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medium. Only one of the manifestations of water being compressible in the case
of underwater earthquakes, namely the T-phase, has been studied relatively well
[Soloviev et al. (1968), (1980); Brekhovskikh (1974); Kadykov (1986), (1999);
Lysanov (1997); Okal (2003); Okal et al. (2003)]. The range of frequencies between
1 and 100 Hz is usual for the T-phase. We note, here, that in this chapter we will be
interested in hydroacoustic phenomena that are related to another frequency range
(
0
.
1 Hz) and are localized in the immediate vicinity of the tsunami source.
The necessity of taking into account non-linear effects during tsunami formation
is related to the fact that in the case of seismic motions of the seabed exhibiting
small amplitudes, the velocities of these motions may turn out to be quite significant.
The non-linear mechanism of tsunami generation was first considered in [Novikova,
Ostrovsky (1982)]. This line of research was further developed in [Nosov, Skachko
(2001a, b); Nosov, Kolesov (2002), (2005) and Nosov et al. (2008)].
∼
3.1 Excitation of Tsunami Waves with Account
of the Compressibility of Water
3.1.1 Preliminary Estimates
If the process is treated from a formal physical point of view [Landau, Lifshits
(1987)], then a liquid can be considered incompressible only when
∆ρ
ρ
1,
where
is the density of the liquid. As it is known, the necessary condition for
the above to be valid is that the motions of the liquid exhibit small velocities, as
compared to the velocity of sound. In the case of stationary motion this condition
is sufficient. The problem of tsunami generation is evidently non-stationary, so one
more, additional, condition must be fulfilled. In the problem of tsunami generation
both conditions are of the following form:
1.
v
ρ
c
.
τ
Hc
−
1
,
Lc
−
1
where
v
is the characteristic mass velocity of motion (of water or of particles of
the ocean bottom),
c
is the velocity of sound in water,
2.
is the duration of bot-
tom displacement,
H
is the ocean depth and
L
is the characteristic horizontal size
of the source. Note that even in those rare cases, when the authors of one or an-
other investigation substantiate application of the theory of incompressible liquids
in the tsunami problem, the second condition is always forgotten. The characteris-
tic values of the indicated parameters are the following:
v
τ
∼
1m/s,
c
∼
1
,
500 m/s,
H
1-100 s. The first condition is seen to be well
satisfied, while the second can be violated in many cases. In the case of a running
displacement (a fault ripped open or a surface seismic wave) the first condition will
also be violated.
For problems concerning tsunami propagation in the open ocean and waves run-
ning up a coast, the first condition remains the same and is quite fulfilled. The
second condition assumes the following form:
T
∼
4
,
500 m,
L
∼
10-100 km,
τ
∼
Hc
−
1
,
c
−
1
, where
T
is
