Effects of finite-bandwidth driver on the parametric instability

A set of model equations for the parametric instability that consist of damped harmonic oscillators coupled with a stochastic oscillatory driver is considered. With the assumption that the autocorrelation time of the stochastic part of the driver is very short, a solution for the average response of the oscillators is derived. This solution holds for arbitrary damping rates, unlike the results of previous authors. Threshold levels and growth rates for the parametric decay and oscillating two-stream instabilities are found as a function of driver bandwidth.