Wakefields, Landau damping and beam instabilities

The concepts of wake potential and impedance are introduced in the example of a single mode cavity. Longitudinal beam dynamics is then briefly discussed and the Vlasov equation derived. With this as a base, the effects of a broad band reactive impedance on the bunch (potential bunch lengthening) and of a narrow band resistive impedance on the bunch stability (Robinson instability) are calculated. The concept of Landau damping is introduced by calculating the response of coasting beam with frequency spread to a transverse excitation (transfer function). The obtained result is used to obtain a transverse beam stability criterion. Finally, the response of a bunched beam is investigated.