SIMPLE-HARMONIC WAVES. DIFFRACTION 251 



due to an electrified disk*, and the normal velocity has the 

 distribution 



_d$ = B 



dn V( 2 -^ 2 )' 



where tn- denotes the distance of any point of the aperture from 

 its centre. The velocity becomes very great near the edge, and 

 is mathematically infinite ( at the edge itself (r = a), but it 

 appears on integration that the parts of the area near the edge 

 contribute little to the total flux, which is 



(7) 



If the incident waves be represented by 



(8) 



the same flux will as in 82 be expressed by 2KC, or 4a(7. 

 Hence, comparing, 



In the other extreme, where the wave-length is only a 

 minute fraction of the dimensions of the aperture, the effect of 

 the screen in modifying the distribution of normal velocity over 

 the latter is practically confined to a distance of a few wave- 

 lengths from the edge, and the corresponding part of the integral 

 in (1) is quite unimportant. In this case, the incident waves 

 being still expressed by (8), we can put d(f>/dn = ikC with 

 sufficient accuracy over the whole area of the aperture, whence 



*-(f^ .-do) 



For the methods of approximating to the value of this integral, 

 by the use of Huygens' or Fresnel's " zones," or otherwise, we 

 must refer to books on Optics. It is found that the amplitude 

 is nearly uniform within the space bounded by a cylindrical 

 surface whose generators are normal to the screen through the 

 edge of the aperture, and is nearly zero in the surrounding 

 region. Near the cylindrical boundary, on either side, we have 



* See Fig. 75, p. 247, which represents the configuration of the equipotential 

 surfaces. 



f The awkwardness of this conclusion may be avoided by giving the screen 

 a certain thickness, and rounding the edges. 



