Stability of the longitudinal bunched-beam coherent dipole mode

The evolution of the coherent synchrotron frequency for the dipole mode with respect to the incoherent band is discussed analytically both in the unstable and stable regions. In the unstable region, Besnier's picture is recovered in the case of a capacitive impedance below transition or inductive impedance above transition. A general plot gathering all the results in both the unstable and stable regions is given. Finally, Sacherer's stability criterion is extended to include the potential-well distortion. This result is then applied to the case of the LHC at top energy.