238 
Sec. WE. 2a Hq. 87-88 
f. Acknowledgement. We should like to repeat at this point that 
the method described above is based on that given in Part II of our pre- 
liminary report, which was developed to a considerable extent by Dr. R. S. 
Halford. The numerical results, tables, etc., are, however, taken froma 
report by Dr. Duncan MacDougall and Dr. L. Epstein of the U. S. Bureau of 
Mines, which uses the modified equation of state. This report, (which 
will be available through the N.D.R.C.) should be consulted for further 
details. 
PART III. PROPERTIES OF SHOCK AND RARERACTION WAVES 
We now return to motions which involve no release of chemical enerzy. 
These motions can be divided into two categories; shock waves and rare- 
faction waves. It is particularly important to be able to compute the 
properties of the shock wave initiated in the surrounding medium by an 
explosion, since this shock wave is one of the factors causing damage. It 
is necessary, however, to understand rarefaction waves before the other 
problem can be treated. In what follows only the one-dimensional case 
will be considered, but it is hoped to report on the three-dimensional 
case later, 
12. Integration of the Inergy Equation 
So long as no discontinuities occur, it will be shown that the 
pressure of a given material point is a definite function of the density 
alone. This function may, however, be different for different material 
points but should not change with time (if one follows the material point 
in its motion) unless a discontinuity occurs. In general the function will 
change from one definite form to another on the passage of a discontinuity. 
If a given material point has a given Pe ) at time t, and another material 
point has the same P( ? ) at any time tp then evidently the two will always 
have the same P( Go), barring discontinuities. 
The proof involves the combination of the equation of state with the 
energy equation (6). From the equation of state and the heat capacity, 
etc., one can obtain E = E(P, C ). Then 
HE). Oke id Oma hin Puls ae (37) 
a gl = 4 CE = av 
at Omid ae at C2 at : 
an equation connecting aP/dt with 4 (C fat and known functions of P and C. 
oP (20) 
