336 MOTION OF THE GAS SPHERE 



B. The 7iiaximum radius. The maximum radius in the first expan- 

 sion is of course determined by the condition that da^ jdi^ = 0, which 

 gives, if the vertical velocity at this time can be neglected, 



(8.62) 1 - a.*3 - ^%^ = 



The internal energy for TNT has, in almost all mathematical calcula- 

 tions, been taken to correspond to the adiabatic relations Py^-^s = ]^^ 

 where the exponent 1.25 and value of k are chosen for best fit to Jones's 

 calculations (see section 8.5). This specialization is unnecessary, and 

 if a more general law PF^ = /c is used for one gram of explosive with P 

 and Y in cgs. units (8.56) gives the result that E{(i'')IY = k*a*-^^^-^\ 

 where 



^ k(rQ)-y(7.SQXlO-')y-\ , 

 (t - 1) 4.19 X 10^ 



The conversion factors appearing in this equation are such that Zo is in 

 feet, k and Q in cgs. units per gram of explosive. The largest root of 

 Eq. (8.62) for the more general adiabatic may be obtained numerically, 

 and is approximately given by 



a^* = 1 - — H — k""^ 



Values of the radius a^ in feet are then obtained by using the scale 

 factor L* of Eq. (8.54). 



It should be noted that the approximations by which a,„* is obtained 

 will often be valid only for the first maximum, as the bubble is in most 

 cases moving with appreciable vertical velocity at later maxima. A 

 correction could be made using measured velocities at the time of later 

 maxima, if desired, but this procedure has not been attempted in any 

 practical calculations. A similar difficulty applies in calculations of 

 bubl)le minima with the further complication that the bubble is not in 

 fact spherical and so such results would be of dubious value. 



C. Migration of the bubble under gravity. The nature of the vertical 

 motion of the bubble presents some difficulties in devising suitable ap- 

 proximation formulas, and attempts of this kind have been confined to 

 determining the migration over the first period, that is, the vertical 

 displacement when the bubble has contracted to its first minimum 

 radius. These approximations have been based on empirical fitting of 

 numerical calculations. One such formula has already been given in 

 section 8.6, in which the migration was found to be roughly propor- 



