Abstract Here's a fun science project for anyone who plays an electric guitar. You'll learn about the physics of vibrating strings, and find out why the tone of your guitar changes when you switch between the different pickups.
To do this project you need to have an electric guitar and guitar amplifier. You'll need to know enough about playing the instrument to produce clear, ringing tones by picking (or plucking) the string.
The goal of this project is to determine how the position, relative to the end of the string, of the pickup on an electric guitar affects the tone of the sound produced when the string is plucked.
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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.
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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).
Sound waves move in a pattern of high density and low density air molecules as they approach a detector. The detector can measures high pressure and low pressure waves and sends the data to a monitor that graphs the sound waves as pressure over time. The graph has high peaks when the air molecules are moving closely together and dips in low troughs when the air molecules are spaced further apart.
Figure 1. Illustration of a sound wave as compression and rarefaction of air, and as a graph of pressure vs. time (Henderson, 2004).
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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).
Graph of high frequency waves has a wavelength that has 6 peaks and 6 troughs. The low frequency wave graph on the bottom has only 3 peaks and 3 troughs in the same amount of time.
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.
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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 an electric 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 at the bridge of the guitar by brass washers wound on to the ends of the strings. Figure 4 has a detail view showing the bridge and the pickups. Each string passes over three separate pickups. The pickups have strong magnets lined up with each string. Around the magnets is a coil of wire with thousands of wraps. When the steel strings vibrate, they cause fluctuations in the magnetic field of the magnets. The changing magnetic field causes electric current to flow in the surrounding coil. The signal is in synch with the vibrations of the strings. It is this signal that is amplified to produce the sound of an electric guitar.
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Figure 4. Detail view of an electric guitar bridge, showing the bridge and pickups. The strings are labeled from the low E to the high e. The white switch at the edge of the photo is used to select which pickup(s) is (are) active. (© Buffalo Brothers Guitars, 2006)
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.
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).
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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.
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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 nodes 1/3 and 2/3 of the way along the string, with antinodes in between. The wavelength of this node will be equal to 2/3 of the length of the string. In this project, you will map out the locations of the nodes and antinodes for a series of harmonics for each string. You will then compare the locations of the nodes and antinodes to the locations of the pickups for each string. Which harmonics will be sensed most strongly (and which most weakly) for each pickup? How will this affect the tone that you hear for each pickup?
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