Comparison of the longitudinal coupling impedance from different source terms

The longitudinal coupling impedance and the transmission coefficient resulting from a thin ring and from a uniform disk are obtained analytically for a resistive cylindrical beam-pipe of finite wall thickness. The impedances are derived and then compared with the well-known corresponding expression for perturbations on a uniform, coasting beam [A. Al-Khateeb, O. Boine-Frankenheim, R.W. Hasse, I. Hofmann, Phys. Rev. E 71 (2005) 026501]. The transmission coefficients from both sources are found to be exactly the same. Differences do appear in the expressions for the electromagnetic fields within the beam region, and therefore leading to different coupling impedances. By applying the results to parameters relevant for the SIS-18 synchrotron at GSI, it is found that the formula from the ring source underestimates the space-charge impedance at all beam energies and it shows a noticeable deviation from the disk formula for all frequencies. Although their mathematical expressions are different, resistive-wall impedances from the two sources are found to be numerically equal. The space-charge impedances become equal asymptotically only in the so-called ultra-relativistic limit.