A Timoshenko-type theory is presented for the dynamics of a cylindrical shell whose two layers are joined by a perfect bond. An assessment of the accuracy of the theory is obtained by solving the problem of axial propagation of an infinite train of axially symmetric waves and comparing the results with those obtained from the three-dimensional elasticity theory. Frequencies of the four lowest modes are accurately predicted by the shell theory for sufficiently long wavelengths and low frequencies. Preliminary comparisons of displacement distributions indicate that the shell-theory displacements are accurate in a more restricted frequency-wavelength regime. Timoshenko shear coefficients are determined by matching simple thickness-shear cutoff frequencies rather than by matching the lower Rayleigh wave speed. This is found preferable by consideration of the shapes of the first-mode phase velocity versus wave-number curves for the two theories.