194 DYNAMICAL THEOKY OF SOUND 



A more complete investigation was instituted by Kirchhoff 

 (1868) in which thermal processes are considered, as well as 

 viscosity. The effects are thereby increased, as already explained, 

 but remain of the same order of magnitude. 



As already stated, it is implied in the above calculation that 

 the diameter of the tube greatly exceeds the quantity h. When 

 on the other hand the diameter is comparable with, or less 

 than h, the walls have relatively a much greater hold on the 

 vibrating mass, and the character of the motion is entirely 

 altered by the friction. In particular, when h is large com- 

 pared with the width the mere inertia of the fluid ceases to 

 have any appreciable influence, the mean velocity over a 

 cross-section being determined by an approximately statical 

 equilibrium between the pressure-gradient (in the direction of 

 the length) and the friction of the walls. We have, then, 



(23) 



where R is a coefficient of resistance, depending on the nature 

 of the fluid, and on the shape and size of the cross-section. 

 Again, by Boyle's law, 



=.pb(l + 5), ..................... (24) 



the isothermal hypothesis being adopted as now the most 

 appropriate, since, owing to the assumed narrowness of the 

 tube, transfer of heat can take place freely. Eliminating p and 

 s between (13), (23), and (24), we find 



du_p d*u 



dt~ RW 



This has the same form as the equation of linear conduction of 

 heat. Assuming that 



u=Ce int+rn ^, .................. (26) 



we have m* = inR/p , and therefore 



m=(l+*). .................. (27) 



if ^ = ^nR/ Po ................... (28) 



Taking the lower sign we obtain 



M-Ck-ws+^ne-wa^ ............... (29) 



or, in real form, u = Ce~ wx cos (nt ^x) ............. (30) 



