  There is a new effect that can limit the lifetime of a weakly nonlinear
wakefield in a radially bounded plasma. If the drive beam is narrow, some of
the plasma electrons fall out of the collective motion and leave the plasma
radially, forming a negatively charged halo around it. These electrons
repeatedly return to the plasma under the action of the charge separation
field, interact with the plasma wave and cause its damping. The lowest-energy
halo electrons take the energy from the wave more efficiently, because their
trajectories are bent by the plasma wave towards the regions of the strongest
acceleration. For correct accounting of the wave damping in simulations, it is
necessary and sufficient to take the simulation window twice as wide as the
plasma.
