483 



THEORY OF GAS GLOBE OSCILLATION IN UNDERWATER EXPLOSIONS 270 



rection for the internal energy of the gas. Thus 

 the minimum excess pressure is negative relative 

 to the original hydrostatic level, as would be 

 expected, and is not critically dependent upon 

 the equation of state of the gas. 



Because of the base line uncertainties in the 

 measurements described in the preceding paper,^ 

 calculations based on Eq. (20) were used to select 

 a baseline to which the pressure-time records 

 were subsequently referred. 



2.5. It is seen from Eq. (17) that the non- 

 dimensional bubble radius ao at which Ap is zero 

 is determined by the condition that (a-d) ■ be zero. 

 Applying this condition and eliminating d from 

 Eqs. (9) and (19), one obtains 



into two parts. 



4ao' = l-(4-37)/tao-'<^-". 



(22) 



2.6. The positive impulse / delivered by the 

 pressure wave is defined b%' 



... 



Apdr, 



(23) 



where the integration is performed over the 

 region lying between the point at which Ap rises 

 to zero during the collapsing phase and the point 

 at which it again falls to zero on the expanding 

 phase, i.e., between points where the non-di- 

 mensional bubble radii are denoted by Ooi and ao2i 

 respectively. 



Because of the radiation of acoustic energy, the 

 pressure-time curve is not symmetrical about the 

 peak pressure ordinate. For convenience, there- 

 fore, the integral of Eq. (23) will be separated 



/= r ApdT+ r Apdr, (24) 



where a„, is the non-dimensional minimum bubble 

 radius, i.e., the radius when Ap is a maximum. 



Elimination of Ap between Eqs. (17) and (24) 

 and conversion of dimensional time t to non- 

 dimensional time gives 



/ = - 



2PaL\C\ 



3R 



f 



d{a'^d) 



where Li and Ci are the scale factors of the first 

 and Lj and d the scale factors of the second 

 bubble oscillation. 



Since d is zero at the instant of bubble mini- 

 mum, Eq. (25) becomes 



2Po 



1 = lLiCi{a'^d)oi+LiC2{a-d)o22- (26) 



3R 



(In Eq. (26), d is understood to represent only the 

 magnitude of the quantity, whereas in Eq. (25) 

 the symbol has an associated algebraic sign.) 



The appropriate expression for (a^d)o can be 

 obtained from Eq. (19) by setting (a'^d)- equal to 

 zero ; the result is 



(a-d)o= 1.732ao 



(7-1)^1' 



r_(7-m| 



(26a) 



Fig. 1. Non-dimensional 

 bubble period, t, versus pa- 

 rameter k for various values 

 of 7. 



b 



AMMATle PAIIAMCTCII.il 



