Another look at coherent longitudinal instabilities of bunched beams

Existing theories predict that, the higher the mode number is of a longitudinal bunched beam instability, the higher is likely to be the frequency (or range of frequencies) of the instability excitation fields. In this paper a different result is found with all modes having a similar upper frequency limit for the fields but with the odd and even order modes having a dissimilar range for the possible frequencies. The relevance for uncoupled bunch, coupled bunch and mode coupling instabilities is discussed