The dispersion equation in plasma oscillations

A theory is developed of longitudinal oscillations in an infinite homogeneous neutral electron gas in which the thermal speeds of the electrons are taken into account, but collisions and the motion of positive ions are neglected. A perturbation method is used and the existence of oscillations which are harmonic in space and time is investigated. The treatment is relativistic and yields a dispersion equation relating the angular frequency $\omega $ and the propagation constant k of such an oscillation. It is shown that to any real positive value of $\omega ^{2}$ in the range extending from zero to an upper limit just beyond $\omega _{0}^{2}$, where $\omega _{0}$ is the plasma frequency, there corresponds a real value of k$^{2}$ which satisfies the dispersion equation; but that no other solution of the dispersion equation exists for which either $\omega ^{2}$ or k$^{2}$ is real. For the case of an unperturbed electron distribution function of Maxwellian type the dispersion equation is expressed in terms of the probability integral and is examined in detail. Some remarks are offered on the initial value problem in which the disturbance is specified at zero time.