CONTEMPORARY ADVANCES IN PHYSICS 313 



A simple, Instructive, and historically famous example is that of the 

 circular hole. 



Diffraction from a Circular Aperture 



If the propagation of waves were rectilinear, their amplitude would 

 be constant along every line passing normally across an aperture. 

 Such however is as far as possible from being the truth, as we can easily 

 learn by evaluating the wave-motion along the "axis" of a circular 

 hole — that is, the line passing through the centre of the hole perpendic- 

 ular to the plane of the screen. Locate the centre of the circle at the 

 origin, so that its axis is the axis of x. Denote by R the radius of the 

 circle, by Xo the coordinate of the field-point P located anywhere upon 

 the axis. All points on any circle centred at the origin being equi- 

 distant from P, we may divide the area of the hole by concentric circles 

 into annular elements-of-area. Denote the radius of such a one by p, 

 its breadth by dp; then for it: 



dS = lirpdp] r^ = xo^ + ^2. cos = Xo/r, (78) 



and the limits of integration are p = and p = R. 



The problem is now stated in full; but it is very much simplified if 

 the distance Xo from screen to field-point is very many times as great 

 as the width R of the aperture ; and this in practice is commonly the 

 case. Then to first approximation, 



r = Xo, cos 9 = 1, (79) 



and these values are close enough to the correct ones to suffice for the 

 multipliers of the sine-function in (77) ; but in the argument of the sine, 

 r is multiplied by m, and a variation of only half a wave-length in r 

 entails a complete reversal of the function; hence in the argument we 

 must proceed to second approximation, and write 



mr = mxo -f mp^/2xo. (80) 



Making these substitutions in (77), we have finally: 



So = — (m/xo) I p sin {nt — mx — mp^/2xo)dp 

 Jo 



= cos (nt — mxo) — cos (nt — mxo — mR^ /2xo) (81 a) 



= V2(l -f cos mR^ 1 2xo) cos (nt — nixo — a). (81 b) 



Interpreted, these equations tell the startling fact that along the 

 axis of the hole the amplitude, far from being constant, varies in a 



