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Simple Guitar Physics
Construction of the Guitar
In order to achieve the specific sounds required for music, guitars have various components that enable them to produce these specialized sounds. The narrow end of the guitar is called the headstock, and is attached to the neck of the guitar. On the headstock there are machine heads, also known as tuning keys, around which the strings are wound. At the point where the headstock meets the neck of the guitar, there is a small piece of material (plastic, bone, etc.) called the nut, in which small grooves are carved in order to guide the strings up to the machine heads. The neck of the guitar runs all the way down the guitar until it meets the body of the guitar at the upper bout, and it contains the fret board of the guitar, containing the frets embedded in it at points along the length of the neck that divide it mathematically. The body of the guitar is a resonating chamber which projects the vibrations of the body through the hole cut on the top of it, called the sound hole. The strings of the guitar run from the machine heads, over the nut, down the neck, body, and the sound hole, and are anchored at a piece of hardware attached to the body of the guitar, called the bridge.
It is these components of the guitar that allow it to produce the specific sounds required to create music. In order to understand music and how guitars produce it, it is first required to understand the physics of sound. Sound is created when a wave motion is set up in the air by the vibration of material bodies. What this means is that when material bodies vibrate, they create a vibrational energy that travels in pressure waves through a medium. All forms of instruments create vibrations in order to produce sound waves that make the music, which is essentially organized sound, and guitars are a type of musical instrument called a string instrument, meaning that they create their sound through the vibrations of a string. On the guitar, the string that vibrates to produce the sound is fixed at both ends, is elastic, and therefore can vibrate . When the guitar string is either strummed or plucked, the string of the guitar begins to vibrate, and since these vibrations are waves, they begin to travel in both directions along the string and are reflected back at each fixed end. These waves will not cancel each other out as they reflect back upon themselves, but instead form a standing wave, which is a situation where crests and troughs remain at fixed positions in the medium while the wave as a whole increases and decreases together. The guitar strings act in such a way that they can satisfy the relationship between wavelength and frequency, represented by the equation v = fλ . This equation can be rearranged to f = v/λ, meaning that the frequency of a wave (f) is dependent on both the speed of the wave (v), and the length of the wave (λ). As well, the speed of the wave traveling on the guitar string depends on the tension of the string (T) and the linear mass density of the string (µ), in fact, “the root frequency for a string is proportional to the square root of the tension, inversely proportional to its length, and inversely proportional to the square root of its linear mass density” . What this means is that waves will travel faster when the tension of the string is higher, which in turn means that the frequency will be higher as the tension is increased (f = v/λ, the v is increasing).
This also means that waves will travel slower on a more massive string, since if the mass is increased, the v will decrease. This relationship between the speed, tension, and mass density can be arranged into a new equation,
When a standing wave vibrates, a combination of reflection and interference occur in such a way that the reflected waves interfere constructively with the incident waves, because the waves have changed phase when they reflected from one of the fixed ends. When this is happening, the medium appears to vibrate in segments, and it is not apparent that the whole wave is traveling. Since a guitar string has two fixed ends, it will act like a standing wave, and therefore when agitated by either being plucked or strummed; the wavelength that the string can produce is twice the length of the string . Since all the strings are the same length, all six strings on the guitar use the same range of wavelengths, however, in order to produce different sound waves required to create music, different amounts of air must be displaced at different frequencies, meaning the guitar strings must be able to vibrate at different frequencies to do so. In order to create different frequencies on the guitar, one of the factors of the equation f = v/λ must be changed, so either the speed, or the length of the wave must be changed. Since the strings on the guitar are attached to the nut and bridge, and when played open have a fixed wavelength, the only other factor that can be changed to produce a different frequency is the speed of the wave, ‘v’. Since the speed of the wave is affected by the tension on the string and the mass density (v = T/µ.), either the tension of the string, or the mass density must be changed in order to create a different frequency. However, if the frequency of the vibration of the guitar string were only changed by varying the tension, then the high strings (needing a higher frequency) would have to be wound tight since the tension required would be fairly high, while the lower strings (needing a lower frequency) would require much less tension, and subsequently be very loose. Since it would be very difficult to play a guitar where the high strings are tight and the low strings loose, guitars are constructed in such a way that the tension of the strings should be equal. Since the only other factor that can be changed while playing all the strings open is the mass density, guitars are constructed so that the tension of the strings, as well as the mass density are increased together. As a result, guitar strings are made so that the higher the frequency required from the open string, the less mass density the string will have, since higher frequencies require a higher tension, and the less mass they have the less tension is needed to achieve the same frequency. Subsequently, the lower the frequency of a string required is, the higher the mass density is, since a lower tension produces lower frequencies, and the more mass the strings contain, the more tension is required. Since in standard tuning the strings on the guitar are a perfect fourth apart on pitch (frequency), except between G and B, the amount that the mass density must be increased so that the tension remains constant can be calculated.
Frets and Intonation
However, music is complex, and many frequencies are required in order to create the correct sound waves that will produce the music. This poses a problem, because although the 6 strings of the guitar are set up in a playing-friendly manner, at this point each individual string can only produce one frequency, and since no different part of the equation to f = v/λ is being changed when an open string is played, which is not nearly enough variation to produce complex music. Therefore, one part of the equation to f = v/λ must be changed while playing a guitar in order to produce a different frequency. However, the speed of the wave cannot be changed, since the two factors (v = T/µ.), the tension of the string and the mass density are not changed significantly enough while playing to affect the speed of the wave enough to change the frequency. As a result, on the neck of the guitar there are little strips of metal called frets, whose function is to decrease the length of the string, which will cause a higher frequency. When a string is pressed down near a fret, the resonant length of the string is decreased, as it no longer stretches from the bridge to the nut but from the bridge to the fret where the string is being held down. This decreases the length of the wave (λ) through decreasing the length of the medium (string), which consequently increases the frequency of the string. Thus, on every string, the guitar player has an option of decreasing the length of the string in about 24 different ways, which will produce 24 different frequencies on each string. Since a guitar has six strings, and each string can have up to 24 frets, the number of notes available from which to choose is greatly increased. As multiple strings can be played together, the guitarist now has many frequencies from which to choose in order to create music on the instrument.
Frets on the fingerboard serve to fix the positions of notes and scales, which gives them equal temperament. Consequently, the ratio of the widths of two consecutive frets is the twelfth root of two , whose numeric value is about 1.059. The twelfth fret divides the string in two exact halves and the 24th fret (if present) divides the string in half yet again. Every twelve frets represents one octave. The position of the bridge saddles, upon which the strings rest, determines the distance to the nut (at the top of the fingerboard). This distance defines the positions of the harmonic nodes for the strings over the fretboard, and is the basis of intonation. Intonation refers to the property that the actual frequency of each string at each fret matches what those frequencies should be according to music theory. Because of the physical limitations of fretted instruments, intonation is at best approximate; thus, the guitar's intonation is said to be tempered. The twelfth, or octave, fret resides directly under the first harmonic node (half-length of the string), and in the tempered fretboard, the ratio of distances between consecutive frets is approximately 1.06, as derived above.
However if a guitar string had only one single frequency that it vibrated on, the guitar would sound quite boring, and there would not be much difference between the guitar and other stringed instruments. Guitars sound different from other stringed instruments because of the different overtones, or harmonics dominant on a guitar. When a guitar string is either strummed or plucked, the string begins to vibrate, and these vibrations are in the form of waves. However, the waves that are created by the vibrations of the string travel in both directions along the string, and continue forward until they are reflected off the fixed ends. When the waves are reflected, they change direction, and travel back the other way through the medium (the string). When the waves are traveling back through the string, they cause interference with the other waves traveling the string that were also caused by the vibration . The standing wave pattern is formed when there is perfectly timed interference of two waves passing through the same medium, to create a situation where the crests and troughs remain at fixed positions. On a guitar string, the waves that are reflected and are traveling in the opposite direction of the other waves on the string create a standing wave. Because of the interfering vibrations on a guitar string, standing wave patterns are created, meaning that there are some points along the string that appear to be standing still, and these points of no displacement are referred to as nodes. As well, there are other points along the medium that undergo vibrations between a large positive and large negative displacement, and are the points that undergo the maximum displacement during each vibrational cycle of the standing wave and are called antinodes. On the guitar string, a number of different patterns of standing waves may be produced, and each pattern will have different number of nodes and antinodes. Standing wave patterns can only be produced within the string of the guitar when it is vibrated at certain frequencies, however there are several frequencies with which the string can be vibrated to produce the different patterns of standing waves, each with a different number of nodes and antinodes. Every different frequency is associated with a different standing wave pattern, and they are referred to as harmonics. The most simple pattern of standing wave that can be produced is one at which the two nodes are at the fixed ends, which is the longest wavelength, and it is called the first harmonic, or fundamental harmonic. Since on a guitar string the waves keep on being reflected off the fixed ends and causing interference with each other, there are many different frequencies, but with any medium fixed at both ends, only certain sized waves can stand. This means that on a guitar string, only certain types of frequencies can stand, so we say that such a medium is tuned. Therefore, the strings on the guitar are tuned in such a way that the second pattern of the standing wave, or second harmonic, can only have half the wavelength and twice the frequency of the first harmonic. The second harmonic is also referred to as the first overtone, and it is these multiple overtones that we hear from the guitar string that make the guitar sound different from other instruments. Similarly, the third harmonic, or the third pattern possibility for the standing wave on a guitar, has one third the wavelength and three times the frequency when compared to the first harmonic and is called the second overtone. The rest of the harmonics follow the same pattern that the nth harmonic has 1/n wavelength and n times the frequency. It is the fundamental frequency (first harmonic) that determines the note that we hear, and the higher harmonics determine the timbre. This means that the simplest standing wave pattern on the guitar string containing only two nodes and two antinodes, determines what musical note we hear, while the more complex standing wave patterns, the other harmonics, determine how that note sounds.
Sound is created when material vibrations cause changes in air pressure and create pressure waves. However, guitar strings are not large enough to move large enough amounts of air to create a sound loud enough to be easily heard by the human ear. Therefore, the body of an acoustic guitar is used to amplify the sounds the strings produce, and the body of the guitar is made up of different components that allow it to do so. The body of the guitar is basically a larger hollow space that is specially constructed to amplify the sound of the strings. The top plate of the body, the piece of wood located on the front of the body of the guitar, is constructed so that it can vibrate up and down relatively easily, and is usually made of light, springy wood, about 2.5 mm thick. Inside the actual body of the guitar there are series of braces that strengthen the plate and the keep the plate flat, despite the movement of the strings that will tend to make the bridge move, since it is attached to the top plate. On the opposite side of the guitar, there is the back plate that does not play as big a role in amplifying the sound, since it is held against the player's body and cannot vibrate much. The sides of the guitar also do not vibrate much in the direction perpendicular to their surface, so they also don’t radiate much sound.
When the strings are plucked or strummed, they begin to vibrate, and these vibrations in the form of waves are transmitted to the bridge of the guitar. Since the bridge is attached the top plate of the guitar, the top plate also begins to vibrate as a result of the vibrations of the string, via the bridge. If the string is vibrating at a high frequency, and subsequently the bridge is vibrating at a high frequency, most of the sound is radiated by the vibrations of the top plate. Since the top plate has a much larger surface area than the string, when the top plate vibrates as a result of the vibrations of the string, the volume of air the top plate is displacing is much larger than that of the string. Therefore, the pressure waves being produced by the top plate will be bigger, and the sound will be louder. For lower frequencies, the strings vibrations are transmitted via the bridge to the top plate, where it is then transmitted to the back plate, then reflected through the sound hole, which is constantly increasing the volume of the pressure waves being produced. In fact, it is not the vibrations of the guitar string that we hear when listening to a guitar, rather the amplification of the vibrations it produces through the body of the guitar.
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