270 Lord Rayleigh on the Passage of Waves 



corresponding to 



Xm — % cos nt cos kx (48) 



The solution (47) applies directly to aerial vibrations in- 

 cident upon a perforated wall, and to an electrical problem 

 which will be specified later. Perhaps the most remarkable 

 feature of it is the very limited dependence of the transmitted 

 vibration on the width (2b) of the aperture. 



Narrow Slit. — Boundary Condition c£ = 0. 



The principal solution is the same as in (18); and the con- 

 ditions for the supplementary solution, to be satisfied over 

 the aperture, are those expressed in (21). In place of (19) 



*" = -JS^> +*=JS*'* ; • (49) 



the values of W m and ty p being opposite, and those of yjr m and 

 y$r p equal at corresponding points. At a distance we have 



dD C +b 



*-s J L*'*' (50) 



in which 



dD ikocl 7T \i _ ikr , 



IS = —\WFr) e (0l > 



There is a simple relation between the value of ty p at. any 

 point of the aperture and that of yfr p at the same point. For 

 in the application of (49) to any point of the narrow aperture, 

 dD/dx = x/r i , showing that only those elements of the integral 

 are sensible which lie infinitely near the point where ^r p is to 

 be estimated. The evaluation is effected by considering in the 

 first instance a point for which x is finite, and afterwards 

 passing to the limit. Thus 



so that (50) becomes 



*=;a J-,** (52) 



It remains only to express the connexion between fa p di/ 

 and the constant value of dyfr p /dx on the area of the aperture ; 

 and this is effected by the known solution for an incompressible 

 fluid moving under similar conditions. The argument is the 

 same as in the corresponding problem where the perforation 

 is circular. In the motion (u) of a lamina of width (2b) 

 through infinite fluid, the whole kinetic energy per unit of 

 length may be denoted by hu\ and it appears from Green's 



