S6J^ SECONDARY PRESSURE WAVES 



Substituting the parameters usually employed for TNT (7 = 1.25, 

 k = 7.83 X 109, rQ = 440) gives 



Wl/3 



(9.13) Pmax-P. = 2590 -'^^ 



For comparison with this result, the peak shock wave pressure Pm 

 is experimentally found to be given by (see section 7.4) 



(Wl/3\1.13 

 ^) 



At the same distance r, the maximum pressure in the first secondary 

 pulse is thus less than 20 per cent of the shock wave peak pressure if 

 the latter exceeds 1,000 Ib./in.^, and later pulses are much weaker, ow- 

 ing to energy losses in successive contractions. One might therefore be 

 inclined to dismiss the effects of secondary pulses as of little importance, 

 particularly when it is considered that the most favorable case for 

 generation of high pressures has been computed. This conclusion is, 

 however, not justified, because the duration of the bubble pressure pulse 

 is much greater and the impulse, or time integral of pressure, is com- 

 parable to the impulse of the shock wave. Both the magnitude and 

 duration of impulsive pressures need to be considered in most situations, 

 and the large impulse of the first secondary wave, plus the possibility 

 of its originating nearer a target than the primary shock wave, may 

 make it comparable in effect to the shock wave. The later pulses are so 

 much weaker that it is doubtful if they are ever of great importance in 

 causing damage, although their magnitude and time of occurrence are 

 of considerable interest in analysis of energy losses (see, for example, 

 section 9.4). 



9.3. Pressures During the Contraction Phase 



The preceding discussion has shown that the interesting times, in 

 which appreciable pressures are developed, occur when the bubble radius 

 a is small. With this condition, the general energy equation (8.23) for 

 incompressible motion can be simplified by neglecting the potential 

 energy of buoyancy of the cavity in comparison with the total energy. 

 We therefore have, using the dimensionless form of the equation of mo- 

 tion introduced by Shiffman and Friedman (102), 



<•"' -m^im 



+ A;*a*-3/4 = 1 



