The eigenvalue problem for the ‘acoustic’ modes in an inviscid, perfect gas with a quiescent state of isothermal, uniform rotation in a circular cylinder is solved asymptotically for A2 ↑ ∞ with (γ — 1) A2 = O(1), where A is the peripheral Mach number and γ is the specific heat ratio. The limit (γ — 1) A2 ↓ 0 leads to a solution in terms of the confluent hypergeometric function for all A, and the resulting eigenvalue equation is solved explicitly for either A2 [lsim ] 1 or A2 [Gt ] 1. Attention is focused on those modes (likely to be of greatest practical importance) for which the peripheral speed of the wave relative to that of the container tends to the sonic speed as A2 ↑ ∞. Viscosity and heat conduction are significant in an inner domain of low density, wherein the solution is expressed in terms of a generalized hypergeometric function.