DIFFRACTION OF RADIO WAVES OVER HILLS 69 
and when the retlection coefficient is —1, the com- 
plete expression reduces to 
2QrnHh, . 2rHhe 
Xz Sin : 
TT = 4E)S sin 
2 AX 
(2) 
The factor S is the shadow loss shown in Figure 3 
as a function of 
= 22122 
u= Bx hogor a: 
The other symbols in the above expressions have 
the following meanings: 
E = field intensity in microvolts per meter, 
Ey = free space field intensity in microvolts 
per meter 
_ 3 V5P x 10° 
Xi + 22 
P = radiated power in watts, 
\ = wavelength in meters, 
H = height of the obstruction in meters, 
hi,h2 = antenna heights in meters, 
%1,%2 = distances as shown in Figure 1 in meters. 
, 
The approximate expression given in equation (2) 
indicates that the field intensity for points well 
beyond the line of sight may be greater than the 
field over a plane earth which is given in equation (1). 
The sine terms in equation (2) indicate interference 
patterns beyond the line of sight which seem to 
offer an explanation for the experimental fact that 
behind hills raising the antenna may cause a loss, 
or lowering the antenna may result in a gain, in 
signal intensity. 
A comparison between theory and experiment is' 
shown in Figure 4. These data, which were taken 
from the previously mentioned NDRC report pre- 
pared by Jansky and Bailey, show me&sured values 
at 116 me for horizontally polarized waves propa- 
gated over the profile shown in the bottom of the 
“COMPUTED FOR. 
tye hg =19 FT aa 
COTTRANSMITTER 2.7 1 
ei ae uu ies 
; S non c= = = 
#/ELD INTENSITY IN DECIBELS BELOW INVERSE DISTANCE FIELD 
Sie FROM TRANSMITTER IN MILES 
Fiaure 4. Theoretical and experimental results in meas- 
uring field intensity of horizontally polarized waves. 
Frequency 116 me. 
drawing. The open circles show the field intensity 
in decibels below the free space value when both 
antennas are 29 ft in height, and the dots give 
similar data for 19-ft antennas. The two dashed 
lines running from upper left to lower right are the 
computed values for smooth earth for 29-ft and 19-ft 
antennas, respectively. The solid line with the inter- 
ference fringes is obtained from equation (2) for the 
case of 29-ft antennas. The correlation between 
theory and experiment is not complete, but at least 
the theory may be a step in the right direction. 
Similar theoretical and experimental results are 
obtained with vertical polarization. 
Thus far the only type of profile considered has 
been one with a single prominent hill, and it is 
natural to ask what happens over profiles containing 
several hills. There are less experimental data avail- 
able on this point than for propagation over a single 
hill, and consequently the remainder of this discus- 
sion is more speculative than the preceding part. 
An ideal profile consisting of two hills of equal 
height is shown in Figure 5. The complete mathe- 
matical solution for this case is difficult, but an 
approximation can be obtained in the following 
manner. The field at any poiut P midway between 
the two hills can be obtained by means of the expres- 
sion for the diffraction over a single hill. The field at 
this point is then propagated over the second hill to 
the receiver. The total received field is obtained by 
mechanical integration, that is, by adding the effect 
(magnitude and phase) of many evenly spaced points 
in the vertical plane midway between the two hills. 
The net result is that the total received field is 
represented more closely by the path ACB than by 
the path A DEB. The energy received over any given 
he 
A B 
Ficure 5. Field intensity computation for a profile of 
two hills by a solid triangle. 
path such as path ADHB decreases rapidly as the 
number of diffractions in that path increases. How- 
ever, for any profile there is always at least one 
path between transmitter and receiver such as path 
ACB that requires no more than one diffraction, 
and the field intensity over this path is usually 
controlling. In other words, the profile consisting of 
two hills can be approximated for computation pur- 
poses by a solid triangle which is formed by a line 
from the base of the transmitting antenna to the base 
of the receiving antenna and lines from the base of 
