Practical Ship Hydrodynamics

Ship seakeeping 113
The (only statistically defined) wave steepness D wave height/wave length
does not depend strongly on the wind velocity, Uc/cp, orω/ωp. The wave
steepness is so large that the celerity deviates noticeably from the theoretical
values for elementary waves (of small amplitude) as described above. Also,
the average shape of the wave profiles deviates noticeably from the assumed
sinusoidal wave forms of elementary waves. However, non-linear effects in the
waves are usually much weaker than the non-linear effects of ship seakeeping
in the seaway.
The significant wave height H1/3 of a seaway is defined as the mean of
the top third of all waves, measured from wave crest to wave trough. H1/3 is
related to the area m0 under the sea spectrum:
H1/3 D 4 p m0 with
� 1 � 2
m0 D
0 0
S ⊲ω, ⊳ d dω
For the above given wind sea spectrum, H1/3 can be approximated by:
H1/3 D 0.21 U2 c
g
The modal period is:
Tp D 2 /ωp
� � 1.65
Uc
cp
The periods T1 and T2, which were traditionally popular to describe the
seaway, are much shorter than the modal period. T1 corresponds to the
frequency ω where the area under the spectrum has its centre. T2 is the average
period of upward zero crossings.
If we assume that water is initially calm and then a constant wind blows
for a duration t and over a distance x, the seaway parameter Uc/cp becomes
approximately:
Uc
cp
D max⊲1, 18
3/10 , 110
3/7 ⊳
is the non-dimensional fetch x, the non-dimensional wind duration t:
D gx/U 2 c ; D gt/Uc
The fetch is to be taken downwind from the point where the seaway is considered,
but of course at most to the shore. In reality, there is no sudden and then
constant wind. But the seakeeping parameters are not very sensitive towards x
and t. Therefore it is possible to estimate the seaway with practical accuracy
in most cases when the wind field is given.
Table 4.1 shows how the above formulae estimate the seaway parameters
H1/3 and Tp for various assumed wind durations t for an exemplary wind
velocity Uc D 20 m/s. The fetch x was assumed to be so large that the centre
term in the ‘max’-bracket in the above formula for Uc/cp is always smaller
than one of the other two terms. That is, the seaway is not fetch-limited, but
either time-limited (for 110 3/7 > 1) or fully developed.
Figure 4.12 shows wind sea spectra for Uc D 20 m/s for various fetch values.
Figure 4.13 shows the relation between wave period Tp and significant wave