  A beam of ultrarelativistic charged particles in a plasma can reach
equilibrium with its own radial wakefield and then propagate with little change
in shape. If some co-moving perturbation appears ahead of the beam, it may or
may not destroy the beam with its wakefield, depending on the phase and
amplitude of the wakefield. We numerically study which perturbations can
destroy a single short bunch or a train of many short bunches at the parameters
of interest for plasma wakefield acceleration in an axysimmetric configuration,
and how fast. We find that there are particularly dangerous wakefield phases in
which the beam can be destroyed by perturbations of very low amplitude. We also
find that perturbations with an amplitude larger than the wakefield of a single
bunch in the train are always destructive.
