154 



SITING AND COVERAGE OF GROUND RADARS 



ANTENNA 



MODIFIED 

 EARTH SURFACE 



Figure 54. Reflection point geometry. 



and neglecting §(nX/l 0,560) compared to A, equa- 

 tion (67) becomes 



?iX 



B 



10,560 



.di 



2d l <S>- 



nX 



(69) 



10,560 

 From the law of sines 



sin 2^ sin (M> + ty d ) sin (^ — V d ) 



r d B 



sin (V + <ff d ) 



When ty and ^ d are small 



B sin 2ft 



Td 



B 



<i> + ^ rf = - 2* , 



and 



* 



^d = - 2* 



rd 



and hence by subtraction 



B - A T 

 *d = ^ • 



»'d 



Since 'i' is a small quantity r d may be taken to equal 

 A + B, and A = di, that is 



B - d, 



*^=B-+T*- 



(70) 



This is the angle of the target with respect to the 

 tangent plane CE as seen from the antenna. The 



angle desired however is y, which is measured with 

 respect to the horizontal at the antenna, GH. As 

 shown in Figure 54 



t = *d + e . 



From equation (61) 



The line of minimum path difference (A = 0) is 

 along the earth's surface from the transmitter to 

 the horizon, and beyond it is along the line of sight 

 tangential to the horizon since the direct and indirect 

 waves are equal in that case. Maximum path differ- 

 ence occurs directly below the antenna and is equal 

 to 2h\. Since the path difference is also nX/2, the 

 maximum value of n is 4/ii/X. In practice the vertical 

 directivity of the antenna limits n to a much smaller 

 value. 



Consider a wave which is reflected from directly 

 under the antenna, and let ho denote the height 

 above the reflector at which the path difference is 

 n\/2. Then 



hi + h — (hi — ho) 



wX 



T 



nX 

 ~2~ 



(71) 



Thus if X is 10 ft the center of the first lobe will be 

 2.5 ft high at zero range. For most purposes the lobes 

 and nulls may therefore be considered to start at 

 the origin. 



