574 SECONDARY PRESSURE WAVES 



very short time after the initial peak. At this time, the pressure is 

 both fairly large and slowly decreasing, and if the integration were ex- 

 tended to the time of h3^drostatic pressure a considerable increase in 

 impulse would result. This increase must in fact be an appreciable 

 fraction of one-half the secondary impulse computed above, for the 

 reason that the slowly decreasing pressures in the tail of the shock wave 

 are due largely to noncompressive flow and hence are to be computed 

 in the same way as the secondary pressures. The proper conclusion 

 from the impulse calculations is therefore that the secondary pulse has 

 an impulse comparable to the shock wave because of its long duration, 

 but is inferior to the shock wave in both peak pressure and energy flux. 

 The question of energy radiated acousticafly in the secondary pulse is 

 of interest in this connection and is considered in the next section. 



The variation of pressure with time during the phase of the motion 

 when the bubble is large and the pressure less than hydrostatic presents 

 no unusual features. It is evident from the slow variation of the func- 

 tion a^ da/dt at these times that the pressure differences are small and 

 smoothly varying. It is interesting, however, that the negative im- 

 pulse is large, owing to the long time interval for which P < Po. In 

 fact, this negative impulse must equal the positive impulse when the 

 pressure exceeds hydrostatic because the integral y* (P — Po) di depends 

 only on the difference in the values of (a^ da/dt) at the two limits of 

 integration. If these are corresponding points in successive cycles, the 

 difference is zero, requiring that positive and negative areas during the 

 cycle be equal. This negative impulse or suction phase is comparatively 

 feeble, as the pressure difference never is greater than hydrostatic. The 

 fact that the total impulse is so large illustrates further the danger of 

 taking large values of impulse at face value without examining the pres- 

 sures and durations in detail. 



9.4. Energy Losses in the Pulsations 



The discussion of fluid motion and pressures during motion of the 

 gas sphere have so far been based on noncompressive theory. The ex- 

 plosive energy left after emission of the shock wave is thus assumed to 

 exist entirely as potential and kinetic energy of flow of the water plus 

 internal energy of the gaseous products. In this approximation, no 

 mechanism has been provided for energy loss during the pulsations, the 

 only change possible being reversible redistribution of the total energy 

 between the gas and the surroimding water. The energ}^ losses actu- 

 ally occurring in the course of the motion cannot therefore be estimated 

 without taking into account, to some degree of approximation, mech- 

 anisms by wliicli energy can be dissipated: the compressibility of the 

 water, by which energy is radiated as a wave and ultimately dissipated 



