TROPOSPHERIC PROPAGATION AND RADIO METEOROLOGY 147 
_ the subscript 1 stand for the transmitter level (of 
height f,). Pass a vertical line through the corre- 
sponding point M, of the M curve. Lay off the 
distance a2/2 to the left of M, for a particular ray, 
1, which emerges from the transmitter at angle a 
with the horizontal. In order to make « and M 
comparable numeri¢ally, the factor 10~® should be 
eliminated from equation (18) above. For this pur- 
pose a? should be measured in the same unit as M, 
that is, in 10-* radian. The distance between M and 
1 at any height,h then is equal to (M — Mj) + a:’/2, 
and by equation (19) the square root of twice this 
quantity is equal to the slope of the-ray at height h. 
Hence, ray 1 starting downward from the transmitter 
is bent more and more toward the horizontal as h 
decreases. At point P this ray becomes horizontal 
and from there on increases in slope with increasing 
height. 
Ray 1’ starting upward from the transmitter at the 
same angle a: continues to curve upward more and 
more rapidly as the height increases. Ray 2 is the 
horizon ray which represents the limit to which rays 
can be directed by refraction. Beyond this lies the 
diffraction region where ray tracing cannot be used. 
To study the field in the diffraction region the original 
wave equation must be used. Ray 3 is reflected from 
the ground and in crossing some of the other rays 
produces the phenomenon of interference. In connec- 
tion with Figure 21 it must be emphasized that the 
height scale is tremendously exaggerated and that 
all the rays shown come from a small group which 
are propagated in a nearly horizontal direction. 
Sometimes it is convenient to express the path of 
the ray in terms of ray curvature. The true curvature 
of a ray as it appears on an undistorted (curved 
earth) ‘diagram is different from the curvature exhi- 
bited by a ray on a plane earth diagram. The true 
curvature of a ray is given by 1/p, where p is the 
radius of curvature, and it can be shown that, for 
nearly horizontal rays, this is related to the gradient 
of n by 
J dn 
aii (20) 
However, the relative curvature of the earth with 
respect to that of a ray is (1/a) — (1/p). Now let 
us § . this equal to the curvature 1/ka of an equiv- 
alenu earth. Then 
1 1 1 
a aaa (21) 
and, introducing equation (20), 
(ios ee (22) 
pee tier on 
dh 
This amounts to a definition of k which is more 
general than the one introduced before on page 140 
but reduces to the latter when the index curve varies 
linearly with height. 
For a plane-earth diagram, M is used in place of n. 
Since 
M = (n + ee 1) 108, 
a 
dM 1 dn 
rai = + («4 + 1) 10". 
Substituting the last equation into equation (22) 
gives 
my lidh 
eS Gaal 
10° (23) 
and shows that k, in its most general form, is propor- 
tional to the slope of the M curve. Reference to 
Figure 20 shows that & assumes negative values for 
a range of altitudes whenever a duct is formed in 
the atmosphere. 
These relations may also be expressed in terms of 
m, where 
ee 
Ti = (24) 
is the ratio of the radius of curvature of a ray to 
the radius of the earth. From equation (22) it follows 
that 
aE eeuitie (25) 
Both & and m vary with height except in the special 
circumstance that the M curve is linear. Table 2 
gives a number of corresponding values of k and m 
and indicates their significance. 
TaBLE 2. Relation of k and m. 
2 a =% =il 
DD; CUD 
on He Or 
He] COL 
2 1 
wr 
1 
2 
U.S. Brit. Moist Zero —\—" 
——"_ stand- rela- Duct 
Standard ard tive formation 
curva- 
ture 
The Duct—Superrefraction 
When the M curve has a negative slope, k is 
negative; the curvature of the rays is concave down- 
ward on a plane earth diagram, and the true curva- 
ture of the rays is greater than the curvature of the 
earth. Hence rays which enter the duct under suffi- 
ciently small angles are bent until they become 
horizontal and then are turned downwards. This 
particular form of refraction is called superrefrac- 
tion. Such rays will be trapped in the duct, oscillating 
either between the ground and an upper level, or 
between two levels in the atmosphere. These condi- 
tions are illustrated by Figure 22 for the case of a 
