OUTLINE OF THEORY 



173 



*•••* Application to Straight Edge 



Since 



V2 2 ' 



E = El 



1+i 



I + C{v) -■^- jSiv) 



(6) 



It will be noticed that the quantities 

 1 



^-■) 



2 + ^'\2 



have a simple geometrical meaning. They are the 

 real and imaginary components, respectively, of a 

 vector drawn from the lower point of convergence 

 (point —1/2, —j/2) of the Cornu spiral to a point 

 on this spiral. The bracket in equation (6) is equal 

 to this vector in magnitude but with opposite phase. 

 The Fresnel formulas and Cornu spiral as given 

 above will assist the reader in establishing the rela- 

 tions of our equations with the classical theory of 

 diffraction as foimd in all textbooks on the subject. 

 For practical purposes the field behind a diffracting 

 straight edge given by equation (6) will be denoted by 



E, 



— = ze 



(7) 



zone. If the receiver is sufficiently deep in the 

 shadow, about v > —1, the following approximate 

 formula holds: 



In Figure 4, the modulus s is plotted as a function of 

 V. In Figure 5, the phase lag j" is plotted in a similar 

 way. (With the above choice of the sign, j" is positive 

 in the shadow.) 



The variable v is given by equation (3). On 

 accovmt of the square root, there is an ambiguity in 

 sign. Closer inspection shows that v must he taken 

 'positive when the receiver is in the illuminated region, 

 above the line of sight; v must he taken negative ivhen 

 the receiver is in the shadow zone. 



When V tends to — oo , the line AP5 (Figure 1) 

 moves far upwards relative to the line TR; the 

 receifver lies deep in the shadow and E approaches 

 zero by equation (6) . When v tends to + co , the line 

 APB moves far downward, and the screen ceases 

 to orm an obstruction, E approaches Eq. At the line 

 of sight (when the point P in Figure 1 coincides 

 with M), V = Fiv) = and £■ = £"0/2. Clearly, the 

 effects of diffraction are not confined to the shadow 

 region but extend considerably into the illuminated 



z = 



0225 



V 



equation (2) may be rewritten, on using equations (4) 



and (5), as _ . 12 



Figure 4. Magnitude of relative field strength E/Eo 

 versus v. 



