Image Effects of Cylindrical Pipes on Continuous Beams

We analyze the image effects of a cylindrical pipe on continuous beams with elliptical symmetry. Differential equations involving the second-order spatial moments of a particle distribution are given. From the moment equations, a set of Kapchinskij-Vladimirskij-type equations is developed, which include the image effects of a cylindrical beam pipe. These equations are used to analyze the image effects for focusing, drift, defocusing, drift channels, sheet beams, and a magnetic quadrupole matching section.