This paper considers an example of the propagation of a finite wave-train in a plasma of relatively simple dispersive properties. In this case, the wave field can be evaluated exactly. The concept of a wave-packet moving at the group speed can therefore be tested rigorously. Various approximations to the field, physically more transparent than the exact solution, are presented and compared with the latter. One of these, relevant in the neighbourhood of the ideal group front, is in fact applicable to a wide range of dispersion laws. The paper demonstrates broadening with time, and eventual disappearance, of the pulse.