Geoscience Reference
In-Depth Information
the displacement, correspond to negative values of the dimensionless velocity. The
large spread of experimental data, due to the amplitudes of bottom displacements,
η
i
, not being strictly equal to each other, did not permit to separate the experimen-
tal dependences for different water depths, so the experimental points in Fig. 2.25
reflect the data averaged over all the indicated water depths
H
.
Motions of the basin bottom in laboratory and theoretical models somewhat dif-
fered from each other. Therefore, one cannot expect perfect coincidence of the-
ory and experiment, which is particularly noticeable, when
v
/
v
0
<
0. This is also
related to the fact that the difference between displacement amplitudes of bottom
wave generators could amount to 30%, while the maximum amplitude of the wave,
running against the direction of propagation of the displacement, is determined by
the amplitude of the largest of
η
i
. Consequently, in connection with the wave am-
plitude being normalized to the quantity
A
0
=(
η
3
)
/
3, the dimensionless
amplitude will certainly be overestimated as compared with the case of identical
bottom displacement amplitudes.
Theory and experiment show that a running displacement can indeed serve as
an effective mechanism for the excitation of tsunami waves. In the case of prop-
agation velocities of bottom displacements close to the velocity of long waves,
(g
H
)
1
/
2
, sharp enhancement occurs of the amplitude and energy of waves running
in the direction of the displacement propagation. It is known that a tsunami ampli-
tude in the open ocean cannot exceed the amplitude of a piston-like displacement
of the ocean bottom. Contrariwise, in the case of a running displacement the wave
amplitude can significantly exceed the bottom displacement amplitude. In the case
of identical residual deformations of the bottom, a running displacement may turn
out to be many times more effective that a piston-like displacement. The energy
transferred by a running displacement to gravitational waves, when
v
=(g
H
)
1
/
2
,
increases in proportion to the square distance covered by the displacement.
η
1
+
η
2
+
2.3.4 The Oscillating Bottom
In the case of established harmonic oscillations of the bottom we cannot directly take
advantage of the general solution, obtained applying the Laplace transformation,
since the oscillations take place at times
t
<
0. But in the case considered this is
not necessary. For established oscillations it is possible to obtain a fully analytical
solution, which does not require numerical calculation of integrals [Nosov (1992)].
Owing to the response of a linear system existing only at the frequency of inducing
oscillations, we know the frequency of excited waves. Therefore, the solution of
the problem is expediently sought in the following form:
F
osc
(
x
,
z
,
t
)
+
∞
A
(
,
k
) sinh(
kz
)
.
= exp
{
i
ω
t
}
d
k
exp
{−
ikx
}
ω
,
k
) cosh(
kz
)+
B
(
ω
(2.87)
−
∞
