THE p-q METHOD 



133 



It may be recalled that expressed in coordinates 

 'p and q 



D 



1 



1 + 



4p-g 

 (1 - V'). 



-1/2 



(13) 



(14) 



then proceed in the following manner. To a fair 

 approximation, we may assimie the extreme range 

 of the lobes to correspond to sin- (12/2) = 1, so that 

 by equation (12) the corresponding distance dmax is 

 given by 



rfma. = ^Gx doil + D). 



(16) 



and the total phase shift, by equations (97) and 

 (29) in Chapter 5, is 



47r h_l g(l - y-'Y 



Q. = 



+ <A'. 



(15) 



Expressing d^^^and D by p and q, the above equation 

 determines the envelope of all lobe maxima. The 

 practical way of doing this is to use a graphical 

 representation of D in p and q coordinates (Figure 17 

 in Chapter 5) and to start by selecting a particular 



l*Dr99 



Figure 4. Curves of constant-divergence factor D and path difference parameter R. (Radiation Laboratory.) 



For horizontal polarization, </> — > 0, so that for this 

 case (which is the one under consideration) all 

 variable quantities in equation (12) have been 

 expressed in terms of p and q. 



6.4.2 



Construction of Range Loci 



Suppose to start with that we want to compute 

 the position of the extreme range of a lobe. We may 



value of D, say D = Dj_. Inserting this in equation 

 (16) gives a corresponding value of d^^^, and insert- 

 ing this value of dmax for d in equation (13) deter mines 

 a straight line in the p,q plane, since dr = y2kahi 

 is known. "\^Tiatever is the value of dmau this line 

 passes through the point q = 1, p = 0. In order 

 to determine the position of the line, only one more 

 point is needed. A convenient point to choose is to 

 take q = 0.9 and compute the corresponding p from 



