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< (g H ) 1 / 2 , then the linear response of the wa-
ter column represents a superposition of induced oscillations (in the source area) and
of gravitational waves, emitted towards the distant zone. If the frequency lies within
frequency of bottom oscillations,
ν
the range (g H ) 1 / 2 <
< c / 4 H , then no gravitational waves arise, and movements
of the water column exist only in the immediate vicinity of the source in the form
of induced oscillations. As the frequency increases up to values
ν
> c / 4 H , a quali-
tative change occurs in the dynamics of the linear response, the water column starts
to behave like a compressible medium.
The three frequency ranges identified above are shown in the 'ocean depth-
frequency of bottom oscillations' plane in Fig. 3.17. It is interesting to note
that the dependences intersect at the hypothetical ocean depth of H = 14 , 350 m,
consequently, in the conditions of the planet Earth the three indicated ranges exist
at any point of the World Ocean. Non-linear effects are manifested only in the case
of sufficiently high velocities of bottom movements, which can be characteristic
of the frequency ranges 'II' and 'III'. Clearly, the low-frequency range 'I' is of no
special interest.
The mathematical model of tsunami generation due to non-linear effects will
be constructed on the basis of Euler's equations, assuming the liquid to be
compressible,
ν
v
p
ρ
+(v ,
) v =
+ g ,
(3.37)
t
∂ρ
+ div (
ρ
v)=0 .
(3.38)
t
will be assumed constant,
while for the compressible liquid we shall consider the variations in pressure to be
proportional to the variations in density, p = c 2
In the case of an incompressible liquid the density
ρ
ρ .
Applying a device used, for instance, in turbulence theory or non-linear acoustics,
we represent movement of the liquid as the sum of a slow (average) movement and
of a fast (oscillatory) movement,
+ v ,
+ p ,
ρ .
v =
v
p =
p
ρ
=
ρ
+
(3.39)
1,000
(Hz)
Fig. 3.17 Character of linear
response (frequency ranges)
of a water column of depth
H to oscillations of the ocean
bottom of frequency
1,000
10,000
H (m)
ν
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