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An analysis of the eigenmodes of a cylindrical metallic waveguide partially filled with a relativistic electron beam, and guided with an ion channel, is presented. The analysis is performed in the rest frame of the beam with self-fields taken into account. Dispersion relations for five families of electrostatic and EM modes are derived and solved numerically to study the characteristics of azimuthally symmetric TM and TE waveguide modes, betatron modes, and space-charge modes. A strong dependence of the frequencies of these electromagnetic-electrostatic waves on the ratio of the radius of the waveguide to the beam radius is revealed. The physical insight given to the electrostatic modes, at large and small wavelengths, equally applies to the Trivelpiece-Gould modes in an electron beam with axial magnetic field guiding. Interactions between the dispersion curves of different modes are used to find the infinite-space dispersion relation from the finite-radius dispersion relation and to explain the transition from a weak to strong ion channel.
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