SECONDARY PRESSURE WAVES 



189 



during the contraction, when the effective mass is 

 greatly decreased, the velocity of rise must increase. 

 Besides gravity, other effects such as proximity to 

 the free surface of the water or to the bottom can 

 cause departures from spherical symmetry. The ef- 

 fects of such surfaces become appreciable when the 

 distance from the bubble to the boundary surface is a 

 few times the maximum radius of the bubble, and are 

 of two sorts. In the first place, the period of the oscil- 

 lation is shortened by proximity to the free surface 

 and lengthened by proximity to an unyielding sur- 

 face; secondly, a free surface repels the bubble and 

 a rigid surface attracts it. This translational motion, 

 which becomes very rapid in the contracted stage, 

 weakens the secondary pressure pulse for the same 

 reason that the rise due to gravity does. Much 

 theoretical work has been done on the period and 

 migration of the bubble,-^'-^ and the results are in 

 generally good agreement with experiment.^* If 

 gra^aty can be ignored, asymmetrical motion due to 

 proximity to free or rigid surfaces obeys the scaling 

 law of Section 8.2, the distance from the surface 

 being changed in the same ratio as other linear di- 

 mensions. The gravity effect, however, does not scale 

 in the same mamier; gravity is relatively more im- 

 portant the larger the charge and the smaller the 

 external hydrostatic pressure pa,. Most features of 

 the motion as affected by gravity can be approxi- 

 mately expressed in a form which is independent of 

 the size of the charge, by using a unit of length A = 

 (W/gpf', a unit of time 'V K/g, and a unit of pressure 



From the foregoing it can be seen that the form 

 and strength of the secondary pressure pulses depend 

 greatly on gravity and on proximity of surface, bot- 

 tom, or objects to the explosion. The peak pressure in 

 the first bubble pulse, for example, has been measured 

 at values as large as 0.25, and as low as 0.06, times 

 the peak pressure in the shock wave.^^'^*'^*'^' By 

 contrast, the impulse Xjpdt contained in any one of 

 the secondary pulses is not very sensitive to these 

 factors. The amount of impulse contained within a 

 few half-widths of the main pressure peak is of the 

 same order as that in a corresponding portion of the 

 shock wave; however, just as was the case with the 

 shock wave, this impulse is considerably less than 

 the amount contained in the "tails," which in the 

 case of the secondary pulses extend to both directions 

 in time. The total impulse in a secondary pulse is 

 probably roughly equal to the amount which would 

 be calculated from the simple theory which assumes 



the water to be incompressible. It can be shown -' 

 that this impulse is 



2' "-•' 



<n40 

 o 



z 



?zo 



a. 

 in 



UJ 

 IT 



<n40 

 o 



?20 



<n 



40 



-20 



X 

 0.20 



o 



z 



.10 



u 



(E 



3 



in 

 <n 



/ = 



'6max 



/■ 



V 



Pl>o 



(30) 



-I 



TIME IN MILLISEC 



-10 



10 



TIME IN MILLISEC 



Figure 8. Typical pressure-time records for the first 

 bubble pulse. 



At all ordinary depths this is five or ten times as 

 great as the impulse in the exponential part of the 

 shock wave ; however, one would expect from theory 

 that the impulse in the tail of the shock wave would 



