It is shown that small acoustic disturbances of the form g(x) exp (i($\omega$ft-m$\theta$-bZ)) can occur for m $\geqslant$ 1 in a polytropic gas rotating as an unbounded Rankine vortex. Here (x, $\theta$, Z) are cylindrical polar coordinates with the Z-axis along the axis of symmetry of the system and $\omega$ is the Mach number at the periphery of the core of the vortex. The main emphasis of the paper is on high rotation rates and most of the results presented relate to the regime $\omega$ > 1. For given $\omega$, b and m only a finite number of such modes (or none at all) exist and, depending on these parameters, the modes can relate to stable waves going opposite to the direction of fluid rotation, to stationary waves, to stable waves going in the direction of fluid rotation or to unstable waves going in the direction of fluid rotation. For m = 1, 2 or for $\omega$ < 1 only stable acoustic modes are possible. Stable acoustic modes propagating in the axial direction relating to m = 0, b $\neq$ 0 can also occur for this configuration.