Geoscience Reference
In-Depth Information
The boundary condition on the bottom, (3.6), permits to determine the coefficient
A
(
p
,
k
):
pk
Ψ
(
p
,
k
)
A
(
p
,
k
)=
−
H
))
,
(3.21)
H
)+
p
2
g
−
1
α
−
1
cosh(
α
(sinh(
α
α
where function
(
p
,
k
) represents the Laplace and Fourier-Bessel transforms of
the space-time law of movements of the bottom,
Ψ
η
(
x
,
t
):
∞
∞
1
2
Ψ
(
p
,
k
)=
d
r
d
t
η
(
r
,
t
)
rJ
0
(
kr
) exp
{−
pt
}
.
(3.22)
π
i
0
0
We shall further consider the behaviour of a free surface, the displacement of
which from its unperturbed level is expressed through the potential as follows:
g
−
1
F
t
(
r
,
0
,
t
)
.
ξ
(
r
,
t
)=
−
(3.23)
With use of formula (3.20) expression (3.23) acquires the following form:
∞
s
+
i
∞
g
−
1
ξ
(
r
,
t
)=
−
d
k
d
pp
exp
{
pt
}
J
0
(
kr
)
A
(
p
,
k
)
.
(3.24)
0
s
−
i
∞
3.1.3 Piston and Membrane Displacements
We shall start exposition of the peculiarities of tsunami formation in a compressible
ocean by considering the axially symmetric problem [Nosov (2000)]. As sources of
acoustic-gravity waves we choose two model displacements of the bottom: the pis-
ton and membrane displacements,
R
)
θ
,
η
0
1
(
t
)
t
−
θ
(
t
−
τ
)(
t
−
τ
)
η
1
(
r
,
t
)=
−
θ
(
r
−
(3.25)
τ
0
1
R
)
η
2
(
r
,
t
)=
η
−
θ
(
r
−
2
,
θ
(
t
)
t
−
4
θ
(
t
−
0
,
5
τ
)(
t
−
0
,
5
τ
)+2
θ
(
t
−
τ
)(
t
−
τ
)
×
(3.26)
τ
η
0
is the same throughout the entire active zone,
exhibiting a circular shape of radius
R
, and is zero outside this region. The duration
of the displacement is
The displacement amplitude
.
We introduce the dimensionless variables (the asterisk '*' will be dropped):
τ
k
∗
=
kH
;
p
∗
=
pHc
−
1
;
α
∗
=
α
H
;
R
∗
=
RH
−
1
;
r
∗
=
rH
−
1
;
z
∗
=
zH
−
1
;
(3.27)
t
∗
=
tcH
−
1
;
τ
∗
=
cH
−
1
;
c
∗
=
c
(g
H
)
−
1
/
2
.
τ
