  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.
