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A method is described for predicting the field strength required to produce continuous wave breakdown of gases with spatially varying properties subjected to electric fields of nonuniform intensity. The proposed method is based on a variational principle derived from the electron continuity equation. A Ritz method is used to generate solutions of the variational problem. The approximate solutions are evaluated by comparing them with several exact solutions which are also developed. The use of a one-term trial function in the Ritz method is found to give very good agreement with the exact solutions. A two-term approximation provides greater accuracy than is presently available from experimental data.

A small-perturbation analysis has been carried out to determine the effect of a nonlinear electrical conductivity on the propagation of microwaves normally incident on a homogeneous, slightly ionized gas. The reflecting properties of the interface separating the ionized gas from free space have also been determined. The measure of the nonlinearity is given by the quantity γ, which is the ratio of the energy acquired by the free electrons from the field to the kinetic energy of the electrons in the absence of an applied field. The solution was obtained by performing a series expansion of the field strength in powers of γ. Terms up to order γ2 were retained. It was found that the nonlinearity can cause significant increases in both the wave number and the attenuation factor of the transmitted signal. The transmission coefficient associated with the interface may also be significantly reduced. The change in the phase shift across the interface due to the nonlinearity was found to be positive or negative, depending upon whether the signal frequency is greater or less than the plasma frequency, respectively.