Page 118 - Introduction to Naval Architecture

SEAKEEPING 105

The resultant force acting on a particle in the surface of a wave must
act normal to the surface. If the wavelength is long compared to the
beam of the ship, and it is these longer waves which will cause the more
severe rolling, it is reasonable to assume that there is a resultant force
acting on the ship normal to an 'effective surface', taking account of all
the subsurfaces interacting with the ship. This is another useful concept
proposed by Froude, who further assumed that the effective wave slope
is that of the subsurface passing through the centre of buoyancy of the
ship. With these assumptions and neglecting the added mass and
damping, the equation of motion takes the form:

where <p' = a sin cat, a being the maximum slope of the effective
wave.
If the subscript 0 relates to unresisted rolling in still water it can be
shown that the solution to the equation for resisted motion takes the
form:

In this expression the first term is the free oscillation in still water and
the second is a forced oscillation in the period of the wave.
When damping is present the free oscillation dies out in time, leaving
the forced oscillation modified somewhat by the damping. In a truly
regular wave train the ship would, after a while, roll only in the period
of the wave. The highest forced roll amplitudes occur when the period
of the wave is close to the natural period of roll when it is said to
resonate. Thus heavy rolling of a ship at sea is mainly at frequencies close
to its natural frequency.

Pitching and heaving in regular waves
For these motions attention is directed to head seas as these are the
more severe case. It is not reasonable to assume the wave long in
relation to the length of the ship and the wave surface can no longer be
represented by a straight line. However the general approach of a
forcing function still applies.
When a ship heads directly into a regular wave train it experiences
hydrodynamic forces that can be resolved into a force at the centre of
gravity and a moment about that point. As with rolling the resulting
pitch and heave will be highest when the period of encounter with the
waves is close to the natural period of motion in that mode. When the