A THERMAL OSCILLATOR IS A curiosity that turns up in physics classes because it is an example of a self-sustained, nonlinear oscillating system. What happens is that one part of the system vibrates continuously because of the transfer of heat. Here I shall discuss four such devices, two of them new and two that were demonstrated in the l9th century. Although each one depends on the periodic redistribution of heat, they are all quite different.

Pierre Welander of the University of Washington has devised a novel thermal oscillator based on water. The opening on each side of a U-tube is fitted with a wide container. The tube and about half of each container are filled with water. The tops of the containers are open. The bottom of the tube is heated; the upper sides are cooled. Soon the water level in the apparatus begins to oscillate between the two sides with a period of between several minutes and several hours.

Suppose the system is initially in complete equilibrium with its two sides containing equal amounts of water at the same temperature distribution. Consider a tiny volume of water lying within a thin slice through a cross section at the bottom of the tube. The volume is pressed from both sides because of the weight of the water in the two sides of the tube.

Two factors determine the weight of a column of water: the height of the column and the density of the water. The density is determined by the temperature: cooler water is denser than warmer water. Since initially the two sides of the system are identical in water level and temperature distribution, the central volume of water is pressed equally from both sides. It remains stationary and all the water is in equilibrium.

Suppose a disturbance from the environment causes a small amount of water to move from the left side to the right. The flow upsets the symmetry of the system; now the two sides differ in both water level and temperature distribution. The motion lowers the level on the left side and moves cooler water downward. It raises the level on the right side and moves warmer water to the right at the bottom of the U-tube.

If the containers on top of the U-tube are wide, the changes in the water levels are initially small. The broken symmetry of the temperature distribution is more important. Since the cooler water at the bottom left is denser than the warmer water at the bottom right, the central volume of water is no longer under equal pressure from the two sides. The left side now pushes more. The uneven pressures tend to push more water to the right.

Although the initial change in water levels is small, the continued flow increases the difference until it is significant. Eventually the right side applies more pressure on the central volume than the left side because of the difference in the water levels. The flow from left to right is slowed and then stopped. The process is gradual, however, and so the heating and cooling of the tube has a chance to reestablish the temperature symmetry. By the time the flow stops the system is again approximately symmetrical in temperature, but now the level is higher on the right side than on the left. The water begins to flow back to the left side, and the cycle is renewed in the opposite direction. The system remains in oscillation.

The oscillating flow continues indefinitely because of the heating and cooling of the tube. If the tube is allowed to reach a uniform temperature, the oscillation soon damps down into a stable situation with equal water levels on the two sides. As long as the heating and cooling are continued, however, the oscillation is maintained. Every time the system passes through the state where the two water levels are equal the imbalance in densities drives the water until the levels are unequal, continuing the oscillation.

The dimensions of the system are critical to the onset of oscillation. The top surface area of each container must be larger than the cross-sectional area of the tube by an amount that is related to the temperature distribution in the water. The ratio of the area of a container to the area of the tube must exceed 5,000 divided by the maximum temperature difference (in degrees Celsius) that is set up in the water by the heating and cooling.

For example, if the system gives rise to a temperature difference of 50 degrees C., the top surface area of the container should be at least 100 times larger than the cross-sectional area of the tube. A smaller temperature difference demands an even larger ratio of the areas. If the containers are unusually large, of course, the oscillation of the water levels may be difficult to perceive.

The requirement for the relative sizes of the areas arises from the interplay of pressure on the central volume at the bottom of the U-tube. Suppose the initial disturbance from the environment moves a small amount of water to the right. The resulting asymmetry in temperature tends to drive more water to the right. This flow is countered by the difference in water levels on the two sides. If the top surface area of the containers is too small, the flow immediately creates a difference in water levels that is more than enough to stop the flow resulting from the asymmetry in temperature. Equilibrium is quickly reestablished.

If oscillation is to begin, the top surface area of the containers must be relatively large. That way the flow driven by the temperature asymmetry does not immediately yield a large difference in water levels. The flow has time to develop. The period of oscillation is set by the relative areas of the containers and the tube, the length of the tube, the viscosity of the water and the acceleration of gravity.

An apparatus similar to Welander's can be made by fusing glass containers on top of a U-tube. Since the tube should be Pyrex to preclude its shattering from being heated and cooled, fusing containers onto the tube seemed too difficult to me. I was also worried about how to mount such a delicate system so that it would stay upright. I chose a U-tube with side arms near the top. Short lengths of rubber hose connected the arms to plastic containers I took from my kitchen. The containers were identical; the top surface area of each one was about 200 times the cross-sectional area of the U-tube.

I taped a transparent ruler to the inside of one container so that I could monitor the water level. The U-tube and about half of each container were filled with water. Around the bottom of the U-tube I wrapped a heating mantle of the type that can be found among the supplies of a standard chemical laboratory. This mantle contains a high-resistance electrical wire that supplies heat. I plugged it into a Variac voltage control so that I could regulate the amount of heat applied to the tube. (I took great pains to make sure there would be no sudden leak that would let water reach the electrical line.)

For cooling I wrapped a thin-walled rubber hose several times around the top of the U-tube. Tap water could then be run through the hose. Although the hose did help to cool the water in the top of the U-tube, I decided a better system was needed. The conduction of heat through the glass wall of the U-tube and the rubber wall of the hose seemed to be too slow.

I had no way of measuring the temperature distribution of the water in the U-tube. After experimenting with several settings on the Variac I did manage to observe small oscillations in the water level of the containers. The level adjacent to the ruler rose and fell by about a millimeter every five minutes. Although the oscillations were not large or always regularly spaced, they were of the type observed by Welander because they disappeared when the heating and cooling systems were eliminated.

Welander told me about a more sensitive monitor of the water level. A small cork is hung from the edge of the container on a short arm made of wire. The cork floats on the water. A vertical straw is mounted on the cork. When the water level changes, the floating cork rotates about the wire arm. The upper end of the straw magnifies the motion.

Welander has also described a different kind of thermal oscillator in which a thin layer of ice forms periodically in a container of water. His experiment was done with about 50 liters of water in a Plexiglas container having a square bottom 50 centimeters on an edge. The container was thermally insulated. Near the bottom a heating element released about 100 watts of heat. Magnetic stirrers constantly mixed the water to distribute the heat. The apparatus was kept in a compartment where the air temperature was maintained at-20 degrees C. The air above the water was also continuously stirred.

Welander monitored the temperature of the water at half of the depth. He also watched for the formation of ice. The surface periodically froze and melted as the temperature of the water varied by about 10 degrees. The water temperature rose to about 10 degrees C during the phase of the oscillation when the top surface was frozen. It fell to almost the freezing point during the phase without ice. The period of oscillation was roughly 50 hours.

You can observe similar oscillations of freezing and melting water at home. Welander suggests using a Thermos bottle with a wide top. Fill it with water almost to the top, insert a heating coil below the surface and place the apparatus in a freezer. Ice periodically forms and disappears, although the oscillations are not as uniform as those Welander observed in the larger apparatus.

I followed his procedure with one change. The heating coil was the immersion heater with which I normally heat water for instant coffee. The heater was suspended in place over the center of the Thermos by several strips of reinforced tape. A problem was that the metal casing of the heater extended upward through the surface of the water. Heat from it surely interfered with the periodic formation of ice.

I plugged the immersion heater into a Variac so that I could experiment with the rate at which heat was released in the

water. The rubber gasket lining the edge of the freezer door compressed enough so that I could close the door over the cord. If it had not closed completely, I was prepared to pack the edge with cloth to keep out room air. I turned the control of the freezer to its coldest setting. Again I took precautions to keep water from spilling onto the electrical supply.

The oscillations in the formation of ice are due to the poor transfer of heat through ice compared with the transfer through liquid water that is being stirred, either mechanically or by convection. The heat supplied to the system by the submerged coil must travel upward because the sides of the container are insulated. The surface of the water is being cooled by air at a temperature below the freezing point of water.

The system can develop into either of two possible steady states. In one of them the surface o f the water remains liquid because the air is not particularly cold and the supply of heat from the coil always prevents freezing. The heat travels upward by conduction and convection in the water. The transfer to the top surface is sufficiently fast to prevent freezing.

In the other steady state the system permanently freezes at the top because the air is quite cold and the rate at which heat is supplied is not particularly high. The heat from the coil travels upward to the ice layer by conduction and convection. There it is conducted slowly through the ice and carried away by the convection currents in the air. The system is in a steady state because although the conduction through the ice is slow, it is still fast enough to keep the temperature of the underlying water from rising high enough to melt the ice.

Oscillations can begin if the air is quite cold and the rate at which heat is supplied by the coil is high. Suppose the water is not frozen. The cold air removes heat from the surface fast enough to initiate freezing. When the ice layer forms, the removal of heat from the system diminishes because ice does not transfer heat as well as stirred liquid water. Thus the temperature of the water below the ice begins to rise. Eventually the water is warm enough to melt the ice.

When the ice disappears, the rate of heat transfer increases and the temperature of the water drops. Later the surface temperature is again at the freezing point and a fresh layer of ice forms.

A different kind of thermal oscillator was discovered in 1805 by an inspector in a smelting factory in Saxony. When a heated bar of one metal is laid across a block of another metal, the bar may oscillate vigorously and noisily. This oscillator was rediscovered independently about 25 years later by Arthur Trevelyan and has been known since as the Trevelyan rocker.

I have seen two kinds-of demonstration of this type of thermal oscillator. In each one the choice of metals is important. In the first demonstration two lead plates are mounted in a vise. The top edges of the plates projecting above the vise are filed sharp so that there is a slight separation between them. Then a brass bar is balanced across the plates. When the bar is heated, it begins to rock back and forth. As it hits the edge of one plate and then the edge of the other there is an almost steady tapping.

In the second demonstration a piece of brass with a triangular cross section has two sharp edges at one angle of the triangle. The two edges rest at right angles on the edge of a block of lead. Extending from the piece of brass on a line with the two sharp edges is a brass rod that has a weight at the other end; near the far end the rod rests freely on a support. When the rod is heated, the sharp edges of the brass piece rock back and forth around the axis of the rod, making a sound that is approximately constant in frequency as they tap on the lead block.

The rod in Trevelyan's device was copper. Brass is more commonly used today. Perhaps certain other metals could serve in place of brass, copper or lead, but the choice must be made with care if the continuous oscillations are to be produced. The oscillations and the associated tone were apparently first explained by Sir John Leslie soon after the rocker was demonstrated by Trevelyan. Recent work has refined the explanation but has not replaced it.

Consider the demonstration in which a brass piece rests on a lead block. The regions of contact are the two sharp edges of the brass piece. As the brass rod and the brass piece are heated by a flame, heat is transferred to the lead through the two edges. The rate of transfer through an edge depends partly on the weight resting on it: more weight provides better contact and a greater transfer of heat. One of the edges initially transfers heat better. Thermal expansion builds up the underlying lead into a tiny mound. The expansion can be fast enough to push the edge upward and off the mound.

Two things happen. Since one edge has been pushed up, the weight of the brass piece is shifted to the other edge. The heat is now transferred to the lead at the second edge. The mound at the first edge cools and shrinks. A fresh mound rises at the second edge. The system oscillates as the cycle of rising and falling mounds repeats.

The rate of tapping depends partly on the dimensions of the rocker. With some rockers the rate is so high that a continuous tone is heard.

Why do some materials not work? One reason is that the block must be of a material that expands when it is heated. Lead expands well, even more than brass and copper do. A second reason is that the vibrations of the rocker can be maintained only if the heat transferred to the block under an edge is not rapidly dispersed. If it is conducted away too fast, the region of the block just under the edge will not have enough time for thermal expansion. Lead conducts just poorly enough to ensure that it is effective for the purpose.

A third reason is that the block must conduct heat fast enough to keep the heat from remaining too long in the region under an edge, otherwise the thermal expansion would persist too long and the mound would subside too slowly. The edge would no longer be lifted off the block.

You might like to reverse the brass and lead so that a lead piece with two sharp edges rests on a block of brass. The heating

procedure cannot be reversed; you must still heat the brass piece. You might also enjoy monitoring the oscillations in either system by flashing a stroboscopic light on the rocker. Although the motion is slight, it can be amplified by attaching a needle or a thin rod to the rocker at right angles to its long axis. The end of the needle moves through a greater distance than the rocker and so makes observation of the motion easier.

Another thermal oscillator was once a standard demonstration in physics classes. A section of wire mesh is pushed about a quarter of the way up inside a vertical pipe. The mesh is bent at the edges so that it remains in place inside the pipe. A burner is set under the pipe so that both the pipe and the mesh get hot. When the burner is removed, the pipe howls for several seconds. The tone is nearly pure because it is the fundamental frequency for acoustic oscillations inside the pipe.

This system was discovered in 1859 by Pieter Leonhard Rijke of the University of Leiden. He thought the sound resulted from thermal contraction of the wire mesh and pipe as they cooled. He believed oscillations accompanying the contractions excite the acoustic wave emitted by the pipe. Rijke was wrong; neither the mesh nor the pipe oscillates.

The correct explanation was supplied by Lord Rayleigh in 1878. The sound from the pipe arises from the forced oscillation of the air inside the pipe. When the burner is in place, it heats the air in the pipe and forces a convection current upward through the mesh and pipe. No sound is emitted at this stage because the air is flowing steadily upward.

When the heat source is removed, the convection current continues for a while because both the pipe and the mesh are hot. The many disturbances of the air in the pipe tend to create an acoustic standing wave in that air. I shall first explain such a standing wave in an unheated pipe and then show how the convection current in the Rijke pipe strengthens it until the pipe howls.

An acoustic standing wave inside a straight pipe can be created by the interference of two sound waves traveling in opposite directions through the pipe. The sound waves must be identical except for their direction of travel. They pass through the pipe, reflect from the open ends and then pass through the pipe again. As they pass through each other their interference forces the molecules of air to oscillate by a small amount parallel to the length of the pipe. Because of the repeated interference of the waves a pattern develops in the oscillation of the molecules. The molecules at some places along the length of the pipe never move. Such a place is known as a displacement node. Molecules at other places oscillate by some maximum amount. Such a place is known as a displacement antinode.

This pattern of repeated interference of two traveling waves is called a standing wave because the positions of the nodes and antinodes in the pipe are stationary. The simplest standing wave in the pipe is termed the fundamental or the first harmonic. Its pattern consists of a displacement node halfway through the pipe and an antinode at each end. Hence the molecules near the open ends vibrate parallel to the length of the pipe by a maximum amount, whereas the molecules near the halfway point vibrate little or not at all.

To excite the fundamental in a pipe the traveling waves of sound must have a frequency equal to the speed of sound divided by twice the length of the pipe. Once a standing wave has been initiated it can grow in strength if more energy is given to the sound waves. The pipe is then said to resonate at the frequency of the fundamental standing wave. The sound emerging from the pipe can be quite loud when resonance is achieved.

An acoustic standing wave inside a pipe also sets up periodic variations in the air pressure along the length of the pipe. Consider the node at the midpoint of the pipe when the fundamental is generated. On each side of the node molecules oscillate parallel to the length of the pipe. During one part of the oscillation the molecules on the opposite sides of the node move toward the node, increasing the air pressure there. Half a cycle later in the oscillations the molecules move away from the node, leaving low pressure there. As the molecules oscillate, the pressure at the node varies above and below the atmospheric pressure outside the pipe. The greatest variation in the pressure inside the pipe is at the displacement node.

In some physics textbooks the nodes and antinodes of an acoustic standing wave refer to the pressure variations. Thus a pressure antinode (a large variation in pressure) is found at a displacement node and a pressure node (no variation in pressure) is found at a displacement antinode. In Rayleigh's explanation of the Rijke tube the key is the relative positions of the wire mesh and the displacement node (the pressure antinode) at the midpoint of the pipe. The pipe must be vertical so that the heat forces a convection through it. The mesh must be in the lower half of the pipe, not too close either to the lower end or to the displacement node at the midpoint.

Assume the various disturbances of the air inside the pipe as it cools somehow initiate weak traveling waves that generate the pipe's fundamental standing wave. Air oscillates through the wire mesh. During part of the oscillation cycle the air must move upward through the mesh and toward the displacement node, building up the pressure at the node. During the other part of the cycle the air must move downward through the mesh and away from the node, leaving low pressure at the node.

Rayleigh argued that the oscillation of the air inside the pipe is aided by a transfer of heat from the hot mesh and pipe to the air. The transfer is efficient and beneficial, however, during only part of the oscillation cycle. When the air moves upward through the mesh toward the node, relatively cool air flows into the pipe through the bottom end. Because the temperature difference between the air and the mesh is then large, the mesh transfers a lot of heat to the air. That air is more buoyant because of its increased temperature. The heat supplied by the mesh effectively pushes the mass of air upward toward the node, increasing- the pressure buildup there and furthering the oscillation.

During the downward motion of the air in the oscillation cycle the heat transferred to the air tends to buoy it upward. This time the push is in the wrong direction to aid the oscillation. Because the push is small, however, it does not seriously hinder the acoustic vibration. The air moving downward is almost as hot as the mesh because it has already been through that region. Hence the transfer of heat from the mesh to the air is small. What transfer does occur can only diminish the oscillation because an addition of heat to the air tends to drive it upward, not downward in synchrony with the oscillation.

The oscillation of the air in the pipe's fundamental standing wave is strengthened by the asymmetry in the transfer of heat during the two parts of the oscillation cycle. The oscillation receives a large push during one part and essentially no push during the other. Rayleigh likened this asymmetry to a force applied periodically to a swinging pendulum. Suppose you push on a pendulum every time it passes through its lowest point. In one part of the swing the push adds energy because you are pushing in the direction of motion. On the next pass of the pendulum through its lowest point, however, you are pushing in the direction opposite to the motion. You remove energy. As long as you apply the same. force in each case the pendulum receives no net supply of energy. Its swing eventually decreases because of frictional losses of energy. To build up the swing you must push only when you can add energy to the oscillation.

There is another advantage in transferring heat to the air while the air moves toward the midpoint of the pipe. During the increase in pressure of the air near the midpoint the temperature and pressure of the air are proportional. When the hot mesh increases the temperature, it also enhances the pressure increase at the midpoint. When the air begins to move away from the midpoint and so decreases the pressure there, any addition of heat to the air would again tend to increase the pressure. Hence the fact that the transfer is small during this stage is advantageous to the oscillation.

You can demonstrate the howling of a Rijke pipe with almost any pipe that will not melt when it is heated. Window screen serves well for a mesh. A double layer may work better than a single one. Heat can be supplied by a laboratory burner, a kitchen stove or even a candle.

Once the howling begins try to move the pipe to a horizontal position and then restore it to the vertical. The natural convection through the pipe is destroyed when the pipe is horizontal; the sound ceases immediately. If the pipe is put vertical again before the mesh has cooled, the howling begins anew.

Bibliography

HEAT-MAINTAINED SOUNDS. E. G. Richardson in Sound: A Physical Textbook. Edward Arnold & Co., 1935.

THEORY OF RESONATORS. John William Strutt, Baron Rayleigh, in The Theory of Sound. Dover Publications, Inc., 1945.

OBSERVATION OF OSCILLATORY ICE STATES IN A SIMPLE CONVECTION EXPERIMENT. Pierre Welander in Journal of Geophysical Research, Vol. 82, No. 18, pages 2591-2592; June 20, 1977

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