The interaction between a tubular beam of charged particles and a dispersive metamaterial of cylindrical configuration has been investigated theoretically. This metamaterial may have negative permittivity and negative permeability simultaneously over a certain frequency range where it behaves like a left-handed metamaterial. The dispersion equation for the eigenmodes spectra of a metamaterial and the coupled modes spectra of the system have been derived and numerically analyzed. It has been found that the absolute beam instability of bulk-surface waves occurs because of peculiarities of the eigenmodes spectra of a left-handed metamaterial. Specifically, the resonant frequency behavior of the permeability causes the emergence of the sections of dispersion curves with anomalous dispersion. It has been demonstrated that the symmetric bulk-surface mode with two field variations along the cylinder radius possesses the maximum value of instability increment. The obtained results allow us to propose the left-handed metamaterial as the delaying medium in oscillators of electromagnetic radiation without a need to provide an additional feedback in the system just as in a backward-wave tube.
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