Prof. Rankine on the Thermodynamic Theory of Waves. 307 

 tion ; it is shown that the value of the square of the mass-velocity is 



*-"*§ (A) 



The linear velocity of advance of the wave is obviously mS. 



Let a second transverse plane advance along with the wave in 

 such a manner that an invariable mass of matter is contained between 

 it and the first advancing plane. The condition of permanence of 

 type of disturbance is, that the distance between those planes shall 



be invariable. Let — be the rate at which that distance varies, being 



positive when the second plane gains on the first plane ; it is shown 

 that this quantity has the following value — 



i = p ~-^^, ( b) 



in which p and 5 respectively are the pressure and bulkiness at the 

 second plane. Hence the condition of permanence of type is ex- 

 pressed symbolically as follows : — 



P-V_ dp_ dV 2 , 

 -g-^--^--^=m (a constant). ...(C) 



This relation between pressure and bulkiness is not fulfilled by 

 any known substance when either in an absolutely non-conducting 

 state (called, in the language of thermodynamics, the adiabatic state) 

 or in a state of uniform temperature. In order that it may be ful- 

 filled, transfer of heat must go on between the particles affected by the 

 wave- motion, in a certain manner depending on the thermodynamic 

 function. The value of the thermodynamic function is 



0=Jchyplogr + x(O+^-» • ..... (D) 



in which J is the dynamical equivalent of a unit of heat, c the real 

 specific heat of the substance, r the absolute temperature, x( r ) a 

 function of the absolute temperature, which is = for all tempera- 

 tures at which the substance is capable of approximating indefinitely 

 to the perfectly gaseous state, and U the work which the elastic forces 

 in unity of mass of the substance are capable of doing at the constant 

 temperature r. The thermodynamic condition to be fulfilled by a 

 wave of permanent type is expressed by 



/«^=0 (E) 



In applying this equation to particular cases, and r are to be ex- 

 pressed in terms of p and s. 



It is shown to be probable that the only longitudinal disturbance 

 which can be propagated with absolute permanence of type is a sud- 

 den disturbance, and that the consequence of the non-fulfilment of 

 the condition of permanence of type is a tendency for every wave of 



X2 



