Propagation of acoustic waves through a system of many vortex rings

Analytic expressions are found for the index of refraction and attenuation length of an acoustic wave that propagates in a flow that can be modeled in terms of many slender vortices in three dimensions. This is done to second order in the Mach number, within a multiple scattering scheme and a Born approximation also carried to second order. As an example, their dependence on wavelength is explicitly computed in the case of the vortices being circular rings, uniformly distributed in space but randomly oriented.