Initial and subsequent growth of the fast ion ("ion hose") instability

The fast ion instability is considered for a distribution of ion bounce frequencies. Because of the spread in bounce frequencies, the instability initially grows exponentially with propagation distance. When the initial growth saturates, the instability grows exponentially with the square root of the propagation distance; the saturated growth equals that calculated when the spread in ion bounce frequencies is neglected. For a broad distribution of ion bounce frequencies, instability may be prevented by a betatron damping rate that exceeds the incoherent betatron frequency shift induced by ions at the tail of the bunch train