PLANE WAVES OF SOUND 165 



Since iia is the maximum particle-velocity, we see that the 

 energy in any region including an exact number of wave-lengths 

 is the same as the kinetic energy of the whole mass when 

 animated with the maximum velocity of the air-particles. If s l 

 be used to denote the maximum condensation, we have 81 = na/c, 

 and the average energy per unit volume may therefore also be 

 expressed by ^potfsf. 



We can also estimate, incidentally, the nature of the approxi- 

 mation involved in the derivation of the equation of motion 59 

 (7). The approximation consisted in neglecting the square of s, 

 or 9f/9a?. Since 81 = 27ra/\, this means that the amplitude a is 

 assumed to be small compared with X/27T, a condition which is 

 abundantly fulfilled in all ordinary sound-waves. 



So far we have traced the course of waves regarded as 

 already existent, without any reference to their origin. As an 

 example, though a somewhat artificial one, of the manner in 

 which waves may be supposed to be generated, imagine a long 

 straight tube, of sectional area , in which a piston is made 

 to move to and fro through a small range, in any arbitrary 

 manner. The origin of x being taken at the mean position 

 of the piston, the forced waves in the tube, to the right, 

 due to a prescribed motion 



?=/(*) (8) 



of the piston, will evidently be given by 



In particular, if % = acosnt, (10) 



we have = acosnU ) (11) 



The rate at which work is being done by the piston on the au- 

 to the right is 



4f/i 2/Y 2 



= p 9 <ona sm nt -\ -- o>sm*nt. ......(12) 



c 



The mean value of the first term is zero, whilst that of the 



second is 



(13) 



