Stability of Uniform Plasmas with Respect to Longitudinal Oscillations

It is possible to relate the dispersion formula for longitudinal oscillations in an infinite, uniform, collision-free plasma with no magnetic field to the complex potential of a line charge distribution on the real axis of the phase velocity ($u=\frac{\omega }{k}$) plane. If the initial velocity distribution integrated over directions orthogonal to the direction of propagation is $f_0\mathrm{}\left(v\right)\mathrm{}$ the plasma is stable if and only if $U\left(u\right)=P\int {-\infty }^{\infty }\frac{{f_0}^{\prime }\mathrm{}\left(v\right)\mathrm{dv}}{v-u}$ is negative at the minima of $f_0\mathrm{}\left(v\right)\mathrm{}$ on the real axis, with unimportant exceptions. In particular it is shown that single-peaked distributions are stable, while those with very sharp (e.g., nondifferentiable) minima or with a zero of $f_0$ between two peaks are not. The charge analogy yields information on the wavelengths for which oscillations can grow and on rates of growth. Examples are given, including the case of two identical interpenetrating hot plasmas. A limited generalization to transverse oscillations is given.