Coupling points of the Booker quartic equation for radio wave propagation in the ionosphere

For a horizontally stratified ionosphere the four roots of the Booker quartic equation can often be used to give four independent W. K. B. type solutions of the electromagnetic equations which govern radio wave propagation, but this is only possible if the four roots are distinct. There are points in the complex height plane, called ‘coupling points’, where two roots of the quartic are equal, and in phase integral methods it is necessary to know their positions, because near them the W. K. B. solutions fail. There are eight coupling points which are important in radio propagation and four of these are of particular interest. Their positions are found from an equation given by Pitteway (1959). Their behaviour, including their loci as the azimuth and elevation of the incident wave are varied, is studied. It is illustrated by a specific example, for temperate northern latitudes, in which the frequency is one-quarter of the electron gyrofrequency and the azimuth of propagation is (magnetic) North to South or near to this. The effect of neglect or inclusion of electron collisions is investigated.