Geoscience Reference
In-Depth Information
In the case of the ocean bottom, longitudinal waves are intended, since in the one-
dimensional case considered transverse and surface seismic waves are not excited.
The maximum period of elastic oscillations of a water column with a free surface
is known to be
T
0
= 4
H
/
c
. Thus, we obtain the following formula for the damping
time:
τ
s
=
T
0
2
D
0
.
We shall consider the density of water and the propagation velocity of sound
in water to be
= 1
,
000 kg/m
3
and
c
= 1
,
500 m/s, respectively. In the case of
rock, making up the ocean bottom, its density and the velocity of longitudinal
waves in it vary within the respective limits 1
,
400
<
ρ
ρ
b
<
3
,
500 kg/m
3
and 1
,
700
<
c
b
<
8
,
000 m/s. Correspondingly, the transition coefficient varies within the limits
0
.
19
<
D
0
<
0
.
95. The lower limits of the indicated ranges correspond to friable
sedimentary rock. Usually, the ocean bottom has a stratified structure. The effective
reflection of elastic waves takes place from the acoustic base—a certain sufficiently
dense and high-velocity column. Thus, for example, in the case of
ρ
b
= 3
,
000 kg/m
3
and
c
b
= 7
,
000 m/s, we obtain
2
T
0
. So, it takes two periods for the energy of
elastic oscillations to be reduced by a factor of
e
, and four periods for the oscillation
amplitude.
τ
s
≈
3.1.2 General Solution of the Problem of Small Deformations
of the Ocean Bottom Exciting Waves in a Liquid
In this section the general solution is constructed for the problem of small deforma-
tions of the basin bottom exciting acoustic-gravity waves in a column of liquid. The
problem is formulated exactly like in Sect. 2.2.1, with the sole exception consisting
in that, here, the liquid is considered to be compressible.
We shall consider an infinite, in plane 0
xy
, column of an ideal compressible ho-
mogeneous liquid of constant depth
H
in the field of gravity (Fig. 2.8). The origin
of the Cartesian reference system 0
xyz
will be located on the free non-perturbed
plane with axis 0
z
directed vertically upward. Motion of the liquid is caused by de-
formations of the bottom of small amplitude (
A
H
). Moreover, we shall assume
the deformation velocity of the bottom, and consequently, the velocity of motion
of particles of the liquid to be small quantities. This makes it possible to neglect
the non-linear term (v
) v in the Euler equation. Dynamical variations of density
ρ
and pressure
p
in the liquid are also small as compared to the equilibrium quan-
tities (
∇
ρ
ρ
0
,
p
p
0
). Assuming the equilibrium flow velocity to be zero, we
substitute the pressure and density, expressed as
p
=
p
0
+
p
,
ρ
,into
the Euler and the continuity equations. Neglecting small quantities of the second
order and introducing the potential of the flow velocity,
F
(v =
ρ
=
ρ
0
+
∇
F
),wearriveat
the wave equation
2
F
∂
∂
c
2
t
2
−
∆
F
= 0
.
(3.4)
where
c
=
∂
p
∂ρ
s
represents the velocity of sound.
