ULTRA-SHORT WAVE PROPAGATION 159 

 is found with the help of (6) to be as follows for the two cases: 



r - ro = ro([l + LJi' - 1) (real case), (12) 



(j' - ro') = - -" ([1 + L]-i - 1) (fictitious case). (13) 



s 

 where L Is defined by the equation 



(i + L) = (iy = e2iM^. (14) 



^ \ ^0 / cos lA 



L is small compared with unity in the cases that we are considering. 

 By expanding each and subtracting, the error caused by assuming 

 that (r — ^o) equals (r' — ro') is found to be 



►7—^ (1 -\- s) + higher order terms in L. (15) 



We have found above that 5 is approximately equal to — 0.75. Taking 

 r^ = 6370 km. and remembering that we are ordinarily not concerned 

 with rays farther above the earth than, say, 5 km., we have from (14) 



whence L > - 0.0006. 



Substituting these values in (15) we find that the error in height is 

 less than 50 cm. This is negligible in the altitude of 5 km. which was 

 assumed and we may consider the equivalence to be proved. 



Appendix IV — Diffraction Calculations 



The method of Huyghens applied to optical diffraction past a 

 straight edge results in the following expression for the received field,. 

 E, in terms of Fresnel integrals. 



where 



■^ = a - jb = C exp ( - jrj), 

 -c-o 



- If-. 7r?>2 



