jf44 THEORY OF THE SHOCK WAVE 



tion to values of P obtained at a fixed point is not important. If the 

 total work done by both the shock wave and the noncompressive flow 

 which the motion increasingly resembles is required, both terms in Eq. 

 (4.39) should be employed. If, however, a measure of the radiated 

 shock wave energy only is required, Eq. (4.39) is not correct as it stands, 

 because of the recoverable changes in kinetic and potential energy in- 

 cluded. 



The second term in Eq. (4.39) is, in the noncompressive approxi- 

 mation, the change in kinetic and potential energy of the fluid flow. 

 Therefore, the natural approximation to a value of the radiated energy 

 is obtained by dropping this afterflow term. Actually, the two types 

 of energy change are not distinct or mutually exclusive, and at no time 

 can the pressure behind the shock front be regarded as the result of 

 either compression or noncompressive flow alone. In cases of interest, 

 however, the difference in variation with time of the two terms in Eq. 

 (4.39) makes their contributions reasonably distinct. The first term 

 increases rapidly during the initial portion of the shock wave for which 

 the pressure P is large and later additions from the tail of the shock 

 wave become less and less important. The afterflow term, on the other 

 hand, increases much more slowly at first because of the integral 

 SPdt' in the integrand, but for the same reason continues to increase 

 appreciably for pressures very much less than the initial peak pressure 

 of the shock wave. The character of the afterflow term is also indi- 

 cated by the factor \/R as compared to the first term, thus being the 

 result of the decrease of afterflow velocity with distance from the gas 

 sphere boundary. 



Returning to the question of the time r to which the integrals in 

 Eq. (4,37) or (4.39) should be evaluated, it is clear that no single answer 

 is possible for all cases. A major part of the radiated shock wave 

 energy is accounted for if r is chosen to be several times the initial time 

 constant of the wave, but the outward energy flux of noncompressive 

 flow increases for much longer times (up to times of the order of one- 

 tenth of the bubble period). 



B. Energy flux density. The work done on a surface fixed in the 

 fluid is not the only way in which energy behind a shock front can be 

 expressed. Another measure which is sometimes useful is in terms of 

 an energy current density, or rate of energy flow through a surface of 

 \mit area. A unit volume of fluid with particle velocity u and internal 

 energy E per gram has energy of amount p{E + \iC^), where p is its 

 density, and the rate of transport through unit area is therefore 

 pu{E + \y?). If we add to this the rate Pu at which pressure P in the 

 fluid can do work on this surface, we obtain for the total energy trans- 

 port as a result of fluid motion through unit area 



