Bunch energy loss in cavities: dependence on beam velocity

Proton or H/sup -/ linacs accelerate particles through the wide range of the particle velocities. For intense beams, and especially when superconducting cavities are used, a detailed knowledge about how the beam energy loss depends on the beam velocity /spl beta/=/spl upsi//c, and what part of the lost energy remains in the cavity, is required to calculate the cavity heat load. Using a frequency-domain approach, we calculate the bunch energy loss as a function of /spl beta/ for a few particular cases. We are mostly concerned about the cavity modes below the beam-pipe cutoff, because they produce most of the cavity heating. While the loss factor due to an individual mode can show peaks at some particular values of /spl beta/, the total loss for important particular cases was found smoothly decreasing as /spl beta/ decreases. Comparison with time-domain results for the ultra-relativistic case, /spl beta//spl rarr/1, is also presented.