Dispersion relations for longitudinal and transverse waves propagated perpendicularly to an externally applied high‐frequency electric field in a fully ionized plasma has been obtained. It has been shown that in addition to&ohgr; ∼ &ohgr;0(&ohgr;0is the characteristic electron frequency), the longitudinal mode may be excited with frequencies&ohgr; ≃ &OHgr;k± &ngr;(&OHgr;k2 = c2k2 + &ohgr;02, &ngr; is the external field frequency); and under certain conditions the excitation near&ohgr;0may be unstable. For the transverse mode instabilities may occur owing to resonance between a backward and a forward moving wave of the same frequency and vice versa. Apparently these are owing to the periodic nature of the dispersion relations found and to the mode‐mode coupling present.