of Aerial and Electrical Waves by a Grating. 305 

 these, we shall have 



|j£+P*=0, (7) 



approximately, and therefore, for x>x\ 



cf) = A sin Zr^ + B cos&.r, (8) 



whilst, for #< — &', 



(f> = A sin /;/£ — B cos kx, (9) 



<f> being, in the gravest mode, an odd function of x. 



In the region between the planes x — +x' the configuration 

 of the lines </> = const, is, on the principles explained by 

 Helmholtz and Lord Rayleigh *, sensibly the same as if in 

 (4) we were to put k = 0. So far as this region is concerned, 

 the problem is in fact the same as that of the conduction of 

 electricity in a bar of metal which has the same form as the 

 actual mass of water, and has accordingly one or more 

 perforations occupying the place of the obstacles. The 

 electrical resistance between the two planes is then equivalent 

 to that of a certain length 2x + a of an unperforated bar of 

 the same section. The difference of potential between the 

 two planes may be taken to be 2(A:A^' + B), by (8), since kx f 

 is small ; and the current per unit sectional area is kA, 

 approximately. Thus 



2{kAx' + B) = (2x' + a)kA, . . . (10) 

 whence 



B/A = ik*, (11) 



and 



cf> = A (sin kx + ^ka cos kx) 7 . . . . (12) 

 for x > x . 



The condition to be satisfied at the end x = \l gives 



cos ^kl— ^kasin ±kl = 0, .... (13) 



which determines k. When, as in the experiments to be 

 described, ka. is a small quantity, this is equivalent to 



GOsik(l + a)=0, (14) 



so that the introduction of the obstacles has the effect of 

 virtually increasing the length of the tank by a. 



The value of a is known in two cases. When the plane 

 ,r = is occupied by a thin rigid diaphragm of breadth a, 

 having a central vertical slit of breadth c, we have t 



2(2, nria — c) , _ 



a= — Jog sec— ~ (10) 



2a v 



+ 



7T 



Theory of Sound, § 313. 



Hydrodynamics, p. 512. The notation is slightly altered. 



