Geoscience Reference
In-Depth Information
where
C
1
,
2
are coefficients determined by the initial conditions,
(
t
) is a periodic
function. The growth increment of the oscillation amplitude in the case of parametric
resonance (
n
= 1) is determined by the following formula [Rabinovich, Trubetskov
(1984)]:
ϕ
3
=
η
ω
0
α
.
8g
The development of the parametric resonance may be hindered by various effects.
But one such effect is always active—the viscosity of water. Gravitational surface
waves in a viscous liquid are characterized by an amplitude damping decrement
expressed by the formula [Landau, Lifshits (1987)]
k
2
,
γ
= 2
ν
where
is the kinematical viscosity of water. Assuming the length of waves, formed
in the case of parametric resonance, to be significantly smaller than the ocean depth
(
kH
>>
1), we obtain from formula (7.19) the relationship between the frequency
and wave number,
ν
2
= 4g
k
. The decrement of viscous damping can now be readily
expressed via the frequency,
ω
4
8g
2
.
ν
ω
γ
=
Equating the increment
α
and the decrement
γ
, we obtain the threshold condition
for development of the parametric resonance,
ν
ω
g
.
η
0
c
=
For development of parametric waves it is necessary for the amplitude of bottom
oscillations to exceed the critical value
η
0
c
. At frequencies peculiar to seismic os-
cillations of the bottom ((0.1-1 Hz), the critical amplitude turns out to be extremely
small,
m. Therefore, one can conclude that the conditions for the para-
metric resonance to develop are fulfilled in the case of any earthquake that is just
felt. A strong earthquake usually lasts for several minutes. Will the amplitude of
parametric waves reach a noticeable value in this time depends on the increment.
The characteristic growth time of the amplitude is defined as the quantity inverse to
the increment,
η
0
c
∼
1
µ
1
α
8g
τ
≡
=
3
.
η
ω
0
The quantity
is seen to be determined by the amplitude and frequency of
oscillations, but its dependence upon the frequency is stronger. Thus, for example,
in the case of an oscillation frequency of 1 Hz and amplitude of 0.1 m, the devel-
opment time of parametric waves will only amount to 3 s. Therefore, during an
earthquake the amplitude has a high probability of reaching a significant value.
Note that to a frequency of 1 Hz there corresponds a wavelength of the order of
10 m (see Fig. 7.12). Standing waves of precisely such lengths have been repeatedly
observed by eyewitnesses of seaquakes.
τ
