Riemannian Geometry of Irrotational Vortex Acoustics

Abstract

We consider acoustic propagation in an irrotational vortex, using the technical machinery of differential geometry to investigate the “acoustic geometry” that is probed by the sound waves. The acoustic space-time curvature of a constant circulation hydrodynamical vortex leads to deflection of phonons at appreciable distances from the vortex core. The scattering angle for phonon rays is shown to be quadratic in the small quantity $\Gamma /2\mathrm{}\pi \mathrm{cb}$, where $\Gamma$ is the vortex circulation, $c$ the speed of sound, and $b$ the impact parameter.