Explicit Inversion of the Helmholtz Equation for Ultra-Sound Insonification and Spherical Detection

  • James Ball
  • Steven A. Johnson
  • Frank Stenger
Part of the Acoustical Imaging book series (ACIM, volume 9)


In this paper we wish to describe a method by which scattering data measured on a closed surface can be used to reconstruct the sound speed as a function of position in the enclosed volume. When the variation of the sound speed from its average value is small, the ray paths become straight lines, and the inversion can then be easily performed. For largervariations of the sound velocity, the paths become curved, and the three dimensional nature of the problem becomes essential. In our analysis we will employ the Rytov approximation, which has the following desirable properties:
  1. 1

    When variations are small, it reduces to the straight line case.

  2. 2

    It reduces to the Born approximation when the scatterer is small compared to the wavelength of the sound.

  3. 3

    It is a three dimensional treatment and includes contributions from none-straight line paths.



Sound Velocity Sound Speed HELMHOLTZ Equation Enclose Volume Spherical Detection 
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  1. [1]
    P. M. Morse and K. U. Ingard, “Theoretical Acoustics”, McGraw-Hill, N. Y. (1965).Google Scholar
  2. [2]
    I. S. Gradshteyn and I. M. Ryzhik, “Table of Integrals, Series and Products,” Academic Press, N. Y. (1965).Google Scholar

Copyright information

© Plenum Press, New York 1980

Authors and Affiliations

  • James Ball
    • 1
  • Steven A. Johnson
    • 2
  • Frank Stenger
    • 3
  1. 1.Department of PhysicsUniversity of UtahSalt Lake CityUSA
  2. 2.Department of BioengineeringUniversity of UtahSalt Lake CityUSA
  3. 3.Department of MathematicsUniversity of UtahSalt Lake CityUSA

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