The calculation of reflexion coefficients for long wireless waves incident obliquely on the ionosphere requires an exact solution of the differential equations governing the propagation of electromagnetic waves in the ionosphere. Equations are developed for the electromagnetic field in a horizontally stratified medium of varying electron density, the presence of a vertical external magnetic field and also the collision frequency of the electrons with neutral molecules being taken into account. Provided certain inequalities hold amongst these ionospheric characteristics, the ionosphere splits up effectively into two regions, in each of which the differential equations of wave propagation approximate to simpler forms. If a model ionosphere is chosen in which the ionization density increases exponentially with height/and the collision frequency is assumed constant over the range of height responsible for reflexion, the equations for the two regions can be solved exactly. The solution for the lower region is expressed in terms of hypergeometric functions, and that for the upper region in terms of generalized confluent hypergeometric functions. Exact expressions in terms of factorial functions can then be deduced for the reflexion coefficients of both regions separately. Moreover, these coefficients can be combined, with due allowance for the path difference between the two regions, to give the overall reflexion coefficients for the effect of the ionosphere as a whole on an incident wave. A suitable definition is given for the apparent height of reflexion in terms of the phase of the reflected wave. The results of the theory are illustrated in graphical form for a particular model ionosphere approximating to the 'tail’ of a Chapman region, and a brief comparison with experimental observations concludes the paper.
Three-dimensional critical phase diagram of the Ising antiferromagnet
CeRh2Si2
under intense magnetic field and pressure