214 WAVES IN AIR. [CHAP. vm. 



conduction and radiation. &quot; In order then that permanency of type 

 may be possible in a wave of longitudinal disturbance, there must 

 be both change of temperature and conduction of heat during the 

 disturbance.&quot; 



Kankine, in the paper referred to, considers it unlikely that 

 the conduction of heat ever takes place in such a way as accurately 

 to maintain the relation (43), except in the case of a wave of sud 

 den disturbance, where we have adjacent portions of the medium 

 at a finite interval of temperature. 



This latter kind of wave is of interest because, as we have 

 seen, any disturbance however started tends ultimately to become 

 discontinuous. The simplest case is that in which there is no 

 variation in the properties of the medium except at the plane of 

 discontinuity. If p, s, u denote the values of the pressure, the 

 bulkiness, and the particle- velocity behind, P, S, U those in front 

 of the discontinuity, the conditions to be satisfied are obtained by 

 making p lt s l} u t p, s, u, and p 2 , s z , M 2 = P, S, U, respectively, in 

 the above formulas. We find 



m = 



S-s 1 



and if further 17= so that the medium is at rest in front of the 

 wave, 





and u = m (S-s) = J (p - P) (S- s), 



the upper or lower sign being taken according as S is greater or 

 less than s, i. e. according as the wave is one of sudden compression 

 or of sudden rarefaction. 



The mathematical possibility of, and the conditions for, a wave 

 of discontinuity were first pointed out by Stokes. 



