Effect of the frequency spread on the transverse beam break-up instability of a bunch in storage rings

We study modifications of the beam break-up instability of transverse coherent oscillations of a single bunch which occur in storage rings due to the frequency spread of incoherent oscillations of the bunch particles. We assume that the frequency spread is produced by the octupole nonlinearity of the lattice focusing. Provided that the bunch wakefields decay longer than the revolution period of particles, the frequency spread results in the Landau damping of the self-consisting modes of the bunch and in the cumulative decoherence of the beam break-up part of the coherent oscillations. We study resulting features of coherent signals in the time-domain.