304 Lord Ray lei gh on the Propagation of 



where 



»'=£{>'^" + v(l-5)}. . . .(15) 



In (14) an arbitrary multiplier and an arbitrary addition to 

 t may, as usual, be introduced ; and, if desired, the solution 

 may be realized by omitting the imaginary part. 



In passing on to consider the influence of walls by which 

 gas is confined upon the propagation of sound, it is here pro- 

 posed to take the case of two dimensions, rather than the tube 

 of circular section treated by Kirchhoff. The analysis, how- 

 ever, is nearly the same. We suppose that sound is propagated 

 in the layer of gas bounded by fixed walls at y = ±y^ so that 

 w = 0, while u, v, 6 r are functions of x and y only. The like 

 may be assumed respecting u', v f , Q x , Q 2 . We suppose further 

 that as functions of x these quantities are proportional to 

 e mx , where m is a c@mplex constant to be determined. The 

 equations (3) for Q 1; Q 2 become 



rf 8 Q I /rfy j = (X 1 -m 2 )Q I , (16) 



d*Q 2 /dy 2 =(\ 2 --m*)Q 2 . (17) 



For u', v equations (6), (8) give 



f =(?-■)•' <»> 



?-(?-■)•'. c 9 » 



mu' + dv'/dy^Q (20) 



These three equations are satisfied if u' be determined by 

 means of the first, and v' is chosen so that 



/_ 



du 



h/fA — m 2 dy i ^ 



a relation obtained by subtracting from (19) the result of 

 differentiating (20) with respect to y. The solution of (18) 

 may be written w' = AQ, in which A is a constant, and Q a 

 function of y satisfying 



f -(!-»')« « 



Thus, by (5), (7), 

 u=AQ—A 1 m(- i<)Qi-A 2 m(^— v)Q 2 , . . . (23) 



/ = A 1 Q l + A 2 Q 2 (25) 



