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A higher harmonic cavity (HHC), used to cause bunch lengthening for anincrease in the Touschek lifetime, is a feature of several fourth generationsynchrotron light sources. The desired bunch lengthening is complicated by thepresence of required gaps in the bunch train. In a recent paper the author andVenturini studied the effect of various fill patterns by calculating the chargedensities in the equilibrium state, through coupled Ha\"issinski equations. Weassumed that the only collective force was from the beam loading (wake field)of the harmonic cavity in its lowest mode. The present paper improves thenotation and organization of the equations so as to allow an easy inclusion ofmultiple resonator wake fields. This allows one to study the effects of beamloading of the main accelerating cavity, higher order modes of the cavities,and short range geometric wakes represented by low-$Q$ resonators. As anexample these effects are explored for ALS-U. The compensation of the inducedvoltage in the main cavity, achieved in practice by a feedback system, ismodeled by adjustment of the generator voltage through a new iterative scheme.Except in the case of a complete fill, the compensated main cavity beam loadinghas a substantial effect on the bunch profiles and the Touschek lifetimes. A$Q=6$ resonator, approximating the effect of a realistic short range wake, isalso consequential for the bunch forms.

Equilibrium of an Arbitrary Bunch Train in the Presence of Multiple Resonator Wake Fields