In this paper we consider adiabatic disturbances of the form h(r) exp [i($\omega_2$ t - m$\theta$ - bZ] that occur in an inviscid compressible fluid rotating as an unbounded Rankine vortex, for m = 0, 1 and 2. Here (r, $\theta$, Z) are cylindrical polar coordinates with the Z-axis along the axis of symmetry of the problem. When the Mach number M at the periphery of the core of the vortex tends to zero the problem reduces to the incompressible fluid model that was considered by Kelvin, who showed that for a given m and a non-zero b there is an infinite spectrum of finite eigenfrequencies. It is shown that when b is small, for m = 2, as M increases the eigenfrequencies, starting from the largest one, come successively to a limiting stage with the following consequences: for waves rotating faster than the vortex core the limiting stage frequency represents a cutoff frequency. For waves rotating slower than the vortex core the limiting stage represents the boundary of stability. At higher M these waves become unstable. When b is sufficiently large there is no limiting stage for the eigenfrequencies and the perturbation is stable. The `critical' values of b given by another author for m $\geqslant$ 2 relate to only one frequency and are not critical for all frequencies. The behaviour of the eigenfrequencies for m = 1 is qualitatively similar to that for m = 2. The frequencies associated with m = 0 are real for all M. For m = 0 there is also a solution representing an acoustic wave, which in the absence of the vortex would be plane and propagating in the axial direction. We also report a set of solutions for b = 0, m $\geqslant$ 1, which appears to have gone unnoticed, and give some details for the case where the vortex is confined in a cylinder.