A History of Research and a Review of Recent Developments

Loads on underwater structures 87
compared with the natural period of the structure, which is more likely when
the structure is very flexible, then the basis of damage assessment would be
the total impulse of the shock load.
We have already seen that the pulsating bubble phenomena associated with
underwater explosions result in the mass movement of water as well as pressure
pulses. The pressure pulse from a bubble at its minimum is greatest when
bubble migration is small. According to ref. [4.16] this meant for example
that for a 300 lb charge of TNT in a mine, the mine should be moored 14 feet
above the seabed. The mass motion of water that accelerates radially outwards
from the expanding bubble is an important factor when considering closerange
underwater damage, but for larger, distant explosions its effect is
considerably diminished. This is because the kinetic energy of the outflowing
water falls away as the fourth power of distance, whereas the energy in the
shock wave is reduced according to the second power of distance.
Christopherson [4.6], in summarizing the work in underwater explosions
in the UK during the Second World War, made the point that underwater
structures such as submarines are virtually unsupported, in that there is no
rigid support to ensure that the natural period of oscillation of the vessel is
short in comparison with the duration time of a high incident pressure. He
quoted the work of G.I. Taylor [4.17] who considered the oblique impact of
an underwater shock wave on targets with a variety of supporting forces. For
the simplest condition, in which a shock wave strikes normally a completely
unsupported underwater plane surface having mass m per unit area, he showed
that on the surface of the plate the total pressure (p) is given by
p/p 1=e –t/θ +φ(t),
(4.14)
where p 1 is the pressure in the incident shock wave, t is the time measured
from the moment of arrival, and θ is the time for the pressure to fall to half its
initial value. If the unsupported plane surface remains in contact with the
water in the incident wave it is possible to formulate its equation of motion,
and from this to show that
(4.15)
where e is the dimensionless parameter ρcθ/m, c is the velocity of sound in
water, and ρ is water density. The maximum velocity of the plane surface
occurs when the pressure has fallen to zero, i.e. when
(4.16)
and at this moment the water begins to exert a retarding force on the plane
surface. The plane surface then loses contact with the water and cavitation
occurs. Substituting for t/ρ in [4.15], and remembering that the net velocity