Two-Stream Instability of Plasma Waves Trapped in a Relativistic Electron Beam of Finite Perpendicular Extent

Effect of finite perpendicular extent of a cold relativistic electron beam on the two-stream instability excited in a warm plasma is investigated. It is shown that if the beam density profile is sufficiently slowly-varying the fastest growing modes are deeply trapped in the beam region and have growth rates comparable to the maximum growth rate of the usual two-stream instability by a uniform beam. The trapping is due to the frequency downshift of the space-charge wave below the Bohm-Gross frequency in the beam region. Both the dispersion relation analysis based on the WKB approximation and the eigenvalue analysis are presented.