PHYSC 3622
Experiment 2.8
6 February, 2016
Surface Waves under Vertical Forcing
Purpose
This experiment will introduce you to the basic applications of imaging techniques.
You will use these techniques to study the standing wave patterns on the surface of a
fluid layer under vertical forcing.
Equipment
Loudspeaker (a car subwoofer), Petri dish holder with a reflection mirror, Overhead
projector lamp with a cooling fan, HP 200DC frequency generator, Dynaco stereo
amplifier (400W), USB video conference camera, Vicam Vidcap (video capture
program), SigmaScan Pro 5.0 (image processing software), Glycerin aqueous solutions.
Background
When a thin fluid layer is rigidly oscillated in the vertical direction, the acceleration
periodically modulates the effective gravity. As a result, the flat fluid surface becomes
unstable at sufficiently large driving and surface waves form with a frequency one half
the driving frequency. The spatial pattern, which is usually seen initially, corresponds
to the linear modes of most closely resonant with this sub-harmonic frequency. The
linearly unstable modes are similar to the modes of a drum. This phenomenon was first
observed and studied by Faraday (1831) and is associated with his name. By varying
the amplitude and frequency of the driving force and by using fluids of different
viscosities, a number of interesting effects have been observed. These include the
emergence of standing wave patterns of different symmetries (parallel strip patterns,
square patterns, and hexagonal patterns) near the onset, secondary instabilities of these
patterns when the amplitude of the periodic driving is increased, and spatial temporal
chaotic states at even larger amplitudes of the driving force. Figures 1 and 2 show two
wave patterns observed in the 20% glycerin solution.
Fig. 1 A surface wave observed at the driving frequency 130.7 Hz.
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PHYSC 3622
Experiment 2.8
6 February, 2016
Fig. 1 A surface wave pattern observed at the driving frequency 144.9 Hz.
References:
(1) S. Ciliberto and J. P. Gollub, "Chaotic mode competition in parametrically forced
surface waves," J. Fluid Mech. 158, 381 (1985).
(2) N. B. Tufillaro, R. Ramshankar, and J. P. Gollub, "Order-Disorder Transition in
Capillary Ripples," Phys. Rev. Lett. 62, 422 (1989).
(3) P.-L. Chen and J. Vinals, "Pattern Selection in Faraday Waves," Phys. Rev. Lett. 79,
2670 (1997).
(4) T. B. Benjamin and F. Ursell, “The stability of the plane free surface of a liquid in
vertical periodic motion,” Proc. R. Soc. London A, 225, 505 (1954).
Procedure
The system is a cylindrical fluid layer about 1 cm deep in a plastic petri dish of
inner diameter 8.5 cm. The fluid samples are mixtures of distilled and de-ionized water
and glycerin. The concentration of the glycerin solutions varies from 20 to 80 wt.%. A
small amount (a few drops) of polystyrene latex spheres of diameter ~0.1 m is added
into the solution. These small particles give the solution a milky look, which increases
the contrast of the surface waves.
The petri dish holder with a reflection mirror is mounted on the cone of a
loudspeaker in a way that allows vertical forcing while still permitting light to be
transmitted vertically through the petri dish. Precautions should be taken to ensure
that the oscillation direction and the cell axis are vertical. A power amplifier and a
frequency generator are used to drive the loudspeaker. The control parameters of the
experiment are the amplitude (0-200 m) and frequency (0-1000 Hz) of the oscillation.
To a good accuracy the motion of the fluid cell is sinusoidal.
The surface deformation of the fluid layer is studied by allowing an expanded
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PHYSC 3622
Experiment 2.8
6 February, 2016
light cone to pass through the fluid layer vertically. The use of an overhead projector
lamp provides strong illumination and better contrast. To avoid the heating effect from
the light source, the projector lamp should be placed ~60 cm away from the fluid
sample. The shadow image of the surface waves is formed on a translucent screen (a
tracing paper) located ~7 cm above the fluid sample. A videoconference camera, which
is mounted 25-30 cm above the screen, is used to record the image of the surface waves.
The camera is connected to a computer through the USB cable. Using the video capture
program Vicam Vidcap, one can directly view the surface wave images on the
computer monitor. With some minor adjustments, you should be able to get clear
pictures of the surface waves.
After setting up the apparatus, you may study how the surface wave patterns
change with the amplitude and the frequency of the vertical driving. The driving
frequency can be easily measured by using an oscilloscope. To measure the oscillation
amplitude, you need to attach a mirror to the oscillating frame and get a laser beam
reflected from the mirror. The oscillation amplitude can be determined directly from
the change of the position of the reflected laser beam using a position-sensitive
photodiode (see Ref. 1 for more details).
Under a low-amplitude oscillation of frequency f0, the standing wave patterns are
well described by the simple linearly unstable modes. In this case, the surface
displacement has the following form:
Slm (r, , t)=Jl (klm r) sin (l+0) sin (2f0 t/2),
where r and  are the polar coordinates, t is the time, 0 is an initial phase angle, and Jl
is the Bessel function of order l. The wave number klm is determined by the boundary
condition that the derivative J’l (klm R)=0. The modes are labeled by the indices (l, m),
where l is the number of angular maxima and m-1 (or m if l=0) is the number of nodal
circles (see Ref. 4 for more details).
Before going to more complicated non-linear wave patterns, you may first try to
find a linear mode. Using the SigmaScan Pro 5.0 program, you can get an intensity
profile of the circular standing wave along the cell diameter. An azimuthal average of
the intensity profile improves the statistics of the final intensity profile over varying
positions. This intensity profile should be compared with the equation discussed
above.
For other standing wave patterns, you may plot their phase diagram as a
function of the driving amplitude A and frequency f 0 (see Refs. 1 and 3 for more
details). These wave patterns remain certain symmetry in a given range of A and f 0.
Another interesting effect you may study is to see how the fluid viscosity and cell size
affect the wave pattern. The fluid viscosity can be changed using the glycerin solutions
with different concentrations. To change the cell size, you need to make a new petri
dish adapter so that a smaller petri dish can be mounted on the oscillation frame.
Questions
Because the parameter space of this experiment is very large, you may want to vary
only a few parameters and keep the other parameters fixed. In you project report, you
need to discuss how these parameters are chosen and compare your results with the
previous results in the literature.
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