[P. Alstr°m] [J. S. Andersen] [W. I. Goldburg] [A. E. Hansen] [M. T. Levinsen] [E. Schr÷der]
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The principle in the Faraday experiment is to vibrate a container with liquid vertically at a fixed frequency and amplitude.
At low vibration amplitudes, the liquid moves as a solid body, but at high vibration amplitudes, surface waves form. The vibration amplitude is denoted by A, the onset amplitude for the waves is denoted by , and an excitation number is defined:
A collection of different wave modes compete and which pattern wins depends on the excitation number, the frequency, and the liquid. It is worth noting that all the modes have the same wave length, and that the wave length is independent of the vibration amplitude.
The wave patterns are visualised by shining light through the container from below and using the wave-crests and troughs as respectively convex and concave lenses. This results in an intensity pattern on a frosted glass plate close to the liquid surface, where crests appear light and troughs appear dark. The image formed on the glass plate is recorded with a 8-bit CCD camera.
On the surface of a vertically oscillated fluid, capillary waves with a clearly discernible wavelength are formed if the amplitude of the oscillations exceeds a critical value. Particles sprinkled on the fluid surface are experimentally found to move in an almost Brownian motion when measured over distances larger than . We observe a cross-over in the diffusive motion from a strongly anomalous diffusion below , to a motion that is closer to being Brownian above . Our observations show that the particle motion is well described by an amplitude-independent fractional Brownian motion, effective at sizes less than , convoluted with an amplitude-dependent fractional Brownian motion, effective on all length scales smaller than the system size.
The velocity field on the free fluid surface is visualised by tracking small floating particles that follow the liquid (mushroom spores on water). The particles are illuminated by shining light from DC lamps at a very small angle relative to the fluid surface. A CCD camera records images of the particles on a VCR.
The motion of particles was analysed by two means.
(i) We measured the particle displacement
over a time along an arbitrary axis x for many initiation times t. From this, the variance
was found, and the diffusivity extracted; . For Brownian motion, the diffusivity is a constant. Ramshankar, Berlin, and Gollub  found the diffusion to be slightly anomalous with a time-dependent diffusivity that obeys a power law on time scales larger than 1 s. This type of motion can be modelled by fractional Brownian motion. We have extended these studies to a larger range of time scales and excitation numbers .
(ii) We measured the yard stick fractal dimension D of the particle tracks from
where is the number of yard sticks of length l needed to cover the track. If the motion can be modelled by fractional Brownian motion then .
We find that all seven curves in figure 1(a) show a cross-over at an -dependent time , with one power-law behavior for large times, , and a steeper power-law behavior at small time scales. The times correspond to the wave length .
Rescaling time, we see that the curves may be collapsed, whereby any -dependence is removed, see figure 1(b).
The separation of the cross-over at from the -dependence is emphasized by plotting the variance V for the various values versus V for a particular value, using the reduced time as the parameter (figure 1(c)). We have chosen as the reference (having ordinary Brownian motion at ). As is observed, there is no sign of ; all curves are straight lines.
This suggests that the diffusion may be modelled by a convolution of two fractional Brownian motions: an -independent motion effective below the wavelength (figure 1(b)), convoluted with an -dependent motion which is effective on all length scales smaller than the system size (figure 1(c)). In physical terms, the former may result from turbulent eddies of size less than the wavelength, while the latter results from turbulent eddies of all sizes less than the system size.
A similar analysis as for the particle displacements may be performed for the yard stick fractal dimension. In figure 2(a) we show the number of sticks of length l needed to cover the particle tracks, normalized by so that long tracks do not dominate the calculations. Again we see a cross-over at length , and we see that the dependence may be removed by a rescaling of the length, , (figure 2(b)). And finally, we can again remove the cross-over by plotting the curves against a particular curve (figure 2(c)).
As suggested by the relation we can compare as determined from the fractal dimension and H as determined from the particle displacements, (figure 3). The curves have a similar shape, and both curves approach (Brownian motion) for large excitation values .
The motion of particles along the x- and y-axes is fractional Brownian with exponent H.
Let H be a number between 0 and 1. If is fractional Brownian, then is a random function, where increments fulfill that
is Gaussian distributed with mean zero and variance one.
Ordinary Brownian motion corresponds to
i.e. the mean distance travelled grows as . In general, however
where means that distributions are equal. Consequently, the following relation is obeyed for the x,y - motion
where D is the fractal dimension of the particle tracks.
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