384 PROPAGATION THROUGH THE STANDARD ATMOSPHERE 
be expressed as 
pees ETE yp BP (28) 
In equation (27), the lag angle @ is measured with 
respect to zero-degree phase shift at reflection. For 
horizontal polarization, @ varies from 180 degrees to 
183 degrees, and from 180 degrees to 3 degrees for 
vertical polarization at 3,000 me over sea water. 
In equation (28), lag angle @’ is measured with 
respect to a 180-degree phase shift (that is, from 
E; reversed), and varies from 0 degree to 3 degrees 
for horizontal polarization and from 0 degree to 
—177 degrees for vertical polarization. 
The resultant field intensity is 
E=8, + pHve 2" +¢) — Ty Se pine JO te +) (29) 
= Ey(1 — pe ae) 
where 
Q=6+¢ =i8+¢--7. 
The absolute value of the received field |E | is 
given by - 
| B| = Boa) (1 = pe?) a = per) 
or a ee ee 
| B| = Bo | 1+ p* ~ 2p cos 2 
= Ep | @ = peo sintZ. (30) 
Equation (30) shows that the received field intensity 
has a maximum of 2H) when 
p= TU 
Q = (2n+ 1)z. . (31) 
The value of £ is a zero when 
p=1, 
Q = 2ntr (32) 
In equations (31) and (32), » includes all integral 
values and zero. Equation (80) may be written as 
E 
= an (1 — p)? + 4p sin’ (33) 
where E is the field at distance d from the trans- 
mitter, and H; is the field strength at unit distance. 
From equation (33), 
A ORR RE NG 
d= Ba = p)?+ 4p sin’ 5 (34) 
In free space where there is no reflecting earth, 
p = 0, and 
E 
= zm (35) 
where dy is the equivalent free-space distance from 
the transmitter at which the field strength EZ would 
be found. Hence equation (34) may be written in 
the form 
d= a a/c — p)? + 4p sin’ 5 (36) 
Divergence 
The divergence factor D is introduced to account 
for the decreased gain produced by the spreading 
Ficure 10. Increased divergence resulting from re- 
flection by a sphere. 
of a wave reflected from a spherical surface. Re- 
ferring to Figure 10, 
Bs _ daz _ [daz da _% [dap (37) 
Ep da; “dada, ™ Nda, 7% 
In calculating the field intensity reflected from a 
spherical earth, the inverse distance attenuation 
factor 1/rg used for the direct wave must be multi- 
plied by the divergence factor D, which is always 
less than unity. 
As a result of this divergence the reflection 
coefficient for a spherical surface is less than that 
for a plane surface as given by 
pD = 7’ (38) 
where p’ is the spherical earth reflection coefficient. 
Equations (30) and (36) may then be written as 
| E| = By Ja — p!)P+ 4p" sin’ (39) 
and poaeneein. 38 SUE Seton ee 
d= ds 4] (1 = p')? + 49" sin’. CD) 
Antenna Gain and Directivity 
The effects of antenna gain and directivity are 
expressed by means of the gain factor G, defined in 
text on p.339, and the antenna pattern factors F1 
and F2, which are the fractions of the maximum radi- 
ation amplitude in the direction of the direct and 
reflected rays respectively. The maximum ampli- 
tude for a transmitting antenna with gain G; is 
