SPHERICAL EARTH 



71 



the fact that the image source (Figure 8) of the 

 reflected wave is at a distance )• + A from the re- 

 ceiver. The reflected wave is attenuated more than 

 the direct wave, according to the free-space attenua- 

 tion ratio (r -{- A)/r. If this is taken into account 

 the ordinary reflection coefficient is replaced by 



In order to express the slant range r^ in terms of the 

 curved distance d to a higher order of accuracy, the 

 cosine law is applied to the triangle, transmitter- 

 receiver-earth center. This gives the equation 



r/ = (ka + hY + (ka + hy 



Fi \r + A/ 



2{ka + h) {ka + /!2) cos 



pD. 



(56) 



(£)■ 



The correction is not necessary when 2/ii/)2 < < d- 

 [see equation (52)]. 



Selecting the relatively important terms of the order 

 d'hju and dHJii -f- /la), as well as powers of d higher 

 than the fourth, the above equation reduces to 



5.5 



SPHERICAL EARTH 



ri 



5.5.2 



*"*"* Measurement of Distance 



The difference between the slant range r^ and the 

 distance measured along the surface of the earth and 

 designated by d in Figure 14 is usually negligible. 

 For a transmitter height of 1,500 meters, the error 

 in assuming r^ = d is 0.04 per cent at a distance of 

 161 km and height of 6,900 meters, and 1 per cent 

 at the same distance but at a height of 22,500 meters. 

 As the transmitter height is increased, the error is 

 increased. 



= d"- -I- Qi, - hrT- + f- (h + h - -^) . (57) 

 ka \ 12ka/ 



Equivalent Heights 



Solving equations (13) and (14) for hi and h-2 gives 



2ka 



h. = ^ 



2ka' 



These results may also be expressed by saying that 

 the distance from the surface of the earth to a plane 

 which is tangent at a distance rfi from the trans- 

 mitter is di-/2ka. 



FiGUBE 14. Geometry for radio wave propagation over a spherical earth. (Vertical dimensions greatly exaggerated.) 



