Pressure of Gaseous Vibrations. 367 



o£ the law of pressure 



/^Const.-a 2 ^ 2 (15). 



P 

 According to this 



/'M=« 2 , /"(*>)= -2a 2 //°o • • • (16), 



and (10) gives 



S( Pl -p o )dt=0 (17). 



The law of pressure (15) is that under which waves of finite 

 condensation can be propagated without change of type t 



In (17) the mean additional pressure vanishes, and the 

 question arises whether it can be negative t It would appear 

 so. If, for example, 



i> = Const.-^5? (is), 



f\p,) = a\ f"(p ) = -Wj P<t , 



aild fGn-^^-iPoJJ 1 ^ ■ • • (19). 



I now pass on to the question of the momentum of a pro- 

 gressive train of waves. This question is connected with 

 that already considered ; for, as Prof. Poynting explains, if 

 the reflexion of a train of waves exercises a pressure upon 

 the reflector, it can only be because the train of waves itself 

 involves momentum. From this argument we may infer 

 already that momentum is not a necessary accompaniment of 

 a train of waves. If the law were that of (15), no pressure 

 would be exercised in reflexion. But it may be convenient 

 to give a direct calculation of the momentum. 



For this purpose we must know the relation which obtains 

 in a progressive wave between the forward particle velocity u 

 (not distinguished in one-dimensional motion from U) and 

 the condensation (p—p )lp Q , usually denoted by s. When 

 the disturbance is infinitely small, this relation is well known 

 to be u = as, in the case of a positive wave. Thus 



u: s=V {dp/dp) (20). 



The following is the method adopted in ' Theory of Sound,' 

 § 351 : — " If the above solution be violated at any point a 

 wave will emerge, travelling in the negative direction. Let 

 us now picture to ourselves the case of a positive progressive 

 wave in which the changes of velocity and density are very 

 oradual but become important by accumulation, and let us 



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