1140 
522 A.B: 
and Co is replaced by the propagation velocity U. 
This is equivalent to treating the fluid at time ¢ 
behind the shock front as though it had just 
passed from its unperturbed state through a 
shock front of corresponding amplitude Af, i.e., 
directly along a Hugoniot curve from 0 to Ap. 
Strictly speaking this is not correct, since the 
particle of fluid under consideration has actually 
passed through a shock front of greater amplitude 
at a preceding time and has returned to the 
pressure Ap along an adiabatic. Also neglected is 
the effect of spherical divergence on the particle 
velocity, u. However, the above substitution is 
introduced as a first approximation, and since the 
correction is quite small for all practical cases, the 
approximation is probably adequate. 
Making the substitution discussed above in the 
first term of Eq. (5) and neglecting $v? and Ay in 
the second term, one obtains, after algebraic 
manipulation: 
4rR? 41 (Ap)? 
E f cheer 
p Ap 
0 0 7 Sees 
poU 
+ if [ae fanaa A (6) 
For comparison, it is convenient to state the 
result yielded by Eq. (5) if 4u? and An are neg- 
lected throughout and Co is not substituted 
by U: 
1 fh 
E=4nR'| it (Ap) 2dt 
poCo “0 
+f [ae f° apaear. (7) 
It will be shown in Section 10 that Eq. (6) 
differs from Eq. (7) only in that the first term 
contains a correction factor which does not de- 
part from unity by more than a few percent at 
pressures as high as 20- or 30-thousand pounds 
per square inch. 
+ Note that the second integral of Eq. (6) may also be 
written in the form: 
mld. pat] 
this form being more useful for purposes of computation. 
ARONS AND D. R. YENNIE 
In acoustic theory Ap varies inyersely as R, the 
distance from the origin. Inspection of Eq. (7) in 
the light of Eq. (2) therefore indicates that the 
first term on the right-hand side represents 
principally “‘radiated’’ energy associated with 
what will be termed the “irreversible energy 
flux,’’ while the second term represents energy 
stored reversibly in the region covered by the 
shock wave. It should be noted that the small 
contributions made by 4x? and Ay to the first 
term are also reversible. 
Ill. THE FIRST TERM FOR THE ENERGY FLUX 
IN THE SHOCK WAVE 
To complete the development of expressions 
necessary for the interpretation of pressure-time 
data, we return to that part of the energy flux as 
given by the first term of Eq. (6): 
ley (Ap)? 
ei f ~ dt 
poro U—(Ap/poV) 
(8) 
The Rankine-Hugoniot conditions give a rela- 
tion for U in terms of Ap at a shock front: 
U=v9(AP,/v9 —2)}, (9) 
where AP, is the excess pressure at the shock 
front and vp and v are the specific volumes of the 
fluid ahead and behind the shock front, re- 
spectively. 
Combination of Eq. (9) with certain thermo- 
dynamic relations and with equation of state 
data‘ makes it possible to calculate U for corre- 
sponding arbitrary values of AP,. Such calcula- 
tions have been made for sea water at quite 
closely spaced values of AP,, and the results can 
be represented empirically by the following ap- 
proximate fit: 
U=C[1+5.6X10-8AP, 
—17X10-"(AP,)?], (10) 
where the excess pressure is expressed in 1b./in.?. 
For the purpose of obtaining a first-order cor- 
rection to the energy flux, the propagation 
velocity may be represented approximately by 
the following linear relation: 
U=C)[1+eAP,], 
U=C,[1+5.3 X10-®AP, ]. (11) 
4J. M. Richardson, A. B. Arons, and R. R. Halverson, 
J. Chem. Phys. 15, 785 (1947). 
