Practical Ship Hydrodynamics

Ship seakeeping 109
If several ω result in the same ωe the contributions of all three frequencies are
added on the r.h.s. of this equation (Fig. 4.6). Correspondingly an encounter
directional spectrum can also be determined. Because of the several possible
contributions on the r.h.s. and the singularity at S e – where the denominator
on the r.h.s. in the above equation becomes zero – the encounter spectrum is
not used in seakeeping computations. However, it is needed for the analysis
of data if these were measured from a ship with forward speed.
s(w) s(w e )
w we 0 w 0
0
0.25 g /u
Figure 4.6 Sea spectrum and corresponding encounter spectrum
4.3.3 Wind and seaway
We distinguish between swell and wind sea. Swell waves have a celerity higher
than the present wind speed (e.g. measured in 10 m height above mean sea
level; only the component in wave propagation direction is considered). Swell
has been excited originally by some stronger winds at some other location
and propagates without significant damping or excitation until it is damped
in shallow-water regions or excited again to wind sea in stronger winds. By
definition, wind sea has celerity less or equal to the wind speed. Due to the
gustiness of wind and other factors, the distinction between swell and wind
sea is not sharp.
Swell, sometimes also wind sea (for winds changing rapidly in time or
space), can change the form of the spectrum considerably. On the other hand,
a rather uniform form of a wind sea spectrum is achieved within 1
2 to 1 hour
if the wind is constant in time and space. The relevant area in this context
extends over a distance of ( 1
to 1 hour)/group velocity of waves in a downwind
2
direction. In the following, we will consider only spectra developed in constant
wind. The spectrum parameters, especially wave height and period, converge
only after many hours or several days to constant values. The form of the
spectrum is determined by the physical processes of:
ž wave generation (e.g. the wind resistance of wave crests)
ž dissipation (wavebreaking; in shallow water also friction at the ocean
bottom)
ž convection (transport of wave energy with group velocity)
ž non-linear interaction between waves of different frequencies and direction
The directional spectrum is described as the product of a one-dimensional
spectrum S ⊲ω⊳ with a function f. f describes the distribution of the wave
energy over the propagation direction assumed to be symmetrical to a main