78 THE DETONATION PROCESS 



the wave profile is proportional to the time t. This is mathematically 

 equivalent to the statement that the pressure, density and velocities 

 behind the front are functions only of the ratio r/R, where R is the dis- 

 tance Dt travelled by the wave following its initiation at r = 0, ^. = 0. 

 That this is true is readily seen from the fundamental equations of 

 motion and of continuity, which are unaltered if t and r (or x) are re- 

 placed by I3t and ^r and the pressure, density, etc. are evaluated at ^t, 

 jSr. This result is an example of the principle of similarity which will 

 be discussed more completely for shock waves in Chapter 4. For the 

 short periods of time before the steady state of the wave has been 

 reached and the exciting cause has ceased, the condition wdll not be 

 fulfilled, nor is it true in the front where dissipative processes cannot 

 be neglected. In cases of interest, however, it is to be expected that 

 the scaling law will be realized to a very good approximation before 

 the disturbance has travelled any appreciable distance. 



The detailed calculation of the properties of detonation waves is a 

 relatively straightforward numerical integration once the conditions at 

 the front and the adiabatic pressure-density relation for the products 

 behind the front are known. These calculations have been made by 

 G. I. Taylor for TNT, using as a starting point the conditions computed 

 by H. Jones, and will be described in section 3.6. 



3.3. The Equation of State foe Explosives 



A necessary preliminary to computation of conditions at and behind 

 the detonation front in an explosive is, of course, knowledge of the equa- 

 tion of state and heat capacity of the products which, together with the 

 heat of reaction and the Rankine-Hugoniot or adiabatic conditions, 

 permit determination of the pressure-density relations. It is evident 

 that the products of detonation, although to a large extent gaseous, are 

 initially confined to the volume of the original solid or liquid. In addi- 

 tion, the products are at temperatures of the order 3,000-5,000 A. and 

 pressures of 50,000 atm. or higher. These circumstances thus present 

 two rather formidable problems : first, the choice of a suitable equation 

 of state for products at much higher temperatures and densities than 

 are ordinarily investigated, and second, determination of the appro- 

 priate distribution of the original atoms among the many possible 

 molecular species in the product (i.e., the final composition). 



A. The ideal gas law. Although we shall be interested in the reac- 

 tions of solid explosives, in which the final state even of gaseous prod- 

 ucts is very poorly described by assuming ideal gas laws, it is of inter- 

 est to consider the equations for ideal gases and some experimental 

 results on explosions of gaseous mixtures, as these results give some 

 indication of what composition products determine the detonation 

 process. For an ideal gas the equation of state is 



