422 Mr. F. J. "W . Whipple on Dif Taction of Plane Waves 



' 2u[U-l) l ~^ ' ' ' • CO 

 where G is a constant. 



On making the substitution and integrating, it is found 

 that 



r 1 7s 2 ~\ r n tt-»oo 



[v 2 -- 2 | 5 ]/=-^[ ( »-i)VY(^-y-«f)] u=i • (8) 



Since by definition yjr' vanishes for large negative values of 

 its parameter, its product by (u — 1) 1/2 is also zero for such 

 values and remains zero as u proceeds to infinity. Ac- 

 cordingly the right-hand side of equation (8) vanishes and 

 therefore 



where 



/= ^j" t(ci _ y _ a?) _^L_, . . . ( 9 ) 



this expression being derived from (3) by substitution of its 



C 

 value h— — -r^ for U. 



2u(u — 1) 1/2 



The integral in (9) assumes a neater form if sec 2 a be 

 written in place of u, when it becomes 



7T 



f=cCf{ct- f/ -^ec 2 oc)du. . . . (10) 

 Jo 



It is convenient to take 1/7T as the value of so that 



TV 



/== - I \jr{d - y-?sec 2 u)doc. . . . (11) 



IT 



JO 



It has been shown that/ 1 satisfies the wave-eqnation ; it 

 reduces to -^^(ct—y) when £ — 0, i.e. on the + axis of ?/. 



It can be seen that e- ~0 at points on the same axis whilst 



'df ° x 



'- reduces to — i^'{ct — y). 



It follows that if we make V=/ in the shadow, and 



V=-^ — / outside the shadow, 



V will satisfy the wave-equation throughout the whole space. 

 This solution provides for waves approaching the screen but 



