Geology Reference
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where
n
(
n
+
1) is the separation constant. Thus,
1
n
(
n
f
2
d
2
f
d
ν
m
2
df
d
ν
+
−
ν
2
−
2ν
+
1)
−
=
0,
(6.44)
1
−
ν
2
and
n
(
n
1
2
d
2
m
2
g
d
μ
d
g
d
μ
+
−
μ
2
−
+
1
)
−
g
=
2μ
0.
(6.45)
1
−
μ
2
Both
f
and g therefore obey the associated Legendre equation and χ has the form
P
n
(ν)
P
n
(μ)exp
i
(
m
φ
+
ω
t
).
(6.46)
2
> 1 and the Legendre
function of the second kind,
Q
n
(ν), is an acceptable solution since the singularities
at ν
=±
For modes with periods less than 12 sidereal hours, ν
1 are excluded, so that χ can also have the form
Q
n
(ν)
P
n
(μ)exp
i
(
m
φ
+
ω
t
)
.
(6.47)
For modes with periods greater than 12 sidereal hours, the square of the Coriolis
frequency satisfies σ
2
< 1. In order to avoid imaginary co-ordinates, we define
1
σ
2
τ
=
2
−
1,
(6.48)
2
< 1, we have 0 <τ<
∞
and again scale the
z
-axis by a factor 1/τ. Then, for σ
,
with
z
=
τ
z
. In the new auxiliary co-ordinates (
x
,y,
z
), the Poincare inertial wave
equation becomes
2
2
2
∂
χ
∂
x
2
+
∂
χ
∂y
χ
∂
z
2
=
∂
2
−
0,
(6.49)
rather than Laplace's equation. In auxiliary cylindrical co-ordinates, the equation
of the container surface is once again
2
z
2
a
2
1
R
2
a
2
+
τ
e
2
=
1.
(6.50)
−
In auxiliary spheroidal co-ordinates (ξ,η), we adopt the co-ordinate surfaces
R
2
k
2
1
z
2
k
2
2
+
2
=
1,
−
1 <ξ<1,
(6.51)
−
ξ
ξ
and
R
2
k
2
1
z
2
k
2
2
+
2
=
1,
−
1 <η<1.
(6.52)
−
η
η
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