9. Secondary Pressure Waves 



Theoretical calculations of the pressures in water following emission 

 of the primary shock wave have been without exception based on the 

 incompressive approximation, in which it is assumed that changes in 

 density of the water with pressure can be neglected. While this as- 

 sumption leads to a fairly accurate description of the main features of 

 gas sphere motion, particularly of the maximum radius and the period 

 of pulsation, it becomes an increasingly poor approximation as the 

 bubble contracts to its minimum size with increasing velocities of flow 

 and higher pressures. This part of the motion is, however, the one 

 which is of interest as far as pressures are concerned, for the secondary 

 pressure waves, often referred to as ''bubble pulses," are emitted while 

 the bubble is at or near its states of- maximum contraction. 



The weakness of noncompressive theory in describing secondary 

 pressure waves will become evident in the discussion of this chapter. 

 However, it may be remarked here that it is exemplified by the require- 

 ments of such theory that changes in pressure at any point be propa- 

 gated instantaneously to all parts of the fluid rather than with the 

 finite sound velocity observed. Differences in pressure in this approx- 

 imation are associated with changes in flow velocity, and can be a 

 reasonable approximation to the true pressures only for steady or slowly 

 changing flow. 



A better approach to the analysis of these pressures is that of sup- 

 plementing the noncompressive theory with a wave theory, and con- 

 siderations of this kind are necessary to estimate the energy radiated 

 near the bubble minimum in the wave of compression. A detailed 

 examination involves great mathematical difficulty, if done at all 

 rigorously. If, on the other hand, the simpler procedure were adopted 

 of attempting to keep the two types of approximation reasonably dis- 

 tinct, serious physical problems would arise in suitably patching to- 

 gether basically incompatible solutions. Moreover, the investigations 

 of Herring and of Penney and Price, described in section 8.7, bring out 

 the more fundamental difficulty that the motion in the critical region of 

 minimum size is dynamically unstable. It is therefore reasonable to 

 expect that turbulent flow with dissipation of energy in the formation of 

 eddies will set in, and this conclusion is supported by comparison of the 

 total energy loss at the minimum with that radiated as a compression 

 wave (see section 9.4). The much more complicated problem then 

 presented has, understandably, not been attempted except in very 

 rough approximation. 



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