12 TECHNICAL SURVEY 
This equation indicates how the angle a between a 
ray and the horizontal changes as a function of M 
which, in turn, is a function of the height, both 
explicitly by equation (4) and implicitly because 'n 
is a function of the height in a stratified atmosphere. 
TYPES OF M CURVES 
An M curve is a diagram in which M as abscissa 
is plotted against the height h as ordinate. Extensive 
experience has led to a classification of M curves 
which is shown in Figure 1. The six types exhibited 
STANDARD TRANSITIONAL SUBSTANDARD 
SIMPLE SURFACE 
TRAPPING ELEVATED S SHAPE GROUND-BASED S SHAPE 
4 
hy oINvi (ON cite h INVERSION, LAYER 
! 
Pa | DUCT ; ~ a) oust 
!\nversion Laver elt 
M M 
Ficure 1. Types of M curves. 
comprise all cases that are of practical interest. 
M curves of a more involved structure are rare. In 
all cases it is assumed in accord with experience 
that at sufficiently high elevations the M curves 
become linear and have, or nearly have, the standard 
slope. 
The height at which these variations in refractive 
index occur may vary from a few feet to several 
hundred or even a few thousand feet though they 
are likely to be found at very low elevations in cold 
climates and at higher elevations in warm climates. 
The meteorological conditions which yield these 
curves will be dealt with extensively in Chapter 5, 
and few indications may suffice here. Ordinarily, 
on going aloft the temperature decreases at a slow 
and fairly steady rate. When, instead, the tempera- 
ture zncreases with increasing height, a phenomenon 
known to meteorologists as a temperature inversion, 
equation (9), Chapter 1, shows that n decreases with 
increasing height. This does not necessarily imply 
that M decreases with height since, by equation (4), 
M contains the term h/a, which increases with 
height. If, however, the variation of temperature is 
sufficiently great, a decrease or inversion of M 
results. Such an inversion produces a duct, a term 
which refers essentially to certain meteorological 
phenomena and whose exact significance is explained 
below. A variation of humidity over the layer has 
an effect essentially analogous to, but distinctly 
more pronounced than, the effect of temperature. 
In this case M increases with height with a decreas- 
ing moisture content and vice versa. Variations of 
humidity are common in the lower atmosphere, and 
they constitute the main cause of refractive index 
variations, with temperature variations frequently 
a contributing factor. ? 
The six cases shown in Figure 1 are as follows: 
the standard case which needs no further comment; 
the transitional case where the moisture or tempera- 
ture variation is not great enough to produce a true 
inversion of the M curve but merely results in a 
nearly constant value of M in the lowest strata; the 
substandard. case in which M increases more rapidly 
with height than in the standard case; and three 
cases of ducts. The simple ground-based: duct or sur- 
face trapping, consists in an M inversion immediately 
adjacent to the ground or sea. There are two types 
of elevated M inversions distinguished by the posi- 
tion of the minimum value of M aloft. If this mini- 
mum is larger than the value of M at the ground so 
that the vertical projection from the minimum inter- 
sects the M curve, it is considered a true elevated, 
S-shaped duct. If this minimum is less than the 
value of M at the ground it is ari elevated M inver- 
sion but a ground-based duct. 
In dealing with these M curves it is universally 
assumed that the stratification is the same over the 
whole length of the transmission path. This is a 
severe restriction, but it has proved indispensabie 
up to date in order to make the problem susceptible 
to mathematical treatment, and it is reasonably often 
fulfilled in practice. 
RAY TRACING 
In order to understand the mechanism of trans- 
mission of radiant energy in a duct the course of 
rays issuing from the transmitter is traced according 
to equation (5). Note that for the small angles with 
the horizontal at which these phenomena occur, 
dh 
Oa dz , (6) 
where x designates the horizontal distance. Hence 
from equation (5) 
ae a Z feria: 4+ 2(M — My) - 10-J-?. (7) 
Since M is a given function of height, equation (7) 
gives in integral form the relation between distance 
and height, where ap is the angle with the horizontal 
of the ray emitted by transmitter and M, is the 
value of M at the transmitter height. 
Practicable graphical methods of ray tracing have 
been developed and used extensively to compute 
actual coverage diagrams. ©: 68 69,71,76,82,98,99 Three 
schematic pictures of ray tracing, showing the main 
