Practical Ship Hydrodynamics

Ship seakeeping 103
Re denotes the real part of a complex quantity; i D p 1; z as in Fig. 4.1;
Oh denotes as usual a complex amplitude; Oh D the complex amplitude of the
wave. h DjOhj is the (real-valued) wave amplitude, i.e. half the wave height
(from wave trough to wave crest). The real part of Oh gives the distance of
the wave trough from the calm-water level at time t D 0atxD 0; the imaginary
part gives the same value at 1
4 period earlier. The deep-water formulae
are applicable with errors of < 1
2 % if the water depth is larger than half a
wave length.
The velocity is obtained by differentiation of the potential, e.g. for deep
water:
vx D ∂
∂x D x D Re⊲ ωOhe kz e i⊲ωt kx⊳ ⊳
vz D ∂
∂z D z D Re⊲iωOhe kz e i⊲ωt kx⊳ ⊳
The complex amplitudes of the velocities have the same absolute value and
a phase shift of 90°. A water particle thus follows a circular track or orbital
motion (from Latin orbis D circle). In water of finite depth, the motion of a
water particle follows an ellipse. The vertical axis of each ellipse decreases
with depth until at the water bottom z D H the motion is only in the horizontal
direction.
If we excite a group of waves (not elementary waves, but, e.g., 10 wave
crests and troughs) in initially calm water we will notice that the front of the
wave crests decay while at the end of the wave packet new wave crests are
formed (Fig. 4.2). The wave packet thus moves slower than the wave crests,
t
C
C/2
Group
X
Figure 4.2 Celerity and group velocity