PLANE WAVES OF SOUND 193 



1 dp L ,- ~ h 



,- ~ 

 or ^ = -- f 11 (!-{)-= } ............. (11) 



dt ' 



To this we must add the relations 



..................... (12) 



The elimination of p and s between these equations leads to 



............. < 14 > 



It is already assumed that the time enters through a factor 

 e int ; and the solution of (14) is therefore of the type 



u=Ce int+mx , ...(15) 



with ra 2 = 



or m 



approximately, on account of the assumed smallness of h/a. 

 For waves propagated in the direction of ^-positive we take 

 the lower sign, and write 



m = in/c' a, (18) 



where c' 



and a = nh/4s7rac (20) 



We have, then u=Ce-* .&<- xlc '\ (21) 



or, in real form, u Ce""* cos n It >j (22) 



The wave-velocity is therefore diminished in the ratio given 

 by (19). The exponential factor in (22) expresses the law of 

 decay of the waves as they advance. If I be defined as in 

 64 (23) it will be found that al is of the order \*/ah. The 

 rate of decay is therefore much greater under the present 

 conditions than in the case of sound waves in the open. 



A formula equivalent to (19) was published without demon- 

 stration by Helmholtz in 1863. The above proof is a variation 

 of that given by Lord Rayleigh in his Theory of Sound. 



L. 13 



