THE CALCt I.VIIOiN OK VERTICAL COVERAGE 



147 



It will be noted that the vertical scale is nearly 

 10 times as great as the horizontal scale, causing a 

 marked distortion in angles and crowding of angles 

 above 10 degrees. The lines of constant altitude are 

 parabolas, owing to the curvature of the earth. Their 

 shape is given by the equation 



rf' 2 • 5,280 



y = h- 



2ka 



(48) 



Here y = the ordinate measured from the horizontal 

 line through zero; 

 h = the height of the curve at zero range, in 



feet ; 

 d = distance along the earth in miles; 

 a = radius of the earth in miles; 

 ka = equivalent earth radius. 



DISTANT TARGET 

 k 



Figure 47. Flat earth ray diagram. 



For standard conditions k is taken as %. At 40° 

 latitude the radius of the earth is 3,960 miles. Sub- 

 stituting in equation (48) gives the convenient 

 relation : 





(49) 



with y and h in feet, and d in miles. 



Thus in Figure 46 a medium bomber coming in 

 at 5,000 ft would first be detected at 108 miles, the 

 signal would increase in strength, reaching a maxi- 

 mum around 96 miles, and then decrease and be lost 

 at 84 miles. In the null region between 84 and 77 

 miles there would be no detection. Similar regions 

 of detection and nulls would be encountered as the 

 plane came in closer. The nulls do not come into 

 the origin when the direct and reflected rays are 

 unequal. This gap filling is secured at the price of 

 shorter lobes. Above 3 degrees the lobes cannot be 

 distinguished from the nulls. 



Only lobes due to the main free space lobe of the 

 antenna pattern are ordinarily plotted, as targets 



higher than about 10 degrees are of little interest 

 to an early warning radar. Because most detection 

 occurs at angles under 2 or 3 degrees, no distinction 

 will be made between slant range and horizontal 

 range. 



The calculation of the coverage diagram will be 

 approached in successive steps. The first step will 

 consist of calculation of the angular position of the 

 lobe maxima and minima. This will be done in three 

 different degrees of approximation corresponding to 

 different situations encountered in practice. The next 

 step is the calculation of the length and shape of 

 the lobes themselves, which is given in a later section. 



15.6.3 



Flat Earth Lobe Angle Calculations 



When the reflection point is so close that earth 

 curvature may be ignored, the rays may be drawn 

 as in Figure 47. The transmitter T has the center 

 of the antenna at height hi above the horizontal 

 reflecting surface. The antenna is assumed to have 

 horizontal polarization; that is, the dipoles are 

 parallel to the reflecting plane and perpendicular 

 to the direct ray r d . The target height is /s 2 . Both hi 

 and Ji2 are several wavelengths or more, and r d is so 

 large that the field at the target falls off as l/r d . 

 The image of the antenna is at T' at a distance hi 

 below the reflector. The length of the ray from 

 T is r. 



The coefficient of reflection is p, and the phase lag 

 at reflection is 4>. The electric field strength due to 

 the combined direct and reflected waves (r d ~ r) 

 may be written as 



E = y ^1 + p 2 + 2 p cos (fb + 8) 



(50) 



where 8 = 2x— = phase lag due to the path differ- 

 A 



ence, 



A = r — r d = path difference of the direct 



and reflected rays, 



Ei = the field strength at unit distance. 



For horizontal polarization and small angles p is 

 unity and 4> is 180 degrees and equation (50) reduces 

 to 



E 



2Ei . 1 



sin - 5 



r 2 



2Ei . irA 



sin — 



r X 



(51) 



