LONGITUDINAL WAVES IN AN INHOMOGENEOUS MAGNETIC FIELD – PLASMA INTERFACE BASED ON THE WKB APPROXIMATION

Longitudinal perturbations are investigated which propagate in an inhomogeneous interface or transition boundary layer, from a plasma region where no magnetic field exists well within its interior, to a static magnetic field region from which the plasma is excluded. The inhomogeneity in this boundary layer is the prime inhomogeneity considered here. A WKB solution is obtained for waves whose wavelength is small compared with the equilibrium characteristic length. The component of the propagation wave number is assumed to vary in the direction in which the interface is inhomogeneous. From the zero-order solution it is shown that this wave number satisfies the same relation with the frequency of perturbation and velocities of the electron and ion beams, as that obtained in counter-streaming of an electron-plus-ion beam against another electron-plus-ion beam in a uniform medium. The first-order solution yields the manner in which the amplitude of the wave varies from point to point in the plasma interface. The physical significance of these modes and the time and space variation of these waves will be discussed. It is shown that the WKB solution is valid in the sheath up to quite small distances from the sheath boundary. Under certain conditions, a wave can exist which increases exponentially in time and whose amplitude increases towards the boundary. The problem of joining the WKB solution appropriately to a perturbed solution outside of the boundary is discussed.