Basic physics of the violin

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Basic physics of the violin
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The distinctive sound of a violin is the result of interactions between its many parts. Drawing a bow across the strings causes them to vibrate. This vibration is transmitted through the bridge and sound post to the body of the violin (mainly the top and back), which allows the sound to effectively radiate into the air. The tension and type of strings, the bow, and the construction of the body all contribute to the loudness and tonal quality of the sound.

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[edit] Strings

The strings of a violin are stretched across the bridge and nut of the violin so that the ends are stationary, allowing for the creation of standing waves. Standing waves will take on a specific value according to the properties of the string. The frequency and harmonics of these standing waves can be adjusted by changing the properties of the string, such as the tension, length, and material it is made from.

[edit] Tension

Tension affects the sound a violin produces in an obvious way. Increasing the tension on a string results in a higher frequency note. The strings of a violin are mounted to a fixed base, called the tail piece, and wrapped around adjustable pegs. A violin is tuned by turning each peg to loosen or tighten the string until it produces the desired frequency. Increasing tension causes the sound waves to travel faster through the string. The frequency, wavelength, and velocity of a vibrating string are related by the equation

f\times\lambda=v \,

Because the length of the violin, which determines the wavelength of the sound waves, is constant, an increase in tension will cause a proportional change in the frequency. Adjusting the tension of the strings is how a violin is tuned. This effect can be heard prior to any concert performance as each instrumentalist adjusts the tension until the frequency matches a reference sound, usually 440Hz.

[edit] Length

The length of the string also influences the frequency of the string, and is the basis for how a violin is played. Violinists shorten the playing length of the string by pushing it against the fingerboard of the violin with a finger. This has the effect of shortening the length of the waves produced by the string. Returning to the above equation, only now assuming velocity is constant (the change in tension from pressing the string is negligibly small for this discussion), shortening the length of the string will result in a higher-frequency note. Another way to view the mechanics of this is to visualize the waves as traveling through the string, bouncing back and forth at a certain speed. If the distance the wave must travel is shortened, the wave will be able to make more trips in a given amount of time. The frequency of sound is measured by how many complete vibration cycles (hertz, formerly "cycles per second") the wave makes in one second.

[edit] Materials

The material that makes up the string will also affect the quality of the note produced. A vibrating string is not a single frequency. Any frequency that is an integer multiple of the longest standing wave can vibrate on the string. These higher frequencies are called harmonics. Harmonics do not change the frequency of the note. They change the timbre. The characteristics of harmonics on a violin can be changed by using different materials.

Violin strings are played under a tension of about 220 newtons (50 lbf). Violin strings were originally made from catgut because of its high tensile strength. Modern strings are now made of steel or synthetic materials, though gut (sheep gut is actually the source) is still used. Gut strings are sensitive to changes in temperature and humidity; their tension will change, requiring frequent re-tuning. Synthetic strings are much less susceptible to environmental changes and produce a more consistent, if less expressive, sound.

[edit] Bridge

The shape of the bridge itself is also important. As the violin has developed, the shape of the bridge has been determined by trial and error. The amplitude (loudness) of the note is altered by how the vibrations are channeled through the bridge. A violin maker aims on making a bridge which will transmit the most vibrational energy from the strings to the body. A thinner bridge transmits sound more efficiently, but is structurally weaker. The openings in the bridge (the "heart" and "kidneys") reduce weight, and provide an arrangement of masses and "springs" that filter and shape the timbre of the sound. A thin bridge is characterized as having a light, less expressive sound, and was most popular during the baroque period. Modern violins use a heavier bridge to accommodate a wider range of dynamics.[citation needed]

The bridge must be strong because it also serves as structural support to the strings. As it holds the strings, it transfers the tension force from the strings to the body. By tightening the string, the applied force on the bridge is increased. The applied force exerted on the bridge is spread out on the face of the violin. This helps to ensure that the violin will not snap when the tension is increased. Modern violins typically show a string break angle over the bridge of 158°.

[edit] Bow

Sound on a violin is generally produced by a bow. The bow is made of flexible wood with a hank of horse hair connecting either end. The type of wood and the hair can change the sound of the instrument as well. The hair is coated with rosin, a pitchy resin that makes the string sticky. Sound is made when the violinist creates enough friction between the hair and the string by means of pressure placed on the bow to pull the string along the direction of the bow's travel. The force pulling the string back will eventually become greater than the force from the bow. When this happens, the string quickly recoils back toward its natural position. This causes vibrations to travel up and down the violin string. This can be observed by slowly drawing a bow across the string. By carefully drawing the bow across the string, one can also produce a continuous note. Displacing the string further will cause higher amplitude vibrations -- more sound. Describing the process mathematically, the force the bow exerts on the string can be approximated by the relationship

f = \mu\times n \,

Where μ is the coefficient of friction for a given material, and n is the force of the bow pressing down on the string. The force the string exerts as it is stretched can be described using hooke's law,

f = k\times x \,

where k is a coefficient measuring the stiffness of the spring (or in our case, the stretchiness of the string) and x is the distance the string has been pulled by the bow. Sound is generated at the precise instant when the restoring force of the spring overcomes the frictional force of the bow. If at that point, the forces are the same:

\mu\times n = k\times x \,

Changing the amount of rosin on the bow affects how sticky it is, which determines μ. During the course of a recital, this would stay essentially constant. Because the string is stretched a very small amount compared to its total length, k is also constant. This leaves the relationship

n \propto x \,

The displacement x, or how far the string gets pulled determines how loud the sound is. A violinist can change the volume of play by adjusting the normal force of the bow. Pressing harder on the string increases the frictional force, which causes the string to displace more, which results in a higher-amplitude sound wave.

[edit] Body

[edit] Structure

The body of a violin must be strong enough to support the tension from the strings, but also light and thin enough to vibrate properly. A violin uses a support structure covered by thin wood panels. The bridge pulls the strings away from the body of the violin. The force is similar to that of a bow being drawn with an arrow. To prevent the violin from bending, it is reinforced throughout the body. The force from the bridge can be so strong that cracks can develop on the face of the violin. This usually indicates that a violin has been poorly cared for. The sound post helps transmit sound to the body of the violin and serves as structural support. The final part of the support system of the instrument is the purfling. Purfling is the thin strips of wood placed in grooves made along the edges of the face and back plates of the violin. This was originally thought to be pure decoration but it may decrease surface cracks and allow the back and front plates to vibrate more freely.

The body of the violin acts as a "sound box" to amplify the sound of the vibrating strings and make them audible. (In reality, there is no "amplification": the vibrating top and back plates of the body simply increase the loudness of the sound since they have a larger surface area.) The construction of this sound box, and especially the arching of the top and back, have a profound effect on the overall sound quality of the instrument. The sound-producing system of the violin body includes the top and back (and to some degree the sides, or ribs), the bass bar that is glued to the underside of the top, and the bridge and sound post.

[edit] Surface

The face and back plates of the violin are thin pieces of wood that enclose the violin and create the sound box. The hollow chamber underneath the strings will resonate with the string above. To picture this, imagine pushing someone on a swing. The person pushing the swing represents the vibrational pulses from the string. The height of each swing represents the amplitude of the sound wave. A single vibrating string carries little energy and does not transmit sound to the air very well, just as a person pushing a swing can't push with a lot of force. However, by pushing with that same force over and over, one can quickly impart a lot of energy to the swing, as any playground experience shows. Similarly, the body of the violin collects the energy of the vibration and produces a larger sound.

[edit] Resonance

In the swing analogy, the swing will go back and forth at a specific rate. The person pushing the swing would apply an impulse of power to the swing each time the swing comes back and starts to go forward. Pushing the same direction at any other rate won't result in the swing's amplitude increasing. Imagine pushing a swing and then trying to push it again at the bottom of the return swing. Clearly, the swing will go highest when the pusher applies force at the same rate that the swing oscillates. That rate is the swing's resonant frequency. Similarly, the body of a violin has particular frequencies that will cause it to resonate more. However, the standing waves are now surface waves. The waves on the plates form two-dimensional patterns based on their frequency. The type of wood and materials used, as well as the construction of the violin will determine which frequencies the violin will resonate the most with.

It should be noted that the resonating chamber does not amplify the sound. No new energy is added to the vibrating string. Instead, the body just allows more surface area for the sound waves to transfer to the air as sound. When the surface vibrates, it changes the pressure of the air next to it. After the vibrations are transferred to the surfaces of the body, the vibrations transfer into the air. A good violin will transfer all possible vibrations to the air. A violin maker then would want to transfer the maximum amount of vibrations through the instrument into the air in the sound box. Small changes in the plates can change the quality of the note. To measure the ability for a plate to transer vibrations, a violin maker dusts the surface of each plate with iron filings and vibrates them at different frequencies, producing different sets of standing waves. The iron filings will move from the areas of high vibration (antinodes) to collect in the areas of low vibration (nodes). By comparing the results to standard filing patterns, violin makers can adjust the plates to produce the best possible sound. The name for these patterns is Chladni patterns.

[edit] Bibliography

  • Hutchins, M. 1962. The Physics of Music. Scientific America, W. H. Freeman and Company, 1974.
  • Hutchins, M. The Acoustics of Violin Plates. Scientific America, vol 245, No. 4. Oct 1981

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