ABSTRACT
This chapter deals with the self-consistent dynamics in the case of waves with small amplitude. It provides the analogue of the Landau and van Kampen description of eigenmodes in Vlasovian plasmas, with the advantage of emphasizing mechanical intuition at each step of the calculations. The waves may be damped or unstable due to their interaction with the particles. Indeed, the linearized dynamics is drastically simplified by the facts that the reference state has zero wave amplitudes. A particle motion can deviate from ballistic only due to the waves; and that a wave can change amplitude and phase only due to interaction with particles. The original approach to beam–wave interaction uses kinetic theory, describing both the plasma and the beam through the Vlasov–Poisson integro-differential system of equations. The mathematical treatment of the self-consistent beam–wave system is parallel to this classical one.
