Advertisement

Basic Concepts

  • Michael D. Collins
  • William L. Siegmann
Chapter

Abstract

The parabolic equation method provides an appealing combination of accuracy and efficiency for many nonseparable wave propagation problems in geophysics. Parabolic wave equations are based on the assumption that range dependence (horizontal variations in the medium) is sufficiently gradual so that horizontally outgoing energy dominates energy that is back scattered toward the source. Appearing in Figs. 1.1 and 1.2 are some examples of range-dependent problems in ocean acoustics and seismology [1, 2]. Parabolic wave equations are derived from elliptic wave equations by expanding about a plane wave propagating in a preferred direction. The simplest expansions provide accurate solutions when the field is dominated by energy that propagates within a small angle of the preferred direction. For higher-order expansions, it is possible to handle wide propagation angles, and the only requirement is that outgoing energy dominates. Elliptic wave equations, such as the Helmholtz equation, correspond to boundary-value problems. Parabolic wave equations correspond to initial-value problems, which are much easier to solve. Solutions are marched outward in range by repeatedly solving systems of equations that involve banded matrices (tridiagonal for the acoustic case).

References

  1. 1.
    W.H. Munk, “Sound channel in an exponentially stratified ocean, with applications to SOFAR,” J. Acoust. Soc. Am. 55, 220–226 (1974).ADSCrossRefGoogle Scholar
  2. 2.
    F.B. Jensen and W.A. Kuperman, “Sound propagation in a wedge-shaped ocean with a penetrable bottom,” J. Acoust. Soc. Am. 67, 1564–1566 (1980).ADSCrossRefGoogle Scholar
  3. 3.
    F.B. Jensen, W.A. Kuperman, M.B. Porter, and H. Schmidt, Computational Ocean Acoustics (American Institute of Physics, New York, 1994).zbMATHGoogle Scholar
  4. 4.
    M.A. Leontovich and V.A. Fock, “Solution of the problem of propagation of electromagnetic waves along the earth’s surface by the method of parabolic equation,” J. Exp. Theor. Phys. 16, 557–573 (1946).MathSciNetzbMATHGoogle Scholar
  5. 5.
    V.A. Fock, Electromagnetic Diffraction and Propagation Problems, Pergamon, New York (1965).Google Scholar
  6. 6.
    J.F. Claerbout, “Coarse grid calculations of waves in inhomogeneous media with application to delineation of complicated seismic structure,” Geophysics 35, 407–418 (1970).ADSCrossRefGoogle Scholar
  7. 7.
    J.F. Claerbout, Fundamentals of Geophysical Data Processing, McGraw-Hill, New York (1976).Google Scholar
  8. 8.
    F.D. Tappert, “The parabolic approximation method,” in Wave Propagation and Underwater Acoustics, edited by J.B. Keller and J.S. Papadakis (Springer, New York, 1977).Google Scholar
  9. 9.
    P.G. Bergmann, “The wave equation in a medium with a variable index of refraction,” J. Acoust. Soc. Am. 17, 329–333 (1946).ADSCrossRefGoogle Scholar
  10. 10.
    J.S. Perkins and R.N. Baer, “An approximation to the three-dimensional parabolic-equation method for acoustic propagation,” J. Acoust. Soc. Am. 72, 515–522 (1982).ADSCrossRefGoogle Scholar
  11. 11.
    A. Bamberger, B. Engquist, L. Halpern, and P. Joly, “Higher order paraxial wave equation approximations in heterogeneous media,” SIAM J. Appl. Math. 48, 129–154 (1988).MathSciNetCrossRefGoogle Scholar
  12. 12.
    B.T.R. Wetton and G.H. Brooke, “One-way wave equations for seismoacoustic propagation in elastic waveguides,” J. Acoust. Soc Am. 87, 624–632 (1990).ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    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).ADSCrossRefGoogle Scholar
  14. 14.
    W.M. Sanders and M.D. Collins, “Nonuniform depth grids in parabolic equation solutions,” J. Acoust. Soc. Am. 133, 1953–1958 (2013).ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • Michael D. Collins
    • 1
  • William L. Siegmann
    • 2
  1. 1.Naval Research LaboratoryWashington, DCUSA
  2. 2.Department of Mathematical SciencesRensselaer Polytechnic InstituteTroyUSA

Personalised recommendations