WATER WAVES IN A CIRCULAR ELASTIC VESSEL:
P. DENISSENKO and D. Y. HSIEH
Department of Mathematics, Hong Kong University of Science and Technology
Presented at American Physical Society / Division of Fluid Dynamics
51st Annual Meeting, November 22-24, 1998, Philadelphia, PA.
Water waves in a vibrating circular elastic vessel are investigated experimentally. The vessel is fitted with a transparent bottom. The refraction pattern of the laser beam shining from below is captured by a photographic paper placed above the vessel. Sequence of axisymmetric capillary waves, circumferential capillary waves, jumping droplets and the large amplitude gravity waves as the amplitude of the excitation increases are studied. Photographs taken with stroboscopic lighting are also presented.
One of the marvels of Chinese ancient scientific discovery is the mythical vessel of Dragon Wash or Fish Wash. In simple terms, the Dragon Wash is a brass wash basin with two hollow handles. With the Dragon Wash half filled with water, when the handles are rubbed with clean wet hands, distinct sound is heard and quite regular capillary wave patterns appear along the edge of the water surface. With more vigorous rubbing, water droplets start to jump very intensively from several zones located near the water edge. (Fig.1) Different types of sound and wave patterns can be obtained by rubbing the handles differently. We have managed to detect two types of capillary waves. Low frequency gravity waves may also be observed after some kind of rubbing.
The mechanical properties of the Dragon Wash have been studied both experimentally and theoretically by Wang [1,2], which concerns mainly the mechanical oscillation of the elastic vessel and does not discuss much about the problem of water waves. Theoretical linear studies have also been carried out by Shen, Sun and Hsieh [3,4] on capillary-gravity waves in a circular basin. These studies indicate that standing wave patterns along the edges could be resonantly generated by horizontal forcing oscillation of the side wall of the basin, if the input excitation would contain components, even though very tiny, of the desired wavelength. Subsequently, Wang  performed experiments using commercially available upright circular cooking utensils and reproduced essentially the same dramatic phenomena as those of Dragon Wash, by horizontal excitation at one point of the side wall. Moreover, when the excitation amplitude is large enough, axisymmetric, large amplitude, low-frequency wave begins to appear. Wang and his co-workers have also made flow field measurements using PIV technique . Nonlinear theories have also been developed to study these phenomena [4,8], using the Krylov-Bogoliubov averaging method and the multiple-scale expansion scheme respectively. It is shown that a low frequency oscillation of modulation can be generated due to the nonlinearity of the system.
There are three main problems relating to the Dragon Wash phenomenon:
1. To explain the mechanism of generation and structure of capillary water waves appeared on the surface.
2. To explain the appearance of jumping droplets.
3. To explain the appearance of the low frequency gravity wave.
A theoretical study has been attempted to deal with the first and third problems . However, all these theoretical studies were based on quite primitive experimental findings. More detailed in-depth experimental studies are needed for verification of or to guide further theoretical investigations.
A circular cylindrical stainless cooking vessel was used to model the Dragon Wash. The bottom of the vessel was cut off and replaced with a transparent plexiglass plate. The dimensions of the vessel are height 140mm, radius 220mm and the thickness of wall 0.4mm. The vessel was filled with water at depth from 50mm to 135mm for various experiments. The experimental setup is shown in Fig.2.
Wall oscillations were excited by exerting horizontal vibration at two symmetrically positioned points at some distance above the water surface.
Wall displacement was measured by the capacitor-sensor, consisting of the steady plate and oscillating wall. The capacitor was loaded with 4 Mhz alternating voltage and the passing current was measured. The system was calibrated for steady wall displacement.
For qualitative investigation of water waves, laser beam was directed to the surface vertically from below. When the amplitude of the waves is small, deviation of laser beam from the vertical direction is proportional to the angle between the water surface in the point where the laser beam crosses it and the horizontal. In particular, beam deviation from vertical $ \beta \approx \alpha (n-1) $ where $\alpha$ is deviation of water surface from horizontal, and $n$ is refraction index of water (Fig.3). Laser beam traces were captured by placing a photographic paper on the beam passage.
For visualization and photographing of wave structure, stroboscopic lighting from below was used as well.
Resonances of the vessel are marked by the singing sounds and the appearance of waves on the water surface. There are many resonances as the frequency of the excitation increases. Let us concentrate on the lowest mode of the resonances, which has 4 nodes on the circumference of the vessel. Observations were made as the amplitude of the excitation was increased.
(1) Axisymmetric Capillary Waves
When the amplitude of excitation is very small, axisymmetric capillary waves with wavelength corresponding to the excitation frequency appear on the surface. The amplitudes of these waves are so small that they are almost invisible to naked eyes. However they can be easily detected by the laser beam refraction on the water surface, as shown by the photograph on Fig. 4a, using stroboscopic light tuned with the excitation frequency. Traces of laser beam are straight for this case (Fig.4b).
The correspondence of the frequencies of the water wave and the excitation can also be determined by the following experiment. Laser beam was to be reflected by a mirror fixed at the wall of the vessel, and then refracted by the water surface. Another mirror was positioned so that the deflection of the laser beam corresponding to wall motion would be parallel to the surface waves wavefront. The setup is shown in Fig. 5. When the amplitude of excitation is small as in this case, laser beam traces were steady and have a single loop. It shows that the frequency of laser beam deflection in both directions have the same frequency. It is interesting to note that for very small amplitude of excitation, the excited water wave is axisymmetric in spite of asymmetry of excitation.
When the amplitude of excitation was increased somewhat, azimuthal modulation of the waves appear as shown in Fig. 6. These azimuthal modulations correspond to the asymmetry of the vessel oscillation which has several (say 4,6,8 etc) nodes in the circumferential direction.
(2) Circumferential Waves
Further increase of the excitation amplitude results in the appearance of another kind of waves. These are waves propagating in the circumferential direction. They are visible to naked eyes. Their amplitude grows immediatly after appearance and become larger than the amplitude of axial waves. Further increase of excitation amplitude shows clearly that the wavefronts of circumferential waves are not straight in the radial direction. With stroboscopic light tuning to half of the excitation frequency, the wavelength of the secondary waves can be measured, which was found to be indeed larger than that of the primary wave. These photographs also revealed very interesting patterns of the wave structure (Fig. 7a). They are curved as if the wavefronts were emanated from the point of excitation. The laser trace now shows a figure of 8 (Fig.7b).
This means that the frequency of this "secondary" wave is about half of that of the"primary" one. Examining carefully at this traces one can recognize patterns typical for phase portraits of systems undergoing period doubling bifurcation. At Fig.7c wave pattern is compared with contour plot of function corresponding to natural wave mode.
With further increase of the excitation amplitude other wavemodes appear and the laser traces become rather complicated and eventually chaotic (Fig. 8a,b).
(3) Jumping Droplets
At higher amplitude of excitation, water droplets start to jump from the edge of the vessel (Fig.9). With water droplets jumping and dropping and splashing on the surface, organized pattern can be hardly recognized. However it was observed that there is strong surface flow towards the center . The jumping of droplets apparently exert some kind of force on the fluid in the surface region.
The jumping droplets are not distributed uniformly along the circumference. They are concentrated around the maximums of wall oscillations. For our chosen mode of resonance, there are 4 regions of concentrated activity (Fig. 1). More regions of concentrated activity appear for higher mode of resonance.
Measurements of maximum wall velocity necessary for droplets to start to jump were carried out for different frequences (Fig.10).
Capacitor sensor was used to measure the wall displacement (Fig.11).
(4) Gravity Waves
For some resonant modes, large amplitude standing gravity waves were observed when the amplitude of excitation was increased further after the appearance of the jumping droplets.
The wave frequency agrees with that of the corresponding gravity wavemode.
It should be remarked that the excitation frequency is high, and those high frequency capillary waves and jumping droplets are still there. However, they are greatly suppressed during that phase of gravity wave, when the water level is higher near the walls than in the center. The gravity wave can be easily seen. It results typical modulations of singing sound. Modulation of wall vibration was also registred by the capacitor-sensor.
Non-axisymmetric gravity wave has also been observed (Fig.12).
It seems that the excitation of this gravity mode is due to the positive feedback between the wave and the large average force exerted by the droplets jumping. Quantitative verification has not yet been made. However, there are some facts pointing to the validity of this assumption.
In Fig.13 dependence of natural frequency of our vessel filled with water as a function of the water level is presented for different vibration modes. It was noted that gravity waves appears when excitation frequency is slightly higher than that of the natural resonant frequency. So, when the water level near the walls is less than that in the center, jumping droplets appears since the excitation frequency become locally resonant. The force induced works toward increasing the difference in the water level, that corresponds to a positive feedback in the system.
The Dragon Wash phenomenon is a fascinating subject as well as a beautifull physical effect. It is challenging both theoretically and experimentally. It is a nonlinear problem with many competing high frequency modes. The break-up of the water surface and the resulting jumping of droplets still awaits in-depth studies. The techniques we developed by using laser beams and transparent bottom seem to be a promising experimental method for the study of these phenomena. Together with PIV techniques for probing the internal flow fields, we may be able to have more precise quantitative measurements.
To a lesser extent, the Dragon Wash phenomena exist not only in circular vessels. Preliminary crude experiments have also been carried out for rectangular and triangular vessels. More elaborate investigations still need to be done.
1. WANG, D., A Research on the Mechanical Properties of Cultural Relics of China, Proc. Sci. & Tech. Archaeology, 193-202 (1991).
2. WANG, D., Study on Mechanical Characteristics of Ancient Cultural Relics, Sci. Conservation & Archaeology 5, 35-39 (1993).
3. SHEN, M.C., SUN, S.M. & HSIEH, D.Y., Forced Capillary-Gravity Waves in a Circular Basin, Wave Motion 18, 401-412 (1993).
4. HSIEH, D.Y., Standing Water Waves in a Circular Basin Proc. Int. Conf. Hydrodynamics, 74-79 (1994).
5. WANG, D., Private Communication (1994).
6. WEI, Q., WANG,D., YAN, B., DU, X. & CHEN,J., A Visualization Study on Water Spray of Dragon Washbasin, Chap.11 in Atlas of Visualization III, edited by The Visualization Soc. Japan, CRC Press, New York (1997).
7. WEI, Q., WANG,D., YAN, B., DU, X. & CHEN,J., Flow Field Measurement of "Dragon Washbasin Phenomena" using PIV Technique. proc. of Internatianal workshop PIV-Fukui (?)-95 edited by The Visualization Soc. Japan, CRC Press, New York (1997).
8. SUN, S.M., SHEN, M.C. & HSIEH, D.Y., Nonlinear Theory of Forced Surface Waves in a Circular Basin, Wave Motion 21, 331-341 (1995).
9. HSIEH, D.Y., Water Waves in An Elastic Vessel, Acta Mechanica Sinica 13, 289-303 (1997).
- Last update: 30 Nov, 1998