  It is demonstrated that the performance of the self-modulated proton driver
plasma wakefield accelerator (SM-PDPWA) is strongly affected by the reduced
phase velocity of the plasma wave. Using analytical theory and particle-in-cell
simulations, we show that the reduction is largest during the linear stage of
self-modulation. As the instability nonlinearly saturates, the phase velocity
approaches that of the driver. The deleterious effects of the wake's dynamics
on the maximum energy gain of accelerated electrons can be avoided using
side-injections of electrons, or by controlling the wake's phase velocity by
smooth plasma density gradients.
