STANDARD PROPAGATION 5 
It results from this equation that a linear gradient 
of refractive index has the same effect on refraction 
as the curvature of the earth, 1/ro. By introducing 
an effective earth’s radius it is possible to eliminate 
the refraction term entirely and to treat the atmos- 
phere as if it were homogeneous. This device was first 
introduced by Schelleng, Burrows, and Ferrell,*4 and 
has since been generally accepted. Some German 
writers have introduced a quadratic function to 
represent the variation of refractive index with height 
in the atmosphere, *** the coefficients of the quadratic 
terms being characteristic of the air mass or type of 
atmosphere involved. This has the advantage of per- 
mitting a close fit with observed refractive index 
curves up to heights of 6 to 8 km. It seems, however, 
that the advantage of the greater analytical simplic- 
ity of the linear refractive index curves far outweighs 
the increased accuracy of the quadratic form, and the 
latter has therefore not found acceptance in this 
country and Great Britain. 
It is customary to designate the effective, or modi- 
fied earth radius by ka where k is a numerical con- 
stant and a replaces ro used above and represents 
the mean radius of the earth. Hence 
(17) 
and by comparison with equation (11) it follows that 
(18) 
since dn/dh = — 0.039 - 10-6 
radius a = 6.37 - 10° meters. 
— 1/4a. The earth’s 
In view of this result coverage diagrams of radar! 
and radio communication sets are commonly drawn 
with a % earth’s radius. In such a diagram the rays, 
which are curved in a “true”? geometric representa- 
tion, appear as straight lines. 
The value k = % does not, of course, represent a 
universal law. It is merely an expression of the fact 
that the rate of decrease of the refractive index with 
height has, in the middle geographical latitudes, a 
certain average value. In arctic climates k as a rule 
is somewhat smaller, lying between 4% and %, while 
in tropical climates k is somewhat larger, between 
4% and %. In temperate and tropical climates, the 
main factor determining the magnitude of k is the 
humidity gradient in the lower atmosphere. In 
Figure 5 is shown a nomogram from which the ap- 
propriate value of 1/k can be read directly as function 
of the gradient of relative humidity and air tempera- 
ature. The table has been computed under the as- 
sumption that the temperature gradient has the 
“standard” value of —0.65 C per 100 m, but the 
value of k is relatively insensitive to variations in 
the temperature gradient. 
Usually the value of k = % is referred to as the 
standard case, but this term is also used to designate 
more generally an atmosphere with a linear refrac- 
tive index distribution where k might differ somewhat 
from 4%. Experience shows that the atmospheric 
conditions under which the refractive index is a 
linear function of height are quite common, but this 
is only one case out of several that may, and do, 
arise in the atmosphere. A full appreciation of the 
(RH), 45C +100 +15¢ +20C +25¢ 
1 10: . 51 .194 .240 . 0455 
Ag a 172 .214 405 
5 6 2354 
jo 40%}.006 .008 .012 .018 -026 .037 .052 .060 .079 .101 .129 .161 .200 .247 .304 
On 35 50%|.005 .007 .010 .015 .021 .031 .043 .050 .066 .084 .108 .134 -167 .206 .253 
60%}.004 .005 .008 .012 .017 .024 .034 .040 .052 .067 .080 .107 .134 .165 .202 
70%|.003 .004 .006 .009 .013 .018 .026 .030 .039 .050 .065 .080 .100 .123 .152 
2 002 .003 .004 .006 .009 .012 .017 .020 .026 .034 .043 .065 .067 .082 .101 
15 90%|.001 .001 .002 .003 .004 .006 .009 .010 .013 .017 .021 .027 .033 .041 .051 
8 
=30 =36 
7 = 
~ .) 
iS 
5 4 
6 = =5. 
> 
= o 
gis = 
> 
Ww 
=) 
48 
g ill 
4 ° 
3- 9 10. 
x 
hoe 
a 
2 ical 1 
Sea REL HUMIDITY GRADIENT ° ° ° ° 
% PER TGONMETE RS ee 2 20 és ec 
LO rar Seo NAS OMIGIMGRE A S317) SST EST Oe eT ays SONRT SOT ONENEN) esta Foes 
Figure 5. Graph: 1/k versus RH gradient and temperature for 100 per cent RH at ground. Add correction tabulated to 
obtain 1/k for RH at ground ~ 100%. 
