200 



12 



the wave begins with a very steep front behind which the pressure falls off, 

 and that the bubbles ahead of the wave are in equilibrium under hydrostatic 

 pressure. The behavior of a bubble under such a wave will vary according to 

 circumstances. Special cases of this phenomenon will now be considered. 



SLOWLY VARYING PRESSURE 



Suppose the pressure falls off slowly as compared with the time of 

 contraction of the bubble under the maximum pressure in the wave. This 

 should be the case when the shock wave from a large charge enters water con- 

 taining small bubbles. In this case 

 the pressure on the bubble is prac- 

 tically steady during the process of 

 compression. 



Estimates of the rapidity 

 of heat exchange between the gas in 

 the bubble and the water Indicate 

 that the gas should follow the adia- 

 batic law, pV^= constant. Since 

 V « R', the new equilibrium radius R 

 under pressure p will be 



Figure 4 - Curve illustrating Behavior 



of a Bubble under a Slowly 



Varying Pressure Wave 



where R^ is the radius under hydrostatic pressure p^. 



[18] 

 For air, y = 1 .U and 



R = i^V'R. 



[I8a: 



Thus the equilibrium radius changes but slowly in comparison with the pres- 

 sure. If p/pq = 200, corresponding to a rise of pressure from atmospheric to 

 3000 pounds per square inch, R/Rq = 1/5-5; if p/po = ^00> corresponding to 

 6000 pounds per square inch, R/Rq= 1/4.2. 



The time required for a bubble of Initial radius Rg to contract 

 when the pressure is suddenly raised to a high value and then held steady may 

 be estimated as half of T as given by Equation [4a] or [Ub] with R^^^ re- 

 placed by Rf). For bubbles in water, 



1 ^_ 0.0023R, 



2 y^ 



:i9: 



where R„ is in Inches and p^ is the applied pressure in atmospheres. Thus, 

 even If ^o is as large as 0.1 inch and the pressure no greater than 150 

 pounds per square inch, so that p = 10, the bubble collapses in less than 



