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Consider a floating body of weight which, in equilibrium, has a submerged volume . Thus, the body's downward weight is balanced by the upward buoyancy force,
: i.e.,
. Let be the crosssectional area of the body at the waterline (i.e., in the
plane ). It is convenient to define the body's mean draft (or mean submerged depth) as
. Suppose that the
body is displaced slightly downward, without rotation, such that its mean draft becomes
, where
.
Assuming that the
crosssectional area in the vicinity of the waterline is constant, the new submerged volume is
, and the
new buoyancy force becomes
. However, the weight of the body is unchanged. Thus,
the body's perturbed vertical equation of motion is written

(191) 
which reduces to the simple harmonic equation

(192) 
We conclude that when a floating body of mean draft is subject to a small vertical displacement it oscillates about its equilibrium position at the
characteristic frequency

(193) 
It follows that such a body is unconditionally stable to small vertical displacements. Of course, the above analysis presupposes
that the oscillations take place sufficiently slowly that the water immediately surrounding the body always remains
in approximate hydrostatic equilibrium.
Next: Angular Stability of Floating
Up: Hydrostatics
Previous: Equilibrium of Floating Bodies
Richard Fitzpatrick
20120427