Purely growing parametric instability in an inhomogeneous plasma

Use of a simple method based on energy balance to rederive the well-known threshold condition for the purely growing parametric instability in a homogeneous medium and to estimate the effects of inhomogeneity in a semiquantitative manner. A method different from that of Perkins and Flick (1971) is then used to calculate the threshold in a more quantitative manner for the instance where the effects of inhomogeneity dominate over those of collisions. The result agrees with that of Perkins and Flick for k sub parallel l much greater than 1 in their terminology. For k sub parallel much less than 1, neither theory is directly applicable and the threshold is obtained by numerical methods. The present method of calculation has the advantages that its range of validity is easily checked, that it provides good physical insight, and that it is easily applicable to electromagnetic instabilities.