Acoustical Physics

, Volume 62, Issue 2, pp 151–159 | Cite as

Setting boundary conditions on the Khokhlov–Zabolotskaya equation for modeling ultrasound fields generated by strongly focused transducers

  • P. B. Rosnitskiy
  • P. V. Yuldashev
  • B. A. Vysokanov
  • V. A. Khokhlova
Nonlinear Acoustics


An equivalent source model is developed for setting boundary conditions on the parabolic diffraction equation in order to simulate ultrasound fields radiated by strongly focused medical transducers. The equivalent source is defined in a plane; corresponding boundary conditions for pressure amplitude, aperture, and focal distance are chosen so that the axial solution to the parabolic model in the focal region of the beam matches the solution to the full diffraction model (Rayleigh integral) for a spherically curved uniformly vibrating source. It is shown that the proposed approach to transferring the boundary condition from a spherical surface to a plane makes it possible to match the solutions over an interval of several diffraction maxima around the focus even for focused sources with F-numbers less than unity. This method can be used to accurately simulate nonlinear effects in the fields of strongly focused therapeutic transducers using the parabolic Khokhlov–Zabolotskaya equation.


focusing diffraction parabolic approximation boundary conditions nonlinear waves medical acoustics ultrasound surgery 


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Copyright information

© Pleiades Publishing, Ltd. 2016

Authors and Affiliations

  • P. B. Rosnitskiy
    • 1
  • P. V. Yuldashev
    • 1
  • B. A. Vysokanov
    • 2
  • V. A. Khokhlova
    • 1
  1. 1.Physics FacultyMoscow State UniversityMoscowRussia
  2. 2.Mechanics–Mathematics FacultyMoscow State UniversityMoscowRussia

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