Geoscience Reference
In-Depth Information
Making use of the general solution (3.24), we obtain expressions describing mo-
tion of the free surface of a compressible liquid in the case of piston-like (
ξ
1
) and
membrane-like (
ξ
2
) displacements of the bottom:
ξ
1
(
r
,
t
)=
θ
(
t
)
ζ
(
r
,
t
)
−
θ
(
t
−
τ
)
ζ
(
r
,
t
−
τ
)
,
(3.28)
ξ
2
(
r
,
t
)=2
θ
(
t
)
ζ
(
r
,
t
)
−
4
θ
(
t
−
0
,
5
τ
)
ζ
(
r
,
t
−
0
,
5
τ
)
+ 2
θ
(
t
−
τ
)
ζ
(
r
,
t
−
τ
)
,
(3.29)
where
∞
s
+
i
∞
0
c
2
R
(
r
,
t
)=
η
exp
{
pt
}
J
0
(
rk
)
J
1
(
Rk
)
ζ
d
k
d
p
)
.
(3.30)
2
π
i
τ
α
sinh(
α
)+
p
2
c
2
cosh(
α
0
s
−
i
∞
As a function of the complex parameter
p
the integrand in (3.30) has two or an in-
finite number (depending on the sign of
2
) of poles located on the axis Im(p)=0.
α
∞
An incompressible liquid (
c
=
) represents a special case of the problem dealt with.
The solution for an incompressible liquid can be obtained by a formal substitution
of
α
→
k
in formula (3.30). The integrand, here, will have only two first-order poles
ic
−
1
(
k
tanh(
k
))
1
/
2
, which permits to perform integration over the parame-
ter
p
analytically. In the case of an incompressible liquid, this results in the function
ζ
p
1
,
2
0
=
±
(
r
,
t
), entering into formulae (3.28) and (3.29), assuming the following form:
∞
d
k
J
0
(
rk
)
J
1
(
Rk
) sin(
tc
−
1
(
k
tanh(
k
))
1
/
2
)
cosh(
k
)[
k
tanh(
k
)]
1
/
2
(
r
,
t
)=
η
0
cR
τ
ζ
.
(3.31)
0
The integrals in formulae (3.30) and (3.31) were calculated numerically for
c
= 8
and
R
= 1, 5 and 10.
Figure 3.2 presents the example of time evolvents, showing the displacement
of a free surface at two fixed points (at the centre of the active zone and outside
it) for compressible and incompressible liquids. The insets show the behaviour of
the free surface of an incompressible liquid at long times. The theory of a com-
pressible liquid is seen to provide a more reliable description of the movement of
the surface from the point of view of moment of time the perturbation arrives at
the given point. Before the long gravitational wave arrives at point
r
= 20, acous-
tic precursors of noticeable amplitude are observed. The main difference in be-
haviour between compressible and incompressible liquids consists in the existence
of 'fast' oscillations of the surface with a prevalent period, equal to 4
H
/
c
. Oscilla-
tions take place against the background of the development of a slower gravitational
wave. The origin of surface oscillations is due to the excitation of standing acous-
tic waves in the natural quarter-wave resonator of a 'column of compressible liquid
with a free surface on the rigid bottom'. The resonator exhibits a set of frequen-
cies:
ν
k
= 0
.
25
c
(1 + 2
k
)
H
−
1
, where
k
= 0
,
1
,
2
,
3
,...
. Precisely the lowest mode
corresponds to the period observed.
It is quite probable that such a resonator plays an important part in the formation
of seaquakes. In the case of depths of several kilometres, usual for oceans, the eigen
