The plane‐strain vibration frequencies of an infinitely long hollow cylinder are calculated exactly with the aid of a high‐speed electronic computer for a range of wall thicknesses and azimuthal node numbers and for a variety of boundary conditions. These latter cover the cases: outer radial surface free, supported, or clamped; inner surface free, supported, or matched to a gas filling the inner cavity. In the last case, we calculate the attenuation of sound in the cylinder wall, in addition to the eigenfrequencies. Our computations of mode frequencies disclose that: (i) the shear‐compressional wave coupling in the solid is very weak (except for low frequency); (ii) for the gas‐filled cylinder, the modes are very close either to the gas quasi‐modes (gas in a rigid‐walled cylinder), or to the solid quasi‐modes (shell with free inner surface), depending on the geometry.