106 TECHNICAL SURVEY 
The method described here overcomes the limitations 
of equation (62) and may be used for the highest sites. 
In Figure 54 is shown the antenna above a curved 
reflecting surface whose radius is taken as % of the 
earth’s radius to allow for atmospheric refraction. 
The tangent plane CH makes an angle @ with the 
horizontal at the antenna, and @ is given by — (di/ka) 
as shown in Figure 49. 
h, = height of the center of the antenna above 
the earth’s surface, in feet. 
hy’ = equivalent height of the antenna, in feet 
— equation (59). 3 
ra = distance from the antenna to the target, ir 
miles. 
A = distance from the antenna to the reflection 
point, in miles. 
B = distance from the reflection point to the 
target, in miles. 
A = path difference, A + B — ra, in miles. 
d = wavelength, in feet. 
6 = angle between the tangent plane CE and the 
horizontal at the antenna, in radians. This 
angle is always negative. 
ka = radius of the modified earth, 5,280 miles. 
Wz = angle between the direct ray ra and the 
horizontal plane CZ, in radians. 
W = angle between the reflected ray A or B and 
the horizontal plane CZ, in radians. 
n = number of half-wavelengths path difference. 
In the triangle A Bra (cosine law) 
ra = WA? + B? + 2AB cos 2v. (64) 
From the definition of path difference: 
A=AtB-nesyrag ee (65) 
A+ B—A = V/A? + B? + 2AB cos QV ; 
squaring and dropping terms that cancel out gives 
2AB — 2AB cos 2¥ — 2BA = 2AA — A’, 
or solving with respect to B, 
AA — 3A? 
= : 66 
EG Seas) Ss ) 
Substituting 
mr 
ae 2 X 5,280’ 
into equation (66) gives 
Ds ail J ( md ) 
10,560 2 \10,560 : 
B= (67) 
mr 
A(1 — cos 2v) = 10,560 
Several approximations will be introduced to 
simplify equation (67): 
A will be taken to equal d; since W is of the order 
of 3° or less. 
From Figure 54 it follows that sin = h,’/5,280A, 
or for small angles, ¥Y = hy! /5,280A. 
‘Substituting for hi’ [equation (59)] it follows that 
Ni ede 
oS Rn (68) 
Using the approximation 
cos 2Y = 1 — 2v?. 
and neglecting 4(n\/10,560) compared to A, equa- 
tion (67) becomes 
—— 
10,560 
By gua ak ee (69) 
2d, v2 — _™ 
10,560 
From the law of sines 
sin2¥  sin(¥ + Wa) _ sin (YW — Wa) 
FG B 7 A ¥ 
B sin 2 
sin (YW + Wz) = a 
rq 
TO TARGET 
MODIFIED 
EARTH SURFACE 
Fiaure 54. Reflection point geometry. 
