15 203 



may be about 0.013 pound-second, so that, even If R(, Is as large as 0.1 inch, 

 Vg according to Equation [20a] equals lUOO Inches per second. Moving at this 

 velocity, the bubble would shrink to nothing In l/l4 millisecond. Since the 

 duration of the shock wave scarcely exceeds l/50 millisecond, the value ob- 

 tained for V(, and for the time of collapse should be roughly correct. 



For heavier charges, however, this analysis ceases to apply. Thus 

 for the same bubble at a distance of l8 inches from 1 ounce of TNT or tetryl, 

 / = 0.2 pound-second and Vg = 21,000 inches per second; at this velocity the 

 bubble would collapse In 1/210 millisecond, whereas the shock wave lasts per- 

 haps l/lO millisecond. In such cases a better estimate of the time of col- 

 lapse Is obtained from Equation [19]- If in this equation Rg = 0.1 inch, 

 p^ = 4000/1 4. 7, representing a peak pressure of 4000 pounds per square inch, 

 the value T/2 = 1/72 millisecond is obtained for the time of collapse. Even 

 this latter value is probably considerably in error, but it serves to confirm 

 the conclusion that the bubble will collapse long before the shock wave has 

 disappeared. 



After collapsing, the bubble will re-expand. If the time of col- 

 lapse exceeds the duration of the shock wave, so that the expansion occurs 

 under the original low pressure, the bubble may overshoot its original size. 

 The time required to reach the original dimensions may be of the same order 

 as the time of collapse; for the shortening of the time that results from the 

 loss of energy by radiation will be offset somewhat by a lengthening due to 

 the fact that the expansion occurs against a lower pressure. 



THE GENERAL CASE 



Between the two simple cases of relatively steady pressure and of 

 Impulsive action there lies an intermediate range in which analytical treat- 

 ment Is laborious. Qualitatively, light can be thrown upon these situations 

 with the aid of estimates based on the formulas pertaining to the simple ex- 

 tremes, but quantitative results can be obtained only by numerical integra- 

 tion. 



The foregoing discussion of the effect of a pressure wave on a 

 single bubble may now be followed by consideration of cases involving more 

 than one bubble. 



A PRESSURE WAVE INCIDENT ON BUBBLY WATER 



A problem of great interest is that of a plane wave of pressure 

 entering at normal incidence a layer of water containing bubbles of air or 

 other gas. 



The principal qualitative features of the effect of the bubbles 

 upon the pressure wave are easily inferred. 



