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Home , Standing wave

Palm Pipes Vibrations in Air Columns

C major scale Introduction C D E F GAB C orange green white blue purple black yellow red This is an experiment which demonstrates that beautiful music may be produced from simple ordinary stuff. It is not essential to have expensive, fancy, or shiny instruments to play your favorite tunes. We used PVC pipe of differing lengths to create a scale. Any scale may be made if one understands a little music theory and, of course, some . When one hits an open end of a pipe on the palm of the hand, a note will be heard. Figure 1 - Palm Pipes Physics Principles

The source of any is a vibrating object. Almost any object can vibrate and thus be a source of sound. In a PVC pipe, the source (air) is set into vibration by hitting the end of the pipe with the palm of the hand. Once disturbed, the air within the tube vibrates with a variety of , but only certain frequencies persist, which correspond to standing . Let us look at standing waves a bit more. Figure 2 - Standing Waves A standing occurs when a continuous wave travels down x fundamental or an object, reflects back, and in- a first terferes with the original wave. Waves of certain frequencies x first or will form standing waves; so b second harmonic called because they do not ap- x pear to be traveling. Both the x x second overtone or points of destructive interfer- c third harmonic ence, called nodes (the dots in x Figure 2), and the points of con- x x third overtone or structive interference, called an- d fourth harmonic tinodes (the X’s in Figure 2) re- x x main in fixed positions. Standing waves occur at more than one . The lowest frequency of vibration that produces a standing wave gives rise to the pattern shown in Figure 2a, and it is the one that determines the pitch heard. It is called the . The standing waves shown in Figures 2b, 2c, and 2d are produced at two, three, and four times the lowest frequency, respectively. The frequencies at which standing waves are produced are the natural frequencies or resonant frequen- cies of the object, and the different standing wave patterns shown in Figure 2 are different resonant modes of vibration. Only standing waves corresponding to resonant frequencies will persist for long. Although a standing wave is the result of the interference of two waves traveling in opposite directions, it is also an example of a vibrating object at .

When not played, a palm pipe is an open tube (both ends are open), but when it is played, it is a closed

1 tube (one end is open and the other end is Figure 3 - Standing Waves in a Closed Tube closed). Consequently, we are concerned with the standing waves created in a L closed tube. These are shown graphically first harmonic = fundamental in Figure 3. The graphs represent the dis- L = 1 / 4 f = v / 4L placement of the vibrating air 1 in the tube. Note that the air molecules [ actual motion of air molecules ] oscillate horizontally, parallel to the tube length, as shown by the small arrows third harmonic L = 3 3 / 4 above the second drawing. Since the f 3 = 3v / 4L closed ends of the tubes are fixed points where the air is not free to move, there fifth harmonic L = 5 / 4 are nodes there, and since the air is free 5 f = 5v / 4L to move at the open ends of the tubes, 5 there are antinodes there. (The antinodes do not occur precisely at the open ends of the tubes, for they actually depend on the diameter of the tubes. This causes end-effects that will be addressed later. For now we will assume that the antinodes are at the open ends of the tubes.) Figure 4 shows which standing waves cannot physically exist in a played palm pipe (closed tube).

Figure 4 - Impossible Standing Waves in a Closed Tube Notice that in Figure 3, the length of the tubes (L) relates to the size of the different wave- L lengths ( λ1, λ 3 , and λ 5 ). A quarter of a can fit in the tube for the funda-

mental frequency (L= λ1/4), three quarters cannot of a wavelength fits for the third harmonic have nodes (L=3 λ 3 /4), and one and a quarter wave- at open ends of lengths fit for the fifth harmonic tubes (L=5 λ 5 /4). The frequencies of these stand- ing waves depend on the (v) in air (343 m/s at 20°C) and the different

These standing waves cannot physically exist in a closed tube ( f = v / λ ). Let us look at the (one end open and one end closed) third harmonic as an example. 3 4L v v 3v L f f = λ 3 ⇒ λ 3 = 3 = ⇒ 3 = 4L = 4 3 λ 3 4L 3

The others follow the same substitution of L for λ (see Figure 3). We have f1 = v/4L , f 3 = 3v/4L , and f 5 = 5v/4L , which form a pattern represented by nv ()2n +1 v f = , where n = 1, 3, 5... OR f = , where n = 0, 1, 2.... 4L 4L

Now we can determine all the resonant frequencies produced by a palm pipe of a certain length; how- ever, we are only interested in the lowest frequency or fundamental frequency. Remember that the lowest resonant frequency is the one that determines the pitch heard. So how do we get a complete scale of notes with differing pitches? We can change the pitch by either changing the speed of sound or the wavelength ( f = v / λ ). The speed of sound will not change significantly unless you drastically change the temperature of the air through which the sound travels or replace the air with some other gas. Thus, we will change the pitch by changing the wavelengths. The wavelengths may be varied by having tubes of differing lengths (L). Recall that f = v/4L for the fundamental frequency, so L = v/4f. If we want a specific note, pitch, or frequency played, then we plug its value into the equation L = v/4f and determine what length we need to make the tube.

The end-effects mentioned earlier must now be discussed. The standing waves in a tube set themselves up as if the tube were a little bit longer than it actually is; approximately 1/4 of the inner diameter of the tube longer for each open end. So we need to subtract this added length from our equation L = v/4f. After we do this for our one open end, we get v 1 L = − D , where D is the inner diameter of the tube. ( D = 0.017 m for this case.) 4 f 4 inner inner inner

Now all we need to know are the frequencies (notes) that we want to play. There are many scales from which to choose, so we chose the most basic to western culture — the C major scale.

Note # of Steps Frequency [Hz] Length [cm] Color Abbrev. ------C — 261.6256 32.4 white W D 1 293.6648 28.8 red R E 1 329.6276 225.6 orange O F 1/2 349.2282 24.1 yellow Y G 1 391.9954 21.5 green G A 1 440.0000 19.1 blue B B 1 493.8833 16.9 purple P C 1/2 523.2511 16.0 black X

The frequencies were based upon an equal-tempered chromatic scale and determined by the follow- ing formula:

1 ()number of half steps 1 2 1 ⎡ 12 ⎤ ⎡ 12 ⎤ ⎛ 6 ⎞ f1 = f 0 ⎢2 ⎥ Here is an example: f D = f C ⎢2 ⎥ = ()262 ⎜2 ⎟ = 294 Hz ⎣ ⎦ ⎣ ⎦ ⎝ ⎠

Construction and Use

Materials Required

Material Cost Source ------1 10' long, thin walled PVC pipe < $1.00 hardware store

1 PVC pipe cutter, $6 - $12 hardware store or a hacksaw less $ or whatever works

3 colored tape, a few $ per roll or can hardware store, or spray paint or craft store or markers sandpaper cheap hardware store

Putting It Together

Measure a specific length and cut the pipe. You might find it helpful to cut the pipe just a tad long (at least the first time) because the cut usually needs some smoothing and/or leveling that requires you to sand it a bit. You will get the feel after the first cut. Then measure and cut the next length. We recommend that you measure and cut, measure and cut, as opposed to measuring each length from successive marks on the pipe and then cutting all at once. Both the length sanded off after each cut and the length due to the thickness of the cutting blade will be lost and make your cut pieces shorter than desired if the latter procedure is followed. Each finished tube may then be labeled, either with the appropriate note, frequency, and/or color code (as suggested in the previous table).

Safety Concerns

Unsanded ends of the tubes tend to hurt the palms of the hands. Younger children might have a tendency to swing (and hit) others with these lightweight batons. Experiments

Other Possibilities

• Play the whole scale in order to hear how each of the notes relates to the others. • Find the corresponding note on the piano (or other instrument) to see that they are the same. • Play the melody of one of your favorite songs. (The song must be in the key of C major, so if it is not already, you must transpose it or find another song.) • Distribute different palm pipes to each person in a large group and have them play their note when it comes up in a song. Displaying the colors or notes on the board or an overhead projector allows a director to point to the color or note when it is to be played. • With the pipes distributed among a group, play a song with more than one part (such as harmony). This will require some coordination between the director and the group. • Join a band and cut an album.

• Find other ways to produce using the palm pipes - for instance by blowing across the tops. References and Acknowledgments

We would like to thank Gene Easter and Bill Reitz for sharing the idea of the palm pipes with us. Information about standing waves, sound waves, resonance, and may be found in any basic physics textbook. Music for songs that you might want to play may be found in any beginner’s music book (something with simplified sheet music). We are confident that you will be able to come up with some great ones of your own. If you do, then we would love to hear about it so that we too can play them, and of course, give you the credit. Happy harmonizing!