106 THE DETONATION PROCESS 



A function F{t) described by the peak approximation is assumed to 

 have an exponential decay of the form F{t) = F{0)e-^/^' and the decay 

 constant di can be computed from the initial variation of F{t) by the 

 relation 



\dl /,_o 



Ip(.o) 



The time variation of F{t) is thus determined in this approximation by 

 the initial value and slope of the function. Kirkwood and Bethe apply 

 this procedure to the enthalpy oo by writing w = a>(0)e~^/^\ With the 

 neglect of dissipation doo = dP/ p^ hence di is given by 



= -p(0)co(0) 





and the initial value of dP/dt can be obtained from Eqs. (3.24) and 

 (3.25). Assuming that initially c = c -\- u, Cg = Cg — u, this gives 



.o 26) ^ ^ , o^(Q) f^ociP) PJO) c{0) + Pg{0) CgiP) 



' PoiO) Cg{0) U{0) C{0) J{0) + Cg{0) Jg{0) 



where the functions J and Jg are given by 



cu\_c — u 2 c — uj^ 



^ I V Cg — u , 3 „ u^ ~\ 



Jo = - , ^a + ^U^ , 



CgU \_ Cg + U 2 Cg -{- U_\ 



With the values for co(0) and di, the variation of aj(a, t) is deter- 

 mined from the initial values and rates of change of quantities at the 

 boundary. The leading term in the enthalpy function G(a, t) = a{t) 

 o){a, t) + ia{t)u^ is thus determined. In numerical applications of the 

 theory, the radius a{t) is taken to have its initial value ao for the times 

 at which the first term aow{a, t) is important. At later times, the 

 kinetic term ia{t)u^ becomes increasingly important and must be con- 

 sidered. 



C. Motion of the gas sphere. In order to investigate the expansion 

 of the gas sphere boundary, Eq. (3.24) may be written 



du , c /3 A c'^ — uc -\- u- a dP , c , 2ii^ 



a -77 + Z ;:— 77 '^ I = 7Z ^^ 77 + Z PT ^ + " 



.(|..) = 



dt c — 2u \2 ) c{c — 2u) pc dt c — 2u c — 2u 



