ELEMENTARY THEORY OF NONSTANDARD PROPAGATION 15 
250 
200 
150 
7) 
« 
w 
- 
w 
2 
100 
50 
° 
STANDARD 
GRADIENT 
f Distance —125 | 
1,0 t°) 1.0 ty) 1,0 0 
FIELD IN ARBITR. UNITS. ARROWS SHOW FREE SPACE FIELD 
Fieure 6. Variation of field strength with height for various distances. 
warm air mass from a very dry cool air mass. Tem- 
perature inversions involving differences of more 
than 10 to 15° C are quite exceptional. As a conse- 
quence of this both the actual height of the M 
inversion as well as the difference AM between the 
maximum of M at the bottom and the minimum at 
the top of the M inversion are limited. The height of 
the M inversion layer may be only a few feet if it is 
close to the ground or sea surface. It frequently is 
of the order of 50 to 100 ft or even larger. Under 
particularly favorable conditions in warm climates, 
elevated WM inversions may have heights of several 
thousand feet. The duct itself can be appreciably 
thicker than the M inversion layer, as may be seen 
from the structure of the last two M curves in 
Figure 1. 
Again, the decrease AM over the height of the in- 
version is limited for the same reasons. For low 
ducts values of the order of AM = 5 to 10 are com- 
mon. Somewhat larger values will sometimes occur. 
The maximum value observed is about AM = 40 in 
high-level inversions at San Diego which originate 
in the singular climatic conditions found there. 
An important consideration for the detailed mathe- 
matical treatment of duct propagation is the shape 
of the knees of the M curve. This, again, depends on 
the physical nature of the atmospheric stratification. 
Very often the inflections are so sharp that a succes- 
sion of two or three straight lines furnishes an excel- 
lent approximation. These are known as bilinear and 
trilinear ducts and are of very common occurrence, 
especially with elevated ducts and a large class of 
ground-based ducts. On the other hand, there are 
also ground-based ducts in which the corners are 
extremely well rounded. 
It follows from the restrictions on the numerical 
values of M that there are severe limitations on the 
angle a for which duct effects can occur. Thus AM = 
10 represents a change of one part in 105 in the re- 
fractive index. Now from equation (5), we have 
by differentiation 
AM - 10° = aha. (9) 
For a complete reversal of a ray we must have 
Aa a, and then @ is proportional to the square root 
of AM. In the above case, where AM = 10, we find 
that a is of the order of 3 - 107%, or about 10 minutes 
of arc. 
Carrying considerations of this type into a little 
more detail it is found that the major effects of 
nonstandard refraction occur only for rays which 
emerge from the transmitter at an angle of less than 
Y% degree. For angles ketween 14 and 114 degrees 
the refractive effects produced by the typical non- 
standard M curves consist merely in minor modifica- 
tions of the standard coverage pattern, while for 
angles above 114 degrees the refractive effects are 
negligible. 
SURVEY OF WAVEGUIDE THEORY 
The ray tracing method presented 9n pages 12-14 
is only a rather rough approximation to the true 
solution of the wave equation. It neglects diffraction, 
which on closer investigation is found to be very 
important. In order to visualize this, the waveguide 
analogue was introduced at an early stage of the 
development. Consider a two-dimensional wave- 
guide consisting, for instance, of two parallel plane 
sheets of copper of infinite extent. The propagation 
of an electromagnetic wave in such a guide is some- 
what analogous to that in a duct. The reversal 
of the vertical component of the rays by refraction 
in the duct corresponds to the reflection by the walls 
in the case of a metallic waveguide. It is well known 
that wave propagation under these conditions can 
be described by the methods of geometrical optics 
