The dispersion effects appearing during the propagation of acoustic waves through the plasma of a weakly ionized gas are studied. The main theoretical results are based on the equation of propagation of sound in the medium with the so-called Rayleigh energy release mechanism, which has been obtained earlier. Unlike the previous investigations, the problem of propagation of a perturbation from a source and not the problem of propagation of the initial perturbation is solved. In particular, the sources of an N-shaped shock wave and a wave in the form of a symmetrical step are analyzed in detail. It is shown that depending on the direction of wave propagation (along or across the electric field in a plasma), it degenerates either into a wave packet with a wave frequency lower than a certain frequency characterizing heating, or into a wave packet with a frequency higher than this value. In addition, a quantitative criterion is obtained, which makes it possible to estimate the plasma parameters for which it will be possible to observe the dispersion of acoustic waves in the plasma.