The method of characteristics is used to study the propagation of acceleration waves in a radiation magnetogasdynamic flow, which is induced by the motion of a piston advancing with a finite acceleration into a constant state of rest. The analysis assumes an azimuthal magnetic field produced by a constant current flowing along the axis. The characteristic solutions have been obtained in the neighborhood of leading forward characteristics. The critical values of the initial amplitude and critical time for breakdown of an acceleration wave at the cusp of the envelope of intersecting forward characteristics have been obtained. It has been shown that the effect of coupling of the radiation and the magnetic field on compression waves, which are emanating from piston movement, is to slow down the motion of a breakdown point, and thus to increase the cylindrical shock formation time, while the effect on expansion waves is to enhance the decay rate.