1869.] Theory of Waves of Finite Longitudinal Disturbance. 81 



lative positions of a series of transverse planes that travel along with a wave 

 by means of the masses of matter contained between them, instead of by 

 their distances apart. 



Let such a transverse advancing plaue coincide with that part of a wave 

 of longitudinal disturbance at which the pressure P and bulkiness S * are 

 equal to those corresponding to the undisturbed condition ; 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 au 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 

 dx 



— 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 fol- 

 lowing value — 



• • • (b> 



in which p and s respectively are the pressure and bulkiness at the second 

 plane. Hence the condition of permanence of type is expressed symboli- 

 callv as follows : 



p-V dp rfP a/ f . 



~S^ = ~-£ =-d§=>»-ta constant). ...(C) 



This relation between pressure and bulkiness is not fulfilled by any 

 known substance, when either in an absolutely non-conducting 1 state 

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

 of uniform temperature. In order that it may be fulfilled, 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 ther- 

 modynamic function is 



o = Jchyplogr + x (r)+g; (D) 



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

 cific heat of the substance, r the absolute temperature, xO") a function of 

 the absolute temperature, which is =0 for all temperatures 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 sub- 

 stance are capable of doing at the constant temperature r. The thermo- 

 dynamic condition to be fulfilled bv a wave of permanent tvpe is expressed 

 by 



/r4=0 (E) 



* The word bulkiness is used to denote the reciprocal of the density. 

 VOL. XVIII. G 



