1137 
Reprinted from Reviews or Mopern Puysics, Vol. 20, No. 3, pp. 519-536, July, 1948 
Printed in U. S. A. 
Energy Partition in Underwater Explosion 
Phenomena* 
A. B. Arons AND D. R. YENNIE 
Woods Hole Oceanographic Institution, Woods Hole, Massachusetts 
and 
Stevens Institute of Technology, Hoboken, New Jersey 
Pressure-time curves, 
continuous from initial shock wave incidence through the second 
bubble pulse, are examined in the light of acoustic theory. Calculations of impulse and reversible 
and irreversible energy flux are made for the various phases of the phenomenon. An estimate 
has been made of the amount of energy dissipation associated with the propagation of the 
shock front. A tabulation of the energy partition is included, and it is shown that substantial 
quantities of energy are radiated or dissipated by mechanisms other than these taken into 
account in this discussion. 
I. INTRODUCTION 
1 
OST of the energy released by the detona- 
tion of an explosive charge is ultimately 
imparted to the surrounding medium and be- 
comes distributed among the various phases of 
succeeding phenomena. It is the purpose of the 
present investigation to examine the distribution 
of this energy in underwater explosions, particu- 
larly from the point of view of making a complete 
interpretation of data now available in the form 
of pressure-time curves. 
In general, underwater explosions are charac- 
terized by the emission of a shock wave followed 
by a series of pressure pulses caused by subse- 
quent oscillations of the gas globe containing the 
products of detonation. A typical pressure-time 
record is reproduced in Fig. 1. 
At the instant following detonation the re- 
leased energy is present in the form of potential 
energy of exceedingly high pressure and tempera- 
ture in the resulting volume of gas. As the gas 
proceeds to expand it transfers energy to the 
water. Part of this energy is ‘“‘radiated"’ in the 
sense that it is not stored as reversible potential 
energy in the water. Rather, it is gradually 
dissipated by conversion into thermal energy 
which elevates the temperature of the fluid 
through which the pressure wave is propagated. 
* This investigation was supported under contract with 
the Navy Department, Bureau of Ordnance. Contribution 
No. 449 from the Woods Hole Oceanographic Institution. 
2 
The remainder of the energy transferred to the 
water is imparted to it as kinetic energy, the 
water being pushed radially outward against the 
opposing hydrostatic pressure. The gas globe ex- 
pansion continues until the energy available to 
this phase of the motion is stored as potential 
energy in the water. At this point the gas bubble 
has attained its maximum radius, and its in- 
ternal pressure has fallen well below that of the 
surrounding hydrostatic level. The potential 
energy now stored in the water is given by 
E,=(4/3)rA m1°Po, 
where A yj is the maximum bubble radius and Po 
is the absolute hydrostatic pressure. This will be 
referred to as a “‘reversible’’ energy, since it is 
returned to the gas globe on the succeeding 
collapsing phase. 
The collapse of the bubble and the following 
expansion are characterized by emission of the 
first bubble pulse, in which part of the energy E, 
is again radiated ‘‘acoustically.”. Thus all the 
potential energy stored in the water is not re- 
turned to the bubble as compressional energy in 
the gas** at minimum bubble size. An additional 
** If the charge is detonated under conditions such that 
the gas bubble undergoes appreciable migration because of 
the influence of gravity or boundary surfaces, a substantial 
part of E, will be imparted irreversibly to vertical motion 
of the water. The condition principally considered in this 
report is that in which the bubble migration is negligibly 
small. This, however, has no bearing on considerations 
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