# Fiddle Physics

## Violins and Fiddles

The violin is the lead instrument in most classical orchestras, and its alter ego the fiddle—which is in fact the same instrument—stars in folk music traditions stretching from the west coast of the Americas through just about every country in Europe and as far east as India. But how well do we understand how this small and oddly shaped wooden box produces its rich sounds? There is still a lot to learn, but several hundred years of acoustical research have yielded some of the physics underlying the sounds that have captivated the human ear for centuries.

Picture showing the exterior construction of the violin, taken from the top.

Picture showing the interior construction of the violin. Courtesy of George Bissinger, East Carolina University

The violin player creates sound by bowing one or more of the four strings. The bow hairs are rubbed in a sticky substance called rosin in order to make the strings stick to the bow. The action of the bow on the string causes the string to undergo a so-called slip-stick motion, in which the string travels with the bow for a time, and then slips in the other direction before being caught by the bow again a moment later. These oscillations happen many hundreds of times per second—196 times for the low G, which is the lowest note on the violin in standard tuning, and into the thousands for very high notes.

An animation showing the slip-stick motion of the bow on the violin string. From Violin acoustics, University of New South Wales. © Heidi Hereth

The strings themselves move almost no air and consequently produce almost no sound. Their vibrations are transmitted to the violin body through the bridge, which is a light piece of wood with two feet that stand on the violin belly. The total tension in the four strings is about 50 pounds, about 20 of which is directed straight down into the bridge. When a string is bowed, a force is created in the direction of the bow’s motion. As the bow pulls the string along, the force on the bridge increases in the direction of bowing. When the string slips, the force reverses direction to be opposite from the bow’s motion. Thus the force transmitted to the bridge takes the form of a sawtooth wave.

Any such periodic wave can be analyzed as a sum of pure tones, such as what we hear when we strike a tuning fork. Thus, when a violinist bows a string, he or she not only produces the fundamental pure tone that  he or she is trying to play, but also many harmonics, which are notes with frequencies that are integer multiples of the fundamental. For example, when a violinist bows "Concert A," which has a fundamental frequency of 440 cycles per second, the sound you hear contains pure tones of frequency not only 440 cycles per second, but also 880, 1320, 1760, and so on. In the case of the sawtooth wave, the fundamental is twice as strong as the second harmonic, three times as strong as the third harmonic, and so on.

An illustration of a sawtooth wave on a string

An illustration of harmonic waves on a string

Almost directly underneath the foot of the bridge on the treble side of the instrument is the sound post, which is a thin cylinder of wood, wedged in between the top and bottom plates of the violin. In addition to supporting the top plate of the violin and coupling the oscillations of the two plates, the sound post essentially anchors the treble foot of the bridge to the top plate. The bridge responds to the driving oscillations from the bowed strings by pivoting about its treble foot, which converts the side-to-side motion of the bow into up-and-down motion of the violin belly. In addition, at certain higher frequencies the bridge seems to either aid or dampen out vibrations of the violin body. Despite its apparent simplicity, the role of the bridge in helping the violin produce sound is still not fully understood.

The top plate has two features that significantly affect the sound output. One—the f-holes—are obvious to anyone who has ever seen a violin. These s-shaped holes connect the air inside the instrument with the air outside, and this oscillating air is responsible for the violin’s lowest resonance. A resonance is a frequency at which the violin naturally tends to vibrate—most of us are familiar with resonances of objects like tuning forks, which, very soon after being struck, vibrate at one frequency and not any other. The violin’s lowest resonance is very similar to the noise you hear when you blow air across the mouth of a bottle (or the f-hole itself—try it!), and tends to fall around 300 cycles per second, near the open D string. The f-holes also play an important role in separating the area where the bridge stands from the rest of the belly, allowing this area to move much more easily in response to vibrations from the bridge.

Illustration of a few of the lowest-frequency Chladni nodes of a single violin plate. From Violin acoustics, University of New South Wales.

A graphic showing the relationship between the input waveform of the violin -- what the player creates with the bow -- and the output waveform -- what we actually hear. Note that both the violin bridge and the body tend to emphasize vibrations at some frequencies while damping others. The output waveform has the same overall shape as the input waveform, but the details are very different. Graphic courtesy of Colin Gough and Physics World.

The second feature of the violin’s top plate, which is less obvious because it is on the underside of the plate, is the bass bar, which serves several purposes—it further strengthens the top plate to withstand the string pressure; it couples the vibrations of the area between the f-holes, which oscillates relatively freely, to the more constrained part of top plate closer to the neck; and it prevents energy from being wasted in higher-frequency vibrations of the body that don’t produce much sound. The sound post and bass bar together break the symmetry of the violin body, thereby allowing the body to oscillate in different ways than if it were symmetrical, and generally increasing the sound output of the instrument.

A well-shaped top or bottom plate will naturally resonate in certain “modes.” These can be beautifully illustrated by placing some sand or other fine particle on the plate and vibrating it at varying frequencies until a mode is found, at which time the sand accumulates in bands. These bands are called nodes, and the sand moves to them because they are stationary points on the plate. The other parts of the plate are moving either up or down at a given time. These patterns are called Chladni nodes after Ernst Chladni, the German physicist who developed the technique of finding such patterns on vibrating plates. Many such modes exist, but makers concentrate on those at the lowest frequencies, tuning the plates either by tapping them to hear if they ring, or flexing them by hand.

The acoustics of the assembled violin body are fantastically complex, as they involve coupled oscillations of the strings, the bridge, the top and bottom plates, the ribs, and the fingerboard. The violin body resonances do not necessarily fall at the same frequency as the notes the player wishes to play, although the first (lowest-frequency) body resonance of instruments generally considered of high quality falls near the fundamental frequency of the open A string, 440 cycles per second. In general, the frequency spectrum of the sound coming out of the violin body is very different from the spectrum going in. For instance, the violin body has almost no response at the fundamental frequency of the open G string, which at 196 cycles per second is the lowest note on the violin in standard tuning. However, many of the harmonics (392, 588, etc) are well represented in the output spectrum, and, amazingly, the human ear never knows the difference. We think we are hearing G—a very acoustically rich version of G, but still G. More amazing still, the frequency spectrum of the sound produced by the instrument can change dramatically with a very slight change in the note being played, as when a player moves his or her finger on the fingerboard to produce vibrato, and yet the ear hears a smooth transition between neighboring pitches. As University of Birmingham (UK) physics professor and concert violinist Colin Gough puts it, "The ear is probably the world's most sophisticated spectrum analyzer."

## Research

Recent research has focused on modeling the acoustic output of the violin using modern technology, often in attempt to determine critical differences between highly prized instruments from the great 17th- and 18th-century luthiers like Stradivari and Guarneri and more recently made instruments. Bernard Richardson, a physicist at the University of Wales in Cardiff, describes two recently developed methods for visualizing the instrument’s vibrations—holographic interferometry and finite element analysis. In holographic interferometry, a laser is used to make a hologram of the instrument as it vibrates in a particular mode. The various images produced interfere with each other, producing light bands at nodes and dark bands at antinodes, which are the areas of the instrument that move with maximum amplitude. Scientists use  a computational method called finite element analysis that breaks down the mathematically complex shape of the violin plates into a number of discrete parts whose motion can be numerically simulated on a computer. Richardson and his colleagues have used such models to build virtual instruments that can be "played" using a computer.

Violin sound field radiation patterns at three different frequencies. Courtesy of George Bissinger, East Carolina University

George Bissinger, a physics professor at East Carolina University, is another researcher who uses modern technology to tease out the secrets of centuries-old violins. Bissinger analyzes old and modern violins using a sophisticated laser system with three independent beams that scan the violin’s surface and measure the motion of each point in three dimensions. The violin is carefully suspended and then induced to vibrate by a small hammer that strikes the corner of the bridge, and an array of 266 microphones in an echo-free chamber records the pressure variations from the sound output. From these measurements, a visualization of the "sound field" can be produced. Bissinger then calculates how the radiation efficiency and damping of the violin’s vibration depend on frequency.

Bissinger has been able to isolate out-of-plane vibrations, which move air and thus create most of the sound, from in-plane vibrations, which dissipate energy within the plates of the violin and produce little or no sound. He has found that features such as the f-holes and bass bar are able to convert in-plane motion to out-of-plane motion, thereby increasing the sound production of the instrument. In addition, he has shown that the top plates of good instruments both old and new produce significantly more sound than the bottom plates—a major advantage to a player trying to be heard over an orchestra. Bissinger has also used computed tomography (CT scans) to determine the density of the wood inside various instruments.

Although neither Bissinger nor anyone else has yet unlocked Stradivari’s secret, if such a secret even exists, the increasingly sophisticated methods these investigators are applying to the violin are yielding ever more precise insight into the motion and sound production of this instrument. Makers are experimenting with computer-assisted design and new materials such as balsa and graphite, and it may not be far in the future when the top players accept today’s best instruments as equal to the Italian violins of old.

New York Times: String Theory: New Approaches to Instrument Design

Physics World: Science and the Stradivarius

Physics Education: Good Vibrations

The University of New South Wales: Violin Acoustics

Acoustical Society of America: 3-D Motion in Stradivari and Guarneri dG Violins

The Why Files: Fiddle Physics

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