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They argued that the higher ultimate steepness values observed by Toffol i et al. ( 2010a )
perhaps relate to transient waves, i.e. those already breaking. That is, short-crested direc-
tional waves start breaking if the steepness (2.53) is reached, but in the course of breaking
they can achieve even higher steepness (2.51) at the front face. The limiting steepness
(2.53) for directional waves is only slightly higher than that of (2.47) for two-dimensional
wave trains; this issue will be further discussed in Chapter 5 .
Stansell &MacFarlane ( 2002 ) specifically investigated the kinematic criterion (2.49) in a
dedicated laboratory experiment. Since three different interpretations of what is the wave's
phase velocity are possible, they verified the criterion with respect to all three definitions:
“The first definition, based on the equivalent linear waves, is constant over the wavelength of the
wave. The second, based on partial Hilbert transforms of the surface elevation data, is local in space
and time giving instantaneous values at all space and time measurements. The third, based on the
speed of the position of the crest maximum, is local in time but not in space”.
The orbital velocity at the crest of breaking and non-breaking waves was measured by
means of a PIV system, and breaking in an intended location was achieved through linear
focusing.
The conclusions were not overly supportive of the classical criterion. The ratio of orbital
velocity to phase velocity was found to be, at most, 0.81 in the plunging breakers and 0.95
in a spilling breaker. In this regard, we would like to make two comments. First of all,
this finding means that when criterion (2.49) is satisfied, the waves are definitely breaking,
and therefore experimental detection techniques, and statistical and theoretical approaches
based on this criterion will underestimate rather than overestimate breaking probabilities.
Secondly, linear focusing is most likely to be a mechanism of secondary importance when
it comes to natural field wave breaking (see Chapters 4 - 6 , 10 ). The breaking caused by
instabilities of nonlinear wave trains appears to be a feasible common mechanism for
field breaking, and its physics is quite different from that of breaking because of linear
superposition. For such a breaking onset, brought about by modulational instability, the
experiments and findings of Stansell & MacFarlane ( 2002 ) need to be revisited.
For such modulational breaking, an unconventional interpretation of the kinematic cri-
terion was given by Tulin & Landrini ( 2001 ). In a comprehensive overview of wave-
breaking investigations conducted by their research group over a period of more than 15
years, the authors, in particular, provided analytical derivation and experimental validation
of their version of this criterion:
... upon passing through the peak of a modulation group, when the orbital velocity at the wave
crest, u c , exceeds the wave group velocity d ω/ dk = c g , then the wave crest and trough both rise, the
front face steepens, the wave crest sharpens, and eventually a jet forms at the crest, leading finally to
splashing and a breakdown of the wave”.
Thus, the criterion is stated for wave breaking due to modulational instability rather than
that due to focusing (see Chapters 4 and 5 about the evolution of nonlinear wave groups
to breaking). This does not signify the ratio of orbital and phase velocities at the point
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