Pressure Pulses in Liquid -Bubble Mixtures 



Because the ambient pressure is prescribed at y' ^ h the bubbles oscillate as If 

 in an infinite fluid with pressure Pq + Ap . The frequency of the oscillations is 

 ojg. The relevant part of the phase variation in (2.34) is given by 



(2.36) 



Comparison of the arguments of the cosines in (2.35) and (2.36) shows that in the 

 time interval between he and 3h/c the pressure pg in the gas bubbles varies 

 much slower at y' = than at y' = h. Then it follows from consideration of (2.5) 

 that, while at y' = h the difference Pg- p varies rapidly, this difference is small 

 at y' = 0, because the term (l/^g^) (32p^/3t - 2>) jj^ (2. 5) is small at y' = 0. 



The average pressure p in the fluid at the plate therefore rises at a slow 

 pace compared with wg^ to the value pg + 2Ap. In the nonlinear case the same 

 occurs qualitatively, but since the gas pressure reaches much higher values 

 (than the value p^ + 2Ap in the linear case), also the average pressure in the 

 fluid can assume high values, as will be made plausible in section 4. 



LINEAR THEORY H 



With a view to the nonlinear case, with which we occupy ourselves in the 

 next section, we deal with the linear case in another way. 



The nondimensional form of (2.4) is (cf . (2.14)) 



Bt' 



+ P„ = P 



(3.1) 



Introducing 



Bp 



3t 



(3.2) 



we integrate (3.1) to obtain 



2 J (p-^) d^, 

 Po 



(3.3) 



while combination of (3.1) and the dimensionless form of (2.8) yields 



p 



T = P-P, 



(3.4) 



We now consider u and p as dependent variables and Pg and y as independent 

 variables. In this case we should write (3.4) as 



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