SPHERICAL EARTH 



131 



If n in equation (2) is allowed to assume all frac- 

 tional and integral values from to 2, sin 7 — > 7 

 may be expressed as 



"^ (10) 



sm 7 = 7 = 



4/h 



Substituting this \'alue into equation (9) gives 



d = 2ylG[(k sin y^j = 2VgIc/o sin (90°«)- (1 1) 



Equation (11) is useful in sketching the lobe contour. 

 It holds only when the reflection coefficient equals 

 — 1 (i.e., p = +1) and the angle 7 is small enough 

 so that equation (10) is valid. 



small null areas, ^rhe shape of the contour is, there- 

 fore, less important and it is sufficient to find the 

 maximimi and minimvun ranges and then to sketch 

 the lobe from the polar sme formula [equation (11)]. 

 On the other hand, the shape of the contour is of 

 great importance when there are few lobes and the 

 null area is large. This is illustrated in Figure 3, 

 where there are only two complete lobes in the 

 region of interest. 



The second point to be noted is the varjdng effect 

 of divergence on the lobe number and angle. It 

 may be seen from equation (89) in Chapter 5 that 

 the divergence factor approaches imity as tan \l/ 



I'/i 



126 160 192 224 



d DISTANCE IN KILOMETERS 



Figure 2. Vertical coverage diagram. 



6.3 

 6.3.1 



SPHERICAL EARTH 

 Lobe Characteristics 



Figures 2 and 3 are typical vertical coverage 

 diagrams for a smooth spherical earth. They illus- 

 trate two important points. 



The first is the dependence of the numljer of lobes 

 on the ratio of transmitter height to wavelength. 

 Figures 2 and 3 show that for hi equal to 75.4 wave- 

 lengths, the lobes are much more closely spaced 

 than for a transmitter height of 32.3 wavelengths. 



When the number of lobes is large, there is little 

 possibility for a target to escape detection in the 



increases (see also Figure 11). The divergence 

 factor is low for small angles and then approaches 

 unity rapidly. This accounts for the reduced range 

 of the first three lobes of Figure 2. For the larger 

 angles, the maximum range is approximately equal 

 to twice the free-space range. When the ratio /ii/X is 

 small, the angle at the first lobe maxima is large, since 

 7 = ?iX/4/ii. In this case, the effects of divergence 

 will be negligible except for the lower part of the 

 first lobe, and the polar sine function derived for the 

 [jlane earth may be used. 



There exists also the intermediate case where the 

 effects of divergence may not be neglected and an 



