PLANE WAVES OF SOUND 187 



limit to the frequency of vibrations which are capable of 

 propagation for more than a very moderate distance. 



The viscosity being small, the rate at which work is done 

 per unit area by the piston in maintaining the wave-system 

 (22) must have sensibly the value |p n 2 a 2 c found in 60. Since 

 the energy in the medium to the right is now finite and on the 

 average constant, this must be equal to the rate of dissipation 

 of energy by viscosity. The equality is easily verified. The 

 dissipation is, by (7), 



= J // ^rf V** cos 2 "(*-*) dx, . . .(25) 



approximately, if we keep only the most important term. 

 Writing 



and taking the mean value with respect to the time, we obtain 



by (23). 



65. Effect of Heat Conduction. 



A further cause of dissipation of energy is to be found in 

 the thermal processes consequent on the alternate expansions 

 and rarefactions of the air. If indeed these succeed each other 

 with sufficient rapidity, the variations are almost accurately 

 adiabatic, as explained in 59 ; but, as was first pointed out by 

 Kirchhoff (1868), the residual conduction of heat is in any case 

 of equal importance with viscosity. On the kinetic theory of 

 gases the coefficients of " thermometric " conductivity (v) and 

 of kinematic viscosity are in fact of the same order of magnitude ; 

 according to Maxwell the relation is v=-^v. For this reason 

 the preceding calculations of the effect of viscosity on air- waves 

 must not be looked upon as more than illustrative. A complete 

 investigation, in which both influences are taken into account, 

 shews that the effect is equivalent to an increase in the 

 kinematic viscosity, but the order of magnitude is unaffected. 



