4-6] EFFECT OF EARTH'S CURVATURE 191 



allowed for by increasing the earth's radius by a factor 4/3 to an effective 

 earth's radius 



a.. ^ ^/2a. (4-43) 



With this factor, and expressing heights in feet and distances in statute 

 miles, the height reductions due to curvature take the simple form 



A/7 1,2 = ^1,2-/2. (4-44) 



For a given value of range R, which is practically the same as the total 

 distance d = d\-\- di measured along the earth's surface, the determination 

 of ^1 (or d^ leads to the cubic equation 



Idr^ - Ud;^ - \lalh^ + ^2) - ^-] dr + la,h^d = 0. (4-45) 



Once this is solved for di, h[ and h'^ may be calculated from Equation 4-42 

 and the remainder of the geometry handled like a plane-earth problem. 

 Since the solution of the cubic is laborious, it is usually simpler to employ a 

 graphical solution by plotting h[ jdi and h'^ jdi versus di. The proper value 

 of di occurs where these two quantities are equal, since this gives equal 

 A^alues of before and after reflection. 



As mentioned above, reflection at a spherical surface reduces the reflec- 

 tion coefficient from the plane earth value p to 



p' = pD (4-46) 



where D is the divergence factor. This is given by 



y^ajsmd 



(4-47) 



(•-^r 



For very small values of 6 the divergence factor causes reduction of the 

 effective reflection coefficient p' given by Equation 4-46 to a small value. 

 In fact, at the horizon (9 = 0) D = 0, so that there p' = 0. However, the 

 representation of the propagation process in terms of only a direct and a 

 reflected ray breaks down as the horizon is approached. Norton^ gives as 



the limit to which Equation 4-46 is restricted: 



h 





, . . . . (4-48) 



d\ 



Practically all airborne radar ranges will be within this limit as long as 

 atmospheric refraction does not depart greatly from the standard condition. 



^K. A. Norton, "The Calculation of Ground-Wave Field Intensity over a Finitely Con- 

 ducting Spherical Earth," Proc. IRE 29, 623-639 (1941). 



