  We study the effect of a passive harmonic cavity, introduced to cause bunch
lengthening, in an electron storage ring. We derive a formula for the induced
voltage from such a cavity with high $Q$, excited by a a sequence of bunches,
allowing for arbitrary gaps in the sequence and arbitrary currents. Except for
a minor term that can be determined iteratively, the voltage is given in terms
of a single mode of the Fourier transforms of the bunch forms, namely the mode
at the resonant frequency of the cavity. Supposing that the only wake field is
from the harmonic cavity, we derive a system of coupled Ha\"issinski equations
which determine the bunch positions and profiles in the equilibrium state. The
number of unknowns in the system is only twice the number of bunches, and it
can be solved quickly by a Newton iteration, starting with a guess determined
by path-following from a solution at weak current. We explore the effect of the
fill pattern on the bunch lengthening, and also the dependence on the shunt
impedance and detuning of the cavity away from the third harmonic of the main
accelerating cavity. We consider two measures to reduce the effects of gaps: 1)
distribution of the gaps around the ring to the greatest extent allowed, and 2)
"guard bunches" with higher charges adjacent to the gaps, compensating for the
charge missing in gaps. Results for parameters of the forthcoming ALS-U light
source are presented.
