834 MOTION OF THE GAS SPHERE 



where x = (d — ho) /id + ho), and d is the distance below the free surface. 

 Using the expression for v 1 + Jo, Shiffman and Friedman have de- 

 veloped a method of evaluating the integral for tm^, based on approxi- 

 mating the integrand by suitable orthogonal polynomials. This pro- 

 cedure, for details of which the original paper should be consulted, gives 

 for the period T* the value 



r* = 2e = i.47(i + "-y^^ ) 



Converting this to dimensional units gives the final result 



Wi/3 r 3 42TF1 

 T = 4.19^ 1 + ^-^^^'^ 



Zo'i' boZc 



/'F{x) l 



1/3 



the constants in the expression being based on the expression (8.56) for 

 the internal energy of TNT, using A;* = 0.20. 



It is convenient to write the period formula in a more general form 

 applicable to other explosives than TNT. To do this exactly would 

 mean recomputing the integral for T for each explosive depth, using the 

 best available expression for E{a'^)/Y in place of /c*a*~^^^, and redeter- 

 mining the limits am* and a*. Fortunately the result is insensitive to 

 these changes, and so the evaluation need not be repeated. In a re- 

 vised formula Friedman (37) has, however, used a value k* = 0.16, 

 corresponding to shallower depths, as being more representative of 

 most available measurements, and the revised expression for the period 

 in nondimensional units becomes 



r* = 1.485 ri + o^i?#M1 



The expression for the period T in time units does depend directly 

 on the explosive and energy conversion. The length and time scale 

 factors L* and C* were defined as 



\4:TrpogZo/ \2gZo/ 



where Y is the energy remaining in the system during the cycle and d 

 is the depth. Expressing F as a fraction r of the total detonation 

 energy Q in calories per gram of explosive, the scale factors L* and C* 

 then become, for W pounds of explosive 



