WKB solution for wave propagation in a time-varying magnetoplasmamedium: longitudinal propagation

The interaction of a right circularly polarized electromagnetic wave of frequency ω₀ with a switched-on-time-varying plasma medium, in the presence of a static magnetic field, is considered. Longitudinal propagation is assumed and ion motion is neglected. The electron density is assumed to vary slowly with time, and the solution is obtained through a WKB approximation. The main effect of switching the time-varying medium is the splitting of the original (incident) wave of frequency ω₀ into three new waves with time-varying frequencies. An exponentially increasing electron density profile is considered to illustrate the solution. The distinguishing feature of the presence of the static magnetic field is the creation of the third wave. The initial value of the instantaneous frequency ω₃ of this wave is equal to the gyrofrequency ω_b, and the final value depends on ω_b, ω₀, and the final value of the plasma frequency. ω₃ decreases with time for the profile under consideration