118 



SITING AND COVERAGE OF GROUND RADARS 



earth, or to compute the distance to the horizon, or 

 the amount a ray would have to be diffracted to clear 

 an intervening hill. The methods described here 

 enable one to solve such problems quickly and 

 simply. 



Distance to the Horizon 



The distance d of the horizon on a spherical earth 

 as seen by an observer at elevation h is given by the 

 well-known formula : 



d 



'4 



3h 

 2 



h = 



3 



(2) 



with d in statute miles and h in feet. This expression 

 makes no allowance for refraction and is commonly 

 used in visual work. 



In radio propagation work the refraction of the 

 standard atmosphere is sufficient to increase the 

 distance of the "radio horizon" to 



d = y/2h 



or 



2 



(3) 



where d is expressed in statute miles and h in feet. 

 This corresponds to the use of an effective radius of 

 the earth equal to ka where k is % and a is 3,960 

 miles. This value of k will be used throughout this 

 report. If it is desired to use other values of k, 

 equation (3) may be written as 



likli , 2d 2 



T or /! =^- 



d 



^ 



Points at heights h\ and hi which are separated by 

 the sea or smooth earth are visible from each other 

 if the distance between them is less than 



a/2/u + \/2h. 



Dip and Rise 



Over land, visibility is determined by the profile 

 of the path involved. Elevations obtained from map 

 contours may be plotted on a profile so as to take 

 the effective earth curvature into account, and visi- 

 bility can then be determined by graphical means. 

 However, construction of such profiles on a curved 

 datum line is tedious, and it is easier to compute 

 the earth curvature and the visibility directly from 

 the map by methods given below. 



In Figure 2 is shown the relations between various 

 heights on the earth's surface. In considering the 

 reference line (sea level) flat as on a map or ordinary 

 profile diagram, use is made of the line HiTH-iT' 

 instead of the curve HiHHiH'. This will be com- 

 pensated for by using a fictitious ray path PiPP^P' 

 instead of the line P1QP2Q'. The deviation of this 

 fictitious path from P1QP2Q' at P is QP = HT and 

 is called the dip. The deviation at P' is Q'P' = H'T 

 and is called the rise. 



In the figure on the left the triangles HH2T and 

 HiKT are similar and 



TH, 

 TK 



HT 



(4) 



or approximately (right-hand figure) 



HT X 2ka = didi . 



Therefore the dip, 



= 5,280 X drfz X 3 = chck 

 Q 2 X 3,960 X 4 2 



Similarly for the rise 



(Ypi _ "1 di 



(5) 



Figure 2. Relations between various heights on earth's surface. Dip and rise. 



