An Impedance Boundary Condition for Computing Beam Coupling Impedances in Complex-shaped Perforated Beam Pipes

The effect of (a huge number of) pumping holes on beam dynamics and stability, described in terms of coupling impedances, is a fundamental issue and has been carefully investigated, both theoretically [1] and experimentally [2]. With the exception of a few very simple pipe geometris, for which the modal expansion of the electromagnetic field is available, beam coupling impedances can be only computed numerically. In [3] we developed a general (reciprocity-based, perturbative) analytic approach for computing the (longitudinal and transverse) beam coupling impedances in complexshaped, heterogeneous pipes, where the local wall properties are described in terms of impedance boundary conditions. In this communication we introduce such a boundary condition for a perforated wall, and discuss a number of relevant generalizations, thus extending the applicability of the aforementioned analytic approach to perforated pipes. In Sect. 2, we review the perturbative formula for computing the longitudinal beam coupling impedance in a complex-shaped, heterogeneous pipe. In Sect. 3 a perforated wall impedance boundary condition is obtained by solving an appropriate canonical boundary value problem. In Sect. 4 we introduce a modified polarizability, to account for the effect of an external co-axial shield, for an infinite pipe or a large ring. In Sect. 5 a few possible refinements are discussed. Concluding remarks follow under Sect. 6.