Longitudinal instabilities of bunched beams subject to a non-harmonic RF potential

The longitudinal instabilities of a bunched beam subject to a non-harmonic rf potential is considered. The treatment is based upon the linearized Vlasov equation. The formalism developed is exact, and correctly describes the effect of the dependence on amplitude of the synchrotron oscillation frequency. The fast blowup limit is discussed, and Wang and Pellegrini's [1980, Proc. X1 Int. Conf. on High Energy Accelerators, Geneva, p554] treatment of the microwave instability is extended to include the case of a non-Gaussian bunch. Next, within the short-bunch approximation the determination is made of the Landau damping of coupled-bunch oscillations that results from the use of a higher-harmonic (Landau) cavity. (author)