108 THE DETONATION PROCESS 



e~'/^i, and terms of order e~"^/^i and te~^/^' should properly be dropped. 

 A further consideration is the fact that the value of co used in the equa- 

 tion should be taken as the excess over the equilibrium value for hydro- 

 static pressure Po in the undisturbed fluid. In the initial phases of the 

 motion the pressures are so large that neglect of Po is unimportant, but 

 an investigation of the later motion of the gas boundary when the pres- 

 sure is small must include the effect of hydrostatic pressure. As dis- 

 cussed in Chapter 8 by simpler methods adequate for the later motion, 

 the hydrostatic term leads to long period radial pulsations of the gas 

 products. 



D. The enthalpy function on the gas sphere. The final step in formu- 

 lating conditions at the gas sphere for use in the Kirkwood-Bethe theory 

 is to construct a suitable approximation to the function G{a, t) = 

 a[co + \u"'] from the results of parts (b) and (c). To do this, Kirkwood 

 and Bethe note that the kinetic term au^J2 as given by Eq. (3.29) has 

 for large t the asymptotic form 



OAi} ^ (1 + aY / / \-''' 



2 A 5u{0)diaY" 



(- 



(■4) 



2ao ) 



and for / = is of course aoU^{0)/2. In the initial phases of the motion 

 the enthalpy term is dominant, and an approximate expression for 

 G{a, t) which reduces properly in the limits of short and long times is 



(3.30) 0(a, t) = ao 



_^ A (1 + ay \ u%0) 



A 5u{0)d,a V 

 V 2ao ) 



+ 



2ao ) / 



aoU\0) (1 + -^2 / / \-4/5 



^-t/e. 



A bu{0)e,a \ 

 \ 2ao ) 



^Vfi + ^Y 



where di, do, and a have the values previously given. 



The constant factor multiplying e~'/^i in Eq. (3.30) is much larger 

 than the factor multiplying (1 + t/d-z)"'^'^ and the two characteristic 

 times 01 and 02 are comparable in value. As a result, an initially rapid 

 reduction of enthalpy, and hence pressure, at the gas sphere is replaced 

 at later times by a much more gradual decay, varying asymptotically 

 at long times as {t/02)~'^^^. This later variation of pressure is essentially 

 an effect characteristic of noncompressive flow, by which the motion is 

 approximated for the small, slowly changing pressures at later times 

 (see section 9.2). This asymptotic result is of somewhat dubious quan- 

 titative value because of the approximations involved. Experimental 



