260 Lord Rayleigh on the Passage of Waves 



On this understanding the equation of waves travelling 

 parallel to x in the positive direction, and accordingly incident 

 upon the negative side of a screen situated at x = 0, is 



(f) = e~ iJcx (4) 



When the solution is complete, the factor e int is to be restored, 

 and the imaginary part of the solution is to be rejected. The 

 realized expression for the incident waves will therefore be 



<£=cos(n£ — lex) (5) 



Perforated Screen. — Boundary Condition d<j>/dn = 0. 



If the screen be complete, the reflected waves under the 

 above condition have the expression cj> = e ikx . 



Let us divide the actual solution into two parts % and ifr, 

 the first the solution which would obtain were the screen 

 complete, the second the alteration required to take account 

 of the aperture ; and let us distinguish by the suflixes m 

 and p the values applicable upon the negative (minus) and 

 upon the positive side of the screen. In the present case we 

 have 



Xm = e-^ + e^, X P = Q («) 



This %-solution makes dx m /dn = 0, dx P /dn = over the 

 whole plane x = 0, and over the same plane ^ TO = 2, ^ p = 0> 



For the supplementary solution, distinguished in like 

 manner upon the two sides, we have 



fm= S* m e ~^ dS ' *> = J]*' ^^ 



(?) 



where r denotes the distance of the point at which ty is to be 

 estimated from the element dS of the aperture, and the inte- 

 gration is extended over the whole of the area of aperture. 

 Whatever functions of position M^, M^ may be, these values 

 on the two sides satisfy (2) , and (as is evident from symmetry) 

 they make dyfr m /dn, d^r p /dn vanish over the wall, viz. the 

 imperforated part of the screen ; so that the required con- 

 dition over the wall for the complete solution (% + ^) is 

 already satisfied. It remains to consider the further con- 

 ditions that <j> and dcf>/dx shall be continuous across the 

 aperture. 



These conditions require that on the aperture 



2+^r w = ^rp, df m /dx = d^r T /dx. . . . (8)* 



* The use of dx implies that the variation is in a fixed direction, while 

 dn may be supposed to he drawn outwards from the screen in both 



cases. 



