Effect of the induced field memory on the bunch lengthening

We study the variations of the solutions to the Haissinski equation versus the duration of the memory of the longitudinal induced fields. For simplicity, we assume that a single bunch in a storage ring interacts with surrounding electrodes, which electromagnetically are equivalent to a parallel LR-circuit. For this particular case, we have found out that controlling the decay time of the bunch induced fields we can substantially affect the length of the bunch as well as the features of Landau damping of coherent oscillations of the bunch.