THE DETONATION PROCESS 97 



initial pressure of 112 kilobars has been assumed. It is to be under- 

 stood that this calculation is purely illustrative. 



B. Plane wave, TNT. Determination of the profile of a plane wave 

 for actual cases is formally much the same as for ideal gases ; namely, if 

 the values of pressure and density at the front are known, the Riemann 

 analysis can be used to determine the particle velocity w as a function of 

 density. The actual adiabatic law must replace the ideal gas relation 

 and, if this has been determined in tabular form, the integration for u 

 must be performed numerically. As a starting point of the calculation 

 for TNT, Taylor chose D = 6,380 m./sec, pi = 2.00 gm./cm.^ The 

 pressure and sound velocity for these values are from Jones' calculations 

 Pi = 150 kilobars, Ci = V(dP)/{dp)p^p, = 4,840 m./sec. Numerical 

 step by step integration of Eq. (3.16) using an adiabatic relation for 

 TNT similar to Fig. 3.2 then yields a series of values of u for various 

 values of p, and corresponding values of c and P are known from the 

 adiabatic. The pressure and particle velocity are therefore determined 

 as a function of c + it or of r/R = (c + u)/{ci + Ui), where as before r is 

 the distance from the source at which the values c and u are realized 

 after a time t and R is the distance for the head of the wave. Taylor's 

 results are plotted as the solid curve in Fig. 3.3. The differences from 

 an ideal gas law (dashed curve) are seen to be consistent with the dif- 

 ference of the adiabatics. 



C. Spherical wave, TNT. Taylor (106) has also computed the pro- 

 file of the wave behind a spherical detonation front, assuming initiation 

 at the center which thereafter leaves the products at rest {u = 0). In 

 the spherical case, the Riemann function is not directly applicable and 

 a more complicated procedure is necessary. The fundamental equa- 

 tions describing the motion behind a spherical front are, from section 

 2.3, 



(3.20) *' + ^^=_l^^ 



dt dr p dr 



dp , dp , du 2pu 



\- u \- p — = 



dt dr dr r 



If, as we assume, it is legitimate to suppose that a stationary det- 

 onation condition is attained before the disturbance has proceeded an 

 appreciable distance, the conditions behind the front will scale in pro- 

 portion to the distance the front has progressed. That this is true is 

 evident on noting that if r and t are both changed by a factor /3 into (3r 

 and ^t, the quantities p, u evaluated at /3r and ^t satisfy the same equa- 

 tions. The conditions at R and t, where R is the distance of the front 

 from the initial point are, however, unchanged at (3R and ^t if the time 



