NSWC/WOL TR 76-155 



bubble flow - and the new amplitude is just the instantaneous 

 excursion of the bubble pressure from the changed outside pressure. 

 We made use of this property of the motion to calculate the motion as 

 the response to a series of pressure steps as shown in Figure 2.3.2. 

 (Appendix B describes the way we estimated successive half-periods 

 in order to construct the approximating step wave.) 



The transition from the positive to the negative pressure 

 phase, i.e., surface cut-off, must be treated as a special case, since 

 in general, this will not occur at an extremum of the motion. If 

 cut-off occurs at a compression, this compression is then the compres- 

 sion of the next half cycle also - and the subsequent expansion is 

 the greatest possible. If cut-off occurs at an expansion and the new 

 outside pressure is lower than that inside, this expansion becomes 

 the compression of the next half cycle, there occurs a 180 degree 

 phase shift, and the subsequent expansion is the smallest possible. 

 In general, cut-off occurs somewhere in between and there is an 

 intermediate phase shift and subsequent expansion or compression. 

 In such cases we calculated this next half cycle at lowered pressure 

 as starting at the time of the previous extremum but at a ficticious 

 amplitude interpolated as a function of the time at which cut-off 

 occurred. (The details of this procedure are described in Appendix B.) 



Finally, at the end of the negative phase, the return to 

 ambient pressure can cut short or even prevent the final expansion. 

 This was handled in an analogous manner to surface cut-off and is 

 also described in Appendix B. 



To approximate the dissipation of energy which in nature must 

 occur during the oscillation, we arbitrarily extracted a fraction — 

 30% — of the remaining oscillation energy at each half cycle of the 

 motion. 



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