438 PROPAGATION THROUGH THE STANDARD ATMOSPHERE 
DIVERGENCE D 
— — — PATH DIFFERENCE R 
07999080 70 60 SO 4509540 
Ficure 6. Curves of constant-divergence factor D and y:ath difference parameter R. (Radiation Laboratory.) 
Construction of 
Path-Difference Loci 
An assumed value of ® in equation (12) determines 
6, the phase shift caused by the path difference, as 
5 = Q+ 2xn, (18) 
where n assumes all integral values and zero. This 
value of 6 determines the path difference A = r — rq, 
Nea, (19) 
But from equation (97) in Chapter 5 
2h; (1 — p?)? 2Qh,? 
Nae pCees = =“! of(p) (20) 
dr Pp dp 
where f(p) is given in Figure 18 in Chapter 5. 
In this calculation, g may be taken as the inde- 
pendent variable. The assumed values of g together 
with A from equation (19) determine f(p) in equa- 
tion (20). For given values of f(p), the correspond- 
ing values of p may be read from Figure 18 in Chap- 
ter 5. The coordinates he and d on the path-differ- 
ence loci are found from equation (98) in Chapter 5 
and equation (13) in Chapter 6, giving 
2, 
ae ha (1 — p+? ) 
L=@ = @ 
and 
Intersections of the path-difference loci with the 
range curves determine points on the lobes. 
THE u-v METHOD 
Outline of Method 
This method makes use of the generalized coordi- 
nates u = h2/h,; and v = d/drp described in text on 
p.391. The curves of constant-divergence factor D 
and path-difference parameter F are plotted on the 
same sheet in Figures 4, 5, and 6. The divergence 
lines are shown in full and the path-difference curves 
are dotted. Envelopes of constant sin?(Q/2) [equation 
(46) in Chapter 5] are constructed and their inter- 
sections with path-difference contours corresponding 
to the assumed values of Q/2 determine points on 
the lobes. 
Construction of Lobes 
(Horizontal Polarization) 
In this method, the divergence factor D is con- 
sidered to be the independent variable. Dividing 
both sides of equation (46) in Chapter 5 by dr gives 
g 
Gey ~)2 2 
Boe (1 — K)? + 4K sin 5 
uh aeeaetatiya apie wot (21 
=VGid y(t sp LES ene (21) 
