136 THEORY OF THE SHOCK WAVE 



= PoAVg + 47rPo r ( - - l) ^o'dvo 



the second step following from the relation pr~ch = poVo-dro. Substi- 

 tution in (4.30) gives 



(4.31) Wo = ^TT f Poro'h[Pm(.ro)] dr, + P,A7, + ^tt f^ r'^Pudt 



J ao J t{R) 



after introducing the enthalpy increment h{P), which is li{P) = E{P) 

 + Pod{l/p). The time integral is assumed to vanish as the shock front 

 passes to infinite distance and hence in this limit we have 



(4.32) Wo = 47r r Poro'h[Pm{ro)] dvo + P.AF, 



In this equation, the term PoAVg represents energy stored in the water 

 by expansion of the gas sphere w^hich, on later oscillations, may be dis- 

 sipated by turbulence of the flow or by radiation of later "bubble 

 pulses." 



Subtracting Eq. (4.32) from (4.31) gives the final result. The 

 second relation for F(R) permits its expression in terms of P„i and R, 

 and is conveniently written 



(4.33) ^ = -PoRVi{P.n) 



dR 



The usefulness of considering the energy flux-time curve lies in the fact 

 that, for a given weight of any particular explosive, its shape changes 

 very little with increasing R. Kirkwood and Brinkley make use of 

 this characteristic of F in formulating the desired fourth relation 

 among the partial derivatives of Pm, Um- As written, F{R) does depend, 

 however, on the strength of the source (weight of charge and its com- 

 position) and it is desirable to normalize the time integral to a value 

 which is independent of this factor. This is done by expressing the 

 integrand as a fraction of its initial value R-PmUm and choosing a reduced 

 time scale for which the initial slope of the integrand has unit value. 

 Inasmuch as the observed energy-time curves are approximately of the 

 form R^PnMme~^"^ it is convenient to use as time unit the initial logarith- 

 mic slope of the curve, defined by 



- = — ( ^ log r^Pu ) 

 M \dt /i=.o 



