Geoscience Reference
In-Depth Information
η
L
(
x
,
t
) it is possible to 'construct' the two principal model laws of deformation of
the basin floor at the tsunami source: a motion of the basin floor involving residual
displacement
η
1
(
x
,
t
)=
η
L
(
x
,
t
)
−
η
L
(
x
,
t
−
τ
)
(2.63)
and a motion of the basin floor without residual displacement
η
2
(
x
,
t
)=2
η
L
(
x
,
t
)
−
4
η
L
(
x
,
t
−
0
.
5
τ
)+2
η
L
(
x
,
t
−
τ
)
.
(2.64)
Complying with the terminology proposed in [Dotsenko, Soloviev (1990)], we
call the two indicated types of motion 'piston' and 'membrane' displacements.
The problem considered is linear, therefore, the solutions for the piston and
membrane displacements can be expressed through the solution for the linear
displacement, making use of the superposition principle:
F
1
(
x
,
z
,
t
)=
F
L
(
x
,
z
,
t
)
θ
(
t
)
−
F
L
(
x
,
z
,
t
−
τ
)
θ
(
t
−
τ
)
,
(2.65)
F
2
(
x
,
z
,
t
)=2
F
L
(
x
,
z
,
t
)
θ
(
t
)
−
4
F
L
(
x
,
z
,
t
−
0
.
5
τ
)
θ
(
t
−
0
.
5
τ
)
+ 2
F
L
(
x
,
z
,
t
−
τ
)
θ
(
t
−
τ
)
,
(2.66)
where
F
L
(
x
,
z
,
t
) is the solution of the problem (2.53)-(2.55) in the case of
η
(
x
,
t
)=
=
L
(
x
,
t
). The perturbation of the free surface and of the velocity component cor-
responding to the piston or membrane displacements is obviously calculated by for-
mulae, similar to (2.65) and (2.66). Only a formal substitution of
η
ξ
L
,
u
L
or
w
L
for
F
L
is required.
Calculation of the intermediate integrals, performed applying residue theory,
results in the following expressions for the linear displacement:
+
∞
1
F
L
(
x
,
z
,
t
)=
−
d
k
2
πτ
−
∞
{−
}
−
exp
ikx
cosh(
kz
)(1
(1+ tanh(
kH
) tanh(
kz
)) cos(
tp
0
))
k
sinh(
kH
)
×
X
(
k
)
.
(2.67)
+
∞
−
1
d
k
exp(
ikx
) sin(
tp
0
)
p
0
cosh(
kH
)
ξ
L
(
x
,
t
)=
X
(
k
)
,
(2.68)
2
πτ
−
∞
+
∞
i
u
L
(
x
,
z
,
t
)=
d
k
2
πτ
−
∞
{−
}
−
exp
ikx
(cosh(
kz
)
[cosh(
kz
)+ tanh(
kH
) sinh(
kz
)] cos(
tp
0
))
sinh(
kH
)
×
X
(
k
)
,
(2.69)
