through Apertures in Plane Screens. 265 



the case is that of plane aerial waves incident upon a parallel 

 infinitely thin plate, whose dimensions are ultimately supposed 

 to be small in comparison with \. The analytical process of 

 solution may be illustrated by the following argument. 

 Suppose a motion communicated to the plate identical with 

 that which the air at that place would execute were the plate 

 absent. It is evident that the propagation of the primary 

 wave will then be undisturbed. The supplementary solution, 

 representing the disturbance due to the plate, must then 

 correspond to the reduction of the plate to rest, that is to a 

 motion of the plate equal and opposite to that just imagined. 

 The supplementary solution is accordingly analogous to that 

 which occurs in the second of the problems already treated. 



Using a similar notation, we have for the principal solution 

 upon the two sides 



X»=X P =e- ax , (30) 



giving when x = 



X™ = Xv — 1 > dxJdx — dxp/dx = — ik. 



The supplementary solution is of the form (19), and gives 

 upon the aperture, viz. the part of the plane x = unoccupied 

 by the plate, yjr m = yjr p = 0, and so does not disturb the con- 

 tinuity of (/>. But in order that the continuity of d<f>/dx may 

 be maintained it is necessary that ty p = ty m • and then the 

 values of ^« and yjr p are opposite at any pair of corresponding 

 points upon the two sides. 



It remains to satisfy the necessary conditions at the plate 

 itself. These are 



^,i%? =0 d Xv , d ±p =0 . 



dx dx ' dx dx ' 



or, since d-*}r m /dx, d-*fr p /dx are equal, 



d^Jdx = dyfr p /dx = ik (31) 



It follows that^ f has the opposite value to that expressed in 

 (25) ; and the realized solution for a circular plate of radius 

 c becomes 



&7TC^ X 



<j> p = cos(nt — kx) + ^j- 7z cos(7it—kr) 1 .... (32) 



%TTC X 



(j> m = cos{nl-kx) + ^ -cos (rd — kr), .... (33) 



the analytical form being the same in the two cases. 



It is important to notice that the reflexion from the plate 

 is utterly different from the transmission by a correspondino- 



