SITING AND COVERAGE OF GROUND RADARS 83 
the phase lag 2rd/\ assumes the form 
Pad = et ce (23) 
Using equations (22) and (23), expanding the sine 
expression of equation (18), it follows that 
dE = fey 1 cos & rt) + sin 2n(s = .) 
- sin(§ ) + cos 2r( 7 - >) dv. (24) 
This expression may now be integrated over a 
certain region of the wavefront, say from v = vo to 
v = v, corresponding to s = s to s = s, giving the 
following expression for the electric field strength 
at P: 
me ip B flo.) sin an = ;} 
— g(v,v0) cos 2r( J — :) | (25) 
where uv 
f(v,v0) = i cos (5 2) dv, (26) 
a g(v,v0) = i! sin e i) dv. (27) 
Equation (25) may be brought into a more con- 
venient form by writing 
_ g(v,0o) 
tan @ = Fone) : (28) 
and R = Vf2(v,v0) + g2(v,00) (29) 
It then follows that equation (26) assumes the form 
TABLE 1. Fresnel integrals. 
v Cc S v C S 
0.00 0.0000 0.0000 2.50 0.4574 0.6192 
0.10 0.0999 0.0005 2.60 0.3889 0.5500 
0.20 0.1999 0.0042 2.70 0.3926 0.4529 
0.30 0.2994 0.0141 2.80 0.4675 0.3915 
0.40 0.3975 0.0334 2.90 0.5624 0.4102 
0.50 0.4923 0.0647 3.00 0.6057 0.4963 
0.60 0.5811 0.1105 3.10 0.5616 0.5818 
0.70 0.6597 0.1721 3.20 0.4663 0.5933 
0.80 0.7230 0.2493 3.30 0.4057 0.5193 
0.90 0.7648 0.3398 3.40 0.4385 0.4297 
1.00 0.7799 0.4383 3.50 0.5326 0.4153 
1.10 0.7648 0.5365 3.60 0.5880 0.4923 
1.20 0.7154 0.6234 3.70 0.5419 0.5750 
1.30 0.6386 0.6863 3.80 0.4481 0.5656 
1.40 0.5431 0.7135 3.90 0.4223 0.4752 
1.50 0.4453 0.6975 4.00 0.4984 0.4205 
1.60 0.3655 0.6389 4.10 0.5737 0.4758 
1.70 0.3238 0.5492 4.20 0.5417 0.5632 
1.80 0.3337 0.4509 4.30 0.4494 0.5540 
1.90 0.3945 0.3734 4.40 0.4383 0.4623 
2.00 0.4883 0.3434 4.50 0.5258 0.4342 
2.10 0.5814 0.3743 4.60 0.5672 0.5162 
2.20 0.6362 0.4556 4.70 0.4914 0.5669 
2.30 0.6268 0.5531 4.80 0.4338 0.4968 
2.40 0.5550 0.6197 4.90 0.5002 0.4351 
aby b 
E=k a+b) EoR sin [2(1 _ 3) - a| . (30) 
For tabulation purposes the quantities f(v,vo) and 
g(v,¥o) are replaced by the Fresnel integrals, defined 
by: 
O() = i cos G *) dv (31) 
and 
SG) = il in (Ee *) ao (32) 
Evidently 
f(v,v0) = Cv) — Cw) , (33) 
and 
g(v,v0) = S(v) — Sv) . (34) 
In the sequel the arguments will be omitted wherever 
it can be done without causing misunderstandings, 
and the above symbols will be written simply as f, 
g, C, and S. 
The Cornu Spiral 
In Figure 20 the two Fresnel integrals are plotted 
against each other, S being the ordinate and C the 
abscissa, for different values of v. The resulting curve 
is known as Cornu’s spiral. The upper positive branch 
(C and S positive) corresponds to points on the 
wavefront above the line. S’P in Figure 19, and the 
lower or negative branch corresponds to the wave- 
front below the line S’P. 
By their definition f and g signify the coordinate 
differences between any two given points on the 
Cornu spiral, and it follows that R, as defined by 
equation (29), represents the corresponding distance 
between these points. 
Differentiating equations (81) and (32) for C and 
S, squaring and adding, it follows that 
(dC)? + (dS)? = (dv)? , (35) 
so that dv is the line element of the spiral, and » 
measures length along the curve from the origin. 
In order to see more in detail how the Cornu 
spiral is built up of contributions from different 
zones we may suppose the half-wave zones on the 
wavefront to be divided into equal areas and the 
contributions of these areas to the field strength 
vectorially combined to obtain the resultant effect 
as in Figure 22. Then as smaller areas are used and 
more zones are summed up the vector diagram 
becomes in the limit the Cornu spiral. This is shown 
in greater detail in Figures 21 and 22. Here the first 
half-period zone of Figure 21 is divided into nine 
parts and the resultant is AB (Figure 22). The 
second half-period gives a resultant BC. The sum 
of the first two half-periods is AC. The sum of all 
