Fast transverse multibunch coherent instabilities in a beam with a quasi-uniform filling

We study effects of the long-range wakefield memory on multibunch transverse coherent instabilities of a beam in a storage ring. We assume that the oscillation rise times substantially exceed the frequency of small synchrotron oscillations of particles. Despite the eigensolution spectrum due to the wake memory and because of essential singularity in the oscillation frequency of the function defining the dispersion equation, the amplitudes of coherent oscillations of the beam bunches non-exponentially depend on time. Such beam break-up like instabilities are always suppressed by Landau damping, or by a single bunch feedback damping system. The instability threshold is determined by an acceptable level of coherent oscillations of the beam.