Hugoniot curves detonation branch

Thus far in the development, the deflagration, and detonation branches of the Hugoniot curve have been characterized and region V has been eliminated. There are some specific characteristics of the tangency point J that were initially postulated by Chapman [7] in 1889. Chapman established that the slope of the adiabat is exactly the slope through J, that is,... 

Along the detonation branch of the Hugoniot curve, the variation of the entropy is as given in Fig. 5.4. For the adiabatic shock, the entropy variation is as shown in Fig. 5.5. 

In his Fig I of the Hugoniot curve, shown on p 789, the JM branch corresponds to overcompressed detonation waves... 

It is easily seen from Figure 2.3 or Figure 2.4 that a unique straight line through the point (1, 1) is tangent to the detonation branch of the Hugoniot curve. This can be proven formally by considering the equation... 

Thus, at J the velocity of the unbumed gases moving into the wave, i.e., the detonation velocity, equals the velocity of sound in the gases behind the detonation wave plus the mass velocity of these gases (the velocity of the burned gases with respect to the tube). It will be shown presently that this solution at J is the only solution that can exist along the detonation branch of the Hugoniot curve for actual experimental conditions. 

---