UNDERWATER EXPLOSION PHENOMENA 
TABLE IV. Energy partition at time of second 
bubble maximum. 
Acoustic radiation in first bubble pulse 120 cal./g 
Potential energy in the water at time of second 
bubble maximum based on measured maxi- 
mum radius 120 
Internal energy of gas at second bubble maxi- 
mum: (B2—120) 60 
Unaccounted for 180 
Total energy associated with first maxi- 
mum (B;) 480 cal./g 
Total loss during emission of first pulse 
(180+ 120) 300 
Energy left for second pulse (Bz) 180 
this portion (95 cal./g) is much smaller than the 
combined uncertainty in the detonation energy 
and in the energy dissipated at the shock front, 
and therefore even its order of magnitude is in 
doubt. 
25. The Bubble Pulses 
From the theory of the bubble pulsation it is 
known that the period is proportional to the 
cube root of the total energy associated with 
the oscillation as defined in Section 24. Since the 
periods of successive oscillations decrease pro- 
gressively, it is evident that energy is lost be- 
tween successive bubble maxima. Using the cube 
root law stated above, it is seen that the energy 
left after the emission of a bubble pulse is 
given by 
Bryi=Br(Tr4i/T,)3, (44) 
where B,=total energy associated with the nth 
oscillation and T,,=period of th oscillation. 
The necessary period data? are summarized in 
Table III. 
Using Eq. (44), the data of Table III, and 
the maximum radius data quoted in Sections 21 
and 22, we obtain the energy partition for the 
first and second bubble pulses as given in 
Tables IV and V. 
Since the total energy associated with an 
oscillation and the energy of acoustic radiation 
are both known to within +3 percent, it is 
important to note the magnitude of the unac- 
counted terms in Tables IV and V. 
A summary of energy partition data is given 
1153 
535 
TABLE V. Energy partition at time of third 
bubble maximum. 
Acoustic radiation in second bubble pulse 15 cal./g 
Potential energy in the water at time of third 
bubble maximum 55 
Internal energy of gas at third bubble maxi- 
mum: (B;—55) 40 
Unaccounted for 70 
Total energy associated with second bubble 
maximum (B2) 180 cal./g 
Total loss during emission of second pulse 
=70+15 85 
Energy left for succeeding pulses (Bs) 95 
in Table VI. It is seen from Table VI that less 
than half the detonation energy is to be found 
in waves of compression, while somewhat more 
than half is lost in dissipative processes. 
It is difficult to ascribe any appreciable portion 
of the unaccounted 345 cal./g to dissipation simi- 
lar to that which was computed for the shock 
front. Figure 11 shows the pressure pulses to rise 
relatively slowly with time, and the resulting pro- 
cess should be very nearly isentropic on both com- 
pression and expansion. Furthermore the second 
pulse rises very much more slowly than the first, 
and yet the unaccounted portion in this pulse is 
an even greater fraction of the total energy loss 
than is the case in the first pulse. 
Because of the shortness of the time intervals 
during which temperature and pressure in the 
gas bubble are high, it is doubtful that losses of 
such magnitude could be attributed to conduc- 
tion or radiation of heat. 
We conclude, therefore, that the unaccounted 
for energy losses are associated with some com- 
bination of the following factors: 
(i) turbulence induced in the water surrounding the 
bubble, 
(ii) chemical or physical changes in the gaseous products! 
(iii) actual loss of gaseous products in the form of smal, 
bubbles in the water, perhaps due to high degree of 
turbulence at the periphery of the gas globe. 
TABLE VI. Summary of energy partition tables. 
Total acoustic radiation (through emission of 
second bubble pulse) at R= W#/0.352 410 cal./g 
Shock front dissipation up to R= W4/0.352 200 
Unaccounted losses 345 
Total energy left at third bubble maximum 95 
