1)11 1 KVCTION OK ItADIO WAV 



127 



Equation (25) may be brought into a more con 

 venient form by writing 



g(v,r») 



tan 6 = 



/(»,»o) ' 



(28) 

 (29) 



and R = \/p{v,v ) + g 2 (v,v a ) . 



It then follows that equation (2(3) assumes the form 



E = W^b) EM Sln [*(* ' t) ' 6 ] ■ m 



For tabulation purposes the quantities f(v,v ) and 

 g(v,vo) are replaced by the Fresnel integrals, defined 

 by: 



C(v) 



IT 



cos I - V 

 \2 



and 



S{v) 



= Jo Sin (^ 2 



dv 



dv 



Evidently 



and 



f(v,v ) = C(v) - C(v ) 

 g{v,v ) = S(v) - S(v ) 



(31) 



(32) 



(33) 

 (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 /, 

 g, C, and S. 



Table 1. Fresnel integrals. 



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 / 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 (31) and (32) for C and 

 S, squaring and adding, it follows that 



(dCY- + (dS)* = (dv)* , 



(35) 



so that dv is the line element of the spiral, and v 

 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 

 half-periods is AZ, which is thus the resultant effect 

 at P of the upper half of the wavefront. A similar 

 result is obtained for the lower half. 



It may be remarked that the superiority of the 

 dimensionless variable v over s shows itself in the 

 fact that one Cornu spiral suffices for all situations 

 of the diffracting edge, while the use of s would have 

 necessitated the construction of a special spiral for 

 each specific set of values, a, b, and X. In Figure 20 

 the values v = 1 and v = 2 are marked and corres- 

 pond to path differences A = X/4 and A = X, 

 respectively. 



Equation (30) shows that the electric field strength 

 in the diffraction region which is due to a certain 

 section of the wavefront is proportional to the 

 corresponding value of R. Hence, it follows that the 



