This paper analyzes the transient state of a bunched electron beam traversing a narrow beam pipe under the influence of transverse resistive-wall effects. Because the electron beam in the wigglers of future free-electron lasers (FEL’s) is required to travel in pipes with small diameters, this study is relevant in determining the beam stability in these FEL’s. The analysis is restricted to the case in which the first mode dominates. First, an integral representation of the single-mode solution is obtained. Second, the transient-state solution is obtained for the case in which the focusing force dominates the resistive-wall force. The maximum transient amplitude is derived for the case in which a single bunch is off-axis. Third, this result is used to compute the maximum transient amplitudes when the bunches enter the system with random and with equal initial displacements, respectively. These analytic expressions of maximum transient amplitudes are useful in predicting when the transient could become significant. They are compared with earlier derivations and with numerical simulations. These expressions are also applied to various proposed FEL’s. Results showed that the transverse resistive-wall instability could not pose a problem in the operation of these FEL’s.

• Received 13 June 1989

DOI:https://doi.org/10.1103/PhysRevA.41.333