MOTION OF THE GAS SPHERE 333 



da* Vl - a*3 - fc*a*-3/4 



dt* a*3/2\/l+/. 



This can be integrated to give the time tm* of the first maximum as 



(8.58) /„.* = I , ^^ c/a* 





Vl - a*3 - ^^*a*-3/4 



The hmits of integration are the maximum and minimum radii 

 a^* and a*, and are, in this approximation, the largest and smallest 

 roots of 



(8.59) 1 - a*3 - A;*a*-3/4 = q 



The parameter k*, which represents the effect of internal energy of the 

 gas, depends on the explosive and depth. Using Taylor's assumptions 

 for TNT as expressed by Eq. (8.57) : k* = 0.0607 Zo'/\ k* is found to 

 vary from 0.146 to 0.223 as the depth below the surface changes from 

 to 150 feet. The roots of Eq. (8.59) depend on k* and hence vary 

 slightly with the depth. The smallest root a* is very nearly equal to 

 /c*^/^, and the largest root a^* is somewhat less than the value unity ob- 

 tained for k* = 0. For k* = 0.20, numerical solutions of Eq. (8.59) 

 give a* = 0.118, a^* = 0.924. The value am* = 0.92 shows that the 

 assumed internal energy gives a maximum radius 8 per cent smaller 

 than the value 1.0 neglecting the internal energy. 



The evaluation of Eq. (8.58) requires insertion of the appropriate 

 form of the function fo, which represents the effect of surfaces. This 

 function, obtained by the method of images in section 8.8, is an infinite 

 series in powers of a*/2ho*. An examination of this series shows, how- 

 ever, that the radical V 1 + /„ can be approximated closely by the lead- 

 ing terms in its expansion which gives the relation 



^'+^'-'+£?'^^^ 



where F{x) = 1 for the rigid bottom only, F{x) = — 1 for free surface 

 only. 



For both rigid bottom and free surface, the first order image theory 

 gives 



(8.60) F{x) 



^''-''hi i2n + \y-.^ -'^^''] 



