The theoretical problem of wave propagation and the definition of the signal velocity in a medium exhibiting resonant, absorptive, anomalous dispersion is investigated. The special case of a dilute medium is considered, such as the upper atmosphere with dispersion due to resonant line structure (e.g., the water vapor or oxygen lines in the microwave region), as opposed to plasma dispersion, which is not resonant. Signal velocity definitions analogous to those first made by Brillouin were derived in another work and are called to question as a result of an exact solution of the problem derived here, using a Bessel function series, which allows exact wave forms to be calculated and displayed. When one represents the signal development by using the simulations, it is found that one cannot realistically separate signal from precursor. The pulse envelopes do have some interesting ringing characteristics which are shown to depend upon the density of absorbers and the resonance linewidth. Qualitative experimental results are presented that verify the simulations. The measurement of this ringing appears to be a more sensitive technique than does differential absorption for determining constituent number densities.