374 
786 
(p,,T) ae 
bs] 
Temperature, T 
nae 
Po 
(po+To) 
—-_ 
Pressure, p 
Fic. 1. Adiabatic and Hugoniot contours in p—T plane. 
velocity (U), and enthalpy increment (AH). 
These relations, when applied to data on the 
equation-of-state and specific heat, make it pos- 
sible to calculate u, U, and AH and to evaluate 
certain other functions applicable to the theory 
of the formation and propagation of shock waves 
originated by explosions.’ 
Precise knowledge of wu and U also makes it 
possible to calculate shock wave pressures in 
cases where the particle velocity or propagation 
velocity can be measured. The purpose of the 
calculations described below was to apply the 
Hugoniot relations to appropriate equation-of- 
state data for sea water in order to provide (a) 
tables of the desired functions up to very high 
pressures (ca. 80 kilobars) for use in the theory 
of propagation of underwater explosion waves,” 
and (b) tables of particle and propagation 
velocity at fairly close pressure intervals in a 
lower pressure region (up to ca. 14 kilobars). 
II. OUTLINE OF THE THEORY AND COM- 
PUTATIONAL PROCEDURES 
In this section we give an account of the 
hydrodynamical and thermodynamical relations, 
and the computational procedures leading to the 
numerical results tabulated in Sections III and 
IV. For the convenience of the reader a glossary 
of symbols is presented in Appendix III. 
When a shock wave advances with velocity U 
into a stationary fluid of unperturbed pressure 
2J. G. Kirkwood and H. Bethe, The Pressure Wave 
Produced by an Underwater Explosion (Dept. of Commerce 
Bibliography No. PB 32182), OSRD Report No. 588, 
Part I. 
RICHARDSON, ARONS, 
AND HALVERSON 
po and specific volume vo, the pressure p, specific 
volume v, and particle velocity u of the fluid 
behind the shock front are determined by the 
Rankine®-Hugoniot! conditions, which express 
the conservation of mass, momentum, and 
energy of an element of fluid passing through the 
front. For the purposes of this paper, these con- 
ditions may conveniently be written 
u=[(p—Po)(vo—») |}, (Zt) 
U=voL(p—po)/(vo—v) }, (2.2) 
AH = (i/2)(p— po) (v+00). (2.3) 
In the last equation, AH is the specific enthalpy 
increment of an element of fluid when it passes 
through the front. The specific enthalpy is de- 
fined as the sum of the internal energy per gram 
and the pressure-volume product, pv. 
Given equation of state and specific heat data 
for the fluid, any three of the variables p, v, U 
and u may be determined as functions of the 
fourth. Here we shall regard p as the independent 
variable. For certain hydrodynamic applications 
we must have, in addition to v, U, and uw as 
functions of p, the sound velocity 
c=(0p/dp)*s; p=1/2, 
the Riemann o-functiont 
(2.4) 
He f [oLe’, SVWelp', STldp’, (2.5) 
Po 
and the undissipated enthalpy 
w= f vLp’, S]dp’, (2.6) 
Po 
where S is the entropy. 
In practice, one must resort to successive 
approximations to effect a reduction of the 
Hugoniot conditions, combined with equation-of- 
state and specific heat data, to a set of relations 
expressing wu, U, and v as functions of p. To this 
3W. J. M. Rankine, Trans. Roy. Soc. London, A160, 
277 (1870). 
4H. Hugoniot, J. de l’ecole polyt. 51, 3 (1887); 58, 1 
(1888). 
+ The Riemann o-function occurs in Riemann’s form of 
the hydrodynamical equations, which, for the case of 
spherical symmetry, may be written (see reference 1): 
[(0/dt) + (c+) (0/dr) \(o+u) = —2cu/r, 
[(@/dt) — (c—u)(8/dar) \(e—u) =0, 
where ¢ is the time and r is the radial coordinate: The 
other quantities have already been defined. 
