279 
aS 
Substituting the known equation for the adiabatic, and putting in a numerical coefficlent 
to express C, in mechanical work units, we obtain 
T 
0. 767 cidt = 40 (Vo t+V = 2,) - 45 loge (v, - 68070 (v, — ,6807) + 7.82(Vv, - v,) 
Tz (14) 
where ur Is the temperature at state Por Vyas 
This equation mignt be solved by trial and error. Thus a value might be assumed for v; 
the equation of state then fixes T, and direct substitution, assuming C, known as a function of 
temperature, would show how closely the equation was satisfied. Subsequent attempts could be 
made nearer and nearer to the true solution. Fortunately, we do not need such great accuracy, 
and for our purcose it is sufficient to take Cc, as unity. Bridgeman's results snow that this is 
a good approximation. 
Then 
c.aT = (T-1,) 
Ty 
This simplification helos considerably in solving (14) by successive approximation. The first 
approximation is to put v = v, in the right hana side. Equate this value to 0.767 (T - T) to 
find < value for T. From Bridjeman's results read off a corresponding value for v. Using this 
value of v, substitute in the right hand side of (14); equate to 0.767 (T- Ts and so on. 
The process converjes rapidly. Of course, in the higher pressure regions, Bridgeman has no data, 
and the best that one can do is to study the trend of his results and make an inspired guess. 
As we have already stated, the pressure in our problem starts very high, but drops rapidly into 
the region covered by Bridgeman, and errors due to faulty approximations in the Initial stages 
cannot seriously affect the later course of the motion. 
For pressures up to about 2 x 107 kgmcm., of roughly 13 tons/inch the shock wave velocity 
and the mass velocity increase linearly with p. 
we find 
U = 1550 + 9p+.., u = 63.5p (1 — p/15,6) + .., 
where the units of velocity are metres oer second, and the units of p are 107 kgm/cm, Thus for 
small p, u and f(o) are the same to first order. 
The following table gives the shock wave velocity, the mass velocity behind the shock 
wave, AT, the rise in temperature from the Initial state to the state behind the snocx wave, and 
8 T, the rise in temperature due to an adiabatic compreesion from the initial state to the pressure 
behind the shock wave, 
| 20 | 2210 3000 | 3560 
ee ere ee Rss 
we have now enough information to describe how the actual calculations are made. 
TABLE 4. 
Section IV ..eee 
