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Scattering of acoustic waves by a vortex


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.

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