Diffusive shock acceleration by multiple shock fronts with differing properties

The effect of diffusive shock acceleration on a distribution of particles is explored for multiple shocks, taking into account adiabatic expansion between the shocks. Specifically, the spectral index is calculated numerically for two cases: a sequence of identical shocks, and a sequence of pairs of shocks with alternating shock strength. How these two cases evolve to the asymptotic limit is examined, and it is shown that the evolution of the paired-shock case can be described by a sequence of identical shocks with shock strength equal to the mean of the two.