NSWC/WOL TR 76-155 



primarily on the kinetic energy at cut-off and not on the outside 

 pressure, this t -value calculated using the higher pressure should 

 still be a good approximation to the time of occurrence of this 

 minimum. ) 



Final Oscillation at Negative Pressure . In calculating 

 the final bubble expansion achieved during the negative pressure 

 phase we only consider those cases in which the final oscillation 

 starts with a minimum radius--only they are of interest. In order 

 to estimate this final expansion we calculate the parameter 



(At/T) = (DTNEG + Phase Corr) / T + 1/2 (B14) 



where DTNEG is the duration of the negative phase — and again refer 

 to Table B-l. 



The parameter (At/T) locates the end of the negative 



phase in terms of bubble oscillation cycles. "Zero" is taken as 



the last expansion occurring during the positive phase. 



If (At/T) Np _< 0.58, we conclude the negative phase is too short for 



any significant expansion to take place. Otherwise, we calculate 



the maximum size as indicated in the table. And, if (At/T) > 0.90, 



the maximum bubble size is taken equal to AMAX. In any event, if 

 (At/T) < 0.90, we also calculate the final oscillation that would 

 occur if surface cut-off had returned the outside pressure directly 

 to ambient. If this results in a greater maximum expansion, we 

 take this value as the maximum size achieved during final expansion. 



Damping . Equation Al describes undamped radial pulsation 

 of an ideal frictionless fluid. While it seems unnecessary to make 

 a detailed study of energy dissipation in the fishes' oscillating 

 swim bladder, it does seem prudent to at least approximately account 



B-8 



