WATER AT 
end, it is expedient first to consider certain 
quantities as functions of pressure and tem- 
perature, p and 7, or pressure and entropy, p and 
S. Before proceeding to a more detailed dis- 
cussion of the calculations it may perhaps help 
to orient the reader if we consider, qualitatively, 
contours of some pertinent quantities in the p-T 
plane. In Fig. 1, the possible states of a given 
fluid just behind the shock front lie along a single 
curve, which we have labeled ‘‘Hugoniot.’’ An 
element of fluid initially in the state (po, To) 
which has attained a state (p, 7) just behind 
the shock front finally returns to a state (po, 71) 
along the adiabatic, so labeled in the figure. Also 
included are the designations of a few points on 
a p—S basis using square brackets according to 
the convention introduced in Part b of this 
section. In general, 7; is larger than 7) because 
of the dissipation occuring at the front. The 
central part of our problem is the determination 
of the Hugoniot curve. 
We shall consider in Part a the calculations 
due to Arons and Halverson® which are intended 
to be accurate in the range of relatively low 
pressure (ca. 0 to 20 kilobars). These results as 
stated in the introduction are intended for the 
determination of the peak pressure of a shock 
wave from measured values of the shock front 
velocity U or particle velocity u. In Part b, we 
shall consider the calculations of Kirkwood and 
Richardson,® the results of which were originally 
intended for the applications of the shock wave 
propagation theory of Kirkwood and Bethe? 
which required data over a higher range of pres- 
sures (ca. 20 to 50 kilobars). 
a. Calculations of Arons and Halverson 
Here we outline the calculations’ suitable for 
the relatively low pressure range (ca. 0 to 20 
kilobars) based upon the equation-of-state and 
specific heat data discussed in detail in Appendix 
I. For the range 0 to 1.5 kilobars, the Ekman 
equation-of-state was used; in the range 0 to 25 
5 A. B. Arons and R. R. Halverson, Hugoniot Calcula- 
tions for Sea Water at the Shock Front, OSRD Report 
No. 6577, NDRC No. A-469. 
§ J. G. Kirkwood and J. M. Richardson, The Pressure 
Wave Produced by an Underwater Explosion, Part IIT, OSRD 
Se No. 813 (Dept. of Commerce Bibliography No. PB 
7J. G. Kirkwood and E. Montroll, Pressure Wave 
Produced by an Underwater Explosion, II, OSRD Report 
No. 676 (Dept. of Commerce Bibliography PB-32183). 
He FRONT 
375 
OF N SHOCK WAVE 787 
kilobars, the Tait equation-of-state, 
v(0, 1’) —v(p, T)/v0, T) = (1/n) logl1+p/B()], 
t=(T—273.16)°C. (2.7) 
In the first case, the initial temperature was 
to=15°C; in the second, to =25°C. In both cases, 
the initial pressure po>=0. Neither of the two 
equations-of-state are complete in the sense that 
vo=v(0, T) must be determined by auxiliary 
thermal expansion data (also discussed in Ap- 
pendix I). 
We express the enthalpy and volume incre- 
ments 
AH=H(p, T)—H(0, To), 
Av=v(p, T) —v(0, To), 
in terms of line integrals, first along an isobar 
from (0, 7) to (0,7) and, secondly, along an 
isotherm from (0, 7) to (p, T) (see Fig. 1). For 
the enthalpy increment we obtain 
AH =A,H+ArH, 
T 
(2.8) 
A,H= 4 G0; Pde ean 
= ha (2.9) 
P du(p’, 
artt= fi Ee 1)-T——— Jap’, 
0 
I na 
where c,(0, 7) is the specific heat extrapolated to 
zero pressure and ¢, is the mean of c, over the 
temperature range AT. 
For the volume increment we obtain 
Av=A,v-+Azrv, ‘ 
A,v=v(0, T) —v(0, To) =BoAT, 
Arv=v(p, Tp) —v(0, Jf), 
where Bp is the mean thermal expansion at zero 
pressure over the temperature range AT. 
From the last Hugoniot condition, Eq. (2.3), 
and Eq. (2.9) we obtain 
T [v(0, To) +(1/2) (Arv) |p —ArHT 
Cp— (1/2) (Bop) 
where ArH is to be calculated by means of the 
third of Eq. (2.9) and the appropriate equation- 
of-state, and where A7v is to be obtained from 
compressibility data. The right-hand side of Eq. 
(2.11) depends, of course, on the temperature T. 
The determination of AT is accomplished by the 
method of successive approximations. A trial 
(2.10) 
- (34) 
