Space charge impedances of a rectangular beam with longitudinal density modulations inside a rectangular chamber

This paper presents the space charge impedances of a rectangular beam inside a rectangular chamber, and the limiting case, e.g., a rectangular beam between parallel plates. The charged beam has uniform density in vertical direction and arbitrary distribution in horizontal direction. The method of separation of variables is used to calculate the space charge potentials, fields, and impedances which are valid in the whole perturbation wavelength spectrum. Comparisons between the theoretical calculations and the numerical simulations are also provided and they match quite well. It is shown that the rectangular beam shape may help to reduce the longitudinal space charge impedances.