Don't You Fret! Standing Waves on a Guitar
AbstractIn this project, you'll investigate the physics of standing waves on guitar strings. You'll learn about the different modes (i.e., patterns) of vibration that can be produced on a string, and you'll figure out how to produce the various modes by lightly touching the string at just the right place while you pick the string. This technique is called playing harmonics on the string. By the way, we chose a guitar for this project, but you can do the experiments using any stringed instrument, with or without frets.
The goal of this project is to investigate which standing wave patterns you can produce on a guitar string by playing harmonics.
Andrew Olson, Ph.D., Science Buddies
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In this project, you'll investigate the physics of standing waves on guitar strings. You'll learn about the different modes (i.e., patterns) of vibration that can be produced on a string, and you'll figure out how to produce the various modes by lightly touching the string at just the right place while you pick the string. This technique is called playing harmonics on the string.
You'll need to understand some basic properties of waves to get the most out of this project. We'll provide a quick introduction here, but for a more complete understanding we recommend some background research on your own. The Bibliography section, below, has some good starting points for researching this project. We especially recommend exploring the "Sound Waves and Music" articles (Henderson, 2004).
What is sound? Sound is a wave, a pattern-simple or complex, depending on the sound-of changing air pressure. Sound is produced by vibrations of objects. The vibrations push and pull on air molecules. The pushes cause a local compression of the air (increase in pressure), and the pulls cause a local rarefaction of the air (decrease in pressure). Since the air molecules are already in constant motion, the compressions and rarefactions starting at the original source are rapidly transmitted through the air as an expanding wave. When you throw a stone into a still pond, you see a pattern of waves rippling out in circles on the surface of the water, centered about the place where the stone went in. Sound waves travel through the air in a similar manner, but in all three dimensions. If you could see them, the pattern of sound waves from the stone hitting the water would resemble an expanding hemisphere. The sound waves from the stone also travel much faster than the rippling water waves from the stone (you hear the sound long before the ripples reach you). The exact speed depends on the number of air molecules and their intrinsic (existing) motion, which are reflected in the air pressure and temperature. At sea level (one atmosphere of pressure) and room temperature (20°C), the speed of sound in air is about 344 m/s.
One way to describe a wave is by its speed. In addition to speed, we will also find it useful to describe waves by their frequency, period, and wavelength. Let's start with frequency (f). The top part of Figure 1, below, represents the compressions (darker areas) and rarefactions (lighter areas) of a pure-tone (i.e., single frequency) sound wave traveling in air (Henderson, 2004). If we were to measure the changes in pressure with a detector, and graph the results, we could see how the pressure changes over time, as shown in the bottom part of Figure 1. The peaks in the graph correspond to the compressions (increase in pressure) and the troughs in the graph correspond to the rarefactions (decrease in pressure).
Figure 1. Illustration of a sound wave as compression and rarefaction of air, and as a graph of pressure vs. time (Henderson, 2004).
Notice how the pressure rises and falls in a regular cycle. The frequency of a wave describes how many cycles of the wave occur per unit time. Frequency is measured in Hertz (Hz), which is the number of cycles per second. Figure 2, below, shows examples of sound waves of two different frequencies (Henderson, 2004).
Figure 2. Graphs of high (top) and low (bottom) frequency waves (Henderson, 2004).
Figure 2 also shows the period (T) of the wave, which is the time that elapses during a single cycle of the wave. The period is simply the reciprocal of the frequency (T = 1/f). For a sound wave, the frequency corresponds to the perception of the pitch of the sound. The higher the frequency, the higher the perceived pitch. On average, the frequency range for human hearing is from 20 Hz at the low end to 20,000 Hz at the high end.
The wavelength is the distance (in space) between corresponding points on a single cycle of a wave (e.g., the distance from one compression maximum (crest) to the next). The wavelength (λ), frequency (f), and speed (v) of a wave are related by a simple equation: v = fλ. So if we know any two of these variables (wavelength, frequency, speed), we can calculate the third.
Now it is time to take a look at how sound waves are produced by a musical instrument: in this case, the guitar. For a scientist, it is always a good idea to know as much as you can about your experimental apparatus! Figure 3, below, is a photograph of a guitar.
Figure 3. Top view of an acoustic guitar.
The guitar has six tightly-stretched steel strings which are picked (plucked) with fingers or a plastic pick to make them vibrate. The strings are anchored beneath the bridge of the guitar by the bridge pins (see Figure 4). Each string passes over the saddle on the bridge. The saddle transmits the vibrations through the bridge to the soundboard of the guitar (the entire front face of the instrument). The soundboard, with its large surface area, amplifies the sound of the strings. (One way to see this for yourself is with the mechanism from a music box. First try playing it while holding it in the air. Then, place it in contact with the soundboard of the guitar and play it again. You'll see that the sound is greatly amplified by the wood.)
Figure 4. Detail view of an acoustic guitar bridge, showing the bridge pins and saddle.
The string vibrates between two fixed points:
- where it is stretched over the saddle of the bridge (Figure 4) and
- near the opposite end of the string, where it passes over the nut(Figure 5).
After passing over the nut, the strings wrap around tuning posts. A worm gear mechanism allows the posts to be turned in order to raise or lower the tension on the string.
Figure 5. Detail view of an acoustic guitar headstock, showing the nut and tuning pins. The top portion of the neck (first fret) is also shown. The strings are labeled, from low "E" to the high "e."
When a guitar string is picked, the vibration produces a standing wave on the string. The fixed points of the string don't move (nodes), while other points on the string oscillate back and forth maximally (antinodes). Figure 6, below, shows some of the standing wave patterns that can occur on a vibrating string (Nave, 2006a).
Figure 6. Standing waves on a vibrating string, showing the fundamental (top), first harmonic (middle), and second harmonic (bottom) vibrational modes. (Nave, 2006a)
The string can vibrate at several different natural modes (harmonics). Each of these vibrational modes has nodes at the fixed ends of the string. The higher harmonics have one or more additional nodes along the length of the string. The wavelength of each mode is always twice the distance between two adjacent nodes.
The fundamental mode (Figure 5, top) has a single antinode halfway along the string. There are only two nodes: the endpoints of the string. Thus, the wavelength of the fundamental vibration is twice the length (L) of the string.
The second harmonic has a node halfway along the string, and antinodes at the 1/4 and 3/4 positions. This standing wave pattern shows one complete cycle of the wave. Thus, the wavelength of the second harmonic is equal to the length of the string.
In addition to the endpoints, the third harmonic has a nodes 1/3 and 2/3 of the way along the string, with antinodes in between. The wavelength of this mode with be equal to 2/3 of the length of the string.
Remember that the relationship between wavelength and frequency depends on the speed of the wave. We can rewrite the equation presented earlier as f = v/λ. If we take the ratio between the frequency, f2, of the second harmonic and the frequency, f1, of the first harmonic, the velocity term cancels out:
You can continue the calculations for the higher harmonics yourself. What is the frequency of the third harmonic, relative to the fundamental?
Now you have enough of an introduction to sound waves and guitars so that you can understand how to predict the locations of the standing wave nodes on the strings. Using this knowledge, you can produce harmonics by lightly touching the strings instead of fretting them. The Experimental Procedure section, below, shows you how to explore this musical aspect of standing waves.
Terms and Concepts
To do this project, you should do research that enables you to understand the following terms and concepts:
- String vibrations
- Standing waves
- Musical intervals
- What is the relationship between the length of a string and the wavelength of the fundamental tone it produces when plucked?
- What is the relationship between the length of a string and the wavelength of the first harmonic it produces?
- How can you figure out the wavelength of the higher (second, third, fourth, etc.) harmonics?
- How can your figure out where on the string to place your finger to produce harmonics?
Waves: the first reference is a good general introduction, and those that follow cover the specific topics indicated by their titles:
- Henderson, T., 2004. "Sound Waves and Music," The Physics Classroom, Glenbrook South High School, Glenview, IL [accessed March 27, 2006] http://www.physicsclassroom.com/class/sound.
- Nave, C.R., 2006a. "Standing Waves on a String," HyperPhysics, Department of Physics and Astronomy, Georgia State University [accessed March 27, 2006] http://hyperphysics.phy-astr.gsu.edu/Hbase/waves/string.html#c1.
- Nave, C.R., 2006b. "Resonances of Open Air Columns," HyperPhysics, Department of Physics and Astronomy, Georgia State University [accessed March 27, 2006] http://hyperphysics.phy-astr.gsu.edu/Hbase/waves/opecol.html#c1.
- FlashMusicGames, 2007. "How Guitar Works," FlashMusicGames.com [accessed July 2, 2007] http://www.flashmusicgames.com/how_guitar_works.html.
Here's a great article about harmonics on guitar strings:
Lehman, S., 1999. "Understanding Harmonics," Harmony Central [accessed July 2, 2007] http://www.harmony-central.com/Guitar/harmonics.html.
Here's a more advanced article on harmonics and the physics of guitars:
Hokin, S., 2002. "The Physics of Everyday Stuff: The Guitar," The Physics of Everyday Stuff website [accessed July 2, 2007] http://www.bsharp.org/physics/guitar.
Here is a table showing guitar and piano note frequencies:
Aubochon, V., 2004. "Musical Note Frequencies: Guitar and Piano," Vaughn's One-Page Summaries [accessed July 2, 2007] http://www.vaughns-1-pagers.com/music/musical-note-frequencies.htm.
Here is the source of the diagram showing the fundamental frequencies corresponding to the 88 keys of the piano:
Irvine, T., 2000. "An Introduction to Music Theory," VibrationData.com Piano Page [accessed July 2, 2007] http://www.vibrationdata.com/piano.htm.
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Materials and Equipment
To do this experiment you will need the following materials and equipment:
- As you can see from the pictures in the Introduction, we used an acoustic guitar for this project
- An electric guitar-or other stringed instrument-can be used instead
- Guitar pick
Electronic tuner (to tell you what note you've played)
One alternative is to use a computer with tuning software, such as enable Tuner by Enable Software®. To use this software, you'll need:
- A Windows-based computer with a 16-bit soundcard
- A microphone
- A big advantage of this tuner is that it displays the frequency of the note that you played
- Another alternative is to use a stand-alone electronic tuner. There are many models, available at most music stores. You want a chromatic tuner with a built-in microphone. It will sense the note that you play and indicate whether you are sharp (above) or flat (below), relative to the closest reference note. If you can find one that displays the frequency of the note you played, that is best.
- One alternative is to use a computer with tuning software, such as enable Tuner by Enable Software®. To use this software, you'll need:
- Sewing tape measure (best if marked in metric units)
- Lab notebook
- Pen or pencil
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- Do your background research so that you are knowledgeable about the terms, concepts, and questions, above.
- Measure the length of each of the strings on your guitar (or other stringed instrument). The string length for each string is the distance between the saddle (see Figure 4) and the nut (see Figure 5).
- For each string, use your string length measurements and your knowledge of standing waves to calcuate the location of the first node point for each of the first three harmonics. (If you want to keep exploring higher harmonics, so much the better!)
- Use your tape measure to figure out where the predicted node points fall on the strings. Some of the node points may fall right over a fret, while others may be just above or below a fret.
Get set up to play and record the notes with your tuner software (or chromatic tuner).
- The experiment is best done with a guitar that is in tune, so the first step is to tune the guitar.
- Place the microphone (or chromatic tuner) close enough to the guitar so that the tuner software (or tuner) registers the note even when you play softly.
Pluck the open high E string. From the readout of the tuner software (or chromatic tuner), write down the frequency of the note played.
- If you are using a chromatic tuner without a frequency readout, write down the note played from the chromatic tuner readout.
You can look up the standard frequency of the note using Figure 7 (if you know how guitar notes correspond to the piano keyboard) or the table of note frequencies in the Bibliography (Aubochon, 2004).
Figure 7. Fundamental frequencies of the 88 notes on the piano. (Irvine, 2000)
- Place your finger lightly on the string at the predicted node point for the first harmonic.
- Pluck the string again.
Make sure the note is clear and ringing, then write down the frequency of the note played. If the note does not sound properly, try again until it does. For the clearest sound, you need light pressure on the string. Tip: make sure that your picking (or plucking) sets the string vibrating parallel to the soundboard of the instrument. If the vibration is perpendicular to the soundboard, the vibration will be quickly muted by your finger.
- If you are using a chromatic tuner without a frequency readout, write down the note played from the chromatic tuner readout.
- You can look up the standard frequency of the note using Figure 7.
- When using a tuner without a frequency readout, you will have to use your knowledge of music and careful listening to determine the correct octave of the note played. For example, the open high E-string on the guitar is E4 (329.63 Hz), while the first harmonic is E5 (659.25 Hz).
- Repeat steps 6-8 for the second and third harmonics.
- Repeat steps 5-8 for each of the other five strings.
- How close do you have to be to the node point in order to produce a clear, ringing harmonic from the string? Try moving up or down the string slightly to find out.
- Measure the actual distance for the first node point for each harmonic for each string.
- For each string, what is the relationship between the position of the first node point for each harmonic and the length of the string?
- For each string, what is the interval between the fundamental and each of the harmonics?
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- To see how to calculate the speed of the wave traveling on each guitar string, see the Science Buddies project Guitar Fundamentals: Wavelength, Frequency, & Speed.
- If you play acoustic guitar, maybe you've noticed that sometimes one (or more) of the guitar strings will vibrate after you've picked a different string. This is called sympathetic vibration. You can investigate the physics behind sympathetic vibrations with the Science Buddies project How to Make a Guitar Sing.
- For an experiment on sympathetic vibrations using a piano, see the Science Buddies project How to Make a Piano Sing.
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