170 



SITING AND COVERAGE OF GROUND RADARS 



15617 Divergence 



The reflected wave is scattered somewhat by being 

 reflected from the spherical surface of the earth 

 instead of a plane surface, and this reduction of field 

 strength is taken into account by the divergence 

 factor. This is dependent on geometrical considera- 

 tions and may be expressed as follows (for 7' < 3°) : 



D = — ' = , (97) 



[/(7) ±/(7 - 26)zpD]d 



(99) 



V- 



wX 



2(5,280) 2 ( 7 ') 3 

 where n is the lobe number, 



X is the wavelength, in feet, 

 7' is the reflection angle, in radians, obtained 

 from equation 60. 

 A convenient chart for obtaining D is given in 

 Figure 71. The parameters are 7' in radians and rik, 

 with X expressed in feet. 



As 7' approaches zero, n also approaches zero, and 

 the equation is indeterminate. At the point of tan- 

 gency of the line of sight and the earth, D is 0.5773. 

 At. low angles the field is modified by diffraction 

 around the curved earth. The lower limit of the angle 

 7' for which the optical treatment is valid is usually 

 given by 



7' > 



, LV > 0.00382 >/x , (98) 



\2tt/to 



where k is % and X is in feet. 



For angles below this limit the theory for diffrac- 

 tion of radio waves around the earth is required for 

 a rigorous solution. However, in practice it is found 

 that angles as low as the first maximum (n = 1) may 

 be treated by the ray theory with little error. 



Applying equation (98) to Example 11 



7' > 0.00382 \/Im = 0.0065 radian. 

 In Table 6 this corresponds to n = 2.25. However 

 using n = 1 and 7' = 0.0036 the divergence factor 

 D is found from Figure 71 to be 0.58. For angles 

 below 7' = 0.0036 it is necessary to estimate D. 

 Experience indicates that a fair minimum value to 

 select for D is 0.5773. For angles much below the 

 first maximum, the optical treatment gives values 

 of field strength which are too low because it neglects 

 diffraction. This may be compensated in part by 

 using values of D between 0.5773 and 1.0. 



where L is the distance to the end of the lobes or 

 the nulls, in miles, do is the maximum range (in 

 miles) at which a given response (usually the mini- 

 mum detectable signal) would be obtained if the 

 antenna were in free space. If the lobe diagram were 

 plotted for some signal level above the minimum 

 detectable, rf and L would be correspondingly 

 smaller. 



Example 20. Lobes for a Medium Height Radar. A 

 200-mc radar using horizontal polarization has an 

 antenna composed of two groups of dipoles spaced 

 three wavelengths between centers, each group 

 having four dipoles spaced X/2, as in Example 16. 

 The antenna is 500 ft high and 3 miles inland as in 

 Examples 11 and 15. It is desired to compute the 

 vertical coverage pattern. 



From previous tests on this type of equipment it 

 is known that the maximum range that would be 

 obtained in free space da is 80 miles. Since the 

 polarization is horizontal, p will be taken as unity. 

 Precision is not required for most of this kind of 

 work, and it will suffice to compute values for equa- 

 tion (99) at each integral value of n and to consider 

 the values for odd n's (with the plus sign) as the 

 average of the lobe. The lobe shape will be taken 

 as sinusoidal and the range at the nulls obtained by 

 using even values of n and the minus sign; 7(7) and 

 f(y — 26) are obtained from Figure 63, by using 

 values of 7 and y — 26 from Example 11 corres- 

 ponding to integral values of n. The values of z are 

 obtained from Example 15. The computations are 

 shown in Table 13. Had cliff edge diffraction been 

 involved f(y) and f(y — 26) would be read from 

 curves as in Example 18 with marked effects on 

 the pattern. 



The lobes are plotted in Figure 46 using equation 

 (74) and the value of L for odd numbered n's. For 

 intermediate values of n the factors are: 



156 18 Lobe Lengths 



The contributions of the direct and reflected waves 

 may now be added to obtain the length of the lobes. 



Using these three points above and below the lobe 

 line and the maximum and minimum values from 

 Table 13 the lobes may be plotted quickly as 

 explained in Example 14. 



