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10
−5
10
−1
10
0
10
1
10
0
10
−5
10
−10
10
−1
10
0
10
1
1
0.5
0
10
−1
10
0
10
1
100
0
−100
10
−1
10
0
10
1
frequency
Figure 4.3 Numerical simulations (see
Figure 4.2
). Dimensionless wave period is 1. Co-spectra of
running steepness and skewness for waves of IMS
=
0
.
26
,
U
/
c
=
2
.
5. (top panel) Steepness spec-
trum. (second top panel) Skewness spectrum. (second bottom panel) Coherence spectrum. (bottom
panel) Phase spectrum (in degrees)
zero, as was observed visually. Thus, the steepness and skewness are in phase, and the
maximum steepness is achieved at the same instant as the maximum skewness.
Similarly,
Figure 4.4
compares the co-spectra of running instantaneous skewness
S
k
(1.2)
and asymmetry
A
s
(1.3)
. Again, the broad peak of the asymmetry spectrum falls in
the 0.4-0.6 range of inverse wave periods. Spectra of skewness and asymmetry are almost
perfectly coherent, with a phase shift of about 90
◦
(asymmetry is leading). The latter means
that the asymmetry reaches its positive maximum (i.e. wave is tilted backwards) when
skewness is approximately zero (wave crest and trough are of equal magnitude) and the
local steepness is half-way through rising from its minimum to the maximum value in an
oscillation. From the positive maximum, the asymmetry begins to decrease and reaches
zero half a wave period later (quarter of the period of the oscillation) - i.e. the wave
becomes symmetric with respect to the vertical. At this point, steepness and skewness are
at their maximum, and it is at this point that the wave may break. Whether the wave breaks
or not, the asymmetry will keep decreasing into negative values (wave is tilting forwards),
while the steepness/skewness start subsiding in quadrature with the asymmetry. It is inter-
esting to look at this moment from the point of view of an observer who encounters the
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