This paper deals with the determination of the wake-function, impedance and energy loss due to surrounding accelerator components of two countermoving beams, i.e., two beams moving in opposite directions. Differently from the classical case, the distance between source and test charges cannot be considered constant. The paper defines a generalized wake-function, independent of the source and test particle’s direction of motion. From this generalized wake-function a wake-potential is derived. The paper focuses in particular on the longitudinal direction. The expression of the energy dissipated by two beams transiting in the same vacuum chamber is obtained. This expression is valid both in the comoving and countermoving beam scenario. Subsequently, the proposed model is benchmarked by calculating the longitudinal co and countermoving wake-functions, the corresponding impedances and wake-potentials for two simple geometries (a circular pipe and a pillbox cavity) traversed by two countermoving beams, and by comparing the results with the ones of the electromagnetic code CST Studio Suite. Finally, for the pillbox geometry, the energy loss of the two countermoving beams and their rf-heating are investigated.

中文翻译：

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确定两个反向移动粒子束的苏醒功能，阻抗和能量损耗

本文涉及确定由于两个反向运动的光束（即，两个方向相反的光束）周围的加速器分量而引起的唤醒功能，阻抗和能量损失。与经典情况不同，源电荷和测试电荷之间的距离不能认为是恒定的。本文定义了广义的唤醒函数，与源和测试粒子的运动方向无关。从这种广义的唤醒功能中，导出了唤醒电位。本文特别关注纵向。得到了在同一真空室内通过的两束光所耗散的能量的表达式。此表达式在同向移动和反向移动光束场景中均有效。随后，通过计算纵向co和抵消运动尾随函数来对提出的模型进行基准测试，通过两个反向移动的光束遍历的两个简单几何形状（圆形管道和药丸盒腔）的相应阻抗和唤醒电位，并将结果与​​电磁代码CST Studio Suite的结果进行比较。最后，对于药盒的几何形状，研究了两个反向运动梁的能量损失及其射频加热。