  The self-consistent interaction between a beam of charged particles and a
wave is considered, within a Vlasov picture. The model is discussed with
reference to the case of a Free Electron Laser. Starting with a spatially
bunched waterbag distribution, we derive, via perturbative methods, closed
analytical expressions for the time evolution of the main macroscopic
observables. Predictions of the theory are shown to agree with direct numerical
simulations.
