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Normal Modes

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Computational Ocean Acoustics

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

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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.

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References

  1. C.L. Pekeris, Theory of propagation of explosive sound in shallow water. Geol. Soc. Am. Mem. 27 (1948)

    Google Scholar 

  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)

    ADS  Google Scholar 

  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. 23–56

    Google Scholar 

  4. K. Aki, P.G. Richards, Quantitative Seismology: Theory and Methods (Freeman, New York, 1980)

    Google Scholar 

  5. I. Stakgold, Green’s Functions and Boundary Value Problems (Wiley, New York, 1979)

    MATH  Google Scholar 

  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)

    ADS  Google Scholar 

  7. W.M. Ewing, W.S. Jardetzky, F. Press, Elastic Waves in Layered Media (McGraw-Hill, New York, 1957)

    MATH  Google Scholar 

  8. R.B. Evans, The existence of generalized eigenfunctions and multiple eigenvalues in underwater acoustics. J. Acoust. Soc. Am. 92, 2024–2029 (1992)

    ADS  MathSciNet  Google Scholar 

  9. H.P. Bucker, Sound propagation in a channel with lossy boundaries. J. Acoust. Soc. Am. 48, 1187–1194 (1970)

    ADS  Google Scholar 

  10. D.C. Stickler, Normal-mode program with both the discrete and branch line contributions. J. Acoust. Soc. Am. 57, 856–861 (1975)

    ADS  Google Scholar 

  11. W.H. Munk, Sound channel in an exponentially stratified ocean with applications to SOFAR. J. Acoust. Soc. Am. 55, 220–226 (1974)

    ADS  Google Scholar 

  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)

    ADS  Google Scholar 

  13. J.H. Wilkinson, The Algebraic Eigenvalue Problem (Oxford University Press, Oxford, 1965)

    MATH  Google Scholar 

  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)

    ADS  MathSciNet  MATH  Google Scholar 

  15. M.B. Porter, E.L. Reiss, A numerical method for ocean acoustic normal modes. J. Acoust. Soc. Am. 76, 244–252 (1984)

    ADS  MATH  Google Scholar 

  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, 1979

    Google Scholar 

  17. M.B. Porter, The KRAKEN normal mode program. Rep. SM-245. SACLANT Undersea Research Centre, La Spezia, Italy, 1991

    Google Scholar 

  18. L.B. Dozier, F.D. Tappert, Statistics of normal mode amplitudes in a random ocean. J. Acoust. Soc. Am. 64, 533–547 (1978)

    ADS  MATH  Google Scholar 

  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)

    ADS  MathSciNet  Google Scholar 

  20. D.F. Gordon, Underwater sound propagation loss program. Rep. TR-393. Naval Ocean Systems Center, San Diego, CA, 1979

    Google Scholar 

  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. 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)

    ADS  Google Scholar 

  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)

    ADS  Google Scholar 

  24. C.A. Clark, K.B. Smith, An efficient normal mode solution to wave propagation prediction. IEEE J. Oceanic Eng. 33, 462–476 (2008)

    ADS  Google Scholar 

  25. C.A. Boyles, Acoustic Waveguides (Wiley, New York, 1984)

    Google Scholar 

  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, 1972

    Google Scholar 

  27. H.M. Beisner, Numerical calculation of normal modes for underwater sound propagation. IBM J. Res. Develop. 18, 53–58 (1974)

    MATH  Google Scholar 

  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. H.B. Keller, Numerical Solution of Two Point Boundary Value Problems (SIAM, Philadelphia, 1976)

    Google Scholar 

  30. R.P. Brent, An algorithm with guaranteed convergence for finding a zero of a function. Comput. J. 14, 422–425 (1971)

    MathSciNet  MATH  Google Scholar 

  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. 321–370

    Google Scholar 

  32. D.H. Lehmer, A machine method for solving polynomial equations. J. Assoc. Comput. Mach. 8, 151–162 (1961)

    MATH  Google Scholar 

  33. L.M. Delves, J.N. Lyness, A numerical method for locating the zeros of an analytic function. Math. Comput. 21, 543–560 (1967)

    MathSciNet  MATH  Google Scholar 

  34. R.W. Hamming, Introduction to Applied Numerical Analysis (McGraw-Hill, New York, 1971)

    MATH  Google Scholar 

  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. 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)

    MathSciNet  MATH  Google Scholar 

  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. 775–780

    Google Scholar 

  38. B. Davies, Locating the zeros of an analytic function. J. Comput. Phys. 66, 36–49 (1986)

    ADS  MathSciNet  MATH  Google Scholar 

  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)

    ADS  MathSciNet  Google Scholar 

  40. J.D. Pryce, Numerical Solution of Sturm–Liouville Problems (Clarendon, Oxford, 1993)

    MATH  Google Scholar 

  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)

    ADS  MATH  Google Scholar 

  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)

    ADS  MATH  Google Scholar 

  43. R.B. Evans, The decoupling of stepwise coupled modes. J. Acoust. Soc. Am. 80, 1414–1419 (1986)

    ADS  Google Scholar 

  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)

    ADS  Google Scholar 

  45. W. Luo, H. Schmidt, Three-dimensional propagation and scattering around a conical seamount. J. Acoust. Soc. Am. 125, 52–65 (2009)

    ADS  Google Scholar 

  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)

    ADS  Google Scholar 

  47. H. Weinberg, R. Burridge, Horizontal ray theory for ocean acoustics. J. Acoust. Soc. Am. 55, 63–79 (1974)

    ADS  MATH  Google Scholar 

  48. L.M. Brekhovskikh, O.A. Godin, Acoustics of Layered Media II (Springer, Berlin, 1992)

    MATH  Google Scholar 

  49. M.B. Porter, Adiabatic modes for a point source in a plane-geometry ocean. J. Acoust. Soc. Am. 96, 1918–1921 (1994)

    ADS  MathSciNet  Google Scholar 

  50. F. Ingenito, Scattering from an object in a stratified medium. J. Acoust. Soc. Am. 82, 2051–2059 (1987)

    ADS  Google Scholar 

  51. P.M. Morse, K.U. Ingard, Theoretical Acoustics (Princeton University Press, Princeton, 1968)

    Google Scholar 

  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. J.A. Fawcett, Coupled-mode modeling of acoustic scattering from three-dimensional, axisymmetric objects. J. Acoust. Soc. Am. 102, 3387–3393 (1997)

    ADS  Google Scholar 

  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)

    ADS  Google Scholar 

  55. 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 

  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)

    ADS  Google Scholar 

  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)

    ADS  Google Scholar 

  58. M.J. Buckingham, Theory of acoustic propagation around a conical seamount. J. Acoust. Soc. Am. 80, 256–277 (1986)

    ADS  Google Scholar 

  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. 123–132

    Google Scholar 

  60. R.B. Evans, Stepwise coupled mode scattering of ambient noise by a cylindrically symmetric seamount. J. Acoust. Soc. Am. 119, 161–167 (2006)

    ADS  Google Scholar 

  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)

    MATH  Google Scholar 

  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)

    ADS  Google Scholar 

  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)

    ADS  Google Scholar 

  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)

    ADS  Google Scholar 

  65. ATOC Consortium, Ocean climate change: Comparison of acoustic tomography, satellite altimetry and modeling. Science 281, 1327–1332 (1998)

    Google Scholar 

  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)

    ADS  Google Scholar 

  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)

    Google Scholar 

  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. 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. 71–91

    Google Scholar 

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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). Normal Modes. In: Computational Ocean Acoustics. Modern Acoustics and Signal Processing. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-8678-8_5

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  • DOI: https://doi.org/10.1007/978-1-4419-8678-8_5

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