Geoscience Reference
In-Depth Information
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
Am
Am
(a)
(b)
(mm)
(mm)
Fig. 7.15 Bifurcation portrait of a column of liquid of thickness
H
on an oscillating bottom in co-
ordinates 'amplitude-frequency' (a) and 'amplitude-acceleration' (b). 1—linear wave formation,
2—dissipative structures, 3—chaotic motion
enhancement of the frequency at a constant amplitude and by smooth enhancement
of the amplitude at a fixed frequency. No difference between the two ways of arriv-
ing at the bifurcation point was revealed.
It is remarkable, that the height of the column of liquid in a vessel influences its
dynamic behaviour weakly: when this quantity is altered by nearly an order of mag-
nitude (from 1 up to 7 cm), the shift of the three described regions in the parametric
plane turns out to be insignificant in the case of transition across boundary 2-3 and
absolutely indistinguishable in the case of transition across boundary 1-2.
The creation of bifurcation boundaries in the 'amplitude-acceleration' plane
permits to assume that precisely the acceleration amplitude of bottom oscillations
serves as the main control parameter in the system investigated, which is especially
apparent in the transition from linear wave formation to structures (the two lower
curves are practically parallel to the
x
-axis). This actually signifies that the bound-
aries of the regions, where different dynamic modes in the 'amplitude-frequency'
plane exist, have the form of inverse quadratic dependences.
The stably reproducible hysteresis character of bifurcation boundary 1-2 must
also be noted. When the oscillation frequency (amplitude) is increased gradually,
structures develop in the system at an acceleration amplitude (0
.
26
0
.
004) g. When
the frequency (amplitude) is reduced, a 'collapse' of the wave structures is observed
at a noticeably smaller value of this quantity—(0
.
19
±
0
.
004) g.
Note, also, that transition from stable dynamic structures on water to a state of
chaotic surface motion takes place at different threshold frequencies for waves with
square and hexagonal cells. Figure 7.16 shows photographs of the state of a vibrat-
ing water surface with square wave cells in the case of cyclic oscillation frequen-
cies 100 s
−
1
(a), 160 s
−
1
(b) and 180 s
−
1
(c); here, also, presented are variations in
the mode of hexagonal cells at frequencies 180 s
−
1
(d), 200 s
−
1
(e) and 220 s
−
1
(f).
Figure 7.17 presents a comparison of measurement results for the parameters of
hexagonal cells (triangles) and of square cells (circles) with theoretical dispersion
laws for gravitational waves (1), gravitational-capillary waves (2), and paramet-
ric waves (3). These results were first obtained and described in [Levin, Trubnikov
(1986); Alexandrov et al. (1986)] and [Levin (1996)].
±
