Approximate longitudinal space charge impedances of a round beam between   parallel plates and inside a rectangular chamber
  This paper presents the approximate analytical solutions to the longitudinal space charge (LSC) impedances of a round beam with uniform transverse distribution and sinusoidal line density modulations under two boundary conditions: (a) between parallel plates (b) inside a rectangular chamber, respectively. When the ratio of beam diameter to chamber height is small, the image charge fields of the round beam can be approximated by those of a line charge, the approximate analytical LSC impedances can be obtained by image method. The derived theoretical LSC impedances are valid at any perturbation wavelength and are consistent well with the numerical simulation results in a large range of ratios of beam diameters to chamber heights. 
Yingjie Li (Michigan State University), Lanfa Wang (SLAC National   Accelerator Laboratory)
physics.acc-ph