Practical Ship Hydrodynamics

102 Practical Ship Hydrodynamics
z
Figure 4.1 Elementary waves
l
x
H
X
y
These are, e.g., the broader wave troughs and steeper wave crests, the higher
celerity of steeper waves which results in a tendency to form wave groups
in natural wind seas: groups of waves with low wave height are followed by
groups of waves with larger wave heights.
For ship seakeeping, the relevant waves are dominated by gravity effects and
surface tension, water compressibility and (for deep and moderately shallow
water) viscosity can be neglected. Computations can then assume an ideal
fluid (incompressible, inviscid) without surface tension. Consequently potential
theory can be applied to describe the waves.
Generally, regular waves are described by a length parameter (wave length
or wave number k) and a time parameter (wave period T or (circular) frequency
ω). k and ω are defined as follows:
k D 2 ; ω D 2
T
The celerity c denotes the speed of wave propagation, i.e. the speed of an
individual wave crest or wave trough:
c D D
T ω
k
For elementary waves, the following (dispersion) relation holds:
k D ω2
g
on deep water k tanh⊲kH⊳ D ω2
g
z
x
c
on finite depth
g D 9.81 m/s 2 and H is the water depth (Fig. 4.1).
The above equations can then be combined to give the following relations
(for deep water):
c D
� g
k
D g
ω D
� g
2
D gT
2
The potential of a wave travelling in the Cx direction is:
D Re⊲ icOhe kz e i⊲ωt kx⊳ ⊳ for deep water
�
�
icOh
kx⊳
D Re cosh⊲k⊲z H⊳⊳ei⊲ωt
sinh⊲kH⊳
for finite depth