Role of the Compressibility of Water and of Non-linear Effects in the Formation of Tsunami Waves - Physics of Tsunamis - page 109

character of the curves presented in Fig. 3.3 for the case of incompressible liquids is
in good agreement with the results of indicated publications and Sects. 2.3 and 2.4.
In the case of an incompressible liquid the characteristic features of the amplitude
dependence on the displacement duration are the following. When a displacement
of the ocean bottom is accompanied by a residual displacement, the dependence is
characterized by the presence of a plateau at small values of the parameter τ and

by a monotonous drop at large values. When no residual displacement accompa-

nies the displacement, the dependence considered has a local maximum that shifts
to the right as the source radius increases. Short displacements of the ocean bottom
without residual displacement result in the formation of very weak surface pertur-
bations outside the source.
Enhancement of the size of the active zone leads to an increase in both the ampli-
tude of 'fast' surface oscillations and the amplitude of gravitational waves. It must
be noted that an increase in the size of the active zone causes a noticeable change in
the amplitude only for values of the parameter R < 5, after which the dependence
reaches saturation.
The specific non-monotonous character of the curves, related to the model of
compressible liquids, is due to the aforementioned resonance properties. As the dis-
placement duration increases, the amplitude of 'fast' oscillations tends to decrease.
It is interesting to note that in the case of large values of parameter

τ

τ

all the depen-
dences (both for compressible and incompressible liquids) behave like τ − 1 .Given

other conditions being equal, the amplitude of 'fast' surface oscillations above
the source may be several times larger than the amplitude of surface displacements
of an incompressible liquid.
The duration of real ocean bottom displacements lies within the range 0.4-40
(1-100 s). On the basis of data presented in Fig. 3.3 it is possible to conclude
that the maximum amplitude of gravitational waves proper in the case of bottom
displacements, involving residual deformation, is weakly sensitive to variations of
the parameter . In the case of displacements of the ocean bottom without resid-

ual deformation the amplitudes of gravitational waves, going beyond the limits of
the source, undergo quite significant changes within the range we are interested in.
For any type of displacement the amplitude of 'fast' surface oscillations depends
strongly on the displacement duration.
Figure 3.4 presents the dependences of amplitudes of gravitational waves and of
'fast' surface oscillations upon the distance from the source centre r . The data cor-
respond to the duration of the ocean bottom displacement,

τ

= 1, which does not
violate the general nature of the conclusions, since the form obtained for the solu-
tion of (3.28) and (3.29) reveals that the parameter

τ

does not affect the decrease
of amplitude with distance. In all cases, the amplitude varies weakly immediately
above the source zone. Outside the source zone the amplitude of gravitational waves
decreases approximately like r − 1 / 2 (corresponding to the known asymptotic es-
timates [Pelinovsky (1996)]), while the amplitude of oscillations drops like r − 2 .
Here, displacements of the ocean bottom with and without residual deformation
lead practically to the same oscillation amplitude, while the amplitudes of gravi-
tational waves differ noticeably. Figure 3.4 permits to conclude that 'fast' surface

τ