Sound- Waves of Finite Amplitude. 325 



those given by Eiemann*, we shall find complete accordance 

 so far as § 6 is concerned, though with § 7 the case is dif- 

 ferent ; and this may be easily explained. We cannot in 

 general investigate the motion of a (frictionless) compres- 

 sible fluid by means of the equations of continuity and 

 momentum, without further making some supposition as to 

 the exchange or non-exchange of heat, and so we usually 

 assume either that the temperature remains constant, or that 

 there is no exchange of heat : in either case (provided the 

 motion is continuous), the pressure is a function of the 

 density only. At a surface of discontinuity there is not only 

 the ordinary heating effect due to compression, but also, as 

 we have seen, a dissipative generation of heat, and so, when 

 applying the equations of continuity and momentum at such 

 a surface, we must know what becomes of this additional 

 heat. Now in all cases Eiemann makes the assumption that 

 the pressure is a function of the density only, and this is 

 necessarily equivalent to an assumption concerning the trans- 

 ference of heat. Throughout most of his treatment of waves 

 of discontinuity Eiemann assumes that temperature is 

 constant and that Boyle's law holds good ; accordingly our 

 § 6 is entirely in harmony with his conclusions, in fact (4) 

 and (5) are only particular forms of equations given by 

 Eiemann. Of course the hypothesis that a portion of gas 

 can be instantaneously compressed to a finite extent without 

 any appreciable change of temperature, is not in accord- 

 ance with experience, hut provided we accept the assump- 

 tion that the temperature remains constant throughout, all that 

 Riemann says concerning the propagation of waves of discon- 

 tinuity under Boyle' 's law will hold good. 



The assumption made in § 7, that there is no appreciable 

 transference of heat, is probably much nearer the truth; but 

 this is not in accordance with any assumption made by 

 Eiemann. When pressure is assumed to be a function of 

 density only, and to vary with it according to the adiabatic 

 law, it is virtually assumed that at the discontinuity just so 

 much heat remains in the gas as would be due to sloiv adiabatic 

 compression, while the further amount of heat which is dissipa- 

 tively produced is completely and instantaneously removed by 

 conduction. But though Eiemann's results may thus be 

 justified by impossible assumptions concerning the diffusion 

 of heat, we may more reasonably, following Lord Bayleigh, 

 regard them as involving a destruction of energy. The real 

 source of error lies in Eiemann's fundamental hypothesis. 

 At the outset he supposes the expansion and contraction of 



* Loc, cit. 



