SECONDARY PRESSURE WAVES 377 



d/dt {a^da/dt) and elimination of t by using da = (da/di) dt then leads to 

 a result equivalent to Eq. (9.27).^ 



According to Eq. (9.27), the fractional energy radiated in the second- 

 ary pulse varies as the square root of the maximum gas pressure, and is 

 therefore largest when the bubble attains its smallest minimum volume 

 without vertical motion. If the value 7 = 1.25 is used, the expression 

 becomes 



Taylor's calculation that at the first minimum P^ = 8300 Ib./in.^ for 

 TNT if there is no migration gives AF = 0.31 F, which is a significant 

 fraction of the total energy. As the pressure P^ varies as or^y, a mini- 

 mum radius 3 times its smallest value, as computed in the example of 

 section 8.5, gives AF/F = 0.03. Hence only 3 per cent of the bubble 

 energy is calculated to be lost in the acoustic pressure pulse in this ex- 

 ample. 



Measurements of periods of successive oscillations of the gas bubble 

 give a direct measure of the total fractional energy loss in successive 

 contractions, as shown in section 8.3. The most extensive data of this 

 kind show that for 200 pound charges of a mixed explosive fired at 

 depths from 60 to 800 feet about % of the energy F of the first pulsation 

 is lost in the first contraction, this figure showing little systematic 

 variation with depth when the effect of the free surface on the period is 

 taken into account. Comparable figures are obtained for other explo- 

 sives and charge weights, and the figure of 30 per cent for acoustic 

 radiation accounts for less than half of the total energy loss during the 

 first contraction. In addition, it should be remembered that this cal- 

 culated figure is the result of rather crude approximations, and experi- 

 mental values obtained by integration of observed pressure-time curves 

 give smaller values (see section 9.6). 



B. Energy loss by turbulence. A detailed accounting for the balance 

 of the energy loss in terms of dissipation in turbulent flow would be a 

 difficult task which has not so far been attempted. It is clear that these 

 losses will be largest near the minimum, particularly while the bubble is 

 in its extreme stage of instability while contracting, and the form is dis- 

 torted from its originally spherical shape. The energy loss by thermal 

 conduction for the short times and small temperature gradients is readily 

 shown to be negligible, even in the gas sphere, if there is no turbulence. 

 The remaining mechanism of viscosity is a perfectly reasonable one, and 

 there appears no reason to doubt that it, together with acoustic radi- 



3 The mathematical equivalence of the two approaches is readily demonstrated 

 by transformation of Eq. (9.26) . 



