NSWC/WOL TR 76-155 



17. Simplifying Assumption . To proceed further with the 

 general case we would now need the bubble radius as a function of 

 time in order to apply the jump equations (A8) , (A9) and (A10) at 

 the instant the outside pressure changes to its new value, p- • 

 Although not particularly difficult, such computations are beyond 

 the limited scope of this study. Consequently, at this point we 

 introduce the simplifying assumption that all pressure changes occur 

 at an instant when A = 0, i.e., either at an extremum or at 

 equilibrium bubble radius after the oscillation has completely 

 damped out. Pressure jumps, p to p , , which happen to occur at 

 bubble extrema give the greatest and also the least amplitudes of 



■f- V» 



oscillation in the n+1 state (V = V.. in Equation A10) . Thus, 



c Mm 



even when the pressure jumps occur at other times than the extrema, 

 we can place exact upper and lower limits on the amplitude of 

 oscillation in each state. 



18. Bubble Motion in State 2 . In accordance with our 



simplifying assumption we will assume the pressure jump, p, to p_ , 



occurs at a time t such that 



c l 



t = N X ^ T. (A34) 



C, 2 1 



where N is any positive integer. Alternatively, t may be taken 



c l 



large enough that the bubble oscillation has damped out, i.e., 



— 1/3y 



A = A, = A. (p,/p.) ' ', the equilibrium radius. 



19. Let V , A , P be the volume, radius and air 

 C l C l °1 

 pressure, respectively, of the bubble at time, t . If the new out- 



C l 

 side pressure p„ is higher than P , the air pressure inside the 

 l c l 



bubble, the new oscillation begins at maximum volume; and if p„ is 



lower than P , the oscillation begins at minimum volume. Or, 

 c 

 1 

 restated we have 



A-ll 



