Exact dispersion relations for torsional waves are derived for a system consisting of an elastic cylindrical core embedded in an elastic medium having different material properties. Dispersion curves are presented as a function of an aspect ratio and ratios of material properties. Cutoff wavelengths dependent on material properties are established. The results are compared with a previous solution obtained using a simplified approximate model. While the approximate model is seen to yield the correct range of phase velocities, the resulting dispersion relations are shown to be inaccurate for relatively short wavelengths in systems for which the surrounding material is markedly stiffer or denser than the core. The approximate model is also shown to be overly restrictive in predicting cutoff wavelengths.