n 



199 



or, if the relatively small term R^^/R^ is dropped, 



4£=/l(|7rp„V) [17a] 



This last formula represents the change in the energy as approximately equal 

 to the change in the work required to produce the cavity of maximum size, 

 which can be calculated without making any assumption concerning the equilib- 

 rium size of the gas globe. 



For the gas globe formed by Number 8 detonators exploded Just far- 

 enough under the surface of the water to avoid blowing through, an average 

 value for the first expansion, as inferred from the periods of oscillation, 

 seems to be about R^^^/Rq = 2.6. This corresponds, by Equation [^a], to 

 ^o/^min = 6.2. For this case, by Equation [l6a], a/E= ^^ ^ ZE " O-l^. 

 The observed decrease in energy during the first contraction, calculated in 

 the manner Just described, is about 40 per cent. 



The discrepancy between 0.7*+ and O.UO is in the right direction and 

 may well be due to compression of the water. At minimum radius. Equation [6] 

 makes p, equal to 6.2* = 1500 atmospheres, which would compress the water by 

 about 7 per cent. An attempt to estimate the amount of compressional energy 

 that would exist in the water leads, however, to a divergent integral, which 

 merely indicates that the non-compresslve approximation to p is inadequate 

 for the purpose. It is clear, however, that, if the gas at minimum radius 

 absorbs only part of the energy of motion, its minimum radius will be greater 

 than it has been calculated to be on the assumption that the gas takes up the 

 whole energy, and the pressure peak will accordingly be lower and will result 

 in a considerably smaller radiant emission of energy. Thus the true value of 

 Q/E may easily be 0.40 instead of O.7U, 



The estimate of the radiated energy has been based, as have all of 

 the preceding formulas, upon the assumption of perfectly symmetrical radial 

 motion. Further losses of the energy of the radial motion may result in ac- 

 tual cases from turbulence caused by departures from radial symmetry, or from 

 conversion of the oscillatory energy into energy of translation due to grav- 

 ity or to the proximity of obstacles. 



A PRESSURE WAVE AND A GAS BUBBLE 



It is of special interest to Investigate the propagation of explo- 

 sive pressure waves through water containing bubbles of air, since a screen 

 of bubbles has been proposed as a protective device. Let it be assumed that 



