The ‘switch-on’ ionizing shock wave, initiated by a strong radial discharge between electrodes at one end of a cylindrical tube placed in a solenoid, moves along the tube with a speed that depends principally on the parameter $\xi = B_{\theta } B_0|(2e_0 \rho _1),$ where Bθ is the jump in the transverse magnetic field caused by the discharge current, B0 is the applied axial field, ρ1 is the upstream (neutral) gas density, and e0 is the ionization energy. This simple result, established by Kunkel & Gross (1962) with the aid of the Chapman-Jouguet hypothesis and for fully ionizing shocks in an infinite-slab plasma, is here extended to cylindrical plasmas and to partially ionizing shocks. The effect of dissociation energy is also taken into account. The degree of ionization is determined by applying the Saha equation downstream of the shock. Most reported experiments fall into the partially ionized region, and are found to be in good agreement with the theory given in this paper.