Practical Ship Hydrodynamics

104 Practical Ship Hydrodynamics
i.e. with a speed slower than celerity c, namely with group velocity cgr:
c for deep water
�
1
cgr D c
2 C
�
kH
sinh⊲2kH⊳
for finite depth
The linearized Bernoulli equation
p C ∂
∂t
gz D p0
cgr D 1
2
and the wave potential give the difference pressure to atmospheric pressure at
a point below the water surface (for deep water):
p p0 D gz g Re⊲Ohe kz e i⊲ωt kx⊳ ⊳
p0 is the atmospheric pressure, the water density, z the depth of the point
below the calm-water surface. The first term represents the hydrostatic pressure
in calm water. The second term represents the pressure change due to the wave.
As with all wave effects, it decays exponentially with depth. The pressure
gradient ∂p/∂z is for the hydrostatic case equal to the specific weight of the
fluid and causes a buoyant lifting force on the immersed body that equals
the weight of the displaced water. This lifting force changes in a wave! The
lifting force is lower in a wave crest, higher in a wave trough. This is called
the Smith effect.
The mechanical energy E per area of the water surface is composed of
potential and kinetic energy. Let be the momentary elevation of the free
surface. Then the potential energy (per area) is:
Epot D g⊲ ⊳ D
2 1 2
g
2
The potential energy is positive both in wave troughs and wave crests and
oscillates in time and space between 0 and gjOhj 2 . The time average is
Epot D 1
4
gjOhj 2
The kinetic energy per area is:
Ekin D
� 1 1
2 ⊲v2x C v2z ⊳ dz D
� 1 1
2 ω2 jOhj 2 e 2kz dz ³
� 1
0
... dz D 1
4
gjOhj 2
Here the formulae for vx and vz have been used and in a linearization the wave
elevation was substituted by 0. The kinetic energy is constant in time and
space. The time-averaged total energy per area for a deep-water wave is then:
E D 1
2
The average energy travels with cgr in the same direction as the wave. For
finite-depth water the average energy remains the same but the kinetic energy
oscillates also in time and space.
The elementary wave was so far described in an earth-fixed coordinate
system. In a reference system moving with ship speed V in the direction of
gjOhj 2