This paper presents an approximate theory (with sample calculations) for the thickness eigenmodes of an infinitely long hollow cylinder of arbitrary wall thickness, and subject to a variety of boundary conditions. Beginning with the earlier observation that the coupling between shear and compression waves in the solid wall is weak, a perturbation theory is deduced by which approximate eigenfrequencies can easily be computed by hand, with accuracy better than several percent provided the number of radial nodes is more than a few. The perturbation method can also be used to compute the attenuation of sound in the cylinder wall.