CALCULATION OF RADIO GAIN 389 
Simplifying, 
1 kahy 
Se —o[ 0 +h 2] + = 0. (74) 
If s << 1, the terms s* and s? may be neglected 
to a first approximation, and 
hy 
Lap Rte Me (75) 
Cah In) 2ka 
If d << dz, hy is well above the line of sight and s 
reduces to 
{ae (76) 
hi+ d 
Equation (76) is the plane earth formula. 
Curves showing s as a function of h2/h; and 
d/dy are given in Figures 19 and 20. These may be 
used for the direct calculation of d; = sd within the 
limits of interpolation. 
Path Difference 
Referring to Figure 14, the path difference A 
for a spherical earth is equal to 
A=r— rq = Vd? + (hi! + hy’)? 
—Vd? + (he! — Iy')?. (77) 
It is usually sufficient to expand the square roots 
and neglect powers and products of hy’ and h,’ 
beyond the second. This gives 
ae he! ; (78) 
which is the same as the plane earth formula when 
hy’ and h,’ are written instead of h, and fz. Equation 
(78) is accurate to within 1 per cent for values of y 
(the angle at the base of the transmitter) less than 
about 8 degrees. The error is less than 10 per cent 
for values of y less than about 24 degrees. When 
equation (78) is not sufficiently accurate, the follow- 
ing may be used: 
A=r-T, = 
hy’ he! 
Ee ? 79) 
Va? + (ha!) + (he!)? 
A=r— Ta 
provided 
1 (fy')? + (he’)? 
2 ad 
All the above equations for the path difference 
depend upon: the distance to the reflection point dj. 
However, the calculation of d, may be eliminated 
by first computing the path difference from the 
plane earth formula and then subtracting the correc- 
tion term A(Az). Thus : 
A= A,— A(A,), (80) 
<<l. 
where 
2hyhe 
= d ) 
Ap 
and where 
550 
ins? A (Ap) , 
is given in Figure 16, plotted against 
d d 1 
dy V2ka Vij + Vig 
If he < My, interchange hz and h; on the curves and 
ordinate of Figure 16. 
The maximum value of A (A,) is 
A (Ap) mex = 5.33 X 107 2B) (81) 
If the plane earth correction factor is negligible for 
the wavelength under consideration, the plane earth 
formula may be used throughout the whole range 
within the optical region, not only for the given value 
of h2/h; but for all lower values of hz/h, with the 
same hy. 
When x >> hy, so that the reflection point is 
much closer to the transmitter than to the receiver, 
a good approximation to A is obtained by replacing 
hy’ by hy and hy! by hz — d?/2ka in equation (78). 
IGE i) imal 
erie ae 
A 
—+—t 
Sep tains EC 
+ 14 
Aa s 6 5 al a 12 
Figure 16. Plane earth correction factor versus ae 
(Radiation Laboratory.) dy, 
Then 
which, to the same approximation, means that 
A & 2h; tan y. (82) 
In general, equation (82) is an improvement over 
the plane carth approximation except close to the 
transmitter and at low heights where h, is not much 
greater than fy. 
