ore 
Chapter 23 
CONVERGENCE EFFECTS IN REFLECTIONS FROM 
TROPOSPHERIC LAYERS* 
AX ELEVATED DUCT may be treated as a concave 
spherical mirror whose radius of curvature is 
a, the effective earth radius. This includes any layer 
that can act as a reflector to radiation incident at a 
sufficiently small angle. The problem is here con- 
sidered as one of geometrical optics only. Ray tracing 
methods are used, and the phases are assumed to 
add randomly. This assumption may introduce an 
error as large as 3 db in the result but is necessary 
to simplify the solution of the problem. If the reflec- 
tion coefficient is other than unity, it must be 
multiplied into the general relation which will be 
given for C=KLM the net convergence factor. 
CONVERGENCE FACTOR 
A bundle of rays leaving a transmitter below the 
reflecting layer is converged on reflection from a 
concave surface. The convergence factor K is the 
ratio of the power density at the receiving antenna 
after convergence to the power density at the 
receiver that would be expected after reflection from 
a plane surface (essentially free space condition). 
Referring to Figure 1, the convergence factor can 
be expressed as 
Ka- & ty oa 
760, + y662’ () 
or ; 
oF ay ‘ 
Sis (1 aR sin ¢ (2) 
where x = distance from transmitter to point of 
reflection, 
y = distance from receiver to point of reflec- 
tion, 
R=2x-+ y = total range, 
a = effective earth’s radius (usually 4,590 
nautical miles), 
¢ = angle of incidence of radiation at reflec- 
tion, 
other angles as shown on Figure 1. 
RECEIVER 
TRANSMITTER 
Ficure 1. Convergence factor K. 
Equation (2) can be deduced from equation (1) 
by remembering that 
af _ 266; 
Baa ae (3) 
and 
Oy S503 So 
asin ¢ 
The form shown in equation (2) is the more useful 
and is similar to the divergence factor for reflection 
at a convex surface that has been in use for some 
time. Equation (2) shows that K can grow quite 
large and even become infinite for certain conditions. 
Curve 1, Figure 2, shows a plot of the absolute value 
ReriticaL IN NAUTICAL MILES 
(0) “200 ee 600 6800 
b IN FEET——> 
1000 1200 OO 1600 1800 
Ficure 2. Value of K for height of layer (6 in ft) versus 
range (nautical miles). 
of K as a function of b, the height of the layer above 
the antennas, for a total range of 80 nautical miles. 
This plot also assumes s = y = 40 miles, which is 
a necessary condition for a smooth reflector. In this 
case, K becomes infinite for a layer 1,100 ft above 
the antennas. Curve 2, Figure 2, shows a plot of 
the layer height 6b necessary to give infinite conver- 
gence as a function of the range (plotted on right- 
hand scale). 
ROUGHNESS EFFECT 
The most apparent difficulty with the picture 
presented so far is that the layers actually are not 
perfectly smooth. In order: to take that fact into 
consideration, it was assumed that the layer was 
composed of a large number of plates set at various 
small angles about the horizontal according to a 
Gaussian distribution. As in other parts of this 
8By Ensign W. W. Carter, USNR Radio Division, Consult- 
ant Group. 
