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 sh
ow 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 wakes dynamics on the maximum energy gain of accelerated electrons can be avoided using side-injections of electrons, or by controlling the wakes phase velocity by smooth plasma density gradients.
