Geoscience Reference
In-Depth Information
is a process, permitted from the point of view of energy. Moreover, the process
considered requires a negligible part of the earthquake energy.
7.3 Parametric Generation of Surface Waves in the Case
of an Underwater Earthquake
One of the most widespread effects, described by eyewitnesses of seaquakes, con-
sists in formation on the ocean surface of standing waves of large amplitude. To
reveal the causes resulting in generation of such waves and to calculate their length
it is quite sufficient to apply linear theory. To determine the amplitude of the waves,
their shape, their type of space symmetry it is necessary to take non-linearity into
account.
Assume that during an earthquake the ocean bottom undergoes periodic move-
ments, in accordance with the following law
η
(
t
,
x
,
y
)=
η
0
cos (
ω
t
)
.
(7.13)
Since the horizontal scale of the pleistoseismic zone significantly exceeds
the ocean depth, we assume the amplitude of oscillations,
0
, not to vary along
the horizontal plane. If the ocean depth satisfies the condition
H
<
c
η
, where
c
is the velocity of sound in water (see Sect. 3.2.1), then the water column be-
haves like an incompressible medium, undergoing induced oscillations, that repeat
movements of the bottom. In a deep ocean elastic oscillations of the water col-
umn at normal frequencies
π
/
2
ω
ν
k
= 0
.
25
c
(1 + 2
k
)
H
−
1
, where
k
= 0
,
1
,
2
,...
, can
arise in the case of any vertical movements of the bottom (not necessarily periodic
movements) (see Sect. 3.1.3). In the second case we shall consider the movements
of a certain upper layer of the ocean, of thickness
h
<
H
, to proceed according
to the law (7.13). We shall choose the thickness of this layer so as to be able to
describe its behaviour as the motion of an incompressible liquid
h
<
c
.
Thus, as initial conditions we have a layer of incompressible liquid with a free
surface, in the field of gravity. The layer undergoes vertical oscillations according
to the law (7.13). We shall show that such a system is not stable, and inside it there
develop standing surface gravitational waves. It must be underlined that in the case
dealt with the layer of liquid is not limited horizontally, therefore, the nature of
the standing waves considered here has little in common with the traditional stand-
ing waves, which arise in a restricted region.
We now pass to a noninertial reference system, the origin of which oscillates in
accordance with the law (7.13). In this case the gravitational field is supplemented
with the periodical in time, but uniform in space, component
π
/
2
ω
2
cos (
η
ω
ω
t
)
.
a
(
t
)=
0
