Geoscience Reference
In-Depth Information
Liu
(
1993
) used a continuous wavelet transform and the complex-valued Morlet wavelet
to obtain a time-frequency wavelet spectrum that effectively provided a localised frequency
energy spectrum for each data point in a given time series. This spectrum can then be used
to define an average wave frequency
, and thus combined with the local wave amplitude
a
, to obtain a local surface acceleration
a
ω
2
which would be compared to a given limiting
ω
γ
fraction
of gravitational acceleration
g
to define the breaking event
(2.60)
.
The classical concept of studying the wave-breaking process led to various usages of a
limiting value of wave steepness beyond which a continuous surface cannot be sustained
(e.g.
Longuet-Higgins
,
1969a
, see also
Sections 2.9
,
3.2
,
3.3
,
3.8
). Thus, for example, the
wave surface will break when its downward acceleration exceeds the limiting fraction
γ
(
2.60
, see also
(2.50)
-
(2.59)
).
As mentioned in
Section 2.9
, the difficulties of applying the acceleration criterion to real
sea waves are the uncertain value of
and the fact that the natural wave fields are multi-
scaled. Because of the latter, the quantity on the left-hand side of
(2.60)
cannot be readily
calculated from a time series of wave data. If it were possible to estimate this quantity,
then such a simple familiar notion could readily be used to identify breaking waves in
the time series. It is, however, not immediately clear how to pertinently resolve the local
amplitude
a
and the instantaneous frequency
γ
ω
from the measured time series
η(
t
)
of real
multi-scaled seas.
Liu
(
1993
) suggested use of the time-frequency wavelet spectrum,
W
η
(ω,
t
)
, to obtain
instantaneous values of effective wave amplitude
a
and frequency
ω
from time series of
surface elevations
η(
t
)
as
∞
2
1
C
1
/
2
ψ
∗
[
ω(τ
−
W
η
(ω,
t
)
=
η(τ)
|
ω
|
t
)
]
d
τ
(3.32)
ψ
−∞
where
C
ψ
<
∞
is the admissibility condition and function
ψ
is the mother wavelet. Here,
we use the Morlet wavelet given by
1
e
−
m
2
e
−
t
2
e
−
imt
/
2
/
2
ψ(
t
)
=
4
(
−
)
,
(3.33)
1
/
π
=
π
√
2
with
m
ln 2 chosen to fit the wavelet shape.
Once a localised frequency spectrum at each time moment
t
i
is known:
i
(ω)
=
W
η
(ω,
/
)
t
=
t
i
,
t
(3.34)
then for each
t
t
i
we can define, based on that spectrum, a characteristic wave amplitude
and frequency at the measurement point. In other words, it replaces the localised spectrum
by an equivalent characteristic monochromatic wave.
As the characteristic frequency, average frequency
=
σ
(Rice, 1954) was chosen:
ω
n
λ
L
ω
p
ω
1
/
2
2
i
(ω)
d
ω
σ
i
=
ω
n
λ
(3.35)
p
i
(ω)
d
ω
ω
L
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