262-263] CONDITION FOR PERMANENCY OF TYPE. 477 



space AB at B over that done by the fluid leaving at A, we find 



p 2 (c - Us) -pi(c- MJ) = Jm {(c - ?O 2 - (c - Ma) 2 } 



+ m(E l -E,} ...... (10), 



where the first term on the right-hand represents the gain of 

 kinetic, and the second that of intrinsic energy ; cf. Art. 23. As 

 in Art. 11 (7), we have 



It is easily shewn that (10) is inconsistent with (2) unless 



^-E^^-pf) .................. (12), 



which is only satisfied provided the relation between p and 

 p be that given by (4). In words, the conditions for a wave 

 of discontinuity can only be satisfied in the case of a medium 

 whose intrinsic energy varies as the square of the pressure. 



In the above investigation no account has been taken of 

 dissipative forces, such as viscosity and thermal conduction and 

 radiation. Practically, a wave such as we have been considering 

 would imply a finite difference of temperature between the 

 portions of the fluid on the two sides of the plane of discontinuity, 

 so that, to say nothing of viscosity, there would necessarily be a 

 dissipation of energy due to thermal action at the junction. 

 Whether this dissipation would be of such an amount as to be 

 consistent, approximately, with the relation (12) is a physical 

 question, involving considerations which lie outside the province 

 of theoretical Hydrodynamics. 



Spherical Waves. 



263. Let us next suppose that the disturbance is symmetrical 

 with respect to a fixed point, which we take as origin. The 

 motion is necessarily irrotational, so that a velocity-potential 

 (p exists, which is here a function of r, the distance from the 

 origin, and t, only. If as before we neglect the squares of 

 small quantities, we have by Art. 21 (3) 



J p dt 



