36S SECONDARY PRESSURE WAVES 



mum near the stabilized position, but is considerably smaller than the 

 optimum value for sufficiently large values of s. The pressure is, from 

 Eq. (9.21), an even function of s, and the curve for negative values of s 

 is therefore the reflection of the one for positive s. The value of s for 

 given initial conditions of charge weight and depth must be obtained by 

 integration of the equation of motion for vertical momentum. Inte- 

 grations of this kind have been carried out in special cases by Shiffman 

 and Friedman for the more general problem where the effects of the sur- 

 face and bottom are included, and their result, given in section 8.10, is 

 compared with experiment in section 9.5. Approximate calculations 

 for various charge weights and depths, neglecting surface effects, are 

 given in graphical form in Road Research Laboratory reports (93). 



The effect of vertical velocity in decreasing the maximum peak pres- 

 sure has been illustrated by Taylor (107) in a calculation for 4.65 pounds 

 of TNT fired 20 feet below the surface. If the vertical motion is neg- 

 lected, the calculated internal pressure in the gas sphere has a maximum 

 value of 8,300 Ib./in.^ Because of the vertical momentum and energy 

 acquired, however, the minimum radius actually calculated is 3.3 times 

 that obtained assuming no migration and the internal pressure of the 

 gas is reduced, by a factor (3.3)^^^^, to about 1 per cent of its optimum 

 value. The peak pressure will not be reduced by as large a factor be- 

 cause of the vertical flow velocity, on which the pressure also depends. 

 Other factors must also be kept in mind. The first is that, although 

 the peak pressure is considerably decreased, the duration is longer and 

 the time integral of pressure, or impulse, is not affected to nearly the 

 same extent (see part B). The second factor is the repulsion of the 

 bubble by the free surface which, although insufficient in this case to 

 prevent net migration, would reduce it somewhat and hence partially 

 neutralize the effect of gravity. The calculation should not, therefore, 

 be taken literally as demonstrating that pressures in the water are re- 

 duced by factors of a hundred in representative cases,^ but rather as an 

 illustration that the peak pressure may be appreciably modified by 

 vertical migration. The actual motion of the gas sphere in cases en- 

 countered in practice is nearly always appreciably modified by the 

 proximity of either the surface or bottom, if not other boundaries. 

 Further discussion of pressure variations with the migration is therefore 

 deferred to section 9.6, where these factors are considered. 



B. The impulse. It has been remarked in various places that the 

 peak value of a transient pressure is not the only criterion for comparison 

 of different conditions, nor is it even necessarily a significant one. If the 

 duration of the wave under consideration is very much longer than other 



2 Statements of tliis kind must be properly interpreted, as they refer to values at 

 equal distances from the source at the time the pressure is developed. This position 

 will, if there is migration, not be the initial position of the charge. 



