26 HYDRODYNAMICAL RELATIONS 



d(T , da , du „ 



[- u h c — = 



dt dx dx 



du , du , da 



\- U he — = 



dt dx dx 



A more symmetrical and useful pair of relations is obtained by addition 

 and subtraction, which gives 



(2.16) ^(^j^u) + (c + u)^{a-{-u) =0 



dt dx 



-{a - u) - {c - u)—-{(r - u) = 

 dt dx 



These equations mean that the quantity (o- + u) will be unchanged 

 in time at a point moving with velocity {c -]- u), and the quantity 

 (o- — u) remains unchanged for a point of velocity — (c — u) . If a 

 pressure wave is considered in which the direction of propagation and 

 particle velocity are in the direction of increasing x, i.e., a wave of com- 

 pression, a change in type of this wave will develop. For, as time in- 

 creases, the propagation of (o- + u) with positive velocity (c -f u) and 

 of (o- — u) with negative velocity —{c — u) will lead to the develop- 

 ment of an increasingly distinct region of increasing x, for which {a — u) 

 approaches zero in virtue of the propagation of this quantity backward 

 in space. The value oi a -\- u will thus approach a value 2u in the ad- 

 vancing front of the disturbance. In this limit both a and u will be 

 propagated with a velocity c -\- u which is greater than the local sound 

 velocity c, as a is, from Eq. (2.15), positive for a final densit}^ p greater 

 than po. The quantity o- is a function of density alone in the absence of 

 dissipative processes. The Riemann form of the dynamical equations 

 thus leads to the conclusion that particle velocity and density (hence 

 pressure also) are ultimately propagated with the velocity c -\- u. This 

 conclusion was tacitly assumed in the preliminary discussion, and the 

 further conclusion that plane waves of compression develop increasingly 

 steep fronts must also follow. 



It is possible to apply the Riemann approach outlined to various 

 special forms of disturbance, as has been done by a number of authors.^ 

 Because the plane wave case is of limited utility in actual problems of 

 interest, these applications will not be considered further here. One 

 simple conclusion is of interest, however, namely that even in the plane 

 wave case we cannot expect a shock wave to advance indefinitely un- 

 changed in intensity if the source of the disturbance does only a finite 



3 See, for example, the report by Kistiakowsky and Wilson (G4) . 



