THEORY OF THE SHOCK WAVE 143 



further motion without some consideration of the time r to which the 

 integration is extended. If tliis time were taken as infinitely large, the 

 integral in Eq. (4.37) would include the energy radiated in the later 

 bubble pulses as well as in the primary wave. A smaller value of r is 

 therefore required to exclude these later contributions. 



The proper value of r and the interpretation of Eq. (4.37) are com- 

 plicated by the fact that as the shock wave passes outw^ard the work 

 done on any boundary is increasingly that of noncompressive flow which 

 gives the surrounding fluid kinetic and potential energy. This energy 

 is returned to the gas products on later contraction of the bubble, and 

 should not be included if one is interested in the energy radiated by the 

 shock wave and lost for further motion. If the integration in Eq. (4.37) 

 is extended to infinite time or over complete cycles of the pulsations 

 this contribution is excluded because each outward flux of energy is 

 cancelled by inward flux during the following contraction. 



The energy radiated in later contractions of the gas sphere would be 

 excluded by taking r to be smaller than the time at which the gas sphere 

 reaches its maximum radius. If this is done, however, the work cal- 

 culated by Eq. (4.37) includes both the energy radiated in the shock 

 wave and the recoverable kinetic and potential energies delivered to the 

 fluid external to the surface of radius r. This combination of forms of 

 energy is closely related to the afterflow term in the acoustic expression 

 for the particle velocity u in spherical motion, which from section 2.2 is 



(4.38) u = ^-^=^ + -^ f {P-Po) dt' 



PoCo PoK J ^^^^ 



where Po is the hydrostatic pressure. The second, or afterflow, term 

 in this equation reduces, in the limiting case of noncompressive flow, to 

 the velocity required by expansion of the gas sphere boundary, and the 

 corresponding kinetic energy is returned to the gas sphere in its recom- 

 pression. 



If an evaluation of the work done at any point in the fluid from the 

 measured pressure-time curves is desired, the particle velocity u can be 

 eliminated from Eq. (4.37) by the use of Eq. (4.38), with the result 



(4.39) ^ = -l-f P2dt+±^f Pit) ( P{t')dt'dt 



if the hydrostatic pressure Po is neglected. Strictly, this equation 

 should be evaluated for a surface fixed in the water and hence moving 

 with it, but except at points close to the gas sphere the resulting correc- 



