  We present an analytical method to compute the wavenumbers and electric
fields of the space-charge-wave eigenmodes supported by a two-stream electron
beam, consisting of a solid inner cylindrical stream and a coaxial outer
annular stream, both contained within a cylindrical metallic tunnel.We extend
the analytical model developed by Ramo to the case of two streams. The method
accounts for the interaction between the two streams with the presence of the
beam-tunnel wall; it can be used to model the complex wavenumbers associated
with the two-stream instability and the plasma frequency reduction effects in
vacuum electronic amplifiers and other vacuum electronic devices.
