Geoscience Reference
In-Depth Information
Thus, if the layer of liquid oscillates as a unique whole vertically with a frequency
ω
, then standing waves arise on its surface, that are characterized by the frequency
ω
/
2. Note that for parametric resonance to be realized the water column does not
have to oscillate with a certain definite frequency. The oscillation frequency can be
arbitrary, but the length of the waves formed depends on it.
If quite high oscillation frequencies (
10 Hz are considered, then effects of sur-
face tension must be taken into account and the general formula must be applied:
g
k
+
α
ρ
∼
k
3
tanh(
kH
)=
n
2
2
,
(7.20)
where
is the surface tension coefficient.
The relationship (7.20) is presented in Fig. 7.12 in the 'wavelength-frequency'
plane. The calculation was performed for
n
= 1 and an ocean depth
H
= 4
,
000 m.
The dotted line shows the dependence in the case of small depths (
H
= 10 m), when
the surface waves start 'to feel the bottom'. Deviation of the dependence from lin-
earity at small wavelengths is explained by the action of surface tension forces.
To characteristic seaquake frequencies of 0.1-1 Hz there correspond wavelengths
from tens up to hundreds of meters, which is in good agreement with the testimonies
of eyewitnesses of these events. The characteristic dimensions of the space struc-
tures (
α
1 cm), observed in laboratory experiments at high frequencies (10-50 Hz),
are also in good accordance with theory.
In accordance with the Floke theorem, the general solution of the Mathieu equa-
tion can be represented in the form
∼
A
(
t
)=
C
1
exp
{
α
t
}
ϕ
(
t
)+
C
2
exp
{−
α
t
}
ϕ
(
−
t
)
,
1,000
1,000
(m)
Fig. 7.12 Relationship between the frequency of bottom oscillations (or oscillations of the water
column) and the wavelength of standing waves, formed on the water surface in the case of para-
metric resonance. The calculation is carried out for
n
= 1 and for ocean depths 4,000 m (solid line)
and 10 m (dotted line)
