Influence of a distribution of deflecting-mode frequencies on the transient dynamics of cumulative beam breakup

A distribution of deflecting-mode frequencies in the constituent cavities of a linear accelerator can lead to Q-independent damping of cumulative beam breakup. A probability density for the deflecting-mode frequencies generates an effective transverse wake function. The effective wake function can be used to calculate the transient dynamics of cumulative beam breakup within the framework of a continuum approximation provided the transverse beam displacement changes little over the correlation length of the deflecting-mode frequencies as the beam moves down the linac. We adopt this approach to show that the damping induced by the effective wake function causes the rate of approach to the steady state to depend strongly on the operative probability density for the deflecting-mode frequencies