1138 
520 Agog Be 
1 [I= l l 
Second 
Pulse 
First 
Pulse 
Calibration 
Step 
ARONS AND D. 
R. YENNIE 
| | ] Fal 
Third Fourth Fifth Sixth Seventh 
Pulse Pulse Pulse Pulse Pulse 
1 kc. Timing Wave-——————_—_ 
Fic. 1. Pressure-time record showing shock wave and bubble pulses. Charge: 0.505-lb. TNT; 
gauge dist: 2.25 ft.; depth: 500 ft. 
loss occurs on re-expansion, resulting in a value 
Axes (for the second maximum in the bubble 
radius) considerably smaller than A 1. 
3 
In addition to the energy observed in waves of 
compression and in reversible interchange be- 
tween the gas globe and the water, energy is also 
being continually dissipated because of various 
other factors such as: 
(i) radiation and conduction from gas globe, 
(ii) viscous loss in the water, 
(iii) turbulent loss in the water. 
4 
An observer at a fixed point (radial distance R 
from the center of the charge) would see a “‘flow”’ 
of energy past his point in the positive or nega- 
tive direction of R, depending upon the phase of 
the explosion phenomenon. Piezoelectric gauge 
measurements provide a record of pressure vs. 
time at just such a point of observation. In order 
to investigate the energy flux, it is necessary to 
have a relation between energy and the variables 
pressure and time. These relationships are de- 
veloped in the following chapter. 
The experimental results to be quoted subse- 
quently were all obtained from measurements 
on TNT. 
Il. GENERAL EXPRESSIONS FOR ENERGY FLUX 
by tS 
sn this report the energy flux will be defined as 
the amount of energy that passes through a unit 
area of surface during a given time interval. In 
the case of spherical symmetry this energy flux 
pertaining to events up to the first bubble maximum, since 
migration never becomes appreciable until the phase of 
bubble collapse. 
will be uniform over the surface of a sphere, and 
the total flow of energy through the spherical 
surface will be given by: 
E=4nR°F. (1) 
The symbol E will be used to designate the 
total energy flow through the surface, and a 
component of the total will be indicated by 
attaching a suitable subscript. The symbol F will 
be used in a similar way to represent the energy 
flux. 
In general, the energy flux will be found to 
consist of a number of components varying 
inversely as some power of the radius: 
F=F,/R"+ F,/R™+F3/R™-+--- etc. (2) 
If the value of an exponent m happens to be 2, 
Eq. (1) shows that the total energy flow, £, 
corresponding to that term will be the same at all 
radii and therefore the energy will be radiated 
away to infinity. If the value of is greater than 
2, the total energy flow will be smaller at larger 
radii, indicating that some energy has been left 
behind in the water. In some cases the energy 
left in the water is in the form of kinetic energy 
or other undissipated forms and therefore should 
be returned ultimately to the gas globe. In other 
cases it represents an energy which has been 
dissipated irreversibly and goes into heating of 
the water. 
6 
The general expression for what might be 
termed the ‘‘useful” total energy flow through a 
sphere of radius R relative to the center of the 
explosive charge is given by the following time 
integral : 
 7Ap 
b=4eR: [ (—+412+4n) pude, (3) 
0 p 
