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steepness of an initially uniform train of steep waves (
Babanin
et al.
,
2007a
,
2010a
).
According to the numerical simulations, if IMS is less than
IMS
threshold
=
0
.
1
(5.11)
the wave train will exhibit all the nonlinear behaviour described in
Section 4.1
, including
oscillations of the steepness, skewness and asymmetry, but will never break. This is because
in the course of the modulational evolution of nonlinear groups no wave will grow to reach
the limiting steepness
(2.47)
. Therefore, the threshold
(5.11)
does not signify any new kind
of nonlinear behaviour of the wave system, but merely the magnitude of the modulation.
In this regard, the modulational instability may lead either to a breaking wave, if limit
(2.47)
is achieved, or to a high non-breaking wave. The latter may even qualify as a freak
(rogue) wave if, by definition, its height satisfies the criterion
H
freak
≥
2
.
2
H
s
,
(5.12)
but its steepness is still below the limiting steepness.
An interesting confirmation of this conjecture can be found in
Hwung
et al.
(
2005
).
In this report, a variety of wave behaviours due to modulational instability were investi-
gated in the supertank of the National Cheng Kung University, Taiwan (see also
Hwung
et al.
,
2007
). In their Figure 2.2.6.13, the authors plotted the ratio of the maximal crest
height
a
max
in the unstable wave train to initial wave amplitude
a
in this train, as a function
of the initial mean background steepness IMS
=
=
ak
(1.1)
of such a nonlinear wave
train. Up to steepnesses of
≈
0
.
12-0
.
13
,
(5.13)
this ratio was growing and then it started to drop.
In view of what we would expect from modulational-instability behaviour, the wave
trains below the mean-background-steepness threshold
(5.11)
(which is apparently slightly
greater in the measurements of
Hwung
et al.
(
2005
), i.e.
(5.13)
) would experience the
instability and produce a high wave, but this wave would not break. Therefore, the higher
the IMS, the higher is the individual tallest wave. This, however, can only be true until these
individual waves start reaching the limiting steepness
(2.47)
and start breaking. From this
background steepness on, the highest wave will be determined by this limiting-breaking
steepness of the individual wave
(2.47)
, and its height will no longer be increasing if IMS
is kept tuned up. On the contrary, the ratio of the highest crest and the initial amplitude
will start decreasing, since regardless of the increase of this initial amplitude, the nonlinear
evolution will always be stopped at exactly the same highest crest by the breaking onset.
In the experiments of
Hwung
et al.
(
2005
), the maximal ratio at steepness
(5.13)
was
a
max
a
≈
.
.
3
5
(5.14)
Since this is achieved at the breaking point where skewness is 1
(5.5)
, then the crest is twice
as high as its trough and the maximal ratio of the largest individual-wave height
H
max
to
the height of initial monochromatic waves is
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