The dynamical problem of a brake-like mechanical system composed of an elastic cylindrical tube with Coulomb's friction in contact with a rigid and rotating cylinder is considered. This model problem enables us to give an example of non-trivial periodic solutions in the form of stick-slip or stick-slip-separation waves propagating on the contact surface. A semi-analytical analysis of stick-slip waves is obtained when the system of governing equations is reduced by condensation to a simpler system involving only the contact displacements. This reduced system, of only one space variable in addition to time, can be solved almost analytically and gives some interesting informations on the existence and the characteristics of stick-slip waves such as the wave numbers on the circumference, stick and slip proportions, wave celerities, tangential and normal forces. It is shown in particular that the stick-slip-separation solutions would occur for small normal pressures or high rotational speeds. Since the analytical discussion becomes cumbersome in this case, a second approach based on numerical analysis by the finite element method is performed. The existence and the characteristics of stick-slip and stick-slip-separation waves are discussed numerically.