Geoscience Reference
In-Depth Information
In choosing the concrete function for describing the space-time law determining
the motion of the bottom, part of the integrals in expression (3.15) can be calculated
analytically.
3.1.2.2 Cylindrical Coordinates
The cylindrical reference frame will be introduced in a standard manner with respect
to the Cartesian reference frame, described in Sect. 3.1.2. The origin of the cylin-
drical reference frame will be located on the free unperturbed surface, axis 0
z
will
be directed vertically upward. As a source of elastic gravitational waves we shall
consider axially symmetric movements of the bottom proceeding in accordance
with the law
(
r
,
t
). The wave equation in the cylindrical reference system exhibits
the following form:
η
r
−
1
(
rF
r
)
r
+
F
zz
=
c
−
2
F
tt
.
(3.16)
The boundary conditions on the surface, (3.5), and on the bottom, (3.6), retain
their form in the cylindrical reference frame.
The solution of the problems (3.16), (3.5) and (3.6) is sought applying the sepa-
ration of variables in the form of the inverse Laplace transformation:
s
+
i
∞
F
(
r
,
z
,
t
)=
d
pR
(
r
,
p
)
Z
(
z
,
p
) exp
{
pt
}
.
(3.17)
s
−
i
∞
Substituting formula (3.17) into (3.16), we obtain equations for determining
functions
R
(
r
,
p
) and
Z
(
z
,
p
):
r
−
1
(
rR
r
)
r
+
k
2
R
= 0
(3.18)
2
Z
= 0
,
Z
zz
−
α
(3.19)
2
=
k
2
+
p
2
c
−
2
. The solutions of equations (3.18) and (3.19) are well known
[Nikiforov, Uvarov (1984)]. Making use of this result, one can represent the general
solution of equation (3.16) as follows:
where
α
F
(
r
,
z
,
t
)
∞
s
+
i
∞
=
d
k
d
p
exp
{
pt
}
J
0
(
kr
)(
A
(
p
,
k
) cosh(
α
z
)+
B
(
p
,
k
) sinh(
α
z
))
,
(3.20)
0
s
−
i
∞
where
J
0
is the zeroth-order Bessel function of the first kind.
With the aid of the boundary condition on the free surface, (3.5), we find the re-
lationship between the coefficients
A
and
B
:
)
−
1
.
A
(
p
,
k
)
p
2
(g
B
(
p
,
k
)=
−
α
