Wake Potentials of a Relativistic Point Charge Crossing a Beam-Pipe Gap: An Analytical Approximation

Although much work has been devoted recently to the numerical calculation of wake potentials, there is at present no analytical expression for the wake of a point charge crossing a gap in a beam-pipe of finite diameter. For a relativistic charge the wake potential can be computed in a simple manner from an integral over the electric field at the radius of the beampipe. In the case of a pill-box cavity, our basic assumptions are that the resonant frequencies and the field at the radius of the beam-pipe are essentially the same as in a cavity without beam hole. The summation over the infinity of modes can still be carried out as far as reflections from the outer cavity wall do not contribute; this is sufficient to compute the total loss for short bunches.