MOTION OF THE GAS SPHERE 



297 



the reduced depth Zo', the internal energy of the gas being neglected. 

 Their results, expressed in Taylor's reduced units of length and time, 

 are plotted in Fig. 8.8, the solid curves representing the radius, and the 

 dashed curves the upward displacements of the center and top of the 

 bubble. These curves apply to any desired charge weight at four 

 depths determined by the charge weight and value of Zo'. The con- 

 versions for several weights of TNT, taken from a table given by 

 Kennard (55), are given in Table 8.3, where the distance in feet and 

 time in seconds corresponding to unit value of the reduced variables of 

 Fig. 8.8 are tabulated, together with the corresponding initial hydro- 



^ If the surface is at atmospheric pressure (33 feet of water) , the explosion is at a depth 33 feet less 

 than the equivalent pressure in feet of water. 



Table 8.3. Weights of explosive and depth-time scales corresponding to different 

 values of Taylor's reduced units. 



static pressure in feet of water. The conversions are based on the 

 formulas L = lOW'^' ft., t/t' = 0.5dW^ sec, obtained from Eq. (8.27) 

 assuming a total energy for TNT of 880 cal./gm., half of which is left 

 for the bubble motion (F = 440 cal./gm.). It is to be noted that none 

 of the curves can apply to the smallest weight (1/16 lb.), unless the 

 pressure at the surface is reduced considerably below atmospheric pres- 

 sure, as total pressures (surface plus water) are listed. 



The qualitative effects of depth on migration and bubble contraction 

 are clearly shown in Fig. 8.8. At the greatest depths (zo = 4), the 

 bubble contracts to a small fraction of its maximum radius and the up- 

 ward migration becomes appreciable only after the first minimum. For 

 smaller depths, the minimum becomes less pronounced and the migra- 

 tion increasingly well developed at the time of the minimum. These 

 features of the motion are readily understood if it is remembered that 

 downward momentum of the water is acquired while the bubble is 

 large, and changes less when the bubble radius and buoyancy are small, 

 but that the upward velocity of the bubble for a given momentum in- 

 creases as the bubble contracts. The larger vertical velocity of flow at 

 shallow depths absorbs more of the available energy, which becomes 

 equal to the total at a larger bubble radius, thus accounting for the very 



