112 TECHNICAL SURVEY 
the transmitter distance, however, no such approxi- 
mation will be made. 
h, = depth of the image antenna below the 
reflecting surface, in feet. 
W = angle of the lobe considered with reference 
to the tangent plane at the reflection point, 
in radians. 
= number of the Fresnel zone. 
= 0 for the center of the first zone. 
+1 for the far edge of the first zone. 
= —1 for the near edge of the first zone. 
= +2 or —2 for the edge between the second 
and third zones. 
n = lobe number. For a given radar station 7 is 
related to the angle Y by the equation 
n = (4hi/d) sin V. 
d\ = wavelength in feet. 
d, = distance from the transmitter to the near 
edge for the Fresnel zone and lobe considered, 
in feet. 
d, = distance from the transmitter to the far edge 
SSsss 
lI 
for the Fresnel zone and lobe considered,, 
in feet. 
d; = distance from the transmitter to the center 
of the first Fresnel zone for a particular lobe, 
in feet. 
In Figure 59 is shown the first Fresnel zone for an 
angle Y with a corresponding value of n. Ray 2 
passes through the center of the first zone and rays 
1 and 3 pass through the near and far edges respec- 
tively. Because of the great distance of the target 
the rays 1, 2, and 8 are parallel. 
For the first zone the path difference between 1 
and 2 is 4/2. For zone m the path difference is 
m/2 (where m = 1, 2, 3, etc.). Since the points 
d, and d, are not equidistant from the target, the 
distance x1 cos Y must be subtracted from ray 2 to 
compensate for the increased path length of ray 1 
above the plane. 
mdr -e. hy 
oS hl. (2 y ~ 71.008 v) 
In the right triangle 
h : 
1? = 2? sin? W + (ey — 2, cos v) ; 
Eliminating 1, from these equations and solving - 
for 1 
—m) cos WV + »/m?n? + 4mXh; sin V 
2 sin? V 
For the far point of the zone 
mr nae sn hy 
sors le (si + Ze cos v) ; 
also 
2 
hy ar Z2 COS v) t 
sin V 
1? = Xe? sin? v + ( 
“By a similar process of elimination of J, and solving 
for 22: 
md cos VW + +/m?d2 + 4mXhy sin V 
te 2 sin? v 2 (IO) 
For the near point of the zone 
hi —m cos V + +/m?d? + 4mdh, sin V 
di, 
~ tanv 2 sin? v 
Since sin YW = nd/4h; and W is small, cos ¥ may be 
taken as unity with the following error: 
up to 214° less than 0.1 per cent, 
up to 10° less than 1.5 per cent, 
up to 15° less than 4.5 per cent, 
anil m Vm? + mn\ 8h: 
a, = (5 +% n? =. 
For the far point 
a= (a+ aR 
These equations may be combined: 
(ea i yn 2 
a= (5 4 MMi fmt) Sia 
2 né Ne (0) 
where the plus sigt gives the far point and the minus 
sign gives the near point. The reflection pojnt is 
obtained by using m = 0 and equation (77) reduces 
to: 
Ah? 
Ga es: (78) 
Thus to obtain the range of the near edge of the 
first Mresnel zone for the first lobe, substitute n = 1, 
m = 1 and use the minus sign in equation (77): 
2 
d, = 0.688 = : (79) 
The far edge of this zone is obtained by using the 
plus sign 
2 
d, = 23.3 he ; (80) 
Equation (77) is in the form 
— hi 
d= ane. (81) 
where 
1 m a/ m? + mn 
r=38(5 +5 + 7 i (82) 
or 
div 
Ty = ree (83) 
If d; 1s taken as the distance of the shoreline, 7; 
may be considered as a characteristic site or terrain 
factor at a particular azimuth and combined with 
