Synchronous quenching due to nonlinear mode coupling in beam-plasma system

A nonlinear theory is presented which shows mode-locking of a naturally excited instability by an externally launched wave whose frequency is around that of the instability. A procedure to remove secular solutions leads to a couple of nonlinear equations which describe slow-time evolutions of amplitudes. A stably stationary solution of these equations is investigated to show qualitative agreement with published experiments; the external field strength at which the instability vanishes increases as the frequency discrepancy between the instability and the external wave |Ω-ω₀| increases, and a relation Aa²+Bb²=1 is obtained between the amplitude of the instability a and that of the externally excited wave b where the ratio B/A depends on the frequency discrepancy in the form (Ω-ω0)⁻².