218 50 



\i?. 



whence 



1 /i?o\'(^-') iRx 



In which, according to Equation [30], R2 might be substituted for R^^; or. If 



y = V3, 



BUBBLE UNDER VARIABLE EXTERNAL PRESSURE 



Up to this point the hydrostatic pressure p^ has been assumed to be 

 constant. Let it now be assumed to vary with the time. Such variation may 

 be caused by the action of a piston upon the water; if compression of the wa- 

 ter is negligible, this pressure will be transmitted instantly to all parts 

 of the water. The theory developed for this case should also hold approxi- 

 mately for the action of a shock wave upon a bubble, provided the bubble Is 

 much smaller than the effective length of the shock wave in the water. 



Examination of the deduction of Equations [1] to [10] on pages '+5 

 and 46 of TMB Report 480 (4) shows that all of these equations remain valid 

 If pg varies with the time (. If In Equation [10] on page 46, R is written 

 for r., the radius of a spherical gas bubble, the equation becomes 



[3 idR\^ ^ ud^Rl _, 



Only the simple case of an Impulsive variation of p^ will be 

 treated here. Let pp take on large values during a very short time (j. In- 

 tegrating the last equation during this Interval, 



Now Pg is nearly constant during the short time t,, hence Jpgdt is very small 

 and may be dropped in comparison with Jp^dt. In the second Integral, R is 

 nearly constant, whereas dR/dt may undergo considerable change; hence, ap- 

 proximately. 



