Experiments on the propagation of microwave ultrasonic pulses through a resonant absorber are described. The principal features of self-induced transparency as described by McCall and Hahn for light pulses are observed. Quantitative measurements of pulse delay, output pulse width, energy, and area are in fair agreement with a calculation by Hopf which accounts approximately for the effect of the single reflection present in most of the experiments. Multiply reflected pulses exhibited linearly cumulative delay times, indicating constant reduced velocity during successive traversals through the absorber. Coherence-induced pulse breakup is shown to be a possible explanation of the pulse distortion and modulation observed in acoustic-paramagnetic-resonance experiments on Fe2+.