Geology Reference
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from these expressions. First, modes of a given azimuthal number
m
are independ-
ent of modes with other azimuthal numbers. This absence of coupling over di
er-
ent azimuthal numbers derives from the axisymmetric nature of the Earth model
assumed in the formulation. In addition, it is evident from expression (8.26) for the
functional that only real trial functions are required and that it is unnecessary to
include their azimuthal dependence in their representation. Second, reversing the
signs of both
m
and the angular frequency σ results in the same functional and
boundary conditions. Thus, we can restrict
m
to zero and the positive integers, if σ
is allowed to range over both positive and negative values. Axisymmetric modes are
standing waves, determined by σ
ff
2
alone. Non-axisymmetric modes drift westward
at the angular rate 2
Ω
σ/
m
for negative σ. Finally, modes that are purely even with respect to the equatorial
plane are decoupled from modes that are purely odd, since
z
appears only as a
product with ∂χ/∂
z
, or as a square, in both the functional (8.26) and the boundary
conditions (8.29). Thus, contributions arising from cross products of even and odd
parts of solutions will be odd in the equatorial plane and their integral will vanish.
Support functions that are purely even or purely odd in the equatorial plane are
therefore used in the representation of trial functions.
The functional (8.26) is conveniently displayed in the matrix representation
Ω
σ/
m
for positive σ and drift eastward at the rate
−
2
⎝
⎠
2
1
−
b
m
2
z
2
)0 0
/(1
−
f
U
T
∗
F =−
2πσ
0
1
0
U
dr
0
dz
1
a
z
2
)
0
0 (1
−
⎝
⎠
⊗
1
b
0
z
0,
z
,1
z
2
U
dr
0
dz
f
U
T
∗
+
2π
−
−
1
a
z
2
1
−
⎝
⎠
⊗
1
b
m
σ
m
σ,σ
z
2
U
dr
0
dz
z
1
f
z
2
U
T
∗
2
z
2
2
z
2
+
2π
σ
−
−
,
−
−
2
ζ
1
−
−
1
a
z
3
−
z
⎝
⎠
1
b
z
100
−
z
00
01
−
f
U
T
∗
−
U
dr
0
dz
2π
m
σ
−
1
a
2
σ
1
2
+
4π
b
σ
−
Σ
S
,
(8.30)
=
χ,
r
0
χ
r
0
,χ
z
with
denotes the outer vector product and
U
T
where the symbol
⊗
χ
r
0
=
∂χ/
dr
0
and χ
z
=
∂χ/
dz
.
In addition to the properties of the functional just described, the factor sin
m
θ
is a general feature of solutions of the subseismic wave equation (Smylie and
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