SHOCK WAVE MEASUREMENTS 233 



creased continuously as indicated in Fig. 7.2. It was accordingly con- 

 cluded that a significant measure of impulse for comparison purposes 

 is obtained by integrating to five times the time constant of a spherical 

 charge of the same weight. Values of time corresponding approximately 

 to this rule have been used for most impulse calculations from data ob- 

 tained at Woods Hole, although for special cases other values have been 

 used. Whatever the criterion that is chosen, its arbitrariness must be 

 recognized in any analysis of experimental data. (This point is dis- 

 cussed, for example, in section 9.3 in comparing shock wave impulse 

 with values for the later secondary, or bubble, pulses.) 



C. Energy flux density.'^ Another significant measure of the shock 

 wave is in the energy flux across unit area of a fixed surface normal to 

 the direction of propagation. The rate of energy transport across unit 

 area in a fluid of density p and particle velocity u is, from section 4.8, 

 given by 



puA 



E + W + - 



where A[£' + Ju^] is the increase in kinetic and potential energy for 

 unit mass of fluid, and P is the pressure. The energy flux density to 

 time t after arrival of the shock wave is then 



J 



E + \u^ + -\dt 



(7.3) Ej = puA 



In the limit of small amplitudes, the internal and kinetic energies 

 can be neglected in comparison with P/p, and for an outgoing spherical 

 acoustic wave the particle velocity u is given by (see section 2.2) 



u{R, n = ^ - ^^ + J_ r (p _ p^) dt 



PoCo Poti J 



The energy flux density in terms of pressure then becomes 



(7.4) E, = j-({p- PoY rf^ + -L r (p _ pj r r (p _ p^^^t] d^ 



O 



The second term, representing the effect of the excess particle velocity 

 or afterflow left by an outgoing spherical wave, rapidly becomes negli- 



^ This quantity is frequently described in the literature as energy flux or simply 

 energy, when in reahty a specific quantity for unit area is meant. 



