Although the linearized theory of small amplitude synchrotron oscillations
and the instability thresholds derived from it have long been understood, t
here is no satisfactory description of the large amplitude highly nonlinear
synchrotron motion of a bunched beam. With an appropriate tuning of the RF
cavity impedance, large amplitude, low frequency, self-sustained relaxatio
n oscillations of this synchrotron motion are generated. This paper present
s detailed experimental data on such behavior, tracking code results that r
eproduce the important characteristics, and a simple analytical model that
explains the key features of the relaxation oscillation: growth of the inst
ability, saturation of the oscillation, breakup of the bunch, and subsequen
t damping of the system back to the beginning of the next cycle of the rela
xation oscillation. [S1063-651X(99)10110-7].