Geoscience Reference
In-Depth Information
Problems of established motion are doubtless expedient for understanding the pe-
culiarities of physical processes taking place during wave generation by running
displacements. But in reality a tsunami forms during a certain finite time interval.
Therefore, we shall further consider models assuming deformations of the basin
bottom to be limited in time.
It must be noted that practically all prototypes of the running displacement (with
the exception of underwater landslides) exhibit velocities superior to the velocity of
sound in water, therefore, the model of an incompressible liquid, considered here,
is often not adequate for describing the process. Nevertheless, the solution of this
problem is certainly not without significance, for the following reasons. Earlier,
the running displacement as a tsunami generator was studied exclusively within
the framework of the theory of long waves [Novikova, Ostrovsky (1979)], which
occupies a lower position than potential theory in the hierarchy of models. The
theory of incompressible liquids is a special case (and limit for
c
) of the more
general theory of compressible liquids. Consequently, the solution of the problem
for an incompressible liquid will be a convenient benchmark in the construction of
a more complex theory, and, moreover, the possibility arises of direct comparison of
solutions of one and the same problem, obtained within the frameworks of different
theories.
Making use of solutions (2.65) and (2.72), obtained within the framework of lin-
ear potential theory, we shall perform comparative analysis of dispersive tsunami
waves excited by piston-like and running displacements of the basin bottom and
subject to dispersion [Nosov (1996)]. We shall also compare such piston-like and
running displacements that form identical residual deformations during the same
time period, which, evidently, is expressed by the condition
b
=
v
→
∞
τ
, where
b
is
the horizontal size of the source,
is the duration of the process at the source and
v
is the propagation velocity of a running displacement. In other words, we are
attempting to compare the efficiency of wave excitation, when the area filled ex-
hibits a rectangular shape and is adjacent to the basin bottom, by two methods: from
below upward and from left to right. Figure 2.21 presents the profiles of waves
calculated at time moment
t
= 50(
H
/
g)
1
/
2
for the value of parameter
b
= 10
H
,
which is characteristic of real tsunami sources. An ordinary piston-like displace-
ment forms identical waves in the positive and negative directions of the
Ox
axis,
while waves, excited by running displacements of the basin bottom, manifest an ex-
plicit asymmetry: a more intense train of waves runs in the direction of propagation
of the displacement. The clearest asymmetry is revealed at propagation velocities
of displacements,
v
, close to the velocity of long waves, (g
H
)
1
/
2
. In the case of
sufficiently large velocities
v
the profiles of waves, corresponding to piston-like and
running displacements, actually become identical.
As a measure of the intensity of wave generation by the two mechanisms inves-
tigated, we shall take advantage of energy (per unit 'canal' width), calculated by
the formula
τ
g
2
d
x
.
W
=
ρ
ξ
(2.86)
