Reflexion of Sound from a Perforated Wall. 227 



If the channel be closed at #= — I, 



A' sin k'l + B' cos 1/1=0, 

 and we may take 



4> = A"cosk'(x + l)e int (11) 



From (11) when x is very small, 



u = dcfy/dx =-/„•' A" sin k f l . e in \ . . . (12) 

 as=-a- 1 d<l>/dt=—ikA." cos k'l.e**, . . (13) 



•so that u 1/ , 



= ry tan kl (14) 



as ik 



Now, under the conditions supposed, where the transition 

 from the state of things outside to that inside, at a distance 

 from the mouth large compared with the diameter of a 

 channel, occupies a space which is small compared with the 

 wave-length, we may assume that s is the same in (6) and 

 (14), and that 



(<t + ct')m in (6)=cru in (14), 



where a represents the perforated area and a 1 the unper- 

 f orated. Accordingly, if we put A = 1, as we may do without 

 loss of generality, the condition to determine B is 



B-l _ a^ k' tan k'l 



B + l~ (o- + <7')cos<9 ik • • • • ( l0 ) 



If there be no dissipation in the channels, A = 0, and k' = k. 

 In this case 



jy _ (cr + <r f ) cos 6 cos kl — ia- sin kl . 



{g -f- cr') cos cos /:/ -f- icr sin kl' 



Here Mod. B = l, or the reflexion is total, as of course it 

 should be. If in (16) cr = 0, B = l, the wall being imper- 

 forated. On the other hand, if cr / = 0, the partitions between 

 the channels being infinitely thin, 



-D _ cos 6 cos kl— i sin kl , . 



cos 6 cos kl ■+■ i sin kl ' ^ ' 

 In the case of perpendicular incidence = 0, and 



B=e~ 2kl , (18) 



the wall being in effect transferred from ^ = to x = — I. 



We have now to consider the form assumed when k' is 

 complex. In (15) 



cos k'l= cos k x l cos ik 2 l + sin k Y l sin ik 2 l, } 



sin k'l= sin l\l cos ik 2 l— • cos kj sin ikd. i 



Q2 



