UNDERWATER EXPLOSION PHENOMENA 
where /= time measured from instant of incidence 
of the pressure wave, ¢;=arbitrary upper limit of 
integration, Ap =excess pressure (p— Po), a func- 
tion of time, Po=absolute hydrostatic pressure 
level at point of detonation, «= particle velocity 
relative to the unperturbed fluid, p=density of 
the fluid at time ¢, An =increase in internal energy 
of a unit mass of fluid relative to the initially 
unperturbed state. 
The term in Ap represents a compressional 
energy capable of doing work against the fluid 
external to the sphere through which it passes. 
The second term represents the kinetic energy of 
the mass of fluid moving past the point of 
observation, while the third term in An represents 
the increment of internal energy of this mass. It 
will be shown that at values of R which are large 
relative to the initial charge radius, the last two 
terms are both very small compared to the term 
in Ap. 
7 
The type of experimental data available for 
making an analysis of the energy flux consists 
mainly of pressure-time curves recorded by 
means of piezoelectric gauges.! 
Such data have previously been available for 
the shock wave to times of the order of 100, where 
6 is the time constant of the initially exponential 
shock-wave decay. Additional data have recently 
been obtained from bubble pulse measurements,” 
making it possible to extend the pressure-time 
curve from 106 through the second bubble pulse. 
If w in Eq. (3) can be expressed in terms of the 
variables Ap and ¢, it becomes possible to evaluate 
energy transfer from the primary pressure-time 
data. A rigorous development would involve the 
exact solution of the hydrodynamical equations 
of motion. Such a solution would involve ex- 
ceedingly laborious and complicated numerical 
integrations which are impractical except in a 
very few special cases. 
Fortunately the compressibility of water is 
sufficiently low to make the so-called acoustic 
approximation useful (and probably adequate) 
for the treatment of pressure-time data in the 
1J.S. Coles, OSRD Report No. 6240. 
2A. B. Arons, J. P. Slifko, and A. Carter, J. Acous. Soc. 
Am. 20, 271 (1948), and A. B. Arons, ibid. 20, 277 (1948). 
1139 
521 
to experimental 
region normally accessible 
measurement. 
8 
The acoustic approximation in a case of 
spherical symmetry yields the following relation- 
ship for the particle velocity, u:4 
t 
Fp Rishi if Apdt, (4) 
0 
where 
po=the density of the unperturbed fluid, 
Co = (0P/dpo) so}, the velocity of sound in the fluid. 
The second term on the right-hand side of Eq. (4) 
is frequently referred to as the afterflow term in 
the particle velocity. 
Combining Eqs. (4) and (3) we find 
7A A 
E=4rR? f (+4044) ae 
op poCo 
th A 1,2 A t 
+f [" p/p) +3u?+ = aaa, 6) 
0 poR 0 
The sum (4u?+An) is small relative to Ap/p, as 
will be shown‘in Section 10; at low values of Ap it 
can be neglected entirely. A correction is justified 
at high values of Ap, but only the first term in 
Eq. (5) requires this correction since the second 
term is initially zero and does not acquire ap- 
preciable value until after the elapse of a certain 
amount of time, during which the pressure is 
decaying very rapidly. 
The Rankine-Hugoniot conditions*** afford 
relationships for u, U, and Ay in terms of Ap and 
the change in density across a shock front. (U 
represents the velocity of propagation of the 
shock.) 
An and wu in the first term of Eq. (5) are then 
replaced by the appropriate Hugoniot relations 
3H. Lamb, Hydrodynamics (University Press, Cam- 
bridge, 1932), p. 490. 
*** The Rankine-Hugoniot conditions are obtained by 
application of the laws of conservation of mass, momentum, 
and energy at the shock front (see reference 3). These con- 
ditions are: 
U=v0((p— Po) /(vo—v))}, 
u=((p—Po)(vo—v))}, 
An=4(Po+p)(v0—»), 
AH =An+A(pv) = 4(p— Po) (vo+v). 
