366 PROPAGATION THROUGH THE STANDARD ATMOSPHERE 
from each other by changing the earth’s curvature 
by an arbitrary factor. From this viewpoint Figure 6 
INTERFERENCE REGION 
DIFFRACTION 
REGION 
Ficure 6. Rays in a plane earth diagram. (Radius 
of curvature of rays is — ka.) 
__EQUIVALENT 
EARTH RADIUS kao 
Ficure 7. Equivalent parabolic earth diagram. 
represents the limiting case of an infinite earth’s 
radius. The plane earth diagram is widely used for 
problems of nonstandard propagation. 
In drawing diagrams for a curved earth of equiva- 
lent. radius ka, it is customary to replace the spher- 
ical earth outline by an equivalent parabola (see 
Figure 7). The equation for the surface reduces from 
the circular form, 
zs? + (hs + ka)? = (ka)?, 
to the parabolic form, 
1 
hs a ae 2ka 25", 
for h;<<<a;. The height h measured from the 
surface of the earth, instead of from the x axis, is 
given by 
SWS i Se Bee 
Qka 
in which h is laid off perpendicular to the x axis and 
not to the earth’s surface. For clarity in drawing 
rays or field-strength diagrams, the vertical scale 
is expanded by an arbitrary factor p, whence 
1 
h = p( h'+——22). 
AG +52) (7) 
This distortion of vertical distances, it can be shown, 
does not distort angles. The parabolic representa- 
tion to be reasonably accurate must be restricted to 
heights in the atmosphere small compared with the 
extent of the horizontal scale. 
Curvature Relationships 
The curvature of a ray is defined.as the reciprocal 
of the radius of curvature p. Let w be the angle 
between the ray and a nearly horizontal x axis. 
By Figure 8, p = — ds/dy, and since y is a small 
B 
Ficure 8. Angular relationships of rays. 
angle we may, to a sufficient approximation, put 
ds = dz, so that 
ence, 
p im dx 
Here the curvature has been defined so that it is 
positive when the ray curves in the same direction 
as the earth; with this system the curvature of the 
earth itself is positive. Referring to Figure 8B, 
1 dy da dod 
Pa a fy ey eas 8 
p dx ie dx ) 
But d@/dz = 1/a, and since @ is a small angle 
da da dh da _ 1 d(@’) 
dx dh dz Gi ah 
Consequently, by equation (2) 
Le ate ee 
Teno tannins a ©) 
From this, the curvature of the ray is equal to the 
vertical rate of decrease of the refractive index. 
Notice that dn/dh is usually negative, so that the 
true curvature of a ray is usually concave downwards. 
A simple relationship exists between m = p/a, 
the ratio of the radius of curvature of a ray to the 
radius of the earth, and k. Combining equations (6) 
and (9) gives 
a 
m 
1 
i ap 1. (10) 
Consider again the special case where dn/dh = 
constant, so that n is a linear function of the height 
(standard refraction). Consider the plane earth 
diagram of Figure 6. The angles between corre- 
sponding curves are the same as in the true diagram, 
Figure 4. Hence, for the plane earth diagram, 
equation (8) becomes 1/p’ = — da/dx, where p’ is 
the radius of curvature of the ray in the plane earth 
representation. It is readily found that 
eel (y 
pp ka 2 dh dh oa (11) 
one syd 
dh ka 
Since M usually increases with height, the curvature 
of rays is concave upwards in this diagram. Again, 
equation (11) shows that when the modified earth’s 
