672 Diffraction of Waves by a Semi-infinite Screen. 



becomes discontinuous in form (though, of course, con- 

 tinuous in value) as we cross the lines co l = } o) 2 = 0; 

 i. e., the lines 6 = tt + u, 6 = ir—a. These lines, with the 

 diffracting plane, divide space into three regions, A, B, 



and (fig. 2). 



A, (o l and tw 2 both positive. 



B, (Oi positive, co 2 negative. 



C, co i and o> 2 both negative. 



§ 5. Now for an arbitrary incident wave we may write 



= \ f " ./!*•)«»• ("V aifi ~ X) dt:, . . . (20) 



provided the form of f is such that Fourier's Double 

 Integral may be applied. The solution for the region A 

 then becomes 



IT 



+ similar term in 7 and co 2 . . . . (21) 



= /(/ s)_i 2 f v f A*)«^f "« : ^- s - 1 w *- x >iiT.'.. 



^"Jo J-eo Jo 



= /'(/9)-- ( ,2 /(/3-2 a ., 2 sec 2 f)<^ 



TT To 



7T 



Hr/(7) { -I C 2 /(7-2«i 2 sec 2 ^)#j 



= yijS)-KA«'i) : F{A7)^( f y.«2)} ( 22 «) 



Similarly for region B, 



^=/(/3)-KA Wl )+^,«2); • • • ( 22 ^) 



and for C, 



0= f/(j3,cD l )^g(j,cd 2 ). ...... (22 c) 



The extension of the method and results to the case where 

 the incident waves travel in a direction not perpendicular to 

 the diffracting edge is easy. 



London, Feb. 9th, 1920. 



