Chapter 10 
DIFFRACTION OF RADIO WAVES OVER HILLS* 
| Fie ole HAS SHOWN that frequencies in the 
VHF (very high frequency) range and higher 
are propagated over hills and behind obstacles more 
easily than has been commonly expected. Hills or 
other obstacles in the transmission path cast shadows 
which may make a radio system unworkable when 
either antenna is located close to the obstacle, but 
recent experiments, notably the work of Jansky and 
Bailey,* have shown that hills and mountains can 
cause constructive interference as well as destructive 
interference. In other words, with proper antenna 
siting, the field intensity beyond the line of sight may 
be higher than is expected for the same distance over 
plane earth. This improvement in field intensity may 
be & to 10 db or more. : 
One attempt to develop a theory for radio trans- 
mission over hills is based on the computed field 
intensity over the solid triangle shown in Figure 1. 
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Fieure 1. Analysis of field intensity over a solid triangle. 
It was reasoned that a good approximation to the 
field over any profile might be obtained from a 
knowledge of (1) the field over a perfectly smooth 
earth, (2) the field over the solid triangle that 
encloses the actual profile, and (3) the field over a 
knife edge equal in height to the highest point in 
the profile. The theory of propagation over a perfectly 
smooth earth is well known; it is the basis of all the 
published theoretical curves on radio propagation. 
The corresponding expressions for the field intensity 
over a solid triangle and over a knife edge are indi- 
cated in a paper by Schelleng, Burrows, and Ferrell,*47 
but some effort is needed to place these expressions 
in a convenient form for computation. 
The method of obtaining an expression for the 
field over a solid triangle is indicated in Figure 1, 
"By K. Bullington, Bell Telephone Laboratories. 
and the same analysis applies to each of the ideal 
profiles shown in Figure 2. The field intensity at any 
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H c i H D 
Ficure 2. Analysis of field intensity over various tri- 
angular profiles. 
point P in the vertical plane through the apex of the 
triangle is assumed to be the sum of a direct ray and 
a ray reflected from the ground which is equivalent 
to a ray from an image antenna. In a similar manner 
the field at point P is propagated to the receiving 
antenna by means of a direct ray and a ground 
reflected ray. By integrating over the plane above 
the apex of the triangle (that is, from y = H to 
y = o and from z = — o to z = o) an expression 
for the total received field is obtained. The complete 
expression is not as complicated as the expression 
for propagation over a smooth sphere, but two simple 
approximations will be sufficient for the present 
discussion. When the height of the hill H = 0 and 
when the ground reflection coefficient is —1, the 
complete expression reduces, as it should, to the well- 
known formula for VHF propagation-over plane earth. 
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When the height of the triangle H is greater than 
three to five times the average height of the antennas 
E = 2K, sin 
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Ficure 3. Shadow-loss factor S. 
