Reflexion of Sound from a Perforated Mall. 231 



increase in amplitude in virtue of the factor k x . The solutions 

 of (27) correspond, of course, to the equality of the ordinates 

 y and y'. It is evident that there are no solutions when y is 

 negative. The most important occur when k 2 is small and 

 2k x just short of it. But to the same small values of k 2 

 correspond also values of 2/( 1 which fall just short of 37r, 

 D7T, &c, or which just exceed 2-7T, 4-7T, &c. More approxi- 

 mately these are 



4 cos mir.k 2 2 , 



2ki = 7mr-\ , .... (32) 



where m = l, 2, 3, &c. 



In order to examine whether these solutions are really 

 available, we must calculate S. By (25) 



ks=kJi-lkA( m J r + 2cosm,r -^ 



\ o /\ 2 mir J 



T . / mir 2 cos mir . k 2 2 \ 



+ k 2 tan I -~- H . 



\ 2 mir / 



If m is odd, we have approximately 



*S-jg(l+V); (33) 



and if m is even, 



a-^ii+V^-i)}. • • (34) 



Since £ is approximately ^??i7r, we see that when m is odd, 

 S is large, and the condition of no reflexion can be satisfied, 

 as when m = l. On the other hand, when m is even, S is 

 small, and here also the condition of no reflexion can be 

 satisfied, at any rate at high angles of incidence. 



It should be remarked that high values of m, leading to 

 high values of k, correspond with overtones of the resonating 

 channels. 



A glance at fig. 1 shows that there is no limitation upon 

 the values of the positive quantities k l and k 2 . And since k k 

 is always greater than k 2 , k, as derived from k\ and k 2 , is 

 always real and positive. 



So far we have supposed that the values of /c b corresponding 

 with small values of k 2 , are finite, as when m = l, 2, 3, &c. 

 But the figure shows that solutions of (27) may exist when A:^ 

 as well as k 2 , is small. In this case we obtain from (31) 



ft^va+jv), (35) 



making W^kf-k^U^ (36) 



