The coalescence of coupling points in the theory of radio waves in the ionosphere

Abstract
When a plane radio wave is obliquely or vertically incident on a horizontally stratified ionosphere, there are certain points in the complex height plane, called coupling points (which include reflexion points) where there is a breakdown in the independent propagation of the four waves, ordinary upgoing and downgoing, and extraordinary upgoing and downgoing. If a coupling point is far enough away from other coupling points and from singularities, it is said to be isolated and the electromagnetic field near it can be expressed in terms of Airy Integral functions. Then the phase integral method can be used with suitably chosen contours, to calculate reflexion coefficients and coupling coefficients. If two coupling points are too close together, however, the procedure needs modification. This paper studies the theory when two coupling points approach coalescence. It is confined to the cases where the same two waves are coupled at the adjacent coupling points, since this is the most important in practice. It is found that there are two kinds of coalescence called coalescences of the first kind C1, and of the second kind C2. For C1 the coupling remains strong when the coupling points move to coalescence. For C2 the coupling gets weaker, and disappears completely at exact coalescence. The electromagnetic fields near a point of coalescence can be expressed in terms of solutions of Weber’s equation, but the form of this equation is different in the two cases. The type C1 is important in the theory of partial penetration and reflexion for frequencies near the penetration frequency of an ionospheric layer. The type C2 is important in the theory of ʻcrossover’ for ion-cyclotron whistlers, and in the theory of the Ellis window and related phenomena. These applications are worked out as illustrations.

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