  Here we will derive the general theory of the beam-breakup instability in
recirculating linear accelerators, in which the bunches do not have to be at
the same RF phase during each recirculation turn. This is important for the
description of energy recovery linacs (ERLs) where bunches are recirculated at
a decelerating phase of the RF wave and for other recirculator arrangements
where different RF phases are of an advantage. Furthermore it can be used for
the analysis of phase errors of recirculated bunches. It is shown how the
threshold current for a given linac can be computed and a remarkable agreement
with tracking data is demonstrated. The general formulas are then analyzed for
several analytically solvable cases, which show: (a) Why different higher order
modes (HOM) in one cavity do not couple so that the most dangerous modes can be
considered individually. (b) How different HOM frequencies have to be in order
to consider them separately. (c) That no optics can cause the HOMs of two
cavities to cancel. (d) How an optics can avoid the addition of the
instabilities of two cavities. (e) How a HOM in a multiple-turn recirculator
interferes with itself. Furthermore, a simple method to compute the orbit
deviations produced by cavity misalignments has also been introduced. It is
shown that the BBU instability always occurs before the orbit excursion becomes
very large.
