CALCULATION OF RADIO GAIN 383 
It has been pointed out in the previous paragraph 
that within the diffraction region for \ < 10 meters 
and propagation over land, there is practically no 
difference in intensity between a horizontally and a 
vertically polarized radiation field. For \ < 1 meter 
there is, similarly, no difference for propagation over 
sea water. When there is a difference, as for low 
antennas, horizontal polarization gives a smaller 
gain; but as the antennas are raised, the two cases 
approach equality. 
PROPAGATION FACTORS 
IN THE INTERFERENCE REGION 
Propagation Factors 
Te factors affecting gain in the region where the 
methods of geometrical optics may be applied are 
discussed in this and the next three sections. 
Spreading Effect 
From the formula of equation (1) in Chapter 2, 
for the field intensity components of the radiation 
field, it follows that for distances from the transmit- 
ter large in comparison to the wavelength, the domi- 
nant term falls off inversely as the distance from the 
transmitter, or 
E= a, (24) 
where #; is the field strength at unit distance. This 
means that the power per unit area in the radiation 
field varies inversely as the square of the distance. 
This spreading effect is the consequence of the fact 
that the energy of the wave is distributed over larger 
and larger areas as the wave progresses away from 
the transmitter. 
Interference 
When a wave travels over a conducting surface, 
constructive and destructive interference occurs 
between the direct wave from the transmitter and 
the wave reflected by the surface. This is illustrated 
in Figure 8, which is drawn for a plane earth. If 
there is no energy lost in reflection, the direct and 
Ficure8. Geometry for radio propagation over a plane 
earth. 
reflected waves are of equal intensity, and their 
resultant varies from zero to twice the free-space 
value, depending upon the phase difference between 
the two components. The reflected wave lags the 
direct wave by an angle 6 + @, where 6 is the phase 
retardation caused by the greater path length 
traversed by the reflected wave and @ is the phase lag 
occurring at reflection. 
Figure 9 shows the vector diagram for the case 
where the phase shift at reflection is @ = 180 degrees. 
EY 
Fiegure 9. Vector diagram showing the addition of 
the direct and reflected waves for@ = 180° and p = 1. 
This condition holds for horizontally polarized 
radiation of frequency above 100 megacycles, re- 
flected from sea water at grazing angles of less than 
10 degrees. The resultant electric field is equal to 
E = Ee + E,? — 2E.H, cos 6 
= \ — E,)2-+ 4EoE, sin’ (25) 
If the reflection is complete, as from a conductor of 
infinite conductivity, 
E, =F, E=2K sin (26) 
Imperfect Reflection 
In general, the strength of the reflected wave H, 
is less than that of the incident wave H;, partly be- 
cause of diffuse reflection and partly because some 
energy is refracted into the surface and absorbed. 
Furthermore, the phase lag usually differs from 
180 degrees, depending upon the frequency and 
grazing angle. This is especially true for vertical 
polarization where the reflection coefficient is a 
critical function of both the grazing angle and fre- 
quency. The ratio 
2, 8p ae 
R B, pe (27) 
is a complex number and defines the reflection co- © 
efficient R, which has an absolute value p and a 
phase angle ¢. 
In equation (27), a lagging angle is considered 
positive. Writing ¢ = 7+ ¢’, equation (27) may 
