BUBBLE WALL VELOCITY FOR A COLLAPSING 



BUBBLE IN WATER, WITH AP = I ATM AND 



(P + 3000 ATM ) p 7 * CONSTANT 



MACH 



NUMBER 



|U|/C 



0.5 - 



0.2 - 



0.001 0.002 



0.005 0.01 0.02 



RADIUS RATIO, R/R 



taken into account (see my comments on M. S. Plesset's paper). For actual cavita- 

 tion bubbles, however, there must be a point in the collapse after which these simple 

 theories no longer apply, either because the bubble is no longer spherical, or because 

 the pressure of the vapor inside starts to increase rapidly as the collapse becomes too 

 rapid to permit vapor pressure equilibrium, or for any of a number of other possible 

 reasons. Such complicating factors, which are very difficult to treat theoretically, 

 provide an effective "cut-off" to the infinite pressure peak given by theory. However, 

 if one is interested in the pressure pulse propagated to some finite distance from the 

 bubble, there is another cut-off which may be more amenable to theoretical treatment. 

 This arises from the well-known tendency of finite compression waves to become 

 steeper as they propagate. In a compression wave having a sharp peak the peak will 

 move faster than the rest of the wave, until a vertical front (shock wave) is formed. 

 Thereafter, the peak will gradually advance into the shock wave and be effectively 

 "lost." The height of the pressure peak is thus significantly reduced as it propagates 

 (in addition to the geometric reduction in the spherical situation), and the pulse at 

 some distance from the bubble may be independent of the very last stages of the 

 bubble collapse. This possibility deserves careful theoretical investigation, using perhaps 

 the methods developed for underwater explosion shocks during World War II. Since 

 I am presently occupied with Air Force instead of Naval problems, I would particu- 

 larly like to encourage someone else with the appropriate theoretical background to 

 undertake such an analysis. 



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