A History of Research and a Review of Recent Developments

Underwater explosions 51
Figure 2.19 Measured and calculated radius of the gas sphere from a detonator one
foot below the surface (from Herring, ref. 2.33).
Since the total energy associated with the radial flow of water (Y) is given
approximately by
(2.19)
it is possible to express the first oscillation period in terms of total energy by
(2.20)
This is a formula attributed to H.F.Willis, a British scientist, and shows that
the period is directly proportional to the cube root of the explosive charge
weight, since for most high explosive the total energy varies with charge weight.
The period also varies as (d+33) –5/6 , where d is the water depth in feet, and this
was verified experimentally by Ewing and Crary [2.32] who brought together
the results of several studies to produce Figure 2.20. It was also shown that to
a close approximation the periods of the first and second bubble oscillations
are related by the formula
(2.21)
where T 1 and T 2 are the periods, and r 2 and r 1 are the fractional energies
remaining after the first and second contractions.
For depths greater than 320 feet T2/T 1 is very nearly constant and independent
of depth, and is equal to about 0.6. This gives an energy ratio of 0.36.