Longitudinal Instabilities with a Non-Harmonic RF Potential

We consider the longitudinal instabilities of a bunched beam subject to a non-harmonic RF potential. Assuming the unperturbed bunch to be described by a Maxwell-Boltzmann distribution, our treatment is based upon the linearized Vlasov equation. The formalism developed is exact, and in particular, correctly describes the effect of the dependence on amplitude of the synchrotron oscillation frequency. We discuss the fast blowup limit, and extend Wang and Pellegrini's treatment of the microwave instability to include the case of a non-Gaussian bunch. Next, within the short bunch approximation, we derive the dispersion relation describing the Landau damping of the coupled bunch modes, resulting from the use of a Landau cavity.