THE DETONATION PROCESS 107 



The particle velocity u is much smaller than the velocity c, and if u is 

 neglected in comparison with c, the equation becomes approximately 



du , ?> ^ a dP , 



a- — h-'W= -, — h^ 



dt 2 pc dt 



The peak approximation is appropriate for P and co, which gives 



If the factor a"^^^ is approximated by ao~'^''^ this equation is directly 

 integrable with the result (noting that u = da/dt) that 



(3.27) a'l'u = ao^'MO) + ao"MO)di (l - -^^ (1 - e-'/«0 

 A second quadrature gives 



5u(iO)e: 



lOi 



t , 2ac 



(3.28) a^'^ = ao^i- (l - ^!iM^^ 1+ ^ + ^^^^^ e-^A 



/2 ^ 5u(0)M 



2a. 



where ..^;fl__^\ 



aow(O) V c(0) V 



5^(Q)6>ia 



3 2ao 2ao 



h — 



5w(0) 1 + a 



The kinetic term au^/2 then is given by combining Eqs. (3.27) and (3.28) 

 with the result 



(3 29) — = i (^^^'^)^ 

 ^ ' ^ 2 2 (a^/2)4/5 



_ aoW2(()) ((1 _^ ^) - ag-^A)2 



^u{0)eia yi' 



A _ bu{0)e,a Y( 



\ 2a. y L 



2ao 



The rather complicated expression (3.29) for au'^/2 is not strictly 

 consistent with the order of the peak approximation for P and a? used 

 in its development, namely that these variables depend on time as 



