Space-time dependent fluid acoustics and the generation of vorticity

Wave propagation through a fluid with material properties that vary in space and time is examined. When viscosity and any underlying fluid flow caused by the space-time variation can be neglected, self-adjoint wave equations are obtained. The governing equations are not conventional pressure acoustics however (the fluctuating velocity is not simply related to the gradient of a potential): for underlying material properties that vary in space, the wave motion cannot be purely dilatational and the wave has associated vorticity. The specific case of an ideal fluid is then considered, and approximate solutions are found when the sound speed variation is caused by the raising of temperature on a fixed wall. At leading order, vorticity is generated at the wavefront, giving the appearance of vorticity propagating at the sound wave speed rather than being tied to the underlying (stationary) fluid flow and boundaries, as would be the case for an isotropic background. The effect of nonlinearities is then briefly examined and it is found, for the case of sound scattering from sound on an otherwise constant background fluid, that the nonlinearities cause the mass flux density vector to develop a solenoidal component but do not lead to the generation of vorticity.