NSWC/WOL TR 76-155 



6. In what follows, we will use both Equations Al and A3. 



Transitions from one outside pressure level, p , to another, p ,., 



^n ^n+1 



are simply described using Equation Al. And, Snay and Christian's 

 results, e.g., those listed in Table I, which were computed from 

 Equation A3, are convenient for computing the bubble behavior within 

 a given state. 



7. Boundary Conditions at Pressure Jumps . We now consider 



a step change in outside pressure from p to p , . . Let V be the 

 c 3 c ^n ^n+1 c 



n 



bubble volume at the instant of change, and y and y , . be the old 



., n J n+1 



and new total energies. For the n state we rewrite Equation Al as 



/4tt p A 3 \a 2 + 4iTp A 3 + 



V s " ) ^ n 



3 / 4tt p A \a + 4iTp A + E(A) =Y (A7) 

 2 I ^J - — sr n n 



Now, since the oscillating system described by (A7) has finite mass, 

 neither A nor A can change impulsively, i.e., 



AA = (A8) 



AA = (A9) 



at each pressure jump. Thus, the first and third terms of (A7) do 

 not change at the jump, and the change in the total enerqy is given 

 by 



Y n + 1 = Y n + ( Pn + l " P n ) V c (A10) 



n 



where V is the bubble volume at the time of the jump, 

 n 



8. Equations (A7) , (A8), (A10) completely specify the 



motion in the n state given suitable initial conditions. In the 



present case the initial or zeroth state is specified by 



A = constant = A. (All) 



P(A.) = constant = p. (A12) 



l *i 



A- 6 



