164 DYNAMICAL THEORY OF SOUND 



second term we may put, with sufficient accuracy, pp = KS, 

 V Q V = V O S, and obtain ^KS Z .V O . The expression %/cs* is there- 

 fore to be integrated over the volume occupied in the undis- 

 turbed state. So far nothing is stipulated as to the hypothesis 

 to which K relates; but it is only in the case of adiabatic 

 expansion that the result can be identified with the potential 

 energy in the strict sense of this term. We then have 



V = ^ K fs 2 dx } ..................... (3) 



where K = <yp 0) per unit area of wave-front. If K refer to the 

 isothermal condition, the expression on the right hand is what 

 is known in thermodynamics as the " free energy." 



It is unnecessary to repeat what has been said in 23 as to 

 the resolution of an arbitrary initial disturbance into two 

 wave-systems travelling in opposite directions. In a single 

 progressive wave-system, say 



?=/((*-), ..................... (4) 



we have by 59 (5) = cs, ........................... (5) 



where denotes the particle- velocity in the direction of propa- 

 gation. Since f has the same sign as s, an air-particle moves 

 forwards (i.e. with the waves) as a phase of condensation passes 

 it, and backwards during a rarefaction. It appears moreover, 

 from (1), (3), and (5), that the total energy is half kinetic 

 and half potential. This also follows independently from the 

 general argument given in 23. 



The case of a simple-harmonic train of progressive waves is 

 specially important. The formula 



.................. (6) 



represents a train of amplitude a, frequency n/2?r, and wave- 

 length X = 27rc/ft. We find 



n (t - -] 



sin 2 n t - - dx 



(7) 



The mean value of the second term under the integral sign is 

 zero, and the average kinetic energy per unit volume is therefore 

 and the average value of the total energy 



