A cylindrical MGD configuration comprising an electrically conducting, infinitely extended (cylindrical) jet which is both permeated and contained by stream-aligned magnetic fields is considered. Axisymmetric perturbations are then generated by exposure to a stationary, weak, azimuthal current source. Ignoring relativistic and dissipative effects, asymptotic solutions complying with a radiation condition are established for the steady state. Generally speaking, the disturbed system propagates a discrete superposition of wave functions, each of which is uniquely obtained for either the upstream or downstream region. In the non-trivial cases, there is no axial wave attenuation. However, transverse attenuations are experienced within the fluid-free enveloping field. The various jet-flow regimes are thoroughly examined. It is found, in particular, that a single stationary wave is produced upstream whenever M < 1, M2 + A2 [ges ] 2 and λβ < 1, while if M > 1 and A > 1 an infinity of discrete stationary waves occurs downstream. M and A being, respectively, the Mach and Alfvén numbers of the jet; λβ is a parameter involving M, A and A0 (an interface Alfvén number).