Scattering of acoustic waves by a vortex

We investigate the scattering of a plane acoustic wave by an axisymmetric v ortex in two dimensions. We consider vortices with localized vorticity, arb itrary circulation and small Mach number. The wavelength of the acoustic wa ves is assumed to be much longer than the scale of the vortex. This enables us to define two asymptotic regions: an inner, vortical region, and an out er, wave region. The solution is then developed in the two regions using ma tched asymptotic expansions, with the Mach number as the expansion paramete r. The leading-order scattered wave field consists of two components. One c omponent 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-f ield limit, the leading-order scattered field takes the form i(pi - theta)e (i/kX) + 1/2 cos theta cot(1/2 theta)(2 pi/kR)(1/2) e(i(kR-pi/4)), where th eta is the usual polar angle. This expression is not valid in a parabolic r egion centred on the positive X-axis, where kR theta(2) = O(1). A different asymptotic solution is appropriate in this region. The two solutions match onto each other to give a leading-order scattering amplitude that is finit e and single-valued everywhere, and that vanishes along the X-axis. The nex t term in the expansion in Mach number has a non-zero far-held response alo ng the X-axis.