An alternative way of passing through the characteristic wave velocity in a waveguide is investigated by a uniformly moving object along an elastic system with a change in parameters such that a transition from subcritical into supercritical object motion is taking place. To demonstrate this way of passing, a uniformly moving constant load along an infinite elastically supported string with a sudden change in density is studied. The results of this investigation are compared with the more usual way of passing by a fast accelerating load along a homogeneous infinite string on the same foundation. For both cases transient solutions for the string vibrations are derived showing the wave processes, which are a combination of the main features of Mach radiation and transitional radiation in the alternative case and Mach radiation and radiation due to a non-uniform load motion in the usual case. It is shown that the alternative case provides a slightly smaller amplification in displacements than the usual case. The alternative way can be more practical since it is known that the faster the critical velocity is passed the smaller the dynamical amplification will be. Evidently a sudden change in the elastic parameters is much more simply organised than a large acceleration of the train.