SECONDARY PRESSURE WAVES 379 



sidered, the kinetic energy of vertical flow at the minimum was esti- 

 mated to be about six per cent of the total.) 



We have already, in section 8.7, discussed the instability of the 

 spherical bubble form assumed in the approximate theories which have 

 led to fairly complete estimates of the motion. The theories of such 

 departures apply strictly only to small perturbations, but do at least 

 make reasonable the observed flattening of the bubble as it contracts. 

 This flattening implies an increase of hydrodynamic inertia to trans- 

 lation, and is thus consistent with the small observed upward velocity 

 of the bubble as compared with calculated results for a spherical form. 

 The energy considerations, however, seem definitely to rule out the 

 possibility of accounting for the decreased migration by this mech- 

 anism alone. The observed migrations during the second contraction 

 are comparable to those during the first, which suggests that the vertical 

 motion is very nearly stopped at the end of the first contraction, the 

 bubble starting its second expansion practically from rest. The only 

 accounting for the energy losses therefore seems to be in turbulence 

 near the bubble of the general type described. If this is true, at- 

 tempts to refine the noncompressive theory w^ithout including effects of 

 this kind are pointless. It should f mother be no surprise to find the ob- 

 served bubble pressures in poor agreement with calculations based on a 

 spherical bubble in an inviscid fluid if the predicted velocities of trans- 

 lation near the minimum are large. 



9.5. Problems in Measurement of Secondary Pulses 



Measurements of pressure in the secondary, or bubble, pulses from 

 underwater explosions present a number of difficulties, both in measure- 

 ment and in proper evaluation of the experimental data. Unless these 

 problems are understood, conclusions as to the significance of experi- 

 mental results are apt to be inaccurate, if not misleading. It is there- 

 fore important to consider the various factors involved in the measure- 

 ments before examining the specific results in detail. 



A. Experimental errors. The nature of the experimental problems 

 can be brought out by comparing the form of the pressure-time curve 

 for the secondary pulse with that for the shock wave. The bubble 

 pressures are very much smaller than the peak shock wave pressure, 

 have a relatively slow initial rate of rise rather than a discontinuous 

 shock front, and have an appreciable fraction of their peak value for 

 much longer times than the shock wave. Time and pressure scales in 

 which the complete bubble pulse is reproduced to a reasonable scale 

 thus represent the shock wave as a very high and narrow spike, as 

 sketched in Fig. 9.4. 



The principal difficulties in piezoelectric gauge recording of bubble 

 pressures are spurious gauge noise, cable signal, microphonic response 



