D. E. Weston 57 
3.5. MEASUREMENTS AND THEORY FOR UNDERWATER EXPLOSIONS 
Experimental results have been arrived at using octave filters followed by 
small analog computers, which compute {v7dt, where v is the input voltage. The 
results in Fig. 3.3 show some measured differences between charges, with an 
average accuracy of +1 db. They show that there is no simple law, that there is 
a real variation with depth, and that there is general agreement with the theory 
(to be described later). It may be noted that at high frequencies the 50-lb results 
are in the region of the w3 or 111/,-db law, and that at low frequencies the de- 
tonator results are approaching the W? or 54-db region. 
For close shots it is possible to estimate the transmission loss theoretically 
and deduce absolute source levels from the measured signals. Some results are 
shown in Table 3.II]; there is little depth dependence above 140 cps. The 
accuracy is a little worse than that for the differences. 
Results for other charge sizes may be obtained by adding in the measured 
differences, and it then becomes possible to test the w'4 and w% scaling laws 
for the whole spectrum by shifting these spectra as in Fig. 3.1. It may be noted 
that the similarity and scaling laws of section 3.3 should apply to the bubble 
pulses as well as to the shock, provided that the depth is constant and there are 
no appreciable secondary effects due to gravity, etc. The result is shown in 
Fig. 3.4, all points lying well on a single line. Some of the minor discrepancies 
can be explained, e.g., the low-frequency 50-lbvalues are low in level because 
bubble migration under gravity suppreses the bubble pulse. The spectral peak 
at the reciprocal of the bubble-pulse time is evident for the detonator results 
in both Fig. 3.3 (70 to 160 cps) and Fig. 3.4 (8 cps). The agreement with theory in 
Fig. 3.4 may be noted. 
So far it has been demonstrated that the scaling laws work, and now the more 
detailed theory for the spectrum will be presented. This comes from a Fourier 
analysis of the shot pressure-time curves given by Arons [1, 2]. The shock wave 
may be represented as a sharp rise in pressure Po, followed by an exponential 
decay with time constant ft). Either bubble pulse may be approximated by two 
similar back-to-back exponentials, with peak pressure P; and time constant ¢, 
for the first bubble. The pulse shapes and the resulting energy-spectrum equa - 
tions are shown in Table 3.IV (where pc is the characteristic impedance). The 
average total spectrum at high frequencies is obtained by incoherent addition of 
the shock and bubble contributions, as shown in Fig. 3.5. At low frequencies the 
shock and bubbles may be replaced by their impulse values I,,/,, and 1, occur- 
ring at time intervals T, and T; = T, + T, together with a steady negative pressure. 
The results are also shown in Table 3.IV and Fig. 3.5. 
TABLE 3.1I]. Free-Field Source Spectrum Levels for a 1-lb TNT Charge at 
60 Fathoms 
Frequency in cps 
Energy flux density in db 
- 13.9 
re 1 erg/cm’—cps at 100 yards ue 
