MOTION OF THE GAS SPHERE 295 



tained for several charge weights. The dashed Hnes indicate the corre- 

 sponding positions of the surface, which for atmospheric pressure is 

 thirty-three feet below the origin of z. For the two largest charges, the 

 first cycle always lies well below the surface, but the bubble intersects 

 the surface before completion of one period for thirty and ten pound 

 charges. The theory of course becomes meaningless after this occurs, 

 as the bubble loses its identity by venting, and is unreliable in any case 

 when the bubble approaches the surface because effects of this prox- 

 imity have been neglected. 



Despite the weaknesses of the theory, the examples do illustrate the 

 necessity for reducing the pressure at the water surface if model experi- 

 ments are to simulate on a reduced scale even the general features of 

 the motion of the gas sphere from larger charges. For example, if a 

 10 pound charge were to reproduce the behavior for a 2,200 pound 

 charge, the hydrostatic pressure at the surface would have to be re- 

 duced to 26 per cent of atmospheric, and the distance do below the sur- 

 face decreased by the same factor. If the surface pressure is not scaled, 

 it is evident that the total hydrostatic pressure near the original depth 

 of explosion changes proportionately much less for small charges than 

 for large, owing to the relatively larger constant pressure above the 

 surface. Hence, the differences in pressure resulting from gravity have 

 a smaller effect and the migrations are much smaller for explosion prod- 

 ucts of small charges. In any given case, the importance of gravity 

 can be estimated by comparing the maximum bubble radius with the 

 equivalent hydrostatic head at the surface. For charges of 1 pound or 

 less of explosive with a maximum radius not exceeding 4 feet and at- 

 mospheric pressure of 33 feet of water at the surface, the variations of 

 pressure around the gas sphere are clearly not a large fraction of the 

 total and the migration under gravity is a small effect. For 300 pound 

 charges, on the other hand, the radius is of the order of 30 feet and the 

 effect of gravity is large. 



8.6. Calculations of Gravity Effects and Comparison with 

 Experiment 



The first calculations of bubble motion under the influence of grav- 

 ity, made by G. I. Taylor on the basis of his nondimensional formu- 

 lation of the equations of motion, have been extended by other workers 

 using more elaborate numerical methods. In addition, a number of 

 approximate formulas and solutions representing such results have been 

 developed, which permit calculation of the various parameters of inter- 

 est with reasonable accuracy and much more simplicity. 



A. Numerical calculations. Comrie and Hartley (24) have com- 

 puted the expansion and migration of the bubble for 4 different values 

 of initial hydrostatic pressure corresponding to values 1, 2, 3, and 4 for 



