194, 5 TECHNICAL SURVEY 
problem, variations are considered only in the plane 
of transmission, since the effect of sideways deviation 
would cancel out. This reduces the problem to one 
of two dimensions only. Each plate is further assumed 
to retain its original curvature. 
A beam falling on a patch of these plates would 
be reflected in such a way as to spread the energy 
at the receiver in a vertical pattern similar to the 
Gaussian distribution of the plates. It is only neces- 
sary to integrate this curve over the width of th 
antenna to find the fraction, L, of the total energy 
that will be useful. Z will be a function of the 
probable value of the deviation of the plates, the 
range, and the antenna width. 
With the rough layer assumption, there will be 
some plates correctly oriented at each part of the 
layer to reflect energy into the receiver. Therefore, 
a third factor, M, must be included that is the ratio 
of 7/8, where y is the total angle subtended by the 
layer that can reflect rays to the receiver. y would 
be limited by the optical horizons. 6 is the angle 
subtended by the receiving antenna when reflection 
is from a plane surface; i.e., essentially, free space 
conditions. 
The net convergence factor C must be the product 
of these three quantities K, L, M. In this case, K 
must be the mean value of K averaged for various 
points of reflection. In order to integrate the expres- 
sion for the mean value of K, it is necessary to substi- 
tute for sin ¢ in equation (2). 
ino= (2424242), © 
which gives 
si 8 (xy)? ie 
a E — Rab + =. 6) 
This expression is easily integrated if the product 
ay is used for the variable and xy; = 1(R — x). 
Example: The preceding developments have been 
applied to the one-way link of the U.S. Navy Radio 
and Sound Laboratory at San Diego, which has been 
extensively studied. High subsidence layers are 
common for this region. The probable value of the 
deviation of a reflecting plate from horizontal was 
taken as 0.1° as an engineering approximation. In 
this case, C equals 43, assuming a reflection coeffici- 
ent of I. If the reflection coefficient is not unity, its 
value as a function of angle of incidence must be 
multiplied into the equation. 
Since K, Z, and M can each vary through con- 
siderable limits, C can vary through a very wide 
range of values. 
CONCLUSIONS 
The statistical treatment of the roughness is not 
always applicable, since a finite number of plates 
would actually be engaged in reflecting energy. 
Hence, the received signal would vary almost ran- 
domly with time as the orientation of the plates 
changed slightly. This could produce marked fading 
and peaks of large amplitude. Primarily, however, it 
would explain signals of the magnitude of free space 
signals or higher. 
