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Recent high-power gyrotrons, capable of producing radio-frequency power above 1 MW, often suffer from parasitic oscillations in the beam tunnel, despite the presence of dielectric loading materials intended to prevent the growth of such modes. Such oscillations affect the operation and the efficiency of these devices significantly. Lately, a variety of dielectric materials have been used, with limited success, in ring-loaded beam tunnels to suppress unwanted oscillations. In this paper, we perform an extended parametric study of the effects of beam-tunnel geometry and lossy material properties on the damping of such oscillations.