Of all the wave motion on earth--sound, light, radio, X-ray, earthquake--only one kind has a spatial and temporal scale easily perceived by people: water waves. Waves on the surface of water ("gravity waves" in physics) are in the neighborhood of a meter long, and come and go in a second or so... perfect for watching and enjoying.
The Drumhead sculpture proposed here (originally in memory of drum-afficionado Prof. Richard Feynman of Caltech) shows off such water waves to their best advantage. But unlike the travelling waves seen on oceans and lakes, the waves in Drumhead are trapped: standing waves which bounce and spin in a way simultaneously familiar and exotic.
Drumhead is a clear, circular pool of water about six feet across and about two feet deep. It has about five small water jets on its bottom (underwater), pointed directly upwards and controlled by a computer to create precisely-timed independent bursts. As the jets burst in different sequences at at different times, the surface of the water begins to move in a variety of peculiar ways, corresponding to the various kinds of possible standing waves.
There are three basic shapes of wave: side-to-side sloshing, dome-shaped (like in a water-cooler after a bubble), and saddle-shaped. Each shape oscillates back and forth with a time ("perioid") of about a second, so they are easy to see by eye... no strobe lights necessary. But the slosh-shaped wave can also appear to "spin," just like water swirled around in a glass... in fact, the saddle-shaped wave can spin as well, an almost paradoxical image for people accustomed to ordinary, well-behaved water.
Since all the waves are generated by timed sequences of water pulses, it should be possible for visitors to try controlling the timing and sequence to see whether they can create interesting wave patterns by themselves... or, failing that, they could at least select which wave pattern to generate automatically.
In general, resonant surface waves of a circular pool are similar to the surface waves of a circular drumhead like a timpani: both are described by "Bessell Function" solutions to wave equations. Some of the solutions (i.e. possible waves) are rocking side-to-side, some have the middle move up and down, and some have different sides moving up and down at the same time. In general, the more parts of the surface moving up and down at the same time, the higher the frequency of the resonance.
But the similarity ends there. First, the boundary conditions are different: drumheads have their edges nailed down, which the edge of the circular water surface is free to move up and down; only the net volume of water is constrained. More subtley, the two wave equations are different: drumheads follow the ordinary, non-dispersive wave equation for elastic membranes, while water waves follow a more complex dispersive equation (which is why long-wavelength tsunamis travel faster than ordinary waves).
I made a prototype of this system (at a smaller scale and using air bubbles rather than water jets) in about 1990, and verified that all the surface modes can be driven by properly timed impulses. There are several parameters which seem to influence the strength and visibility of the resonant waves: the rigidity of the pool walls, the relative depth of the water, the precision of the timer, and the focus of the water jets.
Rigid walls and "Q"
If the circular wall is not rigid, then wave energy is dissipated in moving the wall and the oscillation tends to die down. My model was a 10-gallon water-cooler bottle with side braces, and I obtained a "Q" of about 7 (i.e. a standing wave would lose about 70% of its energy after 7 oscillations). I estimate that a six-foot pool with walls of 1" acrylic or glass, filled to a couple feet to minimize flow-shear losses, ought to have a "Q" of at least 20, which would make dramatically strong waves containing 20 times as much energy as a single pulse of water jets would provide.
Precision of jets
The stronger the resonance and the higher the "Q", the more precisely-timed the jets need to be, because the resonance peak is more sharply tuned. For example, if the saddle-mode has a natural frequency of 2 Hz, and a "Q" of 20, then the resonance peak will be about 0.1 Hz wide, and the jets would need to be timed to less than 50 msec precision to ensure a strong osciallation.
Furthermore, the jets should have a spray-pattern which does not create ripples on the surface of the water...otherwise, the overall resonant motion will be harder to see among all the churning on the surface (a fact I unfortunately discovered when I drove my own scaled-down model with air bubbles: while I was proud of the visible oscillations, my fiance' wryly commented: "It looks like a washing machine!").
Finally, of course, any system which uses water-jets must have a way of "recycling" the water. A pumping system which recirculates water but which has a strong "reserve pressure" with which to allow near-instantaneous jets ought to do the trick.
For more information contact
Bill Softky (Caltech Ph.D., Physics)
650 329-0256 H
650 321-8282 x221 Redwood Neuroscience Inst.