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Thus, a concrete physical mechanism (or a number of mechanisms) that connects the
breaking probability of dominant waves with the bandwidth in the definitions
(2.3)
and
(2.6)
is not fully certain, but links between breaking rates and the wave groups, and there-
fore breaking rates and the spectral-peak bandwidth, are apparent. This is not the case,
however, with the bandwidths of spectral bins other than the spectral peak in
(2.4)
and
(2.5)
. The wave process at those scales is not narrow banded, therefore artificially imposed
bandwidths have no apparent meaning in terms of wave groups and modulation of wave
trains of respective scales. While the choice of bandwidth for higher-frequency parts of the
spectrum
F
is usually done by analogy with the spectral-peak region, the physical anal-
ogy is not applicable in this case, therefore defining the breaking probabilities in the same
terms appears to be a matter of convenience rather than physics (see
Babanin & Young
,
2005
;
Babanin
et al.
,
2007c
, for a discussion of differences in the physics of breaking at
the spectral peak and in the higher-frequency region). These issues will be discussed in
more detail in
Sections 5.3.1
and
5.3.2
.
Another potential complication of definitions
(2.3)
-
(2.5)
must also be clearly stated.
When breaking probabilities according to these definitions are estimated experimentally,
the number of wave crests is counted. For each determined breaker, the frequency
f
(period
T
) of the wave is also extracted, by zero-crossing analysis (e.g.
Manasseh
et al.
,
2006
), the
riding wave removal method (
Schulz
,
2009
) or by some other means. Thus the total number
of breaking waves
n
(
f
)
(
f
)
is found for each frequency. The total number of expected waves
at a frequency is
N
(
f
)
=
t
/
T
=
t
·
f
(2.13)
and expressions
(2.3)
-
(2.4)
can then be used. As previously discussed in this section, in
practice, in order to obtain
n
(
f
)
and
b
T
(
f
)
, the wave frequencies are effectively discretised
into bands
f
are not spectral densities but statistical
quantities, and there are no exact matches between measured wave periods and a given
T
±
f
(2.5)
because
n
(
f
)
and
b
T
(
f
)
=
f
.
It is clear, however, that if the waves of any given period
T
1
/
f
are counted by the
zero-crossings or other means in a wave record of duration
t
, the resulting count
N
c
(
=
1
/
f
)
will
be less than the nominal reference count
N
(
f
)
given by
(2.13)
, because in real seas, waves
of periods other than 1
f
will occupy some part of the duration
t
. In terms of the breaking-
probability definition, it would not matter if the ratios
N
c
(
/
were constant across
the frequency and therefore the crest counts were simply proportional to the reference
count. This is, however, not the case.
Banner
et al.
(
2002
) demonstrated that if the bandwidth
f
)/
N
(
f
)
f
=±
0
.
3
f
c
were chosen
for a central frequency
f
c
in their experiment,
N
c
(
f
)/
N
(
f
)
was about 0.65 at the spectral
peak (
f
c
=
f
p
) and gradually decreased for higher frequencies (
f
c
>
f
p
), asymptoting
to 0.2 at
f
c
/
2. The ratio also depended on the choice of bandwidth, i.e. it would be
different, for example, for
f
p
>
1
f
c
. Therefore, to avoid this additional uncertainty,
the breaking probability
b
T
used in this topic will be based on the reference count
(2.13)
f
=±
0
.
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