CALCULATION OF RADIO GAIN 387 
the ordinary reflection coefficient is replaced by 
Kr B(_1 ) pp. (56) 
Fy Tr + A 
The correction is not necessary when 2h, << d? 
[see equation (52)]. 
SPHERICAL EARTH 
Measurement of Distance 
The difference between the slant range rz and the 
distance measured along the surface of the earth and 
designated by d in Figure 14 is usually negligible. 
For a transmitter height of 1,500 meters, the error 
in assuming rz = d is 0.04 per cent at a distance of 
161 km and height of 6,900 meters, and 1 per cent 
at the same distance but at a height of 22,500 meters: 
As the transmitter height is increased, the error is 
increased. 
Equivalent Heights 
In order to express the slant range ry in terms of the 
curved distance d to a higher order of accuracy, the 
cosine law is applied to the triangle, transmitter- 
receiver-earth center. This gives the equation 
t¢ = (ka + hy)? + (ka + he)? 
— 2(ka + hi) (ka + he) cos (2) c 
ka 
Selecting the relatively important terms of the order 
@hyh, and d‘(hy + he), as well as powers of d higher 
than the fourth, the above equation reduces to 
& & 
rg = @ + (he — hy)? + = (i + hy — =). (57) 
Solving equations (13) and (14) for h and hz gives 
These results may also be expressed by saying that 
the distance from the surface of the earth to a plane 
which is tangent at a distance d; from the trans- 
mitter is d2/2ka. 
Hence for a transmitter of height h; above the 
ground the height above the tangent plane at the 
reflection point, the so-called equivalent height is, to a 
first approximation, 
d. 2 
hy =h— aa ‘ (58) 
and for the receiver the equivalent height is 
d, 
he’ = ho — —. 59 
2 ata (59) 
The equivalent heights are shown in Figure 14, 
which illustrates the geometry of the spherical earth 
having an effective radius of ka. 
Angles 
Referring to Figure 14 and remembering that the 
angles are greatly exaggerated in the figure, it is 
seen that 
7 , 
any (60) 
d, Qy 
ho d 
tan y = —-—, 61 
Me ke VO eH st) 
he — hy d 
t = —- — 62 
an Wa 5 ae (62) 
d 
veyr+—. (63) 
ka 
Angle y must be evaluated in order to determine the 
reflection coefficient. Angles yz and »v determine the 
antenna pattern factors F; and F2, which are shown 
in Figure 11. The angle y is significant in coverage 
calculations and angular approximations. 
Determination of Reflection 
Point (d,) 
Since several equations of the two preceding 
sections depend upon di, it is necessary to be able 
to determine this distance when the transmitter 
and receiver heights are given and the distance 
between them is known. Let 
he : ao), (64) 
and 
ie ae (+0), (65) 
so that 
a,=2G=0, (66) 
2 
endl m=BFBa—9, (67) 
where b= eae ; (68) 
cena 
and c= lgendliz : (69) 
hy + he 
Assume h; > fie and d; > db, so that 6 and c¢ will 
always be positive. This is always possible because 
of the principle of reciprocity. From equation (60), 
or 
HSS =. (70) 
