Parametric instabilities with finite wavelength pump

The general problem of parametric instabilities driven by a finite wavelength pump is investigated. For the particular case of a Langmuir wave pump, it is shown that resonant decay instabilities (forward or backward scattering in the one-dimensional case), with thresholds which vanish in the collisionless limit, can occur only for pump wavenumber k0 greater than the critical value [(m/M)1/2/γ]k/sub D/, where m and M are the electron and the ion mass, respectively, and γ is the specific heat ratio. For smaller wavenumbers, there is always a nonzero threshold, the instability being of modulation character for long wavelengths and almost purely growing for short wavelengths. Frequency locking for small k0 and wavenumber locking for large k0 are demonstrated. The results are generalized to the case where the coupled waves satisfy arbitrary dispersion relations and simple physical interpretations of the instabilities are given