4 TECHNICAL SURVEY 
From the practical viewpoint the following sum- 
mary may give an overall picture of the more out- 
standing features of ground and sea reflection. 
For horizontal polarization over the sea the reflec- 
tion coefficient may be taken as unity and the phase 
shift as 180 degrees for frequencies up to and includ- 
ing the centimeter range, for practically all angles of 
reflection. Over land there is a slight decrease of the 
amplitude of the reflection coefficient with increasing 
angle; for instance, for a frequency of 200 me, at 
an angle of 15 degrees the reflection coefficient has 
decreased to 0.9 or slightly more for moist soil and 
to 0.8 or slightly more for dry soil. These statements 
apply when, the ground or sea surface is reasonably 
smooth. In order to decide whether’a surface is 
smooth or rough, Rayleigh’s criterion, explained 
below, is usually applied. When the surface is rough 
or wavy, irregular scattering predominates and re- 
duces the intensity to a small part of the value 
attained with a smooth surface. 
For vertical polarization the curve of the magnitude 
of the reflection coefficient versus the angle goes 
through a minimum (see Figure 2). When the imagi- 
nary term of tne complex dielectric constant is 
negligible so that the ground behaves like a pure 
dielectric material, the reflection coefficient goes to 
zero at a certain angle (Brewster angle). Ordinary 
soil nearly fulfills this condition. For instance, at a 
frequency of 200 mc the Brewster angle occurs at 
about 12 degrees with moist soil and at about 21 
degrees with dry soil. 
For the ocean surface, and vertical polarization, 
the imaginary part of the dielectric constant cannot 
be neglected, and the reflection coefficient as a func- 
tion of the angle does not vanish at any angle but 
goes through a minimum, the pseudo-Brewster angle. 
The actual variation of amplitude and phase lag is 
represented in Figures 2 and 3 for the smail angles of 
reflection which are most important in practice. 
When the ground is rough the reflection coefhi- 
cient for both types of polarization is reduced to a 
very small value. For 10-cem waves and still more for 
shorter ones, most types of land are rough. A reflec- 
tion coefficient of 0.2 may be taken as representative 
for an average ground covered with vegetation. A 
slightly ruffled sea is a fairly good reflector for 10-em 
waves but appears somewhat rough at shorter wave- - 
lengths. 
STANDARD REFRACTION 
Numerous experiments have resulted in the fol- 
lowing formula for the refractive index of moist air: 
4,800 
1-10 =F (pe + 25) 
where n = the index of refraction, 
p = the barometric pressure in millibars 
(1 mm mercury = 1.3332 mb), 
ll 
é = partial pressure of water vapor in milli- 
bars, 
T = absolute temperature. 
The mixing ratio, s, which is practically equal to 
specific humidity, is connected with e by the relation 
e = 0.00161ps . (10) 
A recent analysis?78 has shown, moreover, that this 
expression for refractive index must, on theoretical 
grounds, be substantially independent of frequency 
down to the shortest waves employed in microwave 
engineering. 
In an average atmosphere temperature, pressure, 
and water vapor density decrease with height, and, 
in the lowest few kilometers where most of the short 
and microwave propagation takes place, it may be 
assumed +o a good approximation that the decrease 
of refractive index with height is linear though the 
rate of decrease is somewhat dependent on the cli- 
mate. In middle latitudes it is given by eas 
a = —0.039 - 10-* per meter . (11) 
Refraction at the boundary of two media is fa- 
miliar from optics and is expressed by Snell’s law: 
Ny COS a1 = Ne COS a , (12) 
where 7 and nz are the refractive indices of the two 
media and a; and a, the angle between the boundary 
and the direction of the ray in the first and second 
media respectively. In the atmosphere the refrac- 
tive index is a continuous function of height, and the 
sudden change of direction at a boundary is then 
replaced by a curvature of the rays. Equation (12) 
can be written F 
TN COS @ = No COS a , (13) 
where 7m and @ are now continuous functions of the 
height and the subscript 0 designates a reference level. 
The above formulas refer to a plane earth. If the 
earth’s curvature is taken into account so that the 
planes relative to which the angle a is measured are 
replaced by spheres about the earth’s center, for- 
mula (13) must be modified; and the mathematical 
analysis shows“‘? that it is replaced by 
Mr COS & = Nolo COS a (14) 
where r is the distance from the center of the earth 
to the level considered. 
If now we set r = 79 (1 + h/ro) where h = r — 79 
and h/ro is a small quantity and, furthermore, if we 
note that with a linear gradient of n 
nam t+ 2h (15) 
we obtain on substituting into (14) and neglecting 
small quantities of the second order 
E + (+ + a) a| cosa = cosao. (16) 
