520 Sir GL Greenhill on the 



Diffraction. 



15. Take the case of pure plane waves, either of light or 



2tt 

 sound, arriving in a state of vibration ccos-— Yt at a 



parallel plane AOB. 



In the absence of diffraction across AB, the waves advance 

 to a parallel plane PNP' at a distance b in the state of 

 vibration 



eooB^-(Vt-J), (1) 



with no change in the amplitude c of vibration, taken as 

 displacement or velocity. 



But suppose a screen is placed in AB, with a thin circular 

 slit AB, of mean radius a and breadth da, through which the 

 waves are diffracted. 



The vibration arriving at any point P of the parallel 

 coaxial circle PP', of radius A, from an element add da of 

 the slit at Q, will be in the slate 



cad$ da cos — (V«-PQ), . . . . (2), 

 PQ 2 = A 2 -2Aa cos 0+a 2 + b 2 , 

 and with A, a small as compared with b, we may put 



FQ = b-^ ( pA 

 cos^(Vf-PQ) = (50B^(V*-ft)cos(-^^cos^ 



+ sin 2 ^(V/-/>)sin( 2 ^cos^. . (3> 



Integrating with respect to 6 round the circular slit AB 

 for the resultant vibration at P, the integral 



and y (4^, 



j»' : ° s (- 



^-^cos 6)d0 = I cos (2^/a-cos d) = 2ttC(» 



2ttA« _ / <?r 2 A 2 a 2 *(3 



putting -^j— *•*, - = -W- = ^ 



2' 



