146 TECHNICAL SURVEY 
ditions which yield these various M curves are 
described on pages 152 -164 
The opposite effect occurs when the M curve takes 
the substandard form (curve Ib in Figure 20). Here 
the lower portion of the M curve has a slope which 
is less than standard. In this event the rays in the 
lower atmosphere are bent downward to a lesser 
degree than in the standard atmosphere or may 
even be bent upward. Depending to some extent 
upon the elevation of the transmitter, the field 
strength in the substandard region may be reduced 
considerably below normal, even to the point of 
producing a radar and communication “blackout.” 
If the M curve is steeper than average in the lowest 
layers, the transitional case arises (curve Ia). Here 
a slight change in the temperature and moisture 
distribution might lead to a curve of type II and 
a duct. 
Rays in a Stratified Atmosphere 
Nonstandard vertical variations of refractive index 
occur frequently in the lower atmosphere. In addi- 
tion there may be gradual variations in the horizontal 
direction. So far, the theory of propagation has not 
reached a stage where such horizontal variations can 
be taken into account. Unless otherwise stated it is 
always assumed that the stratification extends hori 
zontally as far as the coverage of the transmitter 
and that the variation in the M curve is entirely 
vertical. Weather conditions often are sufficiently 
homogeneous horizontally to warrant this assump- 
tion, but there are exceptions, mainly near coasts 
(see pages 152-164) 
Only those rays are affected by the vertical varia- 
tions of refractive index in the lower atmosphere 
which leave the transmitter at a very small angle. 
Both theoretically and practically it has been found 
that the effects of nonstandard refraction are negli- 
gible for rays that leave the transmitter at an angle 
with the horizontal of more than about 1.5°. Rays 
that leave at an angle with the horizontal of less 
than 1.5°, and especially those emerging at angles 
with the horizontal of 0.5° or less, are strongly 
GROUND 
OR SEA LEVEL My 
TRANSMITTER 
affected by nonstandard refraction. This part of the 
transmitter radiation is of paramount importance in 
early warning radar and in communications. For 
such applications of radar as gun-laying or search- 
light control the effects of nonstandard propagation 
are usually negligible because the rays which reach 
the target have emerged from the transmitter at a 
fairly large angle with the horizontal. 
The progress of a ray through the stratified atmos- 
phere is described by Snell’s law, discussed previously 
(p.188) When the angle a between the ray and the 
horizontal is small 
a? 
2 y 
provided @ is expressed in radians. 
Introducing this into Snell’s law for a curved 
earth, equation (6), noting that n + h/a =1+ M- 
10-* and neglecting second order quantities, it is 
seen that 
3 (a? — ay?) = (M — M,)10°. (18) 
cosa = 1 — 
Since a is the angle which the ray makes with the 
horizontal it is equal to dh/dz, the slope of the ray. 
Solving equation (18) for a, 
a = Fe yo? + 20 — MO. (19) 
These relations apply to any two levels provided a 
and ap are the angles at the levels to which M and 
Mo refer. 
Equation (19) provides a technique for tracing the 
paths of rays emitted by a transmitter at various 
angles with the horizontal, and it indicates how their 
passage through the atmosphere is controlled by the 
variations of the modified index. Although this ray 
tracing method is only an approximation of the true 
solution of the wave equation, it can be used, subject 
to certain limitations, for computing quantitatively 
the strength of the field. The approximation breaks 
down when neighboring rays cross each other and 
form caustics. 
The method may be illustrated by the case of 
standard refraction with k = %. As shown in Figure 
21, draw the M curve with a slope ka = 4a/3. Let 
DIFFRACTION 
REGION 
—— DISTANCE x 
Ficure 21. Rays in the standard atmosphere. 
