Motivated by field and laboratory observations of unusual wave propagation, we investigate the behavior of simple model problems involving transport, with competitive adsorption of H+ and a metal cation. In the absence of diffusion/dispersion, the model problem yields two shocks with the velocities expected from classical theory. In the presence of diffusion/dispersion, the solution exhibits an additional feature, a pulse of metal ion concentration that moves rapidly and independently of the metal ion shock. This "fast wave" is associated with the pH shock in the simplest model problem studied here. Theoretical analysis of this problem yields a jump condition which numerical experiments confirm. Diffusion/dispersion is prerequisite for this phenomenon: it causes a flux of metal cation through the pH shock while the two fronts are near each other. One practical implication of this finding is the importance of accurate handling of diffusion and dispersion in numerical simulation of reactive transport problems. Another is that estimates of species migration based on simple theory, such as retardation factors, may fail to capture important features of the actual behavior.