UNDERWATER 
Equation (10) fits the results of reference (4) 
quite closely from zero to 60,000 p.s.i., while 
Eq. (11) represents a rough average fit of a 
straight line in the region from zero to 40,000 
p.s.i. Equation (11) falls slightly helow the true 
values at low and somewhat above at high 
pressures. 
Inserting Eq. (11) into Eq. (8), and carrying 
the result to first-order terms, the following ex- 
pression is obtained: 
1 t 1 
r= [ (ap)|1-(a--) per, (12) 
polo Yo K 
where k = poCo*. 
The same expression may be derived by ap- 
plying first-order corrections directly te the 
general expression given in Eq. (5): 
4 (Ap 
A= f (— +40 +n) out, (13) 
0 p 
where u is given by the Rankine-Hugoniot 
conditions 
u=(Ap/poU), 
and the internal energy increment, Ay, is ap- 
proximately the compressional energy of the 
fluid : 
An =3(Ap)?/kpo, (14) 
« being the bulk modulus. The kinetic energy per 
unit mass is 
3u? = }(Ap)*/po?U?~ $(Ap)*/po?Co? 
= 2(Ap)?/pox =An. 
Equation (12) is verified by substituting these 
relations in Eq. (13) and carrying the results to 
first-order terms. 
In its present form, Eq. (12) is unwieldy be- 
cause it requires integrations of both (Ap)? and 
(Ap)’. The term in (Ap)3, however, introduces 
only a small correction, and it is convenient to 
represent the average value of this correction in 
terms of one convenient parameter, such as the 
peak pressure of the shock wave. To do this, it is 
assumed that the shock-wave pressure varies 
exponentially with time. 
Ap=Pre "9, 
(15) 
EX PLUSLON 
1141 
PHENOMENA $23 
Although this is not a correct representation of 
the whole wave, it is (rue for times up to 1=8, 
from which region most of the correction to the 
energy flux actually stems. Therefore, applying 
this approximation 
) P,,*6 
if (Ap)*dt= i 
0 2 
zr 'P 4p Dn 
uf (Ap)*dt= = Bef (Ap) dt. 
0 3 n 
The first energy flux term then becomes 
1 1 " 
Fr=—-{1-3(a- Jen] f (Ap)*dt. (16) 
polo A ‘ 0 
The correction represented by the second term 
in the bracket is small (the order of a few percent), 
so that even though the correction itself might be 
subject to a large error because of the crudeness 
of various approximations, the final result for 
energy flux should not be greatly in error. For P,, 
in lb./in.?, and F; in in. lb./in.2, Eq. (16) may be 
written : 
1 4 
Fy\= [1-1.6x10-"Py] f (Ap)*dt. (17) 
0 
Polo 
Equation (17) is based on U as given by Kq. (11). 
This equation can now be used in the computation 
of the energy flux given by the first term of 
Eq. (5). 
This approximation can easily be carried to 
second-order terms, although for most applica- 
tions this is an unnecessary refinement. The 
result to second order is 
Fy,= [1—1.8X10-§P,, +4 10-!2P,,?] 
poCo 
xf cane. (18) 
Equation (18) is based on U as given by Eq. (10). 
If pressures are expressed in Ib./in.? and time 
in seconds, F, is obtained in in. lb./in.? by using 
poCo=5.58+0.00657, (19) 
where 7 is temperature of the water in degrees 
centigrade. 
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