Refinements to longitudinal, single bunch, coherent instability theory

For the case of a bunched beam confined to a quadratic potential well, we demonstrate the necessity for considering mode-coupling to correctly obtain the threshold current for the d.c. instability. Further we find the effect upon growth rate and coherent tune shift of evaluating the impedance at a complex frequency. For the case of a bunched beam confined to a cosine potential well, we give an exact analytic expression for the dispersion integral, and calculate (with no approximations), the stability diagram for the Robinson instability taking into account Landau damping. This paper comprises extracts from a lengthy internal report