Ionosphere reflection coefficients by variational technique

One of the important problems of ionosphere work is to predict the nature of the electromagnetic wave reflected by the ionosphere when a wave of known form is incident upon it. In the present paper, this problem is treated by considering a plane ionosphere model for which the electron density and earth's magnetic field are continuous but otherwise arbitrary functions of the height. It is shown that if a plane wave impinges on such a model at an arbitrary angle of incidence, then four complex reflection coefficients suffice to characterize the reflected wave as to its intensity, phase, and polarization. Thus the problem becomes one of the calculating of these four numbers. The attack on this problem is based on an adaptation of a method due to the physicist, J. Schwinger, wherein, with the aid of an exact integral equation for the electric field in the ionosphere, one derives variational formulas for the four reflection coefficients. These formulas may then be used either to calculate the coefficients numerically or to obtain approximate expressions for them in terms of the various physical parameters of the problem. Finally, as a consequence of a certain symmetry property of the variational formulas, a reciprocity theorem for ionospheric propagation is deduced.