MOTION OF THE GAS SPHERE 



'-^j ^./e/sj -^^^^" - 



z' = 2 \ r^\ -^^dt'' = ^t'2 



an expression valid only for small radius and translational velocity. 

 The values of z' and a' for zj = 2, t' = 2.0 are then used for step by 

 step numerical integration of the complete form of Eqs. (8.26). 



Taylor's results are plotted in Fig. 8.7 and show the characteristic 

 features of the vertical migration described in section 8.1: a slow rise 

 until the bubble passes its maximum expansion, followed by an increas- 

 ingly rapid upward movement as the minimum radius is approached. 

 The bubble has a non-zero minimum radius despite the neglect of in- 

 ternal gas pressure which is apparently required to stop the radial 

 contraction. This seeming paradox Taylor resolves by noting that, 

 as the bubble contracts, all the kinetic energy available is taken up by 

 the vertical motion before the radius becomes zero. The computed 

 minimum radius a' is given by a' = 0.21, and the time at which this 

 occurs is t' = 0.64. These figures are of course only approximate, par- 

 ticularly the value of a', as the effect of internal energy has been neg- 

 lected. 



The results plotted in Fig. 8.7 refer of course to the specific initial 

 condition that Zo' = 2.0 and are in reduced units. They therefore corre- 

 spond to a particular relation between charge weight and initial depth 

 do, and, from Eq. (8.27), this relation is 



do + 2.31Po = Zo = 2l--) 

 \Pog/ 



where Y is the energy available, and Pa is the pressure at the surface in 

 lb./in.2. The value of Y depends of course on the weight and kind of 

 explosive. For TNT, the total energy of explosion is approximately 

 1,060 cal./gm., of which approximately 40 per cent remains after emis- 

 sion of the shock wave. The energy Y for TNT becomes Y = 8.4 • lO^^TF, 

 if the charge weight W is expressed in pounds, and the depths corre- 

 sponding to given weights of explosive and reduced depth zj = 2.0 are 

 given by 



do + 2.31Po = 2()W'i' 

 Similarly, the time scale for a weight W and zJ = 2.0 is given by 



Ksec.) = 0.55Tf 1/8 



These relations can be used to find corresponding depths and charge 

 weights, and the auxiliary scales in Fig. 8.7 illustrate the relations ob- 



