183.] EFFECT ON SOUND WAVES IN AIR. 225 



when x 0. Now (23) is satisfied by the sum of any number of 

 terms of the form 



=Ct* + Pr ........................... (25), 



provided 



a 2 = c 2 /3 2 +i//a/3 2 ..................... (26). 



To obtain a solution consistent with (24), we write a = Zirni, whence 



Here &amp;lt;r 2 = 1 + V ^ 



and 2e&quot;is the least positive solution of 



tan 2e = f 

 Hence 



cycr 

 Substituting, and putting (25) in a real form, we find 



(27), 



1 2-7T71 . 



where 7 = -r sine, 



L c\/cr 



and c = c*/cr sec e. 



In (27) we must take the upper or the lower sign according as 

 we consider the waves propagated in the direction of x negative, 

 or x positive. We see that the velocity of propagation is increased 

 by the friction, the increase being greater for higher notes than 

 for lower ones. The amplitude of the waves diminishes as they 

 proceed, the diminution being the more rapid the higher the note. 

 The change in the velocity of propagation depends ultimately on 

 the square of /A, that of the amplitude on the first power of //,, so 

 that the latter effect is much more important than the former. 

 Stokes* however has shewn that in all ordinary cases the diminu 

 tion of amplitude due to friction is insignificant compared with 

 that due to spherical divergence. 



The equation (27) does not constitute the complete solution of 

 (23) subject to the condition (24). In fact we may add any number 



* Camb. Phil Trans. Vol. ix. p. 94. 



15 



