On long-wave sound scattering by a Rankine vortex: Non-resonant and resonant cases

The well-known two-dimensional problem of sound scattering by a Rankine vortex at small Mach number M is considered. Despite its long history, the solutions obtained by many authors still are not free from serious objections. The common approach to the problem consists in the transformation of governing equations to the d’Alembert equation with right-hand part. It was recently shown I.V. Belyaev, V.F. Kopiev, On the problem formulation of sound scattering by cylindrical vortex, Acoustical Physics 54(5) (2008) 603-614] that due to the slow decay of the mean velocity field at infinity the convective equation with nonuniform coefficients instead of the d’Alembert equation should be considered, and the incident wave should be excited by a point source placed at a large but finite distance from the vortex instead of specifying an incident plane wave (which is not a solution of the governing equations).Here we use the new formulation of Belyaev and Kopiev to obtain the correct solution for the problem of non-resonant sound scattering, to second order in Mach number M. The partial harmonic expansion approach and the method of matched asymptotic expansions are employed. The scattered field in the region far outside the vortex is determined as the solution of the convective wave equation, and van Dyke's matching principle is used to match the fields inside and outside the vortical region. Finally, resonant scattering is also considered; an O(M²) result is found that unifies earlier solutions in the literature. These problems are considered for the first time.