MOTION OF THE GAS SPHERE 276 



much of the expansion suggests, as a first approximation, neglecting it 

 entirely. If this is done, it will be seen that the gas sphere must have a 

 maximum radius when da/dt = 0. Calling this value am, we have from 

 Eq. (8.5) 



(8.6) Y = ^ PoaJ 



o 



This relation thus furn'shes an experimental method for determining, 

 to a rather good approximation, the total energy Y associated with the 

 radial flow of water in terms of the maximum radius a« of the bubble 

 and the hydrostatic pressure Po at the depth of the explosion. Experi- 

 mental measurements based on Eq. (8.6) are described in section 8.3. 



Neglecting the internal energy in Eq. (8.5) makes possible separa- 

 tion of the variables, and using Eq. (8.6) to eliminate Y gives the result 



(8.7) . .-.o . , da 



-iWf 



K^)'-']'" 



where ao is the initial radius of the gas sphere at time ^ = 0. The 

 integral is not expressible in terms of elementary functions, but can be 

 transformed to give a sum of incomplete jS-functions Bx(p, q), defined 



by 



B.{p,q) = J'^xp-'il -xy-' 



dx 



the necessary substitution being x = (am/ay. Values of this function 

 have been tabulated for discrete values of p, but unfortunately the 

 values p = 5/6, q = }/2 required here are apparently not included. In 

 general, the solution must therefore be obtained by numerical methods 

 and some of these results are illustrated in section 8.3. In the par- 

 ticular case x = 1, correspondiiig to a = am, the solution is know^n in 

 terms of the factorial or 7-function: Bi (5/6, 3^) = 2.24. These limits 

 a = 0, am correspond to the time required for expansion from zero to 

 maximum radius. The initial radius ao is small compared to am and 

 hence this time is approximately 3^ the period of oscillation T. Substi- 

 tution in Eq. (8.7) gives the approximate result 



"■(i.rKH) = '-""(R)' 



^.gg^ y. ^ 2 /3pAi/2 ^ /5 i\ ^ _ /,^\m 



