114 TECHNICAL SURVEY 
plus sign is used in equation (82) and —v in Figure 
27. Thus, for n = 2 and »v = +2.11, is read in 
Figure 27 the relative intensity z = 0.980 and in 
Figure 28 the phase lag, ¢ = —0.103 radians. Other 
values are listed in Table 9. ; 
TABLE 9. Shoreline diffraction. (Example 15.) 
n ov Sign z fe n ov Sign z & 
0 2.51 + 1.036 +0.080 1/12 0.14 + 0.582 —0.130 
1 2.31 + 1.083 —0.015 | 13 0.089 — 0.459 +0.2 © 
2 2.11 + 0.980 —0.103 |} 14 0.28 — 0.3877 +0.5 
3 1.91 + 0.884 —0.038 |}15 0.49 — 0.308 +0.9 
4 1.71 + 0.938 +0.100 | 16 0.68 — 0.261 +1.4 
5 151 + 1.082 +0.120 ||}17 0.87 — 0.223 +1.9 
6 1.32 + 1.170 +0.030 }18 1.08 — 0.192 +2.6 
7 #112 + 1.156 —0.085 /19 1.27 — 0.170 +3.3 
8 0.92 + 1.073 —0.181 || 20 1.48 — 0.150 +4.2 
9 0.72 + 0.953 —0.255 | 21 1.67 — 0.185 +5.2 
10 0.52 + 0.825 —0.273 |} 22 1.88 — 0.121 +6.4 
11 0.32 + 0.696 —0.224 |} 23 2.07 — 0.111 +7.6 
The width of the seeond zone may be computed 
from equation (88). The effective height for n = 2 
is obtained from Figures 50 and 51 and is 414 ft. 
b= 4x a? = 1,656 ft . 
The Modified Antenna Pattern 
The vertical directivity of the antenna is modified 
by the local terrain. Unless the ground under the 
antenna is an extension of the reflection plane the 
modification of the free space directivity character- 
istics should be taken into consideration in the 
calculation of radar coverage. 
The vertical pattern of the antenna in the absence 
of a reflecting surface is referred to as the free space 
pattern, f,. This is usually given in the instruction 
manual for the set. If this pattern is not available 
or if the antenna has been modified, the vertical 
directivity may be computed by methods given in 
the next section. Local terrain effects are treated in 
some detail as they are in many cases a controlling 
factor. The resultant effect of the local terrain and 
free space pattern is called the modified antenna 
pattern, f(y). It does not include the effect of the 
main reflecting surface. 
Antenna Patterns 
To obtain f,, the relative amplitude of the radia- 
tion from the antenna, as a function of the vertical 
angle y it is only necessary to'take into account the 
path differences of the elements of the array. The 
absolute field intensity and time phase will not be 
‘considered. In Figure 61 is shown an array of four 
horizontal half-wave dipoles spaced a half wavelength 
apart. The radiation from A in the direction 7 may 
be taken as proportional to cos wt. The path differ- 
ence of radiation from B is A/2-sin y. she corre- 
? | rly 
(9) 
Es 
AX 
Figure 61. The four-element array. 
sponding phase difference is 
2 so 4 
—y Xo 8iny = —mrsiny. 
For C and D the phase is —2z sin y and —3z sin y 
respectively. The total field intensity pattern is 
fa = cos wt + cos (wt — 7 sin +) 
+ cos (wt — 2m sin y) + cos (wt — 3m sin) , 
grcuping 
[cos wt + cos (wt — 3m sin 7)] 
+ [cos (wt — m sin y) + cos (wt — 2x sin y)] . (89) 
From the identity 
cos A + cos B = 2cos}(A + B) cosi (A — B) 
equation (89) may be written 
Sf, = 2cos (1 = ar sin v) cos (= sin 7) 
2 
2 cos ( wot — BE Fi cos (= sin 
2 Dis Ce? 
2 cos (0 = 3 sin v) : 
om. 
[cos( sin 1) + cos - 6 sin v)| ; 
ov. : ie 
fa = 4 cos | ot — "Sin y } cos (x sin 7) - 
cos (= sin 
Pieri 
Since only the rms value of this equation is signifi- 
cant, the terms containing wt may be dropped, and 
the result for the four-element array is 
+ 
fa 
wT 
fa = cos ( sin y) cos ( sin v) : (90) 
It is easily verified that this is a special case of the 
general expression for an N element array spaced at 
intervals of nd and excited in phase (not derived here) 
_ sin (Nn z= sin y) 
Pa = Nisinl Gain) et) 
The effect of a reflecting screen may be computed 
by treating it as though it were 4/4 from the dipole 
