and 



0: p 



Van Wijngaarden 

 - n • ^ 



an 



(2.9) 



y =0:-^=0; y' =h: p - po + ApH(t') 



(2.10) 



In (2.10) H(t ' ) is the unit step function. In case of small values of Ap/p^ the 

 linearized equations (2.5) and (2.8) may be used. The boundary at y' = can be 

 removed, provided we extend the mixture till y' = -h and apply at y' = -h the 

 same boundary conditions as hold for y'= h. If Wg -» oo^ we know from acoustic 

 theory that at y ' = the fluid is quiescent until t ' = h/c. At that moment the 

 pulses emitted from y' = h and y' = -h arrive, and the pressure remains 

 Po + 2Ap until at t ' = 3h/c a rarefaction wave of strength 2Ap arrives, etc. 

 Therefore at y' = and ^g -» oo 



Po + 2Ap [H( t ' - h/c ) - H( t ' - 3h/c ) + . . . ] 



(2.11) 



As long as t' < 3h/c the first two terms on the right side of (2.11) are nonzero, 

 which we can write as 



00 



4Ap r sin kh ( 1 - cos kct ' ) 



Pg - Po = ^r J k 



dk 



(2.12) 



or 



P„- Po = 2Ap < 1 



00 00 



1 f sin k (h + ct ' ) If sin 



k (h - ct ') 



dk 



(2.13) 



The expression (2.13) allows the interpretation for Pg- Pq as the integrated 

 result of a continuous progressing wave spectrum with wave number distribu- 

 tion 1/k, each wave traveling with the same velocity c . 



In dealing with finite values of co^ it is convenient to introduce dimension- 

 less variables by 



^ 



(2.14) 



and 



"bV 



(2.15) 



With help of these relations elimination of either p^ or p from (2.5) and (2.8) 

 yields 



il + ^ - J 



3y2 3t23y2 BtV iPgJ 



118 



. 



(2.16) 



