Detection and measurement of wave breaking - Breaking and Dissipation of Ocean Surface Waves - page 101

where ω p is the localised frequency at the local energy peak, ω n is the cutoff frequency,

and λ L is a number that Liu ( 1993 ) introduced to denote the start of the frequency range

covering the wave-breaking process. We generally carry cut-off frequency up to 2.5 times
the peak frequency. The value of 1, for example,

means that we expect waves of peak frequency and higher to be breaking and there-
fore disregard the contribution of those below

λ L usually lies between 0 and 2. λ L =

ω p in the determination of the character-

istic wave.
For the local

in an
initial application. Another approach was employed in Liu & Babanin ( 2004 ) which was
based on considering the case of a simple monochromatic wave that has an acceleration
A

instantaneous amplitude a , Liu ( 1993 )used a i = η ( t i ) − η

2 cos

, in order to infer that an appropriate characteristic amplitude at local
instantaneous time t i should be given as

σ (ω t + ϕ)

a i = A i cos ( p i ). (3.36)

η (

Here, the local amplitude A i is obtained from the analytic envelope signal of t i ) by

means of a Hilbert transform:

A i =| Hilbert (η i ) | , (3.37)

and the local phase p i can be obtained from the wavelet spectrum W η (ω,

)

.
In order to get the phase information of the time series, the mother wavelet to be used
should necessarily be a complex one such as the Morlet wavelet shown above. So the
calculation of the phase is given as

t

tan − 1

W )

η (ω,

t

W )

p (ω, t ) = (3.38)

η (ω,

t

and thus the local phase p i can be obtained from averaging the local wavelet phase spec-
trum at each t

ω n .
Sample results of the average frequency and local amplitude as obtained from (3.35)
and (3.36) respectively, using

= t i over the same range between λ L ω p and

1 (that is, waves of the peak frequency and above are
expected to break ( Babanin et al. , 2001 )), are shown in the middle and bottom panels
of Figure 3.13 for the Lake George waves. There remains to be determined the limiting
fraction

λ L =

as the threshold for wave breaking that can be rendered through assimilation
with the measured data.
Liu & Babanin ( 2004 ) tested the breaking-detection approach suggested by Liu ( 1993 )
and determined factor

γ

on the basis of field data. The data were obtained under a variety
of wind-wave conditions in deep water in the Black Sea and in a finite-depth environ-
ment in Lake George, Australia. The two data sets included time series of surface ele-
vations, with breaking waves marked. Both had been extensively used to study breaking
statistics, and detailed descriptions of the breaking detection procedure as well as of rele-
vant environmental conditions for the Black Sea experiment are given in Babanin ( 1995 ),
Babanin & Soloviev ( 1998a ), Banner et al. ( 2000 ) and Babanin et al. ( 2001 ). The Lake
George field experiment was described in detail in Section 3.5 (see also Young et al. ,

γ