eee 
Chapter 19 
APPROXIMATE ANALYSIS OF GUIDED PROPAGATION 
IN A NONHOMOGENEOUS ATMOSPHERE" 
Ne MILITARY IMPORTANCE of guided or 
“anomalous” propagation in a stratified atmos- 
phere is now well known. Unfortunately, or perhaps 
fortunately, the problem cannot be treated with the 
aid of known and tabulated functions except in some 
special cases because the exact field distribution with 
height is a function of a function, namely a function 
of the distribution of the modified index of refraction. 
For each distribution of this index with height we 
should have a curve for the field distribution. These 
curves will look similar in a general way and yet 
they will differ in detail; but in this particular 
problem we are not much concerned with details. 
Even if we had exact solutions we should still want 
some generalized way of expressing pertinent infor- 
mation. 
An approximate analysis of field distribution in 
terms of master curves, depending on one, or at 
most, two parameters, will be discussed. For example, 
if we have atmospheric conditions favoring forma- 
tion of a guiding layer immediately above the ground 
or sea level, then we can try to represent the field 
distribution with height with the aid of the master 
curve showr in Figure 1. This curve depends on only 
one parameter, H, so chosen that in the layer between 
(14) H and H, the field intensity does not deviate by 
more than 6 db from the maximum. 
(°) : 
QO O02 04 06 08 LO 1,2 
h/t 
Figure 1. Master curve for field distribution with 
height inside a duct. 
This particular curve is chosen for the first trans- 
mission mode, and it has been suggested by the exact 
analysis of guided waves in a homogeneous layer. 
In this case of sharp discontinuity in the index of 
refraction the field distribution curves are sinusoidal 
in the layer and exponential outside. The position 
of the maximum of the sinusoidal portion of the 
curve and the relative rate of decay of the exponen- 
tial part depend on the ratio of the wavelength to 
SBy S. A. Schelkunoff, Bell Telephone Laboratories. 
181 
the thickness of the layer and on the amount of 
discontinuity in the index of refraction. In Figure 2, 
curve 1 is identical with the curve in Figure 1; curve 
2 shows what happens if the wavelength is doubled; 
7a NS ae 
(Ze Ss 
AL ASS 
fe) 
0 O02 04 06 O08 
10 1,2 
h/H 
Ficure 2. Master curves for wavelength d (1), 2A (2), 
and ¥% »d (8). 
and curve 3 corresponds to the case in which the 
wavelengthishalved. If the wavelengthis (8mv/2)/4& 
3.3 times as large as the wavelength corresponding 
to curve 1 or larger, no guided waves are possible 
with the field intensity vanishing at the ground or 
sea level. 
The situation is different if the index of refraction 
is allowed to vary continuously and to diminish 
indefinitely. Suppose, for instance, that the lapse 
rate of the index of refraction is constant. We don’t 
expect any critical wavelength in this case; as the 
wavelength increases we expect the field to spread 
out more and more. In fact, we expect the shape of 
the field distribution curve to remain the same, 
namely to be determined by that solution of 
CE 
Ga = [Bt = one (WN) E 
(1) 
which vanishes at.» = 0. In this equation 
E = the electric intensity; 
h = the height; 
e = the modified dielectric constant; 
w = the radian frequency; 
8 = the phase constant in the direction 
parallel to the stratification. 
We can try to approximate this solution by a curve 
of the type shown in Figure 1 in which case the 
problem is to select a proper value for H. The ques- 
tion may be raised regarding our preference for this 
particular curve rather than for curve 2 or 3 in 
Figure 2. We shall return to this point later; for the 
present we shall merely point out that curve 1 
