1150 
532 
VI. IMPULSE AND ENERGY FLUX ASSOCIATED 
WITH THE BUBBLE PULSES 
20 
The bubble pulses have been defined as those 
parts of the pressure-time curve lying between 
times of successive bubble maxima. In practice, 
the times of bubble maxima are taken to be half- 
way between successive pressure peaks. This 
assumes that the time of expansion of the bubble 
is equal to the time of collapse. According to the 
theory of the bubble phenomenon,’ the period 
(or half-period) is dependent on the amount of 
energy available for the oscillation. Since the 
bubble is continually radiating acoustic energy, 
the bubble expansion has more energy associated 
with it, and therefore actually lasts longer than 
the following contraction. Our approximation 
can be justified, however, because most of the 
radiation occurs in a relatively short length of 
time near the bubble minimum, and during the 
major portion of a cycle the bubble has nearly 
constant energy. The difference between the time 
of expansion and contraction should therefore be 
very small. 
Composite curves ot the first two bubble 
pulses from the series of measurements reported 
in reference (2) are reproduced in Fig. 11. 
The particular composites shown are for a depth 
of 500 ft. and W!/R equal to 0.352, the gauges 
being positioned to the side of the cylindrical 
charges used. The time scale has been reduced 
by the cube root of the charge size, thus: 
z=t/W}. (37) 
21 
As in the case of the shock wave, it is possible 
to determine the nature of the net impulse de- 
livered by a bubble pulse from theoretical con- 
siderations. At the time of a bubble maximum 
the following condition holds at the bubble 
surface: 
(mM 
f Apdt, (38) 
0 
where APy is the pressure in the gas bubble. 
The impulse as measured at Ay would there- 
AP 1 
poCo 
um= 0 
poA m 
‘Bernard Friedman, Theory of Underwater Explosion 
Bubbles, Report IMM-NYU 166, Inst. for Math. and 
Mech., New York University, September 1947. 
A. B. ARONS AND D. R. YENNIE 
fore be: 
Iam = —(AmAP x1/ Co). (39) 
The impulse varies inversely as the radius (allow- 
ing for the time lag due to finite velocity of 
propagation), so that the impulse at radius R 
would be 
Ir=—(Ax?APy/RC)). 
The incremental impulse delivered at a radius Ri 
between the times of first and second bubble 
maxima would therefore be 
ATp=(—An2?AP2/ RCo) 
—(—AmAPm1/RC)). 
(40) 
(41) 
The terms in parentheses are inherently small 
and positive since the AP's are small and nega- 
tive. The first term is smaller than the second in 
magnitude because both Ay2 and APy2 are 
smaller than the corresponding quantities in the 
second term. AZp, which is the net impulse de- 
livered between the first and second bubble 
maxima, should therefore be small and negative. 
The same statement is, of course, true for the 
second and succeeding bubble pulses. The nega- 
tive impulses delivered in this manner should 
ultimately cancel the net positive impulse de- 
livered by the shock wave (see Section 16). 
This treatment neglects the finite amplitude 
of the wave and other effects such as turbulence 
and migration of the bubble. The effect of these 
factors on the impulse is difficult to ascertain, 
but it is believed that the results of the above 
discussion are in any case qualitatively correct. 
Integrations of Fig. 11 show that the positive 
impulse delivered by the first bubble pulse is 
1.076-lb. sec./in.? lb.t, while the net impulse for 
the whole pulse is +0.106-lb. sec./in.? Ib.!. 
Although the net impulse appears to be positive 
in contradiction to Eq. (41), a base line shift of 
the order of 5 lb./in.? in Fig. 11 could make the 
impulse come out zero or even negative. This is 
the order of magnitude of the error in originally 
determining the base line on the photographic 
records. 
If the net volume flow from the time of first 
bubble maximum to second bubble maximum is 
calculated from Eq. (35), 
4oR poe t 
V= if Lf apdt lr, 
Po tM, ) 
(35) 
