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Open Access Publications from the University of California

Scholarly Works (3 results)

  • Article
  • Peer Reviewed
UC San Diego Previously Published Works (1999)

We investigate the scattering of a plane acoustic wave by an axisymmetric vortex in two dimensions. We consider vortices with localized vorticity, arbitrary circulation and small Mach number. The wavelength of the acoustic waves is assumed to be much longer than the scale of the vortex. This enables us to de ne two asymptotic regions: an inner, vortical region, and an outer, wave region. The solution is then developed in the two regions using matched asymptotic expansions, with the Mach number as the expansion parameter. The leading-order scattered wave eld consists of two components. One component arises from the interaction in the vortical region, and takes the form of a dipolar wave. The other component arises from the interaction in the wave region. For an incident wave with wavenumber k propagating in the positive X-direction, a steepest descents analysis shows that, in the far- eld limit, the leadingorder scattered eld takes the form i( − )eikX + 1 2 cos cot ( 1 2 )(2 =kR)1=2 ei(kR− =4), where is the usual polar angle. This expression is not valid in a parabolic region centred on the positive X-axis, where kR 2 = O(1). A di erent asymptotic solution is appropriate in this region. The two solutions match onto each other to give a leading-order scattering amplitude that is nite and single-valued everywhere, and that vanishes along the X-axis. The next term in the expansion in Mach number has a non-zero far- eld response along the X-axis.

    Cover page: Scattering of acoustic waves by a vortex
    • Article
    • Peer Reviewed
    UC San Diego Previously Published Works (2001)
    The general theory of Part I is applied to the the specific case of scattering of a wave incident along the axis of Hill’s spherical vortex. The full asymptotic solution to the initial-value problem is calculated. Results agree with the general approach, showing that the conditions required for the latter to hold apply in the case of Hill’s spherical vortex.
      Cover page: Three-dimensional acoustic scattering by vortical flows. II. Axisymmetric scattering by Hill’s spherical vortex
      • Article
      • Peer Reviewed
      UC San Diego Previously Published Works (2001)
      When an acoustic wave is incident on a three-dimensional vortical structure, with length scale small compared with the acoustic wavelength, what is the scattered sound field that results? A frequently used approach is to solve a forced wave equation for the acoustic pressure, with nonlinear terms on the right-hand side approximated by the bilinear product of the incident wave and the undisturbed vortex: we refer to this as the “acoustic analogy” approximation. In this paper, we show using matched asymptotic expansions that the acoustic analogy approximation always predicts the leading-order scattered sound field correctly, provided the Mach number of the vortex is small, and the acoustic wavelength is a factor of order M−1 larger than the scale of the vortex. The leading-order scattered field depends only on the vortex dipole moment. Our analysis is valid for acoustic frequencies of the same order or smaller than the vorticity of the vortex. Over long times, the vortex may become significantly disturbed by the incident acoustic wave. Additional conditions are derived to maintain validity of the acoustic analogy approximation over times of order M−1, long enough for motion of the vortex to be significant on the length scale of the acoustic waves.
        Cover page: Three-dimensional acoustic scattering by vortical flows. I. General theory
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