13 201 



1/12 millisecond, which is a short time relative to the duration of the pres- 

 sure wave from a large charge. IT Rq = O'l inch and p = 1000 pounds or p^ = 

 67, T/2 is less than l/30 millisecond, which is a short time even for the 

 wave from a pound of explosive. Smaller bubbles will collapse more quickly 

 in proportion to their smaller radius. 



In collapsing, the bubble will overshoot its new position of equi- 

 librium under the increased pressure, and will then re-expand. If the bubble 

 lost no energy, and if the pressure remained constant, then the bubble would 

 actually expand to its initial size, after which it would collapse again; it 

 would, in fact, execute undamped oscillations about its equilibrium radius 

 under the increased pressure. The bubble would be analogous to a mass on the 

 end of a spring; if, when the mass is at rest, a constant force suddenly be- 

 gins to act on it, the mass oscillates about a new position of equilibrium 

 and, in doing so, returns periodically to its initial position. 



The period of the oscillations will be much shorter, however, than 

 those which the same bubble would execute imder normal pressure. Under a 

 pressure of 1000 pounds per square inch, for example, the equilibrium radius 



/I t| 7\ -jV 1 

 is reduced from its value under one atmosphere in the ratio ( -, qAA ) ' = 2~T' 



The period of oscillation, which is proportional by Equation [1] both to p^" 2 



and to the equilibrium radius, would then be 2.7 (Tij~y) = 22 times less 

 than under one atmosphere. Under 3000 pounds per square inch, the period 

 would be 50 times less than under one atmosphere, for the same relative am- 

 plitude of oscillation. 



Compression of the water cannot be Ignored in these cases. Calcu- 

 lation of the pressure in the bubble when at its minimum radius gives fan- 

 tastically high values. This means that, because of compression of the water, 

 the minimum radius will actually be several times larger, and the maximum 

 pressure many times smaller, than the values derived from non-compressive 

 theory. Furthermore, it is certain that much of the kinetic energy acquired 

 by the bubble as it contracts will be radiated away during the phase of ex- 

 treme compression. The oscillations of the bubble about its new equilibrium 

 radius will thus be highly damped. 



An upper limit can easily be set to the amount of energy that can 

 be radiated away by such a bubble in collapsing. The total work done by the 

 applied pressure as the bubble collapses is equal to the product of the pres- 

 sure into the change of volume of the bubble. With any explosive wave of 

 consequence, however, the final volume is relatively small. For example, 

 even if p is only 300 pounds per square inch as against an initial pressure 

 Po ~ ^5. from the relation pV = Pf.Vg'^ the ratio of the corresponding volumes 



