A higher harmonic cavity (HHC), used to cause bunch lengthening for an increase in the Touschek lifetime, is a feature of several fourth generation synchrotron light sources. The desired bunch lengthening is complicated by the presence of required gaps in the bunch train. In a recent paper the author and Venturini studied the effect of various fill patterns by calculating the charge densities in the equilibrium state, through coupled Haïssinski equations. We assumed that the only collective force was from the beam loading (wakefield) of the harmonic cavity in its lowest mode. The present paper improves the notation and organization of the equations so as to allow an easy inclusion of multiple resonator wakefields. This allows one to study the effects of beam loading of the main accelerating cavity, higher order modes of the cavities, and short range geometric wakes represented by low-$Q$ resonators. As an example these effects are explored for ALS-U. The compensation of the induced voltage in the main cavity, achieved in practice by a feedback system, is modeled by adjustment of the generator voltage through a new iterative scheme. Except in the case of a complete fill, the compensated main cavity beam loading has a substantial effect on the bunch profiles and the Touschek lifetimes. A $Q=6$ resonator, approximating the effect of a realistic short range wake, is also consequential for the bunch forms.

中文翻译：

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存在多个共振器尾波场时任意束流列的平衡

若干第四代同步加速器光源的一个特征是高谐波谐振腔（HHC），该谐振腔用于引起束延长，从而延长了Touschek寿命。所需的束长度由于束列中存在所需间隙而变得复杂。在最近的一篇论文中，作者和Venturini通过耦合的Haissinski方程，通过计算平衡状态下的电荷密度，研究了各种填充方式的影响。我们假设唯一的集体力来自处于其最低模式的谐波腔的光束载荷（尾波场）。本文改进了等式的表示法和组织，以允许轻松地包含多个谐振器尾波场。这样一来，人们就可以研究主要加速腔的光束载荷，腔的高阶模态，$问$谐振器。例如，对ALS-U探索了这些效果。实际上，通过反馈系统实现的对主腔中感应电压的补偿是通过通过新的迭代方案调整发电机电压来建模的。除了完全填充的情况外，补偿后的主腔梁载荷对束轮廓和Touschek寿命有很大影响。一种$问=6$ 对于束形，谐振器也近似于现实的短程苏醒的影响，因此也很重要。