286 3 PROPAGATION THROUGH THE STANDARD ATMOSPHERE 
tion of a plane earth yield satisfactory results. Cal- 
culations for propagation over a plane earth are 
given in this section. 
Path Difference 
It follows from equation (29) that when p = 1 
and @ = 180 degrees, the received field depends 
only upon the phase lag caused by path difference. 
Referring to Figure 8, 
: Za ANe 
ramet ge = ay) + (HS ‘ 
oa 1 Sy t(be ey | 
= a1 +3( = ae) ere lp 
(49) 
2 
pa alt Out mp = aie (BBY, 
ue 1 hi + 2 yi +(t By | 
=a[1+5 (44 (mee) +... |, 
(50) 
Qhyhe — hyhe(Ai? + he”) 
aaron al as aP 000 |lp 
=e etl ae I. (51) 
q OP aoe 
If 
hi? + he? 
Oe nd 1 
WP << il, 
h 
ax? ue (52) 
The phase lag caused by the path difference A is 
equal to 
5 ve 22(?hl am Amhyhe ; (53) 
r~A\ d dd 
where } is the wavelength of the radiation. 
Field Strength Equations 
When p = 1 and ¢ = 180°, equation (26) may be 
used. Substituting equation (53) for 6 into equation 
(26) gives 
- (2rhyh 
E = 2K) sin (ee). 54 
0 a7 (54) 
If 6/2 < 10°, sin (6/2) — 6/2 and 
Athyh 
E=E£ : 55 
Cae; (55) 
When /h, or hz equals 0, equation (55) indicates that 
the received field intensity is equal to zero, which is 
contrary to fact. For this case the diffraction field 
must therefore be calculated and included as ex- 
plained on page 380. 
In the general case (p < 1), equation (44) may 
be applied with K = (F2/F1) pD. A refinement may 
be added to the calculation by taking into account 
the fact that the image source (Figure 8) of the 
reflected wave is at a distance r + A from the re- 
ceiver. The reflected wave is attenuated more than 
the direct wave, according to the free-space attenua- 
tion ratio (r+ A)/r. If this is taken into account 
ko 
Ficure 14. Geometry for radio wave propagation over a spherical earth. (Vertical dimensions greatly exaggerated.) 
