Geology Reference
In-Depth Information
Since the current estimate of the eigenvector satisfies (8.76), we have
A
(σ
r
)
x
r
−
σ
r
)
x
r
+
1
≈
+
(σ
r
+
1
0.
(8.80)
The iterative scheme (8.76) allows large changes in the magnitude and direction of
the eigenvector, so that at each step the eigenvector is normalised to unit magnitude,
giving
x
r
·
1. Thus, if det
A
0,
x
r
=
x
r
+
(σ
r
+
1
−
σ
r
)
x
r
+
1
≈
0
(8.81)
and
1
x
r
+
1
·
x
r
,
σ
r
+
1
=
σ
r
−
(8.82)
giving the improved estimate of the eigenfrequency.
Given the solution for the generalised displacement potential χ, the displacement
vector is found from relation (7.16). In spherical polar co-ordinates, the vector
displacement
u
has components (
u
r
,
u
θ
,
u
φ
)givenby,
F
m
m
σ
+
z
2
σ
z
2
r
z
1
z
2
Φ
z
e
im
φ
,
2
ru
r
=
Φ+
−
Φ
r
−
−
(8.83)
F
m
σ
z
2
1
2
m
σ
z
z
1
z
2
σ
2
r
sinθ
u
θ
=
2
2
2
r
σ
−
ζ
+
−
ζ
Φ+
−
Φ
r
z
2
2
1
z
2
σ
z
e
im
φ
,
2
2
−
−
ζ
−
Φ
(8.84)
F
m
m
σ
+
z
2
1
2
1
z
2
r
2
σ
r
sinθ
u
φ
=
−
ζ
Φ+
σ
−
Φ
r
1
2
z
1
z
2
z
e
i
(
m
φ
+
π/2)
2
−
−
ζ
−
Φ
,
(8.85)
with the common factor
F
m
defined by
1
z
2
m
/2
−
F
m
=
2
N
2
z
2
1
(8.86)
4
Ω
−
ζ
2
and the parameter ζ
2
defined by
N
2
σ
2
2
+
N
2
1
.
ζ
=
1
−
σ
=
2
ζ
(8.87)
Using only the first term of the expansion (8.36) as an approximation, the factor
F
m
becomes
1
z
2
m
/2
−
F
m
=
2
σ
1
.
(8.88)
4
Ω
2
σ
2
−
2
In the case of the Poincare inertial wave equation, the parameter ζ
2
reduces to
2
2
ζ
=
1
−
σ
.
(8.89)
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