Scattering of acoustic waves by a superfluid vortex

The scattering of plane acoustic waves by a vortex in a two-dimensional superfluid is examined for small Mach number M. The solution is developed from a systematic expansion of the governing equations in three separate regions: an inner vortical region which scales as the healing length, an interaction region governed by irrotational hydrodynamics and an outer wave region. The solutions in the different regions are matched together. The leading-order scattered field occurs at O(M(2)delta) in the wave region, where delta is the small non-dimensional amplitude of the incoming acoustic wave. The far-field behaviour of the wave-region solution shows that a different form of the expansion is required in the forward scatter direction: this corresponds to the expression previously derived for the acoustic Magnus force.