ULTRA-SHORT WAVE PROPAGATION 139 



below one kilometer or so, where the ionization is negligible, that is 

 essential. 



The radius of curvature of a ray traveling horizontally in the 

 lower atmosphere can readily be calculated if it is known how the 

 refractive index, w, varies from point to point. If // is the altitude 

 above sea-level, the radius of curvature of the ray is simply 



dnjdH 



But since n = \e, where e is the dielectric constant, the radius of 



curvature is 



2 

 P = — 



deIdH' 



provided 7i is not very different from unity. 



In Appendix II the estimation of this radius of curvature is dis- 

 cussed in some detail. While some of the data upon which such a 

 calculation can be based are rather uncertain, it appears that a good 

 first approximation is obtained by assuming the radius of curvature, 

 p, of the refracted ray to be four times the radius of the earth, Tq. 

 As pointed out in the appendix, this varies to some extent with weather, 

 and even as an average value, it may have to be changed when more 

 reliable data on dielectric constants become available. 



On first consideration of the ways in which refraction can be taken 

 into account, it appears that the attempt must complicate an already 

 involved situation. Fortunately, however, refraction is much simpler 

 to calculate than diffraction or reflection. The method is presented 

 rigorously in Appendix III. At this point we shall merely state the 

 result and show its plausibility. 



In ultra-short wave work we are almost always concerned with 



propagation in a nearly horizontal direction. The curvature of the 



ray is 1/p, while that of the earth is 1/ro. We are interested, however, 



in the relative curvature, which we shall call l/r^. If, instead of using 



simple rectangular coordinates, we transform to a coordinate system 



in which the ray is a straight line, the curvature of the earth will 



become I /re, which is 1/ro — 1/p. The equivalent radius of the earth 



would be 



/ 1 

 re = ro[ . 7- 



and is therefore greater than the actual radius of the earth by the 



factor •:; 7-—: which is 1.33. This fictitious radius is therefore 



1 — 1/4 



