DIFFRACTION BY TERRAIN 4.69 
practice, dz >> d, and the angle y between the 
direct ray and the horizontal will be equal to the 
angle between the image ray and the horizontal. 
Then approximately, since the angles are small, 
ho = dha, = dy (Yo Ss Y), (14) 
where d, and a have the significance given themon 
page 462. There whe signs have again been 
chosen so that ho is negative when the receiver 
(target) is in the shadow of the screen with regard 
to the image transmitter. 
The distance from the transmitter to the diffract- 
ing coast depends on the azimuth (Figure 18). 
Therefore, with the designations of the figure, 
Fees (15) 
cos y 
Equation for Field Strength 
he expression for the diffracted field of the 
image transmitter is given by the straightedge 
formula, equation (6), with v given by equation (3). 
Since 1/d: is assumed negligibly small compared to 
1/di, we find, on using equation (14), 
2d, 
joe eone ey 
This may be further simplified by introducing a new 
variable, the lobe number 
eae (17) 
» 
(fy = transmitter height). This quantity is equal to 
1,3,5--- at the interference maxima and equal to 
0, 2,4,--- at the interference minima but is here 
taken as a continuous variable, defined for any value 
of y. In particular for y = Yo we put n = mo. Since 
Yo = hi/di, we have by equation (15) 
ae Ahw, Ah cosy 
n 18 
0 vi (18) 
Equation (16) may now be written 
Ny — n 
a (19 
V2 ) 
The diffraction formula will again be written, in the 
form of equaticn (7), as 
Bagot 
Eo 
where z and ¢ are the functions of v shown in Figures 
4 and 5. ; 
The total field obtained by the interference of the 
direct and reflected ray is 
E=£&(1 —ze""~*), (20) 
where the negative sign in front of the second term 
in parentheses accounts for the 180-degree phase 
shift at reflection, and the phase lag zn corresponds 
to the path difference between the reflected and 
direct rays. 
The absolute value of the field is 
E , 
4 = V(1 — 2)? + 4zsin (an + £). (21) 
Figure 12 in Chapter 5 may be used for the numer- 
ical evaluation of this equation. 
The formula can readily be generalized to the 
case where the reflected ray is weakened by (1) a 
reflection coefficient, R, different from unity, and 
(2) the effect of the earth’s curvature expressed by 
the divergence factor, D, (Chapter 5). If, moreover, 
the phase lag at reflection is not 7 but 7 + ¢’, the 
equation becomes 
= V(1 — 2RD)?+ 42RD sin? 4 (xn + 6’ + 8). 
(22) 
By 
Example 
Assume the following conditions. A radar set of 
200 me (A = 1.5 meters) is sited at a height hy = 15.3 
meters (about 50 feet) and at a distance to a straight 
shore line of dj = 195 meters (about 0.12 mile). 
The ground between the radar and the seashore 
is level but can be considered as rough for prac- 
tically any angle of elevation, on applying the 
criterion of data discussed .The coverage diagram 
will first be determined in the azimuth perpendicular 
to the coast line, where d; = d, or cos y = 1. 
Then by equation (18), m = 3.20. With this value 
of no the variable v is determined by equation (19). 
We shall confine ourselves to integral values of n, 
that is, to those angles which, in the presence of 
simple reflecting ground, correspond to lobe minima 
and maxima. Having obtained v, one then deter- 
mines 2 and ¢ from Figures 4 and 5. The field in 
terms of the free-space field is then obtained from 
equation (21), either by direct computation or by 
means of Figure 12 in Chapter 5. The numerical 
data for the first five lobes are summarized in 
Table 1. The last column of this table contains the 
values of E/E which would be obtained if the magni- 
tude of the reflection coefficient were assumed to be 
zero over land and unity over the sea and if diffrac- 
tions were neglected. 
The same calculations are carried out for an 
TaBLe l. (y = 0°). 
|E/E)| with |E/E| without 
n v Zz diffraction diffraction 
(degrees) 
() Wee, aleall/ () 0.17 0 
1 0.87 1.05 —12 2.05 2 
2 0.48 0.80 —15 0.75 0 
3 0.08 0.54 — 4 1.54 2 
4 -—0.32 0.36 24 0.69 1 
5 —0.71 0.26 11 1.26 1 
6 —1.11 0.19 145 1.16 1 
7 —1.51 0.14 242 0.94 1 
8 —1.90 0.12 5 0.88 1 
9 —2.30 0.10 158 0.92 1 
10 —2.70 0.08 339 0.94 1 
