Civil Engineering Reference
In-Depth Information
′
()
is the time derivative of
the corresponding signal. The frequency of the “carrier”
signal is
In the equation above,
=
ddt
(
)
(
2
r
kt
,
ω
(
kt
, )
()
i
ω
k
=
c
)
2
rkt
,
The instantaneous frequency is linked to the
instantaneous phase
()
by
ϕ
k
,
t
(
)
dkt
ϕ
,
()
ω
(,)
kt
=
ω
k
+
i
c
dt
Tardu [TAR 08b] applies the simplest wavelet - the Haar
wavelet - to the temporal measurements of velocity
u
(
t
)
in
the inner layer. Figure 4.38(a) shows a few typical temporal
plots of the amplitude
(
)
+
+
+
and the instantaneous
rkt
,
(
)
phase
++
of the wavelet coefficient as a function of
time at
y
+
=
ϕ
kt
10
, for
k
+
=
0.057
corresponding to the timescale
26
. It is easy to see that the instantaneous phase is
curiously constant for long periods of time, separated by
intermittent jumps at
B - C
and
D - E
. These periods are
qualified as “laminar” in the chaotic synchronization theory
[BOC 02, PIC 01]. The phase synchronization is imperfect
because the laminar periods are separated by phase jumps,
but are of finite duration. Figure 4.39 shows that the
duration of the laminar periods reaches 100 inner variables
- a value which is near to the intervals separating two
successive ejections at a clearly-defined wavenumber
k
* +
.
The amplitude
T
W
+
=
(
)
, for its part, is arbitrary.
+
+
+
rkt
,
[TAR 10c] and [TAR 11c] apply the amplitude/local phase
decomposition to the 2D wavelets coefficients described and
analyzed in section 4.6. Their results, illustrated in
Figure 4.38(b), confirm the presence of constant phase zones.
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