Radio caustics and cusps in the ionosphere

When a radio signal from a point source enters the ionosphere, the field contains caustic surfaces, and these in turn contain cusp lines. In these regions there is some focusing, and this paper shows how to calculate the spatial dependence of the field. It is expressed as a double integral representing an angular spectrum of plane waves. Evaluation of the integral uses the method of double steepest descents with two complex variables. Near a caustic, two saddle points are close together and an extension of steepest descents given by Chester, Friedman & Ursell (1957) is used. Near a cusp three saddle points are close together and the method is extended so as to use a cusp function given by Pearcey.