"I am fairly sure it does no such thing," Wright wrote. "I think a stone does not behave on water in the way described, because I know it does not on sand-the hard, wet sand at the water's edge....

"When I saw the marks left in the sand by my first pebble, I think I must have been as astonished as old Crusoe on beholding the first footprint of his man Friday! The first bounce of the pebble was only four inches long; the next was nearly seven feet; then came another short hop of only four inches; then a leap of about five feet; then again the four-inch hop, and so on for seven big hops punctuated by the four-inch ones. Each short hop was unmistakably recorded by two neat little marks in the sand. After the seventh repetition the pebble ceased this strange behavior and merely jumped along with successively shorter strides until it stopped. The total number of hops was about 20-my average on hard sand....

"Now I fancy the same thing happens on the water, though in the water there. is no imprint left to tell the story. A proper record with a camera would give us the answer. In the sand, at least, the story is quite clear, and a very pretty story it is-as pretty as the tracks of some little animal in the fresh snow.

"As yet I have no explanation for these facts. I have put the problem before several physicists of high distinction, but so far have received no answer in return. Somewhere there must be an answer for my little riddle. Who will find it?... What scientist, professional or amateur, wants to go down to the beach with all the needful instruments and find the answer to my riddle? I shall be glad to go along if I am wanted; I can throw pretty well."

During the past 11 years more than 10,000 readers have submitted theories to this department that account in one way or another for the behavior of Wright's pebble, but no one has submitted the question to nature by inviting him down to the beach.

The puzzle has now been solved, by test. A few months ago Kirston Koths, a student at Amberst College, set up a synthetic beach in the laboratory and investigated the performance of stone disks by a series of nicely designed experiments. Koths writes:

"I skipped the stone on surfaces of three types. They were a cloth-covered tabletop, water and sand. The plastic tabletop was covered with a woolen blanket and a thin layer of black cloth. This covering cushioned the stone's impact and resembled sand more than the bare tabletop did.

"The beach was simulated by a sandbox consisting of a rectangular frame of wood that was lined with a sheet of polyethylene, placed on the floor and filled with sand. An aluminum tray was used as a shallow tank for making experiments on water. A strip of canvas was suspended from the ceiling between the table and the sandbox as a backdrop to absorb the impact of the skipped stone.

"A stone of irregular shape could not be skipped reproducibly, so I made symmetrical disks of sandstone in a range of diameters. The technique involved grinding a rough stone into a fiat slab with parallel surfaces and then grinding the edges to make a perfect disk. I used an ordinary grinding wheel to make the slabs.

"A small hole was bored through the middle of the slabs with a cement drill. A bolt placed through the hole served as a mandrel for locking the Rat stone in the chuck of an electric drill. The irregular edges of each slab were then ground into a true circle by holding the spinning stone against the grinding wheel. Finally, I painted the stones white so that they would photograph easily.

"I then applied a pair of black stripes that extended from the hole to the edge on one side. On the other side I painted a single stripe from the hole to the opposite edge [see Figure 4]. This distinctive pattern of markings made it easier to determine the orientation of the stone in photographs made during its flight. Other apparatus used in the experiments included a microphone, an amplifier, an oscilloscope, a stroboscope, a still camera and a high-speed motion-picture camera.

"I was not familiar with the best way to photograph a skipped stone. I did not know whether it would indeed make two closely spaced impacts as Wright had predicted. My first experiment was made to learn something about the velocity so that I could adjust the stroboscope to the optimum flash rate.

"I placed the microphone close to the tabletop and adjusted the oscilloscope to begin a horizontal sweep at the instant the microphone picked up sound above a predetermined volume. The spot of light that appears on the face of the oscilloscope tube was adjusted to sweep from left to right at the rate of five milliseconds per centimeter. The intensity of sounds was indicated by vertical excursions of the spot.

"The stone in skipping on the cloth-covered table definitely made two closely spaced impacts as indicated by the oscilloscope. The recorded pattern [below left] shows that the vertical excursions reach maximum amplitude twice near the beginning of the sweep. The sweep does not start until the sound of the first impact approaches maximum amplitude. For this reason only the final portion of the first impact is displayed. The interval between the two impacts spanned roughly .01 second, indicating that a minimum of 150 to 200 flashes per second would be required to record the path and orientation of the stone between impacts.

To make photographs by the repetitive-flash technique the room must be darkened and all background objects should be black. The shutter of the camera is opened. After the scene has been illuminated by a series of flashes the shutter is closed. The film is exposed during each flash and records a picture of the stone's position at that instant.

"The lamp I used emitted flashes that lasted less than half a millionth of a second-about as long as it takes an automobile moving at 60 miles per hour to travel the thickness of its paint. For this reason the images of the stone in 1ight were sharp.

"The stone reflects the light of only a single flash at each of its positions as it passes through the field of the camera. The background, however, reflects the light of all flashes. For this reason light reflected from the background must be kept low to prevent the film from being overexposed if the shutter is kept open for a half-second, the exposure I used.

"Some photographs were made against a background of black velveteen, which reflects only about 1 percent of the light that strikes it. In other experiments I cleared an area of the background about 50 feet in depth. This had the effect of creating a black background because light reflected by 100 flashes from objects at a distance of 50 feet was not sufficient to expose the film significantly.

"The stone was thrown by hand in all experiments. The direction of rotation of the stone was easily determined on the basis of the hand from which it was released. A right-handed throw resulted in clockwise rotation, a left-handed throw in counterclockwise rotation. The distances between hops of the stone were measured from the centers of the marks made in the sand.

"By averaging the results of scores of experiments I reached the conclusion that the length of the short hop is proportional to the diameter of the stone. As the diameter of the stone increased, the length of the short hop increased. I also observed that the points of impact were alternately displaced sideways, creating a wavy pattern. As viewed from above, a clockwise throw made an initial mark in the sand that indicated a deflection to the right. The second impact was deflected to the left, the third to the right and so on. A counterclockwise throw made a similar alternating pattern, beginning with a deflection to the left.

"The initial attempt to photograph a stone skipped on natural sand failed because the light color of the sand overexposed the film. The sand was therefore colored black with a dye designed for use on cloth. When making the pictures, I turned out the lights and activated the stroboscopic flash. Then I simultaneously threw the stone and opened the shutter, using a cable release.

"The behavior of the stone was determined by analyzing the photographs. The center of mass of the stone was marked with ink on each recorded image. Then the centers of mass of the three images immediately before impact were connected by a straight line, with allowance made for the natural parabolic trajectory of the stone. The angle made by this line and the horizontal was measured and recorded as the angle of approach. The angle of departure was similarly determined and recorded. The tilt of the stone-its angle of attitude prior to impact-was determined by measuring the angle of the image that immediately preceded the stone's contact with the sand.

"Normally a stone makes initial contact at the trailing edge and then strikes on the leading edge; a short hop results. The initial impact, which makes the first mark in the sand, exerts a torque that causes the stone to topple forward and strike the sand at the leading edge, where it makes the second mark. Then, depending on the angle of approach, the angle of attitude and the uniformity of the surface, the stone will either continue to topple or will right itself. Following the double impact, the stone makes a long hop.

"Stones that make initial contact with the sand at the leading edge behave in essentially the same way if they skip at all; usually a stone that strikes first on its leading edge will dig into the sand and come to rest If it skips, the sequence is as follows. The leading edge lifts from the sand. The trailing edge then makes contact at or very close to the point of impact of the leading edge. Only one mark is made in the sand before the long hop.

"The lateral displacement of alternate marks made by a skipping stone in the sand arises from the rotation of the stone about its axis. The spinning edge exerts a sideways force on the sand as it makes impact. The sand in turn deflects the stone to the right or the left depending on the stone's direction of rotation and on which edge hits first, the leading and trailing edges having oppositely directed velocity vectors.

"Still another motion is evident in the photographs. When the tumbling force of the torque that is exerted on the stone by impact with the sand is imposed on the spin imparted by the throw, the stone begins to precess, or wobble, like a spinning top that is pushed from the vertical. In this respect the spinning stone is like a small gyroscope.

"The angular-momentum vector is determined by the direction of rotation, the angular-impulse vector by the edge that makes contact with the sand. Adding the two vectors, it is clear that a stone spinning in the clockwise direction and striking at the trailing edge will precess toward the right. Similarly, a stone that has counterclockwise rotation will precess toward the left. The gyroscopic effect thus produced is not great, however, because of the relatively low angular velocity with which the stone is thrown. "In many photographs the linear and angular velocities of the stone as well as the angle of approach were almost identical, but the angles of departure varied greatly. The angle of attitude was the only significant variable. Hence I concluded that the angle of departure in the case of stones skipped on sand must be a function of the angle of attitude.

"The repetitive-flash technique did not work well in the case of stones skipped on water. Splashes obscured the details at the moment of impact. I therefore substituted a high-speed motion-picture camera for the stroboscopic flash and filmed the events at 600 frames per second at an aperture of f/2. The pictures were taken from the side at a distance of 12 feet and from a point slightly above the horizontal. The analysis was made by printing key frames of the film as still photographs [see Figure 7].

"The photographs show that a stone skipped on water does not react like one skipped on sand. When a stone is thrown at an angle of about 20 degrees, the trailing edge makes contact first. The stone planes along the surface of the water and then tilts backward to an attitude of approximately 75 degrees. A crest of water builds up in front. The stone now lifts out of the water and makes a long hop. The entire interaction of the stone and the water is quite complex, so that it may be possible to throw a stone of some other shape in some way that would cause it to make a double hop, as Wright suggested. All the stones I threw, however, made single impacts followed by long leaps."

Music is often spoken of as though it were a branch of thermodynamics. Louis Armstrong is said to blow the hottest trumpet, and Herb Alpert has the reputation of conducting the coolest brass group. Anyone who wants to put such observations to a literal test can do so by means of techniques that measure music in terms of heat. George R. Stibitz of the Dartmouth Medical School has developed an apparatus that transforms sound into heat and measures the resulting temperature with a thermometer. Stibitz writes:

"It is no startling pronouncement to say that sound is a form of energy or that sound turns into heat when it is absorbed. Those who regard the term 'hot music' as nothing more than a colloquialism, however, might be surprised to watch a thermometer rise above the boiling point of water in response to the absorption of sound by a piece of cloth.

"Any object as large as a thermometer radiates an appreciable quantity of energy, particularly at temperatures above 100 degrees Fahrenheit. To raise its temperature above that level requires a fairly powerful sound. A big loudspeaker could radiate sufficient energy, but I turned to a different source of sound for my apparatus. I designed a powerful whistle, which is shaped like the ones used by traffic policemen but is considerably larger.

"The whistle, which is blown by the exhaust of a vacuum cleaner, was made by bending a strip of sheet metal 26 inches long and four inches wide into the shape of a jet directed across the mouth of a cavity. I then soldered end plates to the sides of the bent strip [see Figure 8].

"The whistle radiates sound in all directions, but the loudest sound comes from the opening of the cavity. In order to concentrate the acoustic energy and direct it into an absorbing chamber I constructed a crude horn of the exponential type. It picks up sound waves over an area of 10 square inches and directs them (through an opening of 1/20 square inch) into the absorbing chamber. The whistle and the horn were designed to generate a sound at about 528 cycles per second, one octave above middle C. Unfortunately the human ear is very sensitive to that frequency. A lower pitch would have been more comfortable during my experiments but would have required a larger whistle and horn.

"Essentially the horn is analogous to an electrical transformer. It matches the acoustic impedance of the region in the vicinity of the whistle (about one acoustic ohm) to the impedance of the absorption chamber (100 acoustic ohms). The horn is not truly exponential, but it approximates this form by a series of conical sections four inches long with ends that have a ratio of 2:1 in diameter. The large end of the horn terminates in a cylindrical resonance chamber about three inches in diameter and four inches long. The chamber contains a rectangular opening 15 inches wide and four inches long flanked by a pair of Hat wings three inches wide. The end plates of the whistle are placed close to the wings of the resonance chamber for coupling the source to the load.

"The absorption chamber was designed not only to transform sound energy into heat energy but also to prevent the loss of heat. The body of the chamber is a cylinder of cardboard (cut from a mailing tube) about 1 3/4 inches in diameter and three inches long [see illustration at right]. The cylinder is coupled to the small end of the exponential horn by a short bushing of wood. The low thermal conductance of the wood keeps heat from flowing rapidly into the metal horn.

"I made the bushing of wood dowel by drilling a 1/4-inch hole through the axis. With a little filing the bushing made a snug fit with the end of the horn. The bushing must be cemented to the horn to make an airtight joint. A remarkable amount of acoustic energy can escape through even a few pinholes at the small end of the horn.

"The absorption material consists of a pad of finely woven cloth. It covers the inner end of the wood bushing. The thickness of the cloth pad must be adjusted by trial and error for maximum performance. I found 15 layers of dress-lining material about right.

"The cloth pad is held in place by masking tape. The mailing tube must also be cemented or taped to the dowel. In addition the tube must be insulated with a layer of absorbent cotton or glass wool and covered with sheet rubber of the kind used in rubber balloons. The rubber sheet prevents a steady flow of air from the horn and so minimizes the loss of heat by convection.

"A candy thermometer makes a convenient indicator; an oven thermometer will work as well. Insert the thermometer through a hole in the rubber sheet. Then tie the rubber tightly around the thermometer to prevent the escape of air from the chamber.

"When the operating whistle is brought close to the resonance chamber of the horn, the thermometer should rise 40 or 50 degrees F. after a few minutes of warm-up. The results will depend critically on the relative positions of the horn and the whistle. Another important factor is the acoustic resistance of the cloth pad.

"The apparatus can be modified in a number of ways. For example, a wad of absorbent cotton placed in the hole of the bushing makes an efficient converter, but it is difficult to adjust. The acoustic resistance of the wad must come close to matching the impedance of the horn. The resistance changes greatly with comparatively small changes in the density of the cotton. The density of the cotton is determined, of course, by the pressure with which it is pushed into the dowel.

"A more efficient design can be made by constructing the whistle and horn as a unit, but the adjustment of the coupling between the whistle and the resonance chamber then becomes difficult. My most efficient design created a temperature of 250 degrees F. The model described can easily be adjusted to create temperatures of 130 degrees. I have found that a space of 3/4 inch between the opening of the whistle and that of the resonator works well, depending on the pressure created by the vacuum cleaner, the shape of the whistle and the effectiveness of the pad in the absorption chamber.

"An interesting experiment consists in measuring the amount of power that is converted from sound to heat. The experiment can be done by inserting a small tube of water in the thermometer hole and measuring its rise in temperature. A rise of one degree centigrade per milliliter of distilled water is equal to one calorie, or 4.2 watt-seconds.

"The power delivered by the stream of air can also be measured. Clear plastic tubing bent into the U shape of a water manometer can be used for measuring the drop in pressure across the mouth of the whistle. The rate of airflow can be estimated with reasonable accuracy by means of a pinwheel made of paper. The power expended by the air is estimated by taking account of the rate of flow and the drop in pressure."

Bibliography

ACOUSTIC MEASUREMENTS. Leo L. Beranek. John Wiley & Sons, Inc., 1949.

ANALYTICAL MECHANICS FOR ENGINEERS. Ered B. Seely, Newton E. Ensign and Paul G. Jones. John Wiley & Sons, Inc., 1958.

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