362 SECONDARY PRESSURE WAVES 



the instant of greatest contraction and the function o?da/dt has a 

 rounded sawtooth outhne as indicated. Over most of the cycle its 

 slope, and hence the pressure difference (gauge pressure) is negative, 

 and only near the times of maximum contraction is the slope positive 

 and the pressure greater than hj^drostatic. The pressure builds up 

 very rapidly, however, and the pressure variation is thus in the nature 

 of relatively large, but short-lived, excess pressures near the time of 

 maximum contraction between much longer intervals of negative gauge 

 pressures. 



The pulses of positive pressure are naturally of primary interest, 

 and it is evident that their character must be determined by the internal 

 energy and pressure inside the gas sphere. The pressure in the gas is, 

 however, considerably^ modified by the translational velocity of the 

 sphere and energy associated with it. As a result, the secondary pres- 

 sure pulses vary greatly in magnitude and duration for different veloci- 

 ties of migration at the times of their emission. A further effect of 

 migration is the displacement of the bubble during the cycle between 

 the short intervals of appreciable pressure, which displaces the effective 

 origin of these pulses vertically from the original location of the charge. 

 Thus, if gravity alone is considered, the secondary pulses always ap- 

 pear to be emitted from a point closer to the surface than the origin of 

 the shock wave, the magnitude of the displacement being greater for 

 larger charges. If there are boundaries sufficiently near to modify the 

 flow appreciably, both the total displacement before the compression 

 sets in and the extent to which it is developed are materially affected. 

 The secondary pulses from explosions and their effectiveness at any 

 point in producing damage must therefore be evaluated not only in 

 terms of the charge and distance from it, but also in terms of the depth 

 of water, proximity of boundary surfaces, and the orientation relative 

 to the charge of the point at which the pulse is of interest. 



D. The optimum "peak pressure} From the analysis of the function 

 d/dt (a^ da/dt), it is reasonable to expect that the peak value of pressure 

 due to this term occurs when the bubble has contracted to its minimum, 

 for in this case there is no kinetic energy of radial flow. It is also reason- 

 able to expect that the peak pressure will be maximized when the bubble 

 has no vertical motion, as in this case all the energy is concentrated in 

 compression of the gas products. Anticipating these results, proved in 

 section 9.3, the optimum peak pressure Pmax at a point r is, from Eq. 

 (9.4), 



rdt\ dt Ja^a 



1 The analysis here and in section 9.3 follows closely the work of Shiffman and 

 Friedman (102). 



