Pressure Pulses in Liquid-Bubble Mixtures 



4. NONLINEAR THEORY 



Starting with the Eqs. (2.2), (2.3), (2.6), and (2.7) for p, p^, R, and v and 

 rendering p dimensionless with p , r dimensionless with Rq (x = R Rq), and y' 

 dimensionless with o^/c (cf. (2.15;), it may be shown (Van Wijngaarden (2)) that 

 with the approximation used in section 3 and consisting in reducing (^VBy^)^^^,^^^^^ 

 to (^^/^y^)x=const *-^^ following equation holds for the difference between aver- 

 age pressure p in the fluid and pressure Pg Ro''/R^ or x" "* in dimensionless vari- 

 ables, in the gas bubbles: 



^ (p-x-^) -x(p-x-^) = -i- (■ ^\p-r')d^. (4.1) 



^y ^ -'i 



The boundary conditions for (4.1) are (cf. (2.10)) 



,-_,: p.£L±^ (4.2) 



and 



y--^-- 1=0- (4.3) 



In Van Wijngaarden (2) an approximate solution of (4.1) subjected to (4.2) and 

 (4.3) was given. An exact solution appears, however, to be possible. We define 



G(x, y) = r ^'2(p-r') d^. (4.4) 



Bearing in mind that x and y are independent variables, we can combine (4.1) 

 and (4.4) into 



^!^-x^.O. (4.5) 



3y^3x Bx 



Introducing the Laplace transform of G with 



G 



j e-Sy G(x, y) dy (4.6) 



and denoting the pressure at y = in dimensionless form by n(x), transforma- 

 tion of (4.5) yields 



(S^-x) ^- G = Sx2 

 dx 



Integration with respect to x results in 



n(x) 



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