Transverse stability of bunch trains

The transverse stability of bunches in a bunch train is determined by solving the equations of betatron motion for macroparticles circulating in a high energy storage ring. We ignore multibunch modes that are more likely to be serious with equal bunch spacing, and find that a nonexponential beam breakup instability may develop, which would not be found by the usual instability analysis with an exponential ansatz. In the absence of radiation or other damping mechanisms, the amplitudes of the trailing bunches would grow with a power law and would soon be lost if the first bunches perform a betatron oscillation about the closed orbit. Experimental observations on a large electron-positron collider are also discussed.