Bunched Beam Envelope Equations Including Image Effects from a Cylindrical Pipe

We derive a set of differential equations for the beam envelopes of an axisymmetric, bunched beam inside a perfectly conducting beam pipe. It is found that the beam dynamics are essentially independent of the form of bunch distribution in the free-space situation, however, in the presence of the beam pipe this is no longer the case. Analytic expressions involving infinite summations of Bessel functions are derived for the image potential and image fields of an ellipsoidally symmetric charge distributions in a beam pipe, in particular, the uniform density distribution. We simulate a simple beam transport system to demonstrate the application of these results.