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Inasmuch as
ϕ
(1)
ωn
(
x
) is continuous in the vicinity of
ω
nl
,
the function
ϕ
(1)
ωn
(
x
)
satisfying (5.74) can be found at some suciently small neighborhood of fre-
quency
ω
nl
. The normalization selected here is convenient for further calcula-
tions, because functions
ϕ
(1)
ωn
(
x
)and
ϕ
(2)
ωn
(
x
) are identical at
ω
=
ω
nl
.
Just as for Alfven waves, calculate the inverse Laplace transform, closing
the integration contour in the lower complex half-plane of
ω
and applying
the theorem on residues. The Wronskian of (5.72) is independent of
x
. Since
the solutions
ϕ
(1)
ωn
(
x
)and
ϕ
(2)
ωn
(
x
) depend analytically on
ω
, then
W
ωn
also
depends analytically on
ω
. Poles of the Green function are determined from
W
ωn
= 0
(5.77)
and coincide with eigenfrequencies
ω
nl
. Indeed, if (5.77) is valid for some
ω
∗
,
then
ϕ
(1)
ϕ
(2)
ωn
(
x
)
.
Here const = 1 due to the boundary conditions (5.74), (5.75). Since
ϕ
(1)
ωn
(
x
) = const
·
ωn
(0) =
ϕ
(2)
ωn
(
x
)=0and
ϕ
(1)
ωn
(
l
x
)=
ϕ
(2)
ωn
(
l
x
)=0
,
then
ω
∗
is the eigenfrequency, and
ϕ
(1)
ωn
(
x
)=
ϕ
(2)
ωn
(
x
) is the eigenfunction.
Thus displacement
ξ
xωn
caused by a point source is given by
+
∞
+
iσ
0
1
2
π
ξ
xn
(
x, t
)=
G
ωn
(
x
)
·
exp (
−
iωt
)d
ω.
−∞
+
iσ
0
For
t>
0 close the contour integral in the lower half-plane and obtain an
expression for
ξ
xn
(
x, t
) as the sum of residues
V
nl
(
x
)
V
nl
(
x
)
d
d
ω
−
1
i
∞
ξ
xn
(
x, t
)=
−
W
ωn
|
ω
=
ω
nl
exp (
−
iω
nl
t
)
.
(5.78)
l
=1
Now we should find derivatives of the Wronskian. Substituting the values
of functions from boundary conditions (5.74) and (5.75) into the definition of
W
ωn
at
x
=
l
x
, we obtain
W
ωn
=
ϕ
(1)
ωn
(
l
x
)
.
The equations for
V
nl
and
ϕ
(1)
ωn
are
d
2
V
nl
d
x
2
+
K
(
x, ω
nl
)
V
nl
=0
,
d
2
ϕ
(1)
ωn
d
x
2
+
K
(
x, ω
n
)
ϕ
(1)
ωn
=0
,
k
n
(
ω
)+
ω
2
/c
2
A
. Multiplying the first equation by
ϕ
(1)
where
K
(
x, ω
)=
ωn
and
the other by
V
nl
, integrating over
x
from 0 to
l
x
and subtracting one obtained
−
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