- Wavetrains of impulses in homogeneous excitable media relax during propagation toward constant-speed patterns. Here we present a study of this relaxation process. Starting with the basic reaction-diffusion or cable equations, we derive kinematics for the trajectories of widely spaced impulses in the form of ordinary differential equations for the set of times at which impulses arrive at a given point in space. Stability criteria derived from these equations allow us to determine the possible asymptotic forms of propagating trains. When the recovery after excitation is monotonic, only one stable train exists for a given propagation speed. In the case of an oscillatory recovery, however, many stable trains are possible. This essential difference between monotonic and oscillatory recoveries manifests itself in qualitatively distinct relaxational behaviors.