The propagation of pulses in a dissipative medium is investigated both theoretically and experimentally. The theoretical work is based on the dissipative dispersion integral, and the measurements are made in an air-filled periodic waveguide (i.e., the dispersion is Bloch wave dispersion). The dispersion integral is considered in the context of a sequence of characteristic pulse duration distances. The pulse propagates without distortion up to the smallest characteristic distance, and thereafter undergoes a new variety of distortion as it encounters each subsequent characteristic distance. Several new solutions of the dispersion integral that exhibit a variety of novel propagation properties are found. Pulses that shift in frequency as they propagate, accelerate as they propagate, and propagate at near-infinite group velocity are found analytically and verified experimentally. [Work supported by ONR.]