A Longitudinal Stability Criterion for Bunched Beams

The unstable motion of a bunched beam consists of rigid-bunch (dipole) and higher bunch-shape oscillations of the individual bunches (individual-bunch modes), plus perhaps coupled motion of the different bunches (coupled-bunch modes). Stability is achieved either by decoupling the bunches or by a spread in synchrotron frequencies within a bunch. A stability criterion analogous to the Keil-Schnell criterion for coasting beams is given which includes the effect of a beam interacting with perfectly conducting walls, resistive walls, and resonant structures. Some examples for the CERN accelerators are included.