  The growth rate of long wavelength kinetic instabilities arising due to the
interaction of a collimated beam of relativistic particles and a cold
unmagnetized plasma are calculated in the ultra relativistic limit. For
sufficiently culminated beams, all long wave-length modes are shown to be
Weibel-unstable, and a simple analytic expression for their growth rate is
derived. For large transverse velocity spreads, these modes become stable. An
analytic condition for stability is given. These analytic results, which
generalize earlier ones given in the literature, are shown to be in agreement
with numerical solutions of the dispersion equation and with the results of
novel PIC simulations in which the electro-magnetic fields are restricted to a
given k-mode. The results may describe the interaction of energetic cosmic
rays, propagating into the far upstream of a relativistic collisionless shock,
with a cold unmagnetized upstream. The long wavelength modes considered may be
efficient in deflecting particles and could be important for diffusive shock
acceleration. It is shown that while these modes grow in relativistic shocks
propagating into electron-positron pair plasmas, they are damped in
relativistic shocks propagating into electron-proton plasmas with moderate
Lorenz factors \Gamma_{sh}\lesssim 100. If these modes dominate the deflection
of energetic cosmic rays in electron-positron shocks, it is argued that
particle acceleration is suppressed at shock frame energies that are larger
than the downstream thermal energy by a factor greater than the shock Lorentz
factor.
