102 ; TECHNICAL SURVEY 
From Figure 47 it follows that d,, the distance to 
the reflection point, is given by 
hy 
d, = 
' tan w’ 
or taking tan YW = sin VW = + (which may be done 
‘provided d; is small enough compared to dz and large 
compared to h;) and substituting in equation (57), 
_ 4h? 
d, = ANE (58) 
Here di, hi, and \ must be expressed in the same units’ 
For high sites and distant targets the angle + 
becomes smaller, and the approximation involved 
in equations (57) and (58) requiring d; to be small 
compared to dz becomes worse. In Table 4 are listed 
the minimum values that n\ may have for an error 
of 1 per cent or less in equation (57) at different 
antenna heights. Also is given the minimum value 
TasLeE 4. One per cent error in ‘Y. 
hy,.ft Minimum 7° Minimum 7), ft 
400 1.5 43 
200 ail 15 
100 0.8 6 
50 0.6 2 
15 0.3 0.3 
of y corresponding to nA. Thus equation (57) when 
used on a 100-ft site at 100 mec (A = 9.84 ft) will 
give values which are in error by less than 1 per cent 
for all lobes and for angles above 0.8 degree. If the 
100-ft site operated at 1,000 mc (A = 0.98 ft) the 
minimum value of » would be 6 corresponding to 
the fourth null. The error in y is always positive 
and increases rapidly with antenna height, and at 
a height of 1,000 ft and a frequency of 100 me the 
formula is incorrect for all angles of interest. At 
distances such that the earth curvature drop is 
comparable to h;, equation (57) does not even give 
the correct order ‘of magnitude for y. 
Examples 9 and 10. Flat Earth Lobe Angle Compu- 
tations. Lobe angles for two cases will be computed, 
Example 9, a 200-me set at 15 ft and Example 10, 
a 500-mce set at 50 ft. 
Example 9 Example 10 
A= oo X 3.28 = 4.92 ft x = 1.97 ft 
For n = 1 (first lobe) 
Y= SS X573=47° 7 = 0.564° 
dy -“ = 183 ft d, = 5,080 ft 
In practice some of the lobes listed for Example 9 
may be absent because of nulls in the antenna 
pattern. The angle listed for the first lobe of Example 
10 is slightly over 1 per cent too large. 
TaBLE 5. Lobe angle and distance to reflection point. 
Example 9 Example 10 
n Y, degrees dj,ft ¥,degrees dh, ft 
1 (lobe 1) 4.7 183.0 0.56 5080 
2 (null 2) 9.4 91.5 1.13 2540 
3 (lobe 2) 14.1 61.0 1.69 1693 
4 (null 3) 18.8 45.7 2.26 1270 
5 (lobe 3) 23.5 36.6 2.82 1016 
Lobe Angles Corrected 
for Standard Earth Curvature 
Equation (57) may be modified to include the 
effect of earth curvature approximately and to give 
the lobe angles for the majority of sites with accept- 
able accuracy. 
For antennas several hundred or more feet high, 
d, as given by equation (58) may be large enough 
so that the earth curvature drop is appreciable. 
In Figure 49 is shown a transmitter of height hi 
above the horizontal plane GH. The radius of the 
standard earth is ka. At D, the center of the reflection 
area for the lobe considered, is drawn a tangent 
plane CDE, which intersects hi at a distance h’, 
below the center of the antenna and which will be 
considered the equivalent antenna height. This then 
is the part of h; which determines the angle 7’ 
which the lobe center line CL makes with the tangent 
plane CDE. Subtracting from y’ the angle @ which 
TO CENTER OF EARTH 
Figure 49. Lobe angles corrected for earth curvature. 
the tangent plane CDE makes with the horizontal 
at the base of the antenna GH, the lobe angle y 
referred to the horizontal at the antenna is obtained. 
From equation (49) it follows that 
d;? 
= -=> 59 
hi =h—S. (59) 
Here hy’ and h, are expressed in feet, d, in miles, and 
k is assumed to be 4%. 
This height hi’ is the portion of h; that is effective 
in connection with the plane CDE and when substi- 
tuted in equation (57) gives the angle 7’. 
ee mr _ md 
~ Aas 2\ ° 
7 a(n ~ =) 
2 
(60) 
