THE DETONATION PROCESS 105 



(3.24) 



<" - 1«') 



l^dP _ cic - 2n) du _ c 



pc dt c^ — uc -\- u^ dt a c^ — uc -\- u^ 



A corresponding equation for the gas products is obtained by replacing 

 p, c, CO by pg, Cg, (Jig (whcrc oig = f {cg^/pg) dpg), and noting that the 

 propagation velocity is —Cg for an ingoing wave, with the result 



(3.25) ^- 



1 dP Cg{Cg + 2U) du Cg 



=.(-,- ^«^) 



cA co„ - - wM — 2u^ 



PgCg dt Cg^ + UCg '\- V} dt tt Cg^ + UC g + U^ 



The two equations, (3.24) and (3.25), can be solved for the deriva- 

 tives dP/dt, du/dt at the boundary in terms of the various velocities and 

 related functions in the two media. The resulting differential equations 

 could then be integrated numerically, obtaining at each step new values 

 of the variables. Rather than carry out these tedious calculations, 

 Kirkwood and Bethe make use of the so-called peak approximation in 

 which the falling off of pressure and enthalpy is assumed to be exponen- 

 tial, with a decay constant chosen to give the initial value of dP/dt cor- 

 rectly. Before describing this method, it is worthwhile to consider the 

 approximations in the development so far outlined. 



The motion of the gas products is assumed by Kirkwood and Bethe 

 to be described sufficiently by a single ingoing rarefaction wave. After 

 a finite time, this wave reaches the center of the charge and internal 

 reflection must take place. At sufficiently great times, the boundarj^ 

 conditions in the gas will therefore be modified by successive compound- 

 ing of internal reflections. The first such reflection will occur at a time 

 greater than 2ao/cg(0), where ao and Cg{0) are the initial radius and 

 sound velocity in the gas sphere. Inclusion of these reflections would 

 require an elaborate analysis and is not attempted in the theory out- 

 lined. These effects occur, however, at sufficiently late times that the 

 major portion of the shock wave is emitted, and the reflections make 

 their appearance in the shock wave "tail." This conclusion can be 

 inferred from the solution for dP/dt, which is readily shown to be always 

 negative and decreasing in magnitude, the function P(a, i) thus having 

 a time variation similar to an exponential. The peak approximation 

 already mentioned is therefore a natural one and should be a sufficiently 

 accurate description of the pressure variation at times for which the 

 theory is in any case reasonably reliable. The particle velocity u, how- 

 ever, is found first to increase to a maximum and then fall off. The 

 peak approximation is therefore not suitable in this case, and the motion 

 of the gas boundary requires a separate examination, as described in 

 part (C). 



