Geoscience Reference
In-Depth Information
+
∞
−
η
0
4
d
k
exp
}
cosh(
kH
)
{−
ikx
w
R
(
x
,
z
,
t
)=
π
−
∞
exp
ib
k
+
p
0
v
⎛
g
k
sinh(
kz
)+
p
0
cosh(
kz
)
p
0
−
1
⎝
exp
×
{−
}
ip
0
t
k
+
p
0
v
exp
ib
k
⎞
p
0
v
−
−
1
⎠
.
−
exp
{
ip
0
t
}
(2.74)
p
0
v
k
−
b
/
v
is
fulfilled. At any rate, this fact gives rise to no essential complications in calculations
for time periods inferior to
b
/
v
, since, from a physical point of view, the solution
of the problem involving a running displacement at
t
=
t
0
<
b
/
v
is equivalent to
the solution of a similar problem for
b
=
vt
0
.
We stress that expressions (2.71)-(2.74) are valid only if the condition
t
2.3.2 Piston and Membrane Displacements
As it was already shown above, tsunami waves are generated by motions of
the ocean bottom occurring along the normal to its surface (normal displace-
ments). Motions of the ocean bottom in its own plane (tangential displacements)
are not effective, from the standpoint of tsunami generation. The term 'vertical dis-
placement' is often encountered in the literature. In the case of small slope angles
of the ocean bottom the difference between vertical and normal displacements is,
naturally, insignificant.
The goal of this section consists in the revelation of relationships between
the main parameters of a tsunami wave and the characteristics of the source gener-
ating it—the deformation area of the ocean bottom. The wave parameters of interest
to us comprise its amplitude, length and the energy of the wave perturbation. The
source is characterized by the amplitude and duration of the ocean bottom deforma-
tion, as well as its horizontal extension.
The piston and membrane mechanisms of wave generation, both of impulse and
finite duration, have been investigated analytically [Kajiura (1970); Murty (1977);
Dotsenko et al. (1993), (1995)] and numerically [Marchuk et al. (1983)]. There also
exists a small number of publications devoted to laboratory simulation of the gen-
eration process [Takahasi (1934, 1963); Hammack (1973); Nosov, Shelkovnikov
(1997)]. A review of experimental works can be found in [Levin (1978)].
We shall first deal with elementary results that can be obtained within the frame-
work of linear theory of long waves. The one-dimensional wave equation, describ-
ing displacements of the free surface,
ξ
, in the case of deformations
η
of the ocean
bottom, exhibit the following form:
