SECONDARY PRESSURE WAVES 369 



times of interest (for example, time of deformation and damage of a 

 structure), then the wave is effectively a hydrostatic pressure and its 

 maximum value is of primary interest. If the opposite situation of a 

 relatively short duration applies, then the peak value loses this predomi- 

 nant importance and the duration becomes equally significant. It is 

 therefore necessary to consider the form of the secondary pulse in greater 

 detail. 



The discussion in part (A) of this section shows that the excess pres- 

 sure, primarily determined at points not too near the bubble by 

 d/dt {a? da/dt), is negative and small over most of the cycle of pulsation. 

 The duration of positive pressure, which occurs during the contraction, 

 is conveniently estimated by determining the radius a^ for which 

 d/dt {a^ da/dt) = and the pressure is hydrostatic (except for the 

 Bernoulli term). The condition is equivalent to requiring that 

 (a^ da/dty be maximized, and this quantity is conveniently obtained 

 from the equation of motion as expressed in terms of vertical momentum 

 s by Eq. (9.17). Setting the derivative d/da^ (a*^ da^/dt*Y equal to 

 zero gives 



If the migration and internal energy are neglected by setting s and k^ 

 equal to zero, the solution is a* = {}/iy'^ = 0.63, corresponding to 63 

 per cent of the maximum radius. If values k* = 0.2, s = 0.06, corre- 

 sponding to TNT and a large vertical momentum, are used, the value 

 of a* is 0.61. This shows that the point in the oscillation at which the 

 pressure exceeds hydrostatic is insensitive to the vertical motion, as 

 would be expected from the fact that the bubble is comparatively large 

 at these times in the first cycle, and has not acquired appreciable 

 velocity of translation. 



The fact that the value of a* for zero excess pressure occurs at times 

 in the pulsation which are insensitive to any later migration implies that 

 these times are approximately constant fractions of the period of oscil- 

 lation. This fraction would be nearly constant under any conditions if 

 the effect of migration is primarily to change the scales of length and 

 time proportionately. Examination of numerical solutions show that 

 this assumption leads to reasonably accurate results, and from such 

 solutions the interval between minima and the nearest time of hydro- 

 static pressure is found to be approximately eleven per cent of the 

 period. 



A rough measure of the second pulse duration can be taken to be the 

 time during which the pressure exceeds hydrostatic, which is thus 

 twenty-two per cent of the period if the second cycle of pulsation is as- 



