Waves are everywhere. They break on the shore of the ocean, bring music to your ears, and carry the signal of your favorite radio station. By attaching a string to two small electric motors rotating in the same direction, you can create and play with a special class of waves called standing waves.
- 4-foot (1.2-meter) length of ½-inch PVC pipe
- PVC shears (or hacksaw)
- Electric band saw (or hacksaw)
- Eight 1/2-in PVC 90-degree elbows
- Seven finishing nails, each 1 ½ in (3 cm) long
- 15-in (38-centimeter) length of standard 1 x 3 pine shelving
- Two D-cell batteries
- 20-ohm or 25 ohm potentiometer (optional)
- 5-in (12.5-cm) length of adhesive Velcro strip
- 1/16-in drill bit
- Two 3/4-in (2-cm) segments of 3/8-in diameter wooden dowel
- Two motors, 1.5–3 volts
- Two cable ties
- 18-in (46-cm) length of string (braided string, such as Wellington Braided Nylon Chalk and Mason Line, works significantly better than ordinary twisted string because it won’t unravel; if you use regular twisted string, you may have to retwist it manually)
- Masking tape
- Two 2-ft-long alligator-clip leads
Build the base of the machine:
- Using the PVC shears or hacksaw, cut the ½-in PVC pipe into the following lengths:
- Two 2-in (5-cm) pieces
- Two 4-in (10-cm) pieces
- Three 12-in (30-cm) pieces
- Use the band saw or hacksaw to slice the top portion off two PVC elbows, as shown in the image below. These modified elbows will serve as cradles for the motors.
- Assemble the PVC elbows and pipe pieces as shown in the image below (click to enlarge) to create a base for your string machine. Note: Do not glue the PVC pieces together! You will need to adjust the joints when the String Machine is running, and the lack of glue allows the machine to be taken apart for storage.
Build the battery holder:
- Hammer the seven nails into the board as shown in the image below to form a holder for the two D-cell batteries. Notice that one of the nails is placed between the batteries.
- Make sure that the nails on each end and between the batteries are firmly in contact with the batteries. Bend the nails slightly if necessary.
- Cut the Velcro strips in two 2-in (5-cm) pieces and one 1-in (2.5-cm) piece.
- Separate each set of 2-in Velcro strips and peel off the adhesive backing. Press one side of the strip on each end of the wood board and the matching side of the strip on the short PVC pieces of the base so the board can be securely attached to the base.
- If you have a potentiometer, use the 1-in (2.5-cm) Velcro strip to attach it to the board next to the batteries.
Attach the motors and string:
- Using your 1/16-in drill bit, drill a hole in the center of the end of each dowel that is 3/8-inch (1 cm) deep. (The hole will be slightly smaller than the diameter of the motor shaft.)
- Hold one of the motors so that the back is touching a hard surface and the long shaft in front is sticking straight up from the motor. Press the dowel down firmly onto the motor shaft so that the shaft fits into the hole—it’s a tight fit, but you want it to be tight so it won’t slip. Note: Do not try to push the motor shaft down into the dowel—there is a possibility that you may pop the motor right out of its casing. Repeat this step for the other motor and other dowel piece.
- Use the cable ties to secure each motor to one of the cradles you made when you sliced off the tops of the two elbows. Make sure the wooden dowels are pointing towards each other.
- Attach the string to each of the wooden dowels by laying the end of the string along the dowel and wrapping masking tape around both the sring and the dowel.
- Adjust the spread between the ends of the motor arms so that the string hangs in a loose curve.
Connect the power:
- Connect the alligator clip leads as shown in the image below, leaving one clip unattached.
- If you are using the potentiometer, connect one clip to the middle contact and the other clip to either of the outer contacts. If you aren’t using the potentiometer, just connect the two clips together. Simple Speed Control: The potentiometer allows you to control and vary the speed. If you want to omit the potentiometer, you can get still two different speeds by attaching the alligator clip to either the end nail of the battery holder (two-battery speed) or to the middle nail (one-battery speed).
- Briefly touch the remaining unattached clip to its contact point to see if both motors are turning in the same direction. If they aren’t, reverse the connections on one of the motors. (If neither motor turns, try turning the potentiometer knob. If that doesn’t get them turning when you touch the clip to the contact point, check all the electrical connections carefully and make sure your batteries are working.)
- Make the final connection and adjust the potentiometer (if present) to obtain an intermediate speed.
Adjust the tension in the string (by moving the motor arms closer together or farther apart) and the motor speed (by turning the potentiometer knob) until you obtain a relatively stable pattern in the string. See if you can get the pattern to change to a different stable pattern by gently “pinching” the string (hold your thumb and forefinger close to the string and slowly compress the pattern without actually making your fingers touch). Play with the machine for at least a few minutes to see how many different behaviors you can produce in the string.
While the machine is running, gently pull the motor ends of the two arms apart to increase the tension in the string. At some point the string should snap into its simplest behavior mode, which looks something like the movement of a high-speed jump rope. (Once you have this pattern, moving the arms back together a little usually helps to stabilize it; you can try further adjusting string tension or motor speed to get the best and most stable pattern.)
Notice that the string moves very little near the ends, but quite a lot in the middle. In wave language, a place in a wave with little or no movement is called a node, and a place with maximum movement is called an antinode. Gently pinch the string (or press down on it with a pencil) near the middle. With a little practice (and perhaps some adjustment of string tension or motor speed), you should be able to make the string jump into a mode with three nodes (one at each end and one in the middle), and two antinodes (one in the middle of each loop) as shown.
Now try pinching the string about one-third of the way across. With a little practice, you should be able to make the string jump into a more complicated pattern with three loops. This pattern, shown in the diagram below, has four nodes and three antinodes.
Put the string back into the mode with three nodes and two antinodes as shown above. Spread the fingers of one hand slightly apart and wave this hand back and forth in front of your eyes so you are looking at the string through your fingers. Can you make the string seem to stand still? If not, reduce the motor speed and try again. (Closing one eye may also help.) Eventually you may be able to see a single wave, rather than the blurred pattern.
As the dowel turns on the motor shaft, the end of the string that is taped to the dowel moves in a circle. If you think of circular motion as a combination of vertical and horizontal motion, you can visualize the string as being shaken up and down at the same time as it is being shaken right and left. The shaking of the string causes wave pulses to travel along the string.
You and a partner can use a jump rope to produce wave pulses very similar to those in the string. If you hold one end tightly in place while your partner shakes the rope, the pulses will bounce off your end and travel back. These returning pulses will travel through the ones your partner continues to make. Since any particular piece of the rope can only be in one place at one time, the two waves traveling in opposite directions combine with each other, adding together to produce a single overall shape for the rope. This is still true even if you both shake the rope to produce wave pulses. If you both shake the right way, you can produce an overall shape that is stable and exhibits nodes and antinodes. This stable overall shape is called a standing wave. The various stable patterns you produced with your String Machine are also standing waves.
The simplest standing wave that you produced was the fundamental, or first harmonic. The fundamental is one-half of a whole wave, or one-half a wavelength.
The second standing wave you made, the one with three nodes, is a second harmonic wave, and it is a whole wavelength. The third wave you made, with four nodes, is a third harmonic wave, which is one-and-a-half wavelengths.
For waves in a string, standing wave formation normally depends on a number of factors, including the frequency with which the string is shaken and the tension in the string. If any of these factors were changed, then the standing wave pattern would change. In your String Machine, however, it's possible to change the frequency (motor speed) and tension (spread between ends of the motor arms) without changing the pattern because of the circular motion of the string. This motion introduces forces on the string not present in waves generated in a single plane, as in the simple case of the jump rope being shaken up and down.
When you “strobed” the string by waving your hand back and forth in front of it, you were able to get successive views of the string over short time intervals. If the time it took for adjacent spaces between your fingers to change places in front of your eyes were exactly the time it took for the string to go through one whole cycle (or any whole number of its cycles) then you would see the wave in the same position each time, and it would appear to be standing still.
On occasion—possibly due to slightly different motor speeds—the moving string may seem to develop sub-patterns within the main pattern, which will vary slowly in a regular way. This behavior is the result of waves with slightly different frequencies interacting in a complex way to produce a regular alternating or oscillating pattern called a beat. Perhaps the simplest and most well-known example of a beat is the loud-soft-loud-soft tone produced when musical instruments (or two different strings on the same instrument) are being tuned. The two instruments are in tune—each producing the same frequency note—when the variable tone is no longer noticeable.
What is the maximum number of loops you can produce in the string? You may try maintaining a continuous loose pinch or exerting continuous light pressure with your finger or a pencil.
If you have a strobe light, try strobing the string on your String Machine—it’s spectacular! A digital camera can produce images of a “frozen” string as well.
Build a motorized color wheel, with transparent pie-shaped segments of a few different colors. If you shine a bright light through the rotating color wheel and use it to illuminate the string machine while the string is in motion, the results can be sensational.
Standing waves are at the heart of musical instruments. In wind instruments, standing waves are set up in an air column. At some locations the density of air molecules is alternately very high and very low, creating large fluctuations in pressure. These locations are the antinodes of the air-pressure standing wave. At other locations, small pressure changes mark the nodes of the standing wave. When the length of the air column is changed, as with keys on a clarinet or the slide on a trombone, a different standing wave is formed, and you hear a different note. Likewise, standing waves are formed on stringed instruments and drumheads.
Harmonics and Overtones
The second harmonic is also called the first overtone, the third harmonic the second overtone, and so on.