Two-dimensional resistive-wall impedance with finite thickness: Its mathematical structures and their physical meanings

When the skin depth is greater than the chamber thickness for relativistic beams, the two-dimensional longitudinal resistive-wall impedance of a cylindrical chamber with a finite thickness decreases proportionally to the frequency. The phenomenon is commonly interpreted as electro-magnetic fields leaking out of the chamber over a frequency range. However, the relationship between the wall current on the chamber and the leakage fields from the chamber is unclear because the naive resistive-wall impedance formula does not dynamically express how the wall current converts to the leakage fields when the skin depth exceeds the chamber thickness. A prestigious textbook [1] re-expressed the resistive-wall impedance via a parallel circuit model with the resistive-wall and inductive terms to provide a dynamic picture of the phenomenon. However, there are some flaws in the formula. From a fundamental standpoint, this study highlights them and provides a more appropriate and rigorous picture of the longitudinal resistive-wall impedance with finite thickness. To demonstrate their physical meaning, we re-express the longitudinal impedance for non-relativistic beams, as well as the transverse resistive-wall impedance including space charge impedance based on a parallel circuit model.