Detection and measurement of wave breaking - Breaking and Dissipation of Ocean Surface Waves - page 73

The linear-superposition technique was also employed by Nepf et al. ( 1998 ) and Wu &
Nepf ( 2002 ) in their laboratory comparison of two- and three-dimensional wave breaking.
The experiment was conducted in a wave basin where water fronts were generated by 13
independently programmed and driven paddles. If moved concurrently, the paddles would
create a long-crested wave, but the crest could have been tapered laterally, and this was
done by applying a cosine window to the signal which controlled the motion of the set of
paddles. While the method of converging a wave packet due to phase dispersion was the
same as in other tests which used such a superposition, the tapering allowed creation of
short-crested waves and thus investigated the influence of the crest's three-dimensionality
on wave breaking.
Here, we would like to point out that short-crestedness and directionality of the waves
are often confused and used interchangeably in the literature, although this is not the same
feature. Indeed, if 2D-Fourier-like analysis is applied to two-dimensional wavy surfaces,
in the case of short-crested waves the outcome will be an angular distribution of wave
energy. Such an angular distribution, however, will not be a

-function even if the short-
crested waves are strictly unidirectional. The opposite is also true: that is the superposition
of long-crested waves having come from different directions will be decomposed by the 2D
Fourier transform into a finite-width directional spectrum. This is because Fourier analysis,
when applied formally, will treat the former situation also as a superposition of long-crested
harmonic waves which they are not, and therefore the wave energy placed by such analysis
at oblique angles is just a noise in the Fourier sense due to the short-crestedness.
This may seem to be an abstract mathematical argument, but it has a principal physical
consequence for wave breaking. Modulational instability, which can lead to wave breaking,
still exists in two-dimensional wave trains with three-dimensional wave crests, even though
the breaking onset is set back from steepness

δ

44 in
2D waves with 3D wave crests ( Melville , 1982 ; Babanin et al. , 2009a , 2010a ). The proper
directionality, however, can influence the modulational instability and therefore the wave
breaking for this reason, if the directional spreading of the wave field is broad enough
( Onorato et al. , 2009a , b ; Waseda et al. , 2009a ; Babanin et al. , 2011a ).
In this context, Nepf et al. ( 1998 ) and Wu & Nepf ( 2002 ) conducted experiments with
short-crested rather than directional waves. The three-dimensional structure of the crest
was created by either focusing or diffracting the waves generated separately by the 13
paddles. Surface elevations were measured by an array of six wave gauges deployed on
a carriage which could traverse the array to a measurement point near the wave-breaking
location. An extensive set of geometric (wave steepness and shape), kinematic (2.49) and
dynamic (up-frequency spectral energy shift) criteria were investigated.
It was found that the imposed directionality of the crest could either increase (focusing
waves) or decrease (diffracting waves) the wave steepness at breaking onset (as opposed to
the lateral modulational instability of the wave crest which increases the steepness at the
incipient breaking (e.g. Melville , 1982 )). Shape parameters at the breaking were altered in
a similar manner. The kinematic and dynamic criteria, on the contrary, were not affected
by the crest directionality. Since the dynamic criterion considered is sensitive and difficult

= 0 . 29 in strictly 2D waves to = 0 .