  Starting from the Vlasov-Maxwell system, an exact relativistic hydrodynamic
closure for a special type water bag distributions satisfying the Vlasov
equation has been derived. It has been shown that the hydrodynamic equations
are fully equivalent to the original Vlasov-Maxwell equations. The linear
stability of the exact hydrodynamic closure has been studied as well. It has
been shown that all basic features of the small signal gain can be derived from
the fluid dynamic description developed here. Satisfactory agreement with
previously reported results has been also found.
