Geoscience Reference
In-Depth Information
15.8 Describe the concept of spectra in the plane.
15.9 Discuss how and why spectra in the plane can be more useful than one-
dimensional spectra.
15.10 Discuss some of the spectral implications of the approach to isotropy at small
scales.
Problems
15.1 Show that the autocorrelation function is an even function of its argument:
ρ(t)
=
ρ(
−
t)
.
15.2 For any two real, random variables
u
and
v
one can write
u
u
2
−
2
v
v
2
≥
0
.
Expand this expression and rearrange the result
to prove Schwartz's
inequality,
u
2
v
2
.
uv
≤
15.3 Show that
b
u(t)dt
2
b
b
u
2
b
a
b
u(t
)u(t
)dt
dt
=
ρ(t
−
t
)dt
dt
.
=
a
a
a
a
15.4 Use the stationarity of
u(t)
as expressed by
2
du(t)
dt
2
d
2
dt
2
u
2
(t)
2
u(t)
d
2
u(t)
dt
2
=
0
=
+
to show that the variance of the derivative is
du(t)
dt
t
=
0
2
u
2
d
2
ρ(t)
dt
2
=−
.
15.5 Referring to
Figure 15.1
,
use Taylor's hypothesis to compare the time scales
τ
and
λ
of the autocorrelation function of a turbulent velocity signal by relating
them to the energy-containing scale
and the Taylor spatial microscale.
15.6 Use the Fourier-Stieltjes representation to relate the spectrum of
∂u/∂t
to
the spectrum of
u(t)
.
15.7 Use the Fourier-Stieltjes representation to assess the reliability of the finite-
difference approximation to the derivative of
u(t)
,
∂u
∂t
(t)
u(t
+
t)
−
u(t
−
t)
.
2
t
