UNDERWATER EXPLOSION PHENOMENA 
where n =7.15 and B(S) =44,400 Ib. /in.?. v; is the 
final specific volume after return to hydrostatic 
pressure, and 1 is the specific volume immediately 
after the passage of the shock wave. This equa- 
tion of state leads to the following formula‘ for 
the dissipated enthalpy increment: 
Bu, V1 be n+1 Vy wl v 
10) 1 
2 v n—1 v Vv} 
B(v1—v WW 
pee) “1(“) -1| (26) 
2 v 
The term v appearing here is the specific volume 
of the fluid before the arrival of the shock front. 
The last term on the right is relatively small and 
may be neglected for shock pressures under 
40,000 Ib. /in.?. 
Figure 2 shows a plot of / as a function of the 
shock pressure, as computed from Eq. (26). 
13 
In a spherical wave the energy dissipated be- 
tween two spherical shells is given by 
Re 
Be ‘i R°H(AP.)dR, (27) 
Ri 
where h(AP,) is the dissipated enthalpy incre- 
ment at pressure AP,, and AP, is the excess shock 
pressure at distance R from the origin. 
This integral can be evaluated in the low pres- 
sure region by use of Eq. (25) and at higher 
pressure by use of Eq. (26), providing experi- 
mental data are available, giving AP, as a func- 
tion of the radial distance R. 
Reliable pressure-distance data, based on 
piezoelectric measurements, are available up to 
pressures of 20 to 30 thousand Ib./in.2. A few 
experimental points based upon measurement of 
spray dome velocities are available at higher 
pressures, but values for the region between the 
surface of the charge and the 30,000 lb./in.? 
pressure level must be based principally upon 
theoretical calculations such as those of Kirk- 
wood, Bethe et al.,5 or Brinkely and Kirkwood.® 
Available estimates of the pressure in the 
water at the surface of the charge range from 30 
6S. R. Brinkely and J. G. Kirkwood, Phys. Rev. 71, 606 
(1947). 
1143 
525 
to 50 kilobars. The calculations cited® * both lead 
to values close to 36 kilobars. 
In Fig. 3 the pressure-radius similarity curve 
for TNT is shown extrapolated back to two 
arbitrary values at the charge surface. The value 
of 36 kilobars or 520,000 Ib./in.? is considered to 
be the order of magnitude of the actual peak 
pressure. The similarity curve extrapolated to 
1,000,000 Ib. /in.? is given for purposes of compari- 
son as a possible upper limit of error. Isolated 
experimental values from dome velocity measure- 
ments are plotted on the same figure. 
Using values from Fig. 3, the integrand of 
Eq. (27) is shown plotted in Fig. 4. For the solid 
curve of this figure the empirical relation 
x?h =1230x-!*8 (where x=R/W}) satisfactorily 
represents the integrand between the limits of 
x =0.136 and «=10. The dashed curve represents 
an upper limit to the integrand based on a peak 
pressure of 1,000,000 Ib./in.? at the charge. This 
curve is drawn to indicate the possible extent of 
the error in calculating the energy dissipation. 
From the empirical relation given above, the 
energy dissipated between any two spherical 
surfaces may be readily calculated: 
Ep 72 
—=Arpy f 1230x-!"38dx 
WwW 
= 
=4,300,000[ (W!/R,)° 8 — (W/R2)°8 ] 
in.-lb. 
Ib. chg.” 
(28) 
where W is charge weight in. lb. and R is in ft. 
ve 
DISSIPATED ENTHALPY INCREMENT h (LB) 
Ce 0 Ci o 
Pak PRESSURE HL BsiM®) 
Fic. 2. Dissipated enthalpy increment, h, versus shock front 
pressure, AP,. 
