A Common Transverse Feedback Damper for Two Beams during a Stacking Cycle

During the stacking cycle at a proton storage ring, it often occurs that coasting and bunched beams coexist, rotating in the same direction. The problem of the two beams oscillating coherently with each other is formulated and discussed. We demonstrate that when the tune distribution is dominated by the sextupole contribution, one common transverse dipole feedback system is enough to damp both the coasting beam and the bunched beam instabilities.