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The theory of free‐electron‐laser instability for a tenuous relativistic electron beam propagating parallel to a longitudinal wiggler field has been extended to include the influence of finite radial geometry on its stability properties. The growth rate of the instability shows a sensitive dependence on (ν/γ)1/3, where ν is the Budker parameter and γ is the beam energy. The growth rate increases with increasing beam intensity. However, the instability bandwidth with respect to the beam location inside the waveguide decreases for beams with somewhat higher ν/γ values. The instability is broadband for several harmonics excited.