  Beam stability is an essential requirement for particle accelerators.
Longitudinal coupled-bunch instabilities (CBI) are driven by beam interaction
with long-range wakefields induced in the resonant structures with narrow-band
impedance. Single-bunch loss of Landau damping (LLD) is mainly determined by
short-range wakefields excited at any geometry change of the beam pipe
(broadband impedance) and leads to undamped bunch oscillations. Up to now, to
define the threshold beam intensity or impedance, these two effects were
evaluated separately. We developed an approach to numerically solve the
stability problem in a more general case and derived a new analytical
threshold. We have shown that LLD can modify the mechanism of multi-bunch
instability and reduce the CBI threshold below the LLD threshold. This effect
explains the existing observations in the CERN SPS and should be considered for
future accelerators, such as HL-LHC, EIC, FCC, and others.
