A. B. ARONS AND D. R. YENNIE 
Fic. 6. Composite pres- 
sure-time curve for tail 
of shock wave. Explosive: 
(es 
TNT; charge depth: 500 
ft.; distance from center 
of charge: R=W1/0.352 
Legend: @ time constant of 
initial shock-wave decay. 
e@, X, ©0.5-, 2.5-, and 12.0- 
Ib. charges based on meas- 
EXCESS PRESSURE 
urements of reference 2. 4 
Points from  shock-wave 
composites obtained by J 
S. Coles et al., Woods Hole. 
REDUCED TIME T (t/e) 
=tmu1, Um1=0 and from Eq. (30) 
mi AyiAp 
0 Cc 
0 
(31) 
Since Ap is negative in this region, Ji is 
inherently positive. Its magnitude is very small 
compared to that of the total positive or negative 
impulse. 
17 
Combining Eqs. (6) and (17), the complete 
expression for total energy flow to time ¢ be- 
comes 
4rR? 
E= 
[« —1.6X10-®P,,) ii (Ap) *dt 
0 
iG t 2 
= A ; 
HRS? 
As previously indicated, the first term in the 
bracket increases monotonically with increasing 
time of integration and represents energy radi- 
ated acoustically, while the second or afterflow 
term represents energy which is stored reversibly 
in the water and returned at intervals to the gas 
bubble. This term attains a maximum at time ¢ 
corresponding to the end of the positive phase 
and then decreases, becoming virtually zero at 
t=tm, since it involves the squaring of the small 
residual impulse given by Eq. (31). In the later 
stages of the positive phase at distances fairly 
close to the charge, the afterflow term predomi- 
nates over the irreversible term. 
Although in the limit of low pressures the 
poCo 
afterflow term represents the contribution of 
essentially incompressive flow consequent upon 
the bubble expansion, it cannot be regarded as a 
purely incompressive term throughout the in- 
tegration. Incompressive and compressive effects 
are not dissociable in the acoustic approximation, 
and in the region just behind the shock front 
the afterflow term represents principally a com- 
pressive contribution due to the radial divergence 
of the flow initiated by passage of the wave of 
compression. 
18 
Figures 5, 6, and 7 are composite pressure- 
time curves for TNT at the depths of 250 and 
500 ft. and at a distance from the charge given 
by W!/R=0.352. Several charge sizes have been 
plotted on the same curve by scaling the time in 
terms of 6, the time constant of initial shock 
wave decay which is given bytt 
6=0.060W1(W1/R)-°8 
=0.0725W? millisec. at W#/R=0.352. (33) 
Using this scale factor, the reduced time 7 is 
defined by 
7=t/0. (34) 
t is, of course, a dimensionless quantity. Any 
other scale factor proportional to W#, such as 
the bubble period at the given depth, might 
equally well have been used. 
From the pressure-time curves in Figs. 5, 6, 
and 7, certain quantities (listed below) have been 
+t Equation (33) isan empirical fit of TNT data obtained 
at Woods Hole by J. S. Coles et al. 
