Application of the magneto-ionic theory to radio waves incident obliquely upon a horizontally-stratified ionosphere

A study is made of the propagation of wave-packets incident obliquely upon a slowly varying plane-stratified ionosphere, allowing for the full effect of the earth's magnetic field. Propagation of a magneto-ionic component is described in terms of the vertical component q of the phase-propagation vector, so as to avoid using a refractive index which depends in a complicated way on an unknown angle of refraction. The fundamental formula of the theory is an algebraic quartic equation for q, the coefficients in which depend on electron-density N, wave-frequency, direction of earth's magnetic field, and direction of incidence upon the ionosphere. The four roots of the quartic for q correspond to the upgoing/downcoming ordinary/extraordinary magneto-ionic components. It is shown that, to plot the four roots of q as a function of N for fixed values of the other parameters, it is easier to restate the fundamental formula as a cubic equation for N as a function of q. Using this cubic equation, simple limiting curves between which the actual curves necessarily lie are deduced, thereby facilitating plotting in a particular case. Methods for mass-production of such propagation-curves are devised but not applied on a large scale. Formulas for attenuation, lateral deviation, horizontal range, and equivalent paths of magneto-ionically split wave-packets are derived and illustrated graphically. The amount to which a wave-packet can be permanently deviated out of its plane of incidence by the earth's magnetic field can be extremely large, but probably not in cases of practical importance in commercial radio communication.