422 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 



1.2 



0.6 



t, = V2r SIN -^ 



Fig. 2.2 — The approximate value | e~'^ + *S'i | of | £ | for hp and of \ H \ for 

 vp at a great distance r behind a half-phme. Here, as in all of our work, the wave- 

 length X is 2-K. For an arbitrar}^ wavelength replace r by 2Trr/\, etc. 



(p~r » 1, exp ( — ^.^•) + S>i has the asymptotic expressions [see Equations 

 (7.7) and (7.8)] 



+ ^1 



c(/-)/2sin|, 



^ < (shadow) 



+ c(r)/2sin|, 



^ > 



c{:r) = i"\2',rr) 



'-"'e-'\ 



(2.4) 



(2.5) 



These expressions lead us to picture the field to the right of the half- 

 plane as the sum of the incident wave and a wa\'e, c(r)/<p, spreading out 

 from the diffracting edge^° (for the small ^'s of interest, 2 sin (p/2 ^ <p 

 even though (pr » 1). In the illuminated region these two waves inter- 

 fere with each other to give the oscillations around unitv shown in Fig. 



1° See, for example, R. W. Wood, Physical Optics, 3rd edition, p. 220 (Mac- 

 Millan, 1935). Curves of equal phase and amplitude have been computed and 

 plotted )i3' W. Braunbeck and G. Laukien, Einzelheiten zur Halbebenen-Beugung, 

 Optik, 9, pp. 174-179, 1952. 



