EDGE WAVES ARE POORLY UNDERSTOOD structures that form a spokelike pattern around a solid object vibrating in a liquid. They were first recorded by the eternally curious Michael Faraday on July 1, 1831. He had set up a long, rectangular piece of wood (he called it a plate) fixed in place at the top and extending a short distance into a bowl of water. When he made the plate vibrate, he noted that "apparently permanent elevations formed, beginning at the plate and projecting directly out from it to the extent of 1/3 or 1/2 an inch or more, like the teeth of a very short coarse comb."

The next day Faraday, who is usually remembered for his pioneering work on electromagnetic theory, produced edge waves by generating vibrations in a large glass container nearly filled with water. Wetting his finger, he drew it around the rim until the container wall vibrated. Extending radially from the wall were the ridges of the edge waves. You can see the same pattern in a glass holding wine or some other liquid when you rub a finger along the rim in order to generate a hum.

In discussing the nature of edge waves I draw on modern studies by C. J. R. Garrett, then at the Institute of Geophysics and Planetary Physics of the University of California at San Diego, and J. J. Mahony, B. J. S. Barnard and W. G. Pritchard of the University of Essex. In addition Robert E. Apfel of Yale University recently sent me a manuscript describing his investigations of edge waves. I begin with Faraday's experiments because they are easily reproduced in the kitchen. Then I shall describe my own experiments employing a wave maker that is standard equipment in high school physics classes.

In Faraday's first experiment he made the plate vibrate by tapping its center with his finger or by touching the center with a glass rod that vibrated when it was stroked with a wet finger. Waves developed as the lower end of the plate oscillated parallel to the water surface.

Water waves come in two general types. If the motion is governed by gravity, they are called gravity waves. They have long wavelengths. If the motion is controlled by surface tension, they are called capillary waves. The wavelengths are short. The waves spreading from Faraday's vibrating plate were normal capillary waves.

In addition four ridges of edge waves formed perpendicular to the face of the plate. Faraday quickly noted that although the ridges were fixed in position, they were not always present. Instead they formed and shrank rapidly enough to give the illusion of being continuous. Actually adjacent ridges were out of phase with each other: when one grew larger, the other shrank.

Faraday's next experiment involved vibration in a large glass nearly filled with water. As he rubbed a wet finger along the rim the finger skipped periodically, setting up vibrations in the glass wall to generate edge waves. At any instant the waves were either absent or weakest in four regions of the rim. One region was immediately next to the finger. The other regions were 90, 180 and 270 degrees from the finger. Between these regions the waves were strong. As the finger moved around the rim the locations of strong and weak edge waves moved accordingly.

Faraday also generated edge waves by making a tuning fork vibrate and dipping it slightly into the water. When the fork vibrated too vigorously, the water surface erupted into a confusion of drops. When the vibration was too weak, only normal capillary waves were created. Their interference sometimes yielded a stationary pattern between the two limbs of the fork, but this pattern did not constitute edge waves. Edge waves formed when the fork vibrated with proper vigor. They extended out from the fork and sometimes even obliterated the interference pattern of the normal capillary waves.

The vibrating fork also generate edge waves in ink, mercury, warm oil and thin (liquid) jelly. In mercury they did not last long because the density of the metal quickly damped the vibration of the fork. Edge waves did not appear in cold oil, apparently because the material is quite viscous.

Faraday then returned to the arrangement in which a vertical object vibrated horizontally. He set up a wood rod that extended from a vise down into a bowl of water. It generated edge waves when he made it vibrate. As Faraday lowered the rod deeper into the water the waves were fewer but more pronounced. When the rod vibrated too vigorously, the water surface broke up into drops.

Next Faraday attached a board to the lower end of the rod. The face of the board measured seven by three inches. When the rod was made to vibrate, the board oscillated horizontally through the water. Faraday spotted many ridges of edge waves across both faces of the board.

Adjacent ridges on a face oscillated out of phase with each other. The ridges on opposite sides of the board seemed to be unrelated. When the board dipped only slightly into the water, it vibrated rapidly and there were many ridges. When the board was deeper, the vibration was slower and there were fewer ridges.

Faraday investigated the flow of water within the region of edge waves by sprinkling lycopodium powder and tiny bits of cork over the water surface. Neither test revealed any repeated pattern of motion, implying that there were no established currents in the waves.

Edge waves can also form from vertical oscillations. Faraday arranged a horizontal wood rod with one end in a vise. He glued a cork to the other end so that it extended down into a bowl of water. The rod was set to vibrating so that the cork oscillated vertically. A "beautiful store of ridges formed all around it, running out 2, 3 or even 4 inches." When the cork was lowered farther into the water the ridges became weaker.

Apfel described how he observed edge waves in a glass of wine whose rim he stroked with a wet finger. The finger periodically slips and sticks on the rim, producing vibrational waves in the glass wall. Most of the waves travel through the glass, accomplishing little, but one, wave has a frequency that forces the wall into a strong, repeating pattern of radial oscillations. This wave is said to have the resonant frequency for the wineglass.

During resonance the glass wall has four regions, equally spaced around the rim, that are lacking in vibration Between them the glass oscillates vigorously along a radius. The motion against the wine at these points creates prominent edge waves.

The edge waves will be visible only if the glass is nearly full, because only the upper portion of the glass wall vibrates strongly during resonance. The bottom of the wall is held almost rigid by the stem of the glass.

The resonant frequency of a wineglass depends on the structure and size of its wall and on how much wine is in the glass. An empty glass has a high resonant frequency. As wine is added the additional mass slows the oscillation of the glass walls, decreasing the resonant frequency.

The pattern of strong and weak edge waves moves along with the finger, making observations difficult. To keep the pattern stationary Apfel mounted a wineglass on a rotating turntable (a record player). To make the pattern clearer for photographs he painted the glass black. With the glass nearly full of water he held a wet sponge against the glass wall to set up the vibrations. For another photograph Apfel replaced the wineglass with a dessert dish containing ethylene glycol. The dish was 20 centimeters wide and had a resonant frequency of about 200 hertz.

Barnard and Pritchard did experiments with edge waves in a water tank 30.6 centimeters wide, 16 centimeters deep and 2.7 meters long. They generated waves by means of a flap hinged along the bottom of one end of the tank. It extended above the water surface. A 9 sloped beach at the other end of the tank absorbed capillary waves reaching it so that they could not reflect back into the region of edge waves. The flap was made to oscillate by an electromagnetic vibrator driven by an amplified signal from an electronic oscillator operating at low frequency. Great care was taken in stabilizing the oscillation frequency so that the creation of edge waves was repeatable.

At certain frequencies the ridges of edge waves appeared along the exposed face of the flap. As in the experiments by Faraday the ridges were perpendicular to the face. Faster oscillations generated more ridges. The highest ridges were closest to the flap; away from it the height diminished.

Edge waves did not appear in any place except in the vicinity of the flap. Strangely, they did not form as soon as the flap began to oscillate. (The normal capillary waves did.) When they did form, sometimes after as long as a minute, they grew slowly at first and then rapidly. The ridges oscillated vertically and their maximum height varied periodically. Typically the maximum height of a particular ridge varied from maximum to minimum and back every 50 seconds or so.

The recent theoretical work by Garrett and Mahony explains some of these experimental results. They concluded that edge waves can be generated only by a wave maker such as an oscillating paddle or a flap. The waves are not generated by the train of normal capillary waves propagating away from the wave maker, otherwise they would be seen whenever capillary waves move over a water surface.

The pressure field created by an oscillating paddle sets up an instability in the water surface. The production and propagation of the normal capillary waves is one way the instability can be relieved. Apparently the creation of a stationary pattern of edge waves is another means. The ridges of the edge waves extend only a short distance from the paddle, limited not by dissipation of their energy but by the extent of the pressure field created by the paddle.

The edge waves are said to be an example of parametric resonance, a term derived from the type of differential equation describing their motion. A characteristic of such a resonance is that one motion (here the edge waves) has half the frequency of another motion (here the paddle). The paddle pushes against the water in front of it, creating a region of stress. The stress is partially relieved by the formation of the edge waves oscillating at half the frequency of the paddle.

Each push by the paddle feeds more energy into the edge waves. They soon reach a limit, however, and then the energy flow is reversed.

This periodic transfer of energy between systems is also characteristic of parametric resonance. The transfer is the reason the maximum ridge height in a pattern of edge waves varies. The height is greatest when the paddle is energizing the wave, least when the wave is energizing the paddle.

I first did some simple experiments with edge waves in a wineglass filled with water. I could excite the waves by carefully rubbing the rim with a wet finger. To ensure that they would remain in sight longer, I excited the glass with a speaker driven by an amplified signal from an audio oscillator. The base of the glass was taped to the top of a table. The open cone of the speaker touched the rim. In this way I could control the frequency and strength of the oscillations.

When I tuned the audio oscillator to the resonant frequency of the glass circular capillary waves immediately spread inward from the edge of the glass. Some 30 or more seconds later edge waves rose up out of the water along the edge, extending inward about a centimeter. They seemed to grow slowly at first, then much faster. They ,q were quite fragile, disappearing when I shook the table slightly while taking notes. The waves soon reappeared. When the audio oscillator operated at a low level, the edge waves surrounded a center of relative calm. As I increased the signal strength they extended into the center of the liquid, forming a complex design of ripples.

I then turned the speaker cone to face the ceiling and placed on it a watch glass partially filled with water. No edge waves appeared, probably because I could not set up a resonance in the watch glass.

Next I tried to generate edge waves with a massaging apparatus that is essentially a simple oscillator. I put the oscillator in a bowl of water, circular capillary waves propagated from it almost immediately and a pattern of edge waves also sprang up around it. This pattern migrated slowly clockwise, apparently because the oscillator vibrated asymmetrically.

I did my remaining experiments with an oscillator I salvaged from an old water-wave apparatus of a type commonly used in classrooms to demonstrate wave interference. I removed the motor and its housing and clamped them to a ring stand. A vertical rod ran from the shaft of the motor. When I ran the motor on direct current, the lower end of the rod oscillated horizontally.

To make a paddle for the rod to drive I cut a flat rectangle out of sheet metal and sprayed it with black paint so that water waves could be seen against it. I glued a nail to the top of the piece at the center and ran the point of the nail through a small hole in the lower end of the rod hanging from the motor. A threaded bolt held the nail in place. I lowered the clamp holding the motor until the piece of sheet metal dipped a few millimeters into a rectangular container of water (a glass baking dish). When I turned on the power, the piece of sheet metal acted as a paddle as it oscillated horizontally. By increasing the voltage from the power supply I could increase the frequency at which the paddle oscillated.

I must warn you that this arrangement can be quite dangerous. Be careful to keep the motor dry and to keep all electrical connections well away from the water. If the paddle starts to throw drops of water out of the container, shut off the current immediately.

In my experiments I increased the current gradually until the motor began to respond. Then I turned the current up slightly more and examined the face of the paddle for edge waves. Since the waves do not appear immediately, I waited several minutes before I increased the current further.

Eventually the paddle oscillated rapidly enough to make an edge-wave pattern rise out of the water surface. Four ridges appeared along the face of the paddle and extended away from it about a centimeter. The same pattern appeared on the other side of the paddle. The next increase in the oscillation frequency created five ridges across the face of the paddle.

I sprinkled ground black pepper on the surface of the water to aid a search for circulation patterns. Like Faraday I found none. I examined the edge waves from the side (with the paddle moving left and right in my view). The nearest ridge resembled a squared-off hump that extended away from the blur of the paddle. I probed the waves with several items including another strip of sheet metal. They were quite stable even when an obstacle was placed directly across all of them.

In order to make the edge waves more visible I turned off the room lights and turned on a stroboscopic light. When the flash rate matched the paddle frequency, the paddle was frozen in place. Since the edge waves oscillate at half the paddle frequency, they continued to be seen in motion. If there are four ridges in front of the paddle, two of them can be seen during a flash. If the leftmost ridge is one of them, the ridge normally just to the right of it is missing, the next one to the right is present and the rightmost one is missing. In the next flash the roles are reversed.

Although the oscillation of the ridges should be visible in stroboscopic light, it was too fast for me to follow. I could not solve the problem by halving the flash rate. The paddle was again frozen in place, but each flash revealed the same set of ridges.

I finally chose a flash rate slightly less than half the paddle frequency. Each of the successive flashes revealed a slightly changed set of ridges. The set I saw first gradually diminished as the other set grew up out of the water. Once the new set reached its maximum height it began to diminish as the original set reappeared. This variation was slow enough for me to follow.

With the flash rate set I finally understood how the ridges form. They result from a combination of the normal capillary waves and the edge waves. Figure 6 shows two cases where the paddle is fully forward and the crest of a capillary wave is along its face. In the first case the edge wave along the face is at a maximum on the left side of the paddle. This maximum combines with the crest of the normal wave to produce a ridge on the paddle. Another ridge is created somewhat farther to the right.

In the second case the paddle has retreated and then advanced again so that the crest of the next normal wave lies along its face. Again the ridge pattern results from a combination of the edge wave and the normal wave. Since the edge wave oscillates at half the frequency of the normal wave, it has not gone through its full cycle. On the left side of the paddle the edge wave is now at a minimum. The ridges resulting from the combination of the two types of waves are displaced from the ridges that appeared in the previous advance of the paddle. On the next advance the initial ridges are re-created.

I made several more paddles to replace my flat one. I curved one into a semicircle, gluing the head of the nail to the convex portion. Edge waves appeared on both sides of the paddle. Another paddle was S-shaped so that on the front side (opposite to the nail) the paddle presented both convex and concave surfaces. A strong ridge appeared in the concave portion and a weaker one in the convex portion. In another flat paddle I cut a small notch in the center of the lower edge. Still the edge waves formed.

Another paddle formed three sides of a rectangle, with the nail glued to the center of the convex side. Edge waves appeared on both sides of the middle section but not along the areas that slipped through the water with little cross section.

My final paddle was a small metal hoop from my kitchen. When the motor drove the hoop horizontally through the water surface, edge waves appeared along the interior wall and also at some places along the exterior wall. They were not well formed because the motor, laboring to drive such a heavy object, was unstable.

I then remounted the flat paddle and investigated the onset of edge waves by slowly increasing the voltage of the power supply in order to increase the frequency of the oscillation. With each increase I paused because edge waves develop slowly.

Once the waves appeared I decreased the voltage and thereby the oscillation frequency. The edge waves persisted, even when the voltage was much lower than the amount required to initiate them. Apparently the paddle frequency needed to maintain edge waves is lower than the frequency needed to initiate them, probably because a good deal of energy is required to generate them When I decreased the voltage somewhat more, the waves slowly died out, taking approximately as long to disappear as they do to appear when the voltage is higher.

I checked the periodic exchange of energy between the edge waves and the paddle while the edge waves were well established. I turned on the room lights so that I could see the ridges continuously. Then I estimated the time required for them to change from the maximum to the minimum height on the paddle face and back. A typical time for this was 14 seconds.

Next I poured corn oil on the water as I monitored the edge waves. The first drops spread into a thin layer. Thereafter the corn oil collected in large pools until eventually it was several millimeters thick.

The edge-wave pattern did not seem to change until several minutes after I had added the last oil. Then the pattern disappeared. At first I thought the oil was too viscous to support edge waves. Then I figured they probably disappeared because of the raft of small water drops and air bubbles churned up by the paddle. I checked this notion by waiting until the raft had had time to disintegrate into a smooth layer of oil. When I turned on the paddle, edge waves soon appeared.

I also wondered if edge waves could form at the interface of water and oil. By carefully extending the baking dish out over the edge of a table I managed to examine the interface from the bottom. I could find no evidence of an edge-wave pattern there, even when the edge waves on the top surface of the oil were quite prominent.

In my final experiments I poured a can of STP Oil Treatment over the water. This fluid is highly viscous. Edge waves did not appear. Either the fluid was too viscous or the motor was too weak to move the paddle vigorously enough through such a thick medium.

You might enjoy searching for edge waves on other liquids. Perhaps you could find waves at the interface of two immiscible liquids. The task may prove to be difficult, however, because such an interface will not support much wave motion.

The diffraction patterns I asked readers to identify last month spelled MORISN. They were intended to suggest the name of Philip Morrison, the book editor of SCIENTIFIC AMERICAN.

Bibliography

ON CROSS WAVES. C. J. R. Garrett in Journal of Fluid Mechanics, Vol. 41, Part 4, pages 837-849; May 15, 1970.

CROSS-WAVES, PART 1: THEORY. J. J. Mahony in Journal of Fluid Mechanics, Vol. 55, Part 2, pages 229-244; September 26, 1972.

CROSS WAVES, PART 2: EXPERIMENTS. B. J. S. Barnard and W. G. Pritchard in Journal of Fluid Mechanics, Vol. 55, Part 2, pages 245-255; September 26, 1972.

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