138 



COVERAGE DIAGRAMS 



The values of p, 4> (or cp') maj' now be read from 

 the reflection curves in Chapter 4. 



Equation (46) in Chapter 5 may now be applied 

 \vith K = {Fi/Fi)pD where D assumes the same 

 A-alues as in the approximate solution. Equation (21) 

 determines the value of d from which %i — d/dr 

 may be calculated. This value of u = d/d^ is laid 

 off on the original divergence contour in Figures 4, 

 5, or 6. This determines v. The assumption under- 

 lying this procedure is that the divergence factor is 

 not appreciably affected by the change in coordinates 

 caused by imperfect reflection and an unsymmetrical 

 antenna pattern. The corrected phase difference 5' 

 is found from 



6.5.2 



Basic Relations 



5' = Q - 0' 



(26) 



and the path difference A' and parameter R' from 



(28) 



R' = -- 

 hidx 



The intersections between the path-difference con- 

 tours and the distance envelope determine points 

 on the coverage diagram. 



The above method should be applied even for 

 horizontal polarization when the directivity of the 

 antenna is such that Fo/Fi 5^ 1. This follows from 

 the concept of generalized reflection coefficients of 

 Section 5.3.1. 



LOBE-ANGLE METHOD 

 (HORIZONTAL POLARIZATION) 



6.6.1 



Outline of Method 



In this method the angles of lobe maxima are 

 determined by modrfjdng the plane earth formula, 

 equation (2). In this equation, hi is replaced by hi', 

 which is the equivalent height above a plane tangent 

 to the earth's surface at the reflection point, as 

 shown in Figure 8. 



The value of hi' is given in equations (58) and (60) 

 in Chapter 5. The maximum and minimum distances 

 from the transmitter base to a point on the lobe 

 are calculated by equation (46) in Chapter 5, using a 

 modified divei-gence factor to be descr'bod in Section 

 6.6.5. 



Referring to Figure 8 and assuming di « f/2, 

 7' < 10° and \p^y', the following relations hold. 



Figure 8. Lobe angles corrected for earth curvature. 



(29) 

 (30) 



(31) 



2ka 

 The basic equations of the lobe-angle method are 

 , n\ n\ 



and 



di = 



\ 2ka/ 



(hi - "ly 



V 2ka/ 



(32) 



)i\ 



(33) 



* *^ Reflection-Point Curves 



The elimination of rfi from equations (32) and (33) 

 is most con^-enientlj' accomplished bj' graphical aids, 

 which maj^ be used in the following way. 



1. From equation (33), a curve may be plotted 

 sliowing di as abscissa and n as ordinate for a given 

 transmitter height and wavelength. This is illus- 



