SECONDARY PRESSURE WAVES 369 



culties of a more inclusive single solution. The analysis does, however, 

 illustrate the relative parts played by shock pressure and "kinetic" 

 pressure of the outward flow. 



B. Afterflow energy. If the vertical motion of the bubble is neg- 

 lected, Eq. (9.4) for the excess pressure P — Po becomes 



^•='i("=f)-i©'(f)' 



The second term is simply — Ju^ from the equation of continuity for 

 noncompressive flow 



dr 

 ~dt 



r2 \dt ) 



This term, which decreases rapidly with distance, represents the 

 Bernoulli pressure —ipoU^ for flow velocity u. 



The outward flow has associated with it kinetic energy, and is often 

 referred to as the kinetic wave to distinguish it from the shock wave of 

 compressional energy. The energy of the noncompressive flow is often 

 described as afterflow energy, and as the "schubenergie" in German 

 research. The relation of this energy to the pressure as given by Eq. 

 (9.9) is made evident by calculating the work done by pressure P on a 

 spherical shell of radius r. This is given by 



, Tf = 47r I r'^Pudt 



Substitution for P in terms of particle velocity u and distance r gives 



W = 4:Trpo I -T-^'^ - -^*' r^'^^dt + iirPo I r^udt 



By rearrangement and use of the relation u dt = dr, we obtain 



W = 47rp, I (^ uhMr + urHu\ + 47rPo f 



= 27rp. r di^li^r') + 47rP, r d (^0 



Hr 



