Bunch energy loss in cavities: dependence on beam velocity

Proton or H^- 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 β=υ/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 β 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 β, the total loss for important particular cases was found smoothly decreasing as β decreases. Comparison with time-domain results for the ultra-relativistic case, β→1, is also presented