Pressure Pulses in Liquid- Bubble Mixtures 



The essential point in the present theory (see also Van Wijngaarden (2)) is 

 that in (2.2) we insert for p the average pressure in the mixture at the location 

 of the bubble, so that for each bubble the "pressure at infinity" is the average 

 pressure in the volume element in which the bubble is one of the large number 

 of constituents. 



Insertion of (2.3) in (2.2) and linearization about the equilibrium radius R^ 

 results in 



^^.P=P. (2.4) 



Recognizing from (1.1) that the factor in the first term of this equation is l/<^ 

 we may also write 



1 ^'Pg 

 :ji^^PE=P- (2.5) 



For the average quantities p and v we now write down the continuity and 

 momentum equations: 



^ = 4TrnR2 H (2.6) 



By' 3t 



and 



P By' 3t' ^^•'' 



In formulating the momentum equation, (2.7), convective acceleration terms are 

 neglected because for a small amount of gas (as is assumed to be the case) the 

 average velocity v will be small. 



When small changes in the bubble radii are involved (resulting from a small 

 value of Ap) we may linearize (2.6). Then upon crossdifferentiation of (2.6) and 

 (2.7) we obtain 



-^=AS|. (2.8) 



By'^ c^ Bt' 



where c^ is given by (1.2). 



For Rq -♦ the bubble frequency ^g tends to infinity. In that case (2.5) re- 

 duces to p = Pg and (2.8) to the well-known acoustical wave equation. Hence the 

 mixture, as described in the present model, behaves in the limit of infinite 

 bubble frequency as a homogeneous fluid with sound velocity c 



The boundary and initial conditions of the problem posed at the beginning of 

 this section are 



117 



