300 MOTION OF THE GAS SPHERE 



the term {4:Tr/3)d'h' representing hydrostatic buoyancy being negli- 

 gible. This equation can be solved graphically for a', and the corre- 

 sponding vertical velocity U' = —dz'/dt' obtained from Eq. (8.29). 

 Results obtained in this way are in only moderately good agreement 

 with experiment. 



The displacement of the bubble at the time of the first minimum is a 

 quantity of some importance, as it determines the position of the source 

 of the bubble pulse. An empirical relation obtained from the numeri- 

 cal integration is that, to within about ten per cent, the rise Ah' is pro- 

 portional to the period of oscillation if these quantities are both ex- 

 pressed in Taylor's nondimensional units, the relation being 



Ah' = 1.05 r = 1.19 Zo'-'i' 

 If the rise Ah is expressed in feet, the formula becomes 



(8.32) Ah = 81 ^^ 



^ 5/b 



where the charge weight W is in pounds, the hydrostatic depth Zo in 

 feet, and the numerical factor is obtained from Taylor's estimate of 

 available energy for TNT. 



It is to be noted that all the formulas of this section which involve 

 numerical factors are restricted to the case of the energy for TNT. 

 Approximate corrections for other explosives can be made either on the 

 basis of calculated detonation energies, assuming the same fraction re- 

 mains for the bubble motion, or by the observed ratio of bubble periods. 

 The formulas and calculations are both somewhat special in that they 

 assume a specific adiabatic law for the expansion products based on 

 Jones' calculations for TNT. The constant c and exponent (— ^) ap- 

 pearing in the formulas are the result of these calculations and are not 

 necessarily exactly correct for TNT or appropriate for other explosives. 

 The differences resulting from better calculations would, however, 

 probably not be large compared with other errors inherent in the theory 

 and experimental uncertainties. 



C. Co7nparison with experiment. A comparison of theoretical cal- 

 culations for the effects of gravity alone on bubble motion with experi- 

 ment is made difficult by the fact that these effects are appreciable only 

 for relatively shallow explosions in which the migrations are large. A 

 shallow explosion, however, implies the presence of a free surface of the 

 fluid above and near the bubble, which introduces further changes in 

 the motion. As a result, clear-cut examples of purely gravitational 

 motion are not readily achieved. The importance of gravity compared 

 with surfaces increases for a larger scale of bubble dimensions and sur- 

 face distance, because variations in hydrostatic pressure over its sur- 



