Three-dimensional acoustic scattering by vortical flows. I. General theory

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