A Discussion on the early days of ionospheric research and the theory of electric and magnetic waves in the ionosphere and magnetosphere - Phase integral methods for studying the effect of the ionosphere on radio propagation

Abstract
The phase integral method is a form of ray theory, extended to use complex values of the space coordinates. Its application to radio propagation studies was pioneered by T. L. Eckersley who showed how to use it for calculating (a) the reflexion coefficient of the ionosphere, (b) the propagation constant for radio waves guided by the Earth’s surface and by the ionosphere or troposphere, and (c) the coefficient for coupling of an ordinary and an extraordinary wave in the ionosphere. The method involves the evaluation of integrals along suitably chosen contours in complex space. It is approximate but often capable of high accuracy and often quicker to use than more exact methods. Its justification is based on the physical principles of analytic continuation and of uniform approximation. For reflexion and coupling problems in a horizontally stratified ionosphere, the contours used for the phase integrals are determined by those real or complex heights called ‘reflexion’ or ‘coupling’ points, where two roots of the Booker quartic equation are equal. The study of the behaviour of the governing equations near these points shows when failure of the phase integral method may be expected.

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