Modes of Oscillation of Trains of Bunches

In storage rings which operate with trains of bunches at high currents the observation of the motion of these bunches is quite important. An important reason is that the storage ring is often attempting to operate at or above the instability threshold for coupled bunch instabilities of the beam(s) and these observations are necessary to help to identify the cause of the instability and its possible cures. With single bunches or equally spaced bunches the observation of beam position signals in the frequency domain has been analyzed and gives fairly simple and understandable results. With trains of bunches the frequency spectrum is more complicated, but is still capable of providing useful information. This paper will present a basis for understanding measurements of the frequency spectrum of beam signals for trains of bunches.