The instabilities of a cylindrical electron stream driven by shear in the velocity parallel to the guiding magnetic field (dv11/dr ≠ 0) are analyzed. Quadratic forms are developed that bound the growth rates and provide necessary conditions for instability. An efficient numerical method for calculating the eigenfrequencies directly from the (singular) differential equation of a system with arbitrary velocity/density profiles is described. This numerical method is used to study the unstable modes of an unneutralized beam with velocity shear arising from space-charge potential depression. The computed growth lengths indicate that these modes are of practical importance in the propagation of high perveance electron beams.
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Rome et al. (1972) studied this question.
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