156 



BELL SYSTEM TECHNICAL JOURNAL 



ro = radius of the earth = 6370 km. 



Appendix III — Effect of Refraction 



In the following it will be shown that the transformation given in 

 the text gives the proper path for the ray and the proper phase rela- 

 tions. The latter are more conveniently treated by determining the 

 "phase time," or the time required for a given phase to traverse the 

 path. As a matter of fact the ray paths and phase times are not 

 exactly the same in the two constructions but it will be shown that for 

 one distribution of refractive index, n, which closely resembles that 

 actually encountered, the error is negligibly small. 



As indicated, this analysis is based on the customary ray treatment 

 of refraction through a medium with a continuously varying refractive 

 index. This simple ray theory is known not to be exact but in the 

 present case we shall always be dealing with very small gradients, a 

 condition in which the error becomes very small. 



A good summary of the relations which we shall use is given in 

 "The Propagation of Radio Waves," by P. O. Pedersen on pp. 154 

 and 155. The nomenclature is indicated in Fig. 17. 



The length of the element of path, hh\ equals 



rdO 

 sin if 



(1) 



and the time required for the wave to traverse it is 



dt = 



rdd 



V sin ip 



nr dd 

 c sin ^ 



(2: 



2' It is inferred from a statement made by Jouaust, Froc. I. R. E., Vol. 19, p. 487, 

 Mar. 1931, that for his experiments between France and Corsica p/ro would have 

 to be 5 or less in order for the direct ray to be unobstructed. Our figure of 4 therefore 

 would indicate that his is an "optical" path. We do not believe, however, that 

 an optical path is a necessary or sufficient condition for strong signals, although it 

 certainly does help to make them probable. 



