166 DYNAMICAL THEORY OF SOUND 



This is exactly the mean energy contained in a volume we of 

 the space occupied by the wave-train (11). The result may 

 perhaps at first sight appear to be a mere truism. It may be 

 argued that in each unit of time fresh waves are generated 

 which occupy a length c of the tube, and that the piston must 

 as a matter of course supply the corresponding amount of 

 energy. It must be remembered, however, that an infinitely 

 long train of waves of the type (11) would take an infinite time 

 to establish, and that in the case of a finite train the suggested 

 line of argument would require us to examine into what is 

 taking place at its front. In the present instance the result 

 would, it is true, be unaffected, but the case would be altered 

 if the wave-velocity were different for different wave-lengths, 

 as it is for example in dispersive media in optics, in deep-water 

 waves in hydrodynamics, and in the case of flexural waves on a 

 long straight bar ( 45). There is then a distinction between 

 the wave- velocity (for a particular wave-length) and the "group- 

 velocity" which determines the rate of propagation of energy. 



In the above problem force must be applied to the piston in 

 order to maintain the vibration (8) against the reaction of the 

 air. If the piston be free, the store of energy which it origin- 

 ally possessed will be gradually used up in the generation of 

 air-waves. Suppose, for example, that the piston is attached 

 to a spring, and that in the absence of the air the period of its 

 free vibrations would be ^TT/U. Under the actual conditions, 

 its equation of motion will be of the form 



M(Z + n^) = -(p- Po )co, (14) 



where the variable part of the pressure alone appears, since the 

 constant part merely affects the equilibrium position. From 

 the general theory of progressive waves we have 



p-p^KS^tcg/c, (15) 



and the equation (14) becomes 



5 + ?+^ = o (16) 



This is of the form discussed in 11, and the solution is 



f .-=(7<?-'/ T cos(rc'+ e), (17) 



