press, Elastic Waves in Layered Media (McGraw-Hill, New York, 1957)MATHGoogle Scholar
  8. 8.
    R.B. Evans, The existence of generalized eigenfunctions and multiple eigenvalues in underwater acoustics. J. Acoust. Soc. Am. 92, 2024–2029 (1992)ADSCrossRefGoogle Scholar
  9. 9.
    H.P. Bucker, Sound propagation in a channel with lossy boundaries. J. Acoust. Soc. Am. 48, 1187–1194 (1970)ADSCrossRefGoogle Scholar
  10. 10.
    D.C. Stickler, Normal-mode program with both the discrete and branch line contributions. J. Acoust. Soc. Am. 57, 856–861 (1975)ADSCrossRefGoogle Scholar
  11. 11.
    W.H. Munk, Sound channel in an exponentially stratified ocean with applications to SOFAR. J. Acoust. Soc. Am. 55, 220–226 (1974)ADSCrossRefGoogle Scholar
  12. 12.
    C.T. Tindle, The equivalence of bottom loss and mode attenuation per cycle in underwater acoustic. J. Acoust. Soc. Amer. 66, 250–255 (1979)ADSCrossRefGoogle Scholar
  13. 13.
    J.H. Wilkinson, The Algebraic Eigenvalue Problem (Oxford University Press, Oxford, 1965)MATHGoogle Scholar
  14. 14.
    M.B. Porter, E.L. Reiss, A note on the relationship between finite-difference and shooting methods for ODE eigenvalue problems. SIAM J. Numer. Anal. 23, 1034–1039 (1986)MathSciNetADSMATHCrossRefGoogle Scholar
  15. 15.
    M.B. Porter, E.L. Reiss, A numerical method for ocean acoustic normal modes. J. Acoust. Soc. Am. 76, 244–252 (1984)ADSMATHCrossRefGoogle Scholar
  16. 16.
    F.B. Jensen, C.M. Ferla, SNAP: The SACLANTCEN normal-mode acoustic propagation model. Rep. SM-121. SACLANT Undersea Research Centre, La Spezia, Italy, 1979Google Scholar
  17. 17.
    M.B. Porter, The KRAKEN normal mode program. Rep. SM-245. SACLANT Undersea Research Centre, La Spezia, Italy, 1991Google Scholar
  18. 18.
    L.B. Dozier, F.D. Tappert, Statistics of normal mode amplitudes in a random ocean. J. Acoust. Soc. Am. 64, 533–547 (1978)ADSMATHCrossRefGoogle Scholar
  19. 19.
    I. Tolstoy, J. May, A numerical solution for the problem on long-range sound propagation in continuously stratified media, with applications to the deep ocean. J. Acoust. Soc. Am. 32, 655–660 (1960)MathSciNetADSCrossRefGoogle Scholar
  20. 20.
    D.F. Gordon, Underwater sound propagation loss program. Rep. TR-393. Naval Ocean Systems Center, San Diego, CA, 1979Google Scholar
  21. 21.
    C.L. Bartberger, The computation of complex normal mode eigenvalues in underwater acoustic propagation. in Computational Acoustics: Algorithms and Applications, ed. by D. Lee, R.L. Sternberg, M.H. Schultz (North-Holland, Amsterdam, The Netherlands, 1988)Google Scholar
  22. 22.
    S.J. Levinson, E.K. Westwood, R.A. Koch, S.K. Mitchell, C.V. Sheppard, An efficient and robust method for underwater acoustic normal-mode computations. J. Acoust. Soc. Am. 97, 1576–1585 (1995)ADSCrossRefGoogle Scholar
  23. 23.
    E.K. Westwood, C.T. Tindle, N.R. Chapman, A normal mode model for acousto-elastic ocean environments. J. Acoust. Soc. Am. 100, 3631–3645 (1996)ADSCrossRefGoogle Scholar
  24. 24.
    C.A. Clark, K.B. Smith, An efficient normal mode solution to wave propagation prediction. IEEE J. Oceanic Eng. 33, 462–476 (2008)CrossRefGoogle Scholar
  25. 25.
    C.A. Boyles, Acoustic Waveguides (Wiley, New York, 1984)Google Scholar
  26. 26.
    A.V. Newman, F. Ingenito, A normal mode computer program for calculating sound propagation in shallow water with an arbitrary velocity profile. Memo. Rep. 2381. Naval Research Laboratory, Washington, DC, 1972Google Scholar
  27. 27.
    H.M. Beisner, Numerical calculation of normal modes for underwater sound propagation. IBM J. Res. Develop. 18, 53–58 (1974)MATHCrossRefGoogle Scholar
  28. 28.
    A.B. Baggeroer, Prüfer transformations for the normal modes in ocean acoustics. in Proceedings of the Shallow Water Acoustics Conference 2009 ed. J. Simmen, E.S. Livingston, J.-X. Zhou, F.-H. Li (American Institute of Physics, New York, 2010)Google Scholar
  29. 29.
    H.B. Keller, Numerical Solution of Two Point Boundary Value Problems (SIAM, Philadelphia, 1976)Google Scholar
  30. 30.
    R.P. Brent, An algorithm with guaranteed convergence for finding a zero of a function. Comput. J. 14, 422–425 (1971)MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    J.H. Woodhouse, The calculation of eigenfrequencies and eigenfunctions of the earth and the sun. in Seismological Algorithms, ed. by D.J. Doombos (Academic, London, 1988), pp. 321–370Google Scholar
  32. 32.
    D.H. Lehmer, A machine method for solving polynomial equations. J. Assoc. Comput. Mach. 8, 151–162 (1961)MATHGoogle Scholar
  33. 33.
    L.M. Delves, J.N. Lyness, A numerical method for locating the zeros of an analytic function. Math. Comput. 21, 543–560 (1967)MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    R.W. Hamming, Introduction to Applied Numerical Analysis (McGraw-Hill, New York, 1971)MATHGoogle Scholar
  35. 35.
    H.V. Hitney, J.H. Richter, R.A. Pappert, K.D. Anderson, G.B. Baumgartner Jr., Tropospheric radio propagation assessment. Proc. IEEE 73, 265–283 (1985)Google Scholar
  36. 36.
    P.S. Dubbelday, Application of a new complex root-finding technique to the dispersion relations for elastic waves in a fluid-loaded plate. SIAM J. Appl. Math. 43, 1127–1139 (1983)MathSciNetMATHCrossRefGoogle Scholar
  37. 37.
    P. Cristini, Implementation of a new root finder for KRAKEN. in Proceedings of the Fourth European Conference Underwater Acoustics, ed. by A. Alippi, G.B. Cannelli (Italian National Research Council, Rome, 1998), pp. 775–780Google Scholar
  38. 38.
    B. Davies, Locating the zeros of an analytic function. J. Comput. Phys. 66, 36–49 (1986)MathSciNetADSMATHCrossRefGoogle Scholar
  39. 39.
    C.T. Tindle, N.R. Chapman, A phase function for finding normal mode eigenvalues over a layered elastic bottom. J. Acoust. Soc. Am. 96, 1777–1782 (1994)MathSciNetADSCrossRefGoogle Scholar
  40. 40.
    J.D. Pryce, Numerical Solution of Sturm–Liouville Problems (Clarendon, Oxford, 1993)MATHGoogle Scholar
  41. 41.
    W.A. Kuperman, F. Ingenito, Attenuation of the coherent component of sound propagating in shallow water with rough boundaries. J. Acoust. Soc. Am. 61, 1178–1187 (1977)ADSMATHCrossRefGoogle Scholar
  42. 42.
    R.B. Evans, A coupled mode solution for acoustic propagation in a waveguide with stepwise depth variations of a penetrable bottom. J. Acoust. Soc. Am. 74, 188–195 (1983)ADSMATHCrossRefGoogle Scholar
  43. 43.
    R.B. Evans, The decoupling of stepwise coupled modes. J. Acoust. Soc. Am. 80, 1414–1419 (1986)ADSCrossRefGoogle Scholar
  44. 44.
    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)ADSCrossRefGoogle Scholar
  45. 45.
    W. Luo, H. Schmidt, Three-dimensional propagation and scattering around a conical seamount. J. Acoust. Soc. Am. 125, 52–65 (2009)ADSCrossRefGoogle Scholar
  46. 46.
    A.D. Pierce, Extension of the method of normal modes to sound propagation in an almost-stratified medium. J. Acoust. Soc. Am. 37, 19–27 (1965)ADSCrossRefGoogle Scholar
  47. 47.
    H. Weinberg, R. Burridge, Horizontal ray theory for ocean acoustics. J. Acoust. Soc. Am. 55, 63–79 (1974)ADSMATHCrossRefGoogle Scholar
  48. 48.
    L.M. Brekhovskikh, O.A. Godin, Acoustics of Layered Media II (Springer, Berlin, 1992)Google Scholar
  49. 49.
    M.B. Porter, Adiabatic modes for a point source in a plane-geometry ocean. J. Acoust. Soc. Am. 96, 1918–1921 (1994)ADSCrossRefGoogle Scholar
  50. 50.
    F. Ingenito, Scattering from an object in a stratified medium. J. Acoust. Soc. Am. 82, 2051–2059 (1987)ADSCrossRefGoogle Scholar
  51. 51.
    P.M. Morse, K.U. Ingard, Theoretical Acoustics (Princeton University Press, Princeton, 1968)Google Scholar
  52. 52.
    J.J. Bowman, T.B.A. Senior, P.L.E. Uslenghi (eds.), Electromagnetic and Acoustic Scattering by Simple Shapes (North-Holland, Amsterdam, 1969)Google Scholar
  53. 53.
    J.A. Fawcett, Coupled-mode modeling of acoustic scattering from three-dimensional, axisymmetric objects. J. Acoust. Soc. Am. 102, 3387–3393 (1997)ADSCrossRefGoogle Scholar
  54. 54.
    N.C. Makris, A spectral approach to 3-D object scattering in layered media applied to scattering from submerged spheres. J. Acoust. Soc. Am. 104, 2105–2113 (1998)ADSCrossRefGoogle Scholar
  55. 55.
    R.N. Baer, Propagation through a three-dimensional eddy including effects on an array. J. Acoust. Soc. Am. 69, 70–75 (1981)ADSCrossRefGoogle Scholar
  56. 56.
    R. Doolittle, A. Tolstoy, M.J. Buckingham, Experimental confirmation of horizontal refraction of CW acoustic radiation from a point source in a wedge-shaped ocean environment. J. Acoust. Soc. Am. 83, 2117–2125 (1988)ADSCrossRefGoogle Scholar
  57. 57.
    K.D. Heaney, W.A. Kuperman, B.E. McDonald, Perth–Bermuda sound propagation (1960): Adiabatic mode interpretation. J. Acoust. Soc. Am. 90, 2586–2594 (1991)ADSCrossRefGoogle Scholar
  58. 58.
    M.J. Buckingham, Theory of acoustic propagation around a conical seamount. J. Acoust. Soc. Am. 80, 256–277 (1986)ADSCrossRefGoogle Scholar
  59. 59.
    R.B. Evans, Three dimensional acoustic scattering from a cylindrical inclusion in a waveguide. in Computational Acoustics: Scattering, Gaussian Beams and Aeroacoustics, vol. 2, ed. by D. Lee, A. Cakmak, R. Vichnevetsky (North-Holland, Amsterdam, 1990), pp. 123–132Google Scholar
  60. 60.
    R.B. Evans, Stepwise coupled mode scattering of ambient noise by a cylindrically symmetric seamount. J. Acoust. Soc. Am. 119, 161–167 (2006)ADSCrossRefGoogle Scholar
  61. 61.
    M.I. Taroudakis, A coupled-mode formulation for the solution in the presence of a conical sea-mount. J. Comput. Acoust. 4, 101–121 (1996)CrossRefGoogle Scholar
  62. 62.
    G.A. Athanassoulis, K.A. Belibassakis, All-frequency normal-mode solution of the three-dimensional acoustic scattering from a vertical cylinder in a plane-horizontal waveguide. J. Acoust. Soc. Am. 101, 3371–3384 (1997)ADSCrossRefGoogle Scholar
  63. 63.
    H. Schmidt, J. Glattetre, A fast field model for three-dimensional wave propagation in stratified environments based on the global matrix method. J. Acoust. Soc. Am. 78, 2105–2114 (1985)ADSCrossRefGoogle Scholar
  64. 64.
    F.B. Jensen, On the use of stair steps to approximate bathymetry changes in ocean acoustic models. J. Acoust. Soc. Am. 104, 1310–1315 (1998)ADSCrossRefGoogle Scholar
  65. 65.
    ATOC Consortium, Ocean climate change: Comparison of acoustic tomography, satellite altimetry and modeling. Science 281, 1327–1332 (1998)Google Scholar
  66. 66.
    W.A. Kuperman, G.L. D’Spain, K.D. Heaney, Long range source localization from single hydrophone spectrograms. J. Acoust. Soc. Am. 109, 1935–1943 (2001)ADSCrossRefGoogle Scholar
  67. 67.
    S.V. Burenkov, Distinctive features of the interference structure of a sound field in a two-dimensionally inhomogeneous waveguide. Sov. Phys. Acoust. 35, 465–467 (1989)MathSciNetGoogle Scholar
  68. 68.
    W.A. Kuperman, G.L. D’Spain, H.C. Song, A.M. Thode, The generalized waveguide invariant concept with application to vertical arrays in shallow water. in Ocean Acoustics Interference Phenomena and Signal Processing, ed. by W.A. Kuperman, G.L. D’Spain (American Institute of Physics, Melville, New York, 2002)Google Scholar
  69. 69.
    S.D. Chuprov, Interference structure of a sound field in a layered ocean. in Ocean Acoustics, Current State, ed. by L.M. Brekhovskikh, I.B. Andreevoi (Nauka, Moscow, 1982), pp. 71–91Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Finn B. Jensen
    • 1
  • William A. Kuperman
    • 2
  • Michael B. Porter
    • 3
  • Henrik Schmidt
    • 4
  1. 1.NATO Undersea Research CentreLa SpeziaItaly
  2. 2.Marine Physical Lab.Scripps Institution of OceanographyLa JollaUSA
  3. 3.Heat, Light, and Sound Research, Inc.La JollaUSA
  4. 4.Massachusetts Institute of Technology (MIT)CambridgeUSA

Personalised recommendations