Van Wijngaarden 



G = 



-j: 



n(^) - r 



d^ 



Inversion then yields, using (4.4), 



<f2(p-^-4)d^ 



cosh 



y(x)'' 



'-][ 



n(<f) - r' 



d^ 



(4.7) 



(4.8) 



For the unknown function n(x) we can now derive a relation, since (4.2) has not 

 been used yet. Doing this now, thereby writing 



Ap + Po 



(4.9) 



we infer from (4.8), by differentiation and substitution, 



n(x)- X- 



p- X 



\ sinh 



cosh 



\(x) 



1/ 2 



x^'' ^ cosh^ 



k(x) 



(p-r')^"'d^ 



(4.10) 



While (4.10) describes how the pressure in the mixture changes at the wall 

 with the pressure in the bubbles, an equation for the rate of change of the bubble 

 radii can be obtained in the following way. Inserting p instead of p^, in (1.3), we 

 obtain 



Using the boundary condition at y = ^ we infer from (4.8) 



(4.11) 



r 



n(<f)-r' 



X 



j ^\p-r') d^ 



(4.12) 



d^ = 



cosh 



\(x)- 



Using this result, combination of (4.8) and (4.11) yields 



_p_ /BR 

 Po \Bt 



2 cosh 

 ^ cosh 



y(x) 



>V(X)' 



I ^\p-r*)d^ 



(4.13) 



The expressions (4.10) and (4.13) contain the information necessary for our 

 present purpose. First we note that for values of x near unity (4.10) reduces to 

 (3.7), if in the last expression we take y = 0. It can easily be shown that also for 

 X near unity (4.13) can be integrated to the relation (3.8). It is interesting to 

 note that while in the linear case cosh \ occurs in the expressions for the gas 

 pressure, in the nonlinear case k is multiplied with x^''^ which expresses the 



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