When a bow is drawn across a violin string, the string doesn't just get dragged along. It sticks to the rosined bow hair and is carried sideways, then suddenly slips and flies back, then sticks again, hundreds of times a second. This simulation models a violin string as a chain of small masses connected under tension and bows it with a simple model of rosin friction, so you can watch that stick–slip cycle in slow motion.

Bow holding the string (sticking) String slipping under the bow Velocity of each mass Net force on each mass
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Bowing

Green under a slider marks the values that give clean Helmholtz motion (a good tone) with the other settings as they are.

Examples
String
Preset Pitch:

The damping is given as the time \(\tau\) for a free vibration to die down to \(1/e\) of its size from these losses alone. A note dies away faster than that, because the simulation also loses energy at short wavelengths (which rounds off sharp corners) and into the bridge.

Bridge
The bridge's moving mass is 1 g.

What you're seeing

The string is held at the nut (left) and rests on the bridge (right). It's divided into 120 segments, with a point mass \(m\) at each joint. Each mass is pulled by the tension \(T\) in the two segments on either side of it, and when the string is straight those pulls cancel. When it's bent, they don't: for small displacements, the net sideways force on mass \(i\) is \[ F_i = \frac{T}{h}\,\bigl(y_{i+1} - 2y_i + y_{i-1}\bigr), \] where \(y_i\) is its displacement and \(h\) is the segment length. That force is proportional to how sharply the string bends at that point, so a straight stretch of string feels no net force and keeps moving at whatever speed it already has. Only the bends accelerate. Turn on "Forces" to see this: the force is concentrated at the corners. With many segments, these equations become the wave equation, with waves traveling at \(c = \sqrt{T/\mu}\), where \(\mu\) is the mass per unit length.

The "String" settings choose \(T\), \(\mu\), and the length \(L\). A wave takes \(2L/c\) to travel to one end of the string, back to the other, and return, so the open string's pitch is \[ f_0 = \frac{c}{2L} = \frac{1}{2L}\sqrt{\frac{T}{\mu}}. \] Tightening the string raises the pitch; a heavier or longer string lowers it. The violin presets all have the same length, and they get their different pitches mostly from their different densities: the G string is more than six times heavier per meter than the E string.

The bow acts on the one mass under it. The rosin gives it two kinds of friction:

  • Sticking. While the string moves with the bow hair, static friction holds it there with any force up to \(\mu_s N\), where \(N\) is the bow force (how hard the bow presses on the string) and \(\mu_s\) is the coefficient of static friction (0.8 by default).
  • Slipping. Once the tension pulls harder than that, the string breaks loose. Sliding friction is weaker, and it gets weaker still the faster the string slides, falling toward \(\mu_d N\) (with \(\mu_d = 0.3\) by default). That drop is what lets the string fly back so freely.

The second chart shows the friction. While the string sticks, friction is whatever force it takes to keep the string moving with the bow, which stays below the upper dashed line, \(\mu_s N\). When the Helmholtz corner (below) reaches the bow, the force needed to hold the string jumps above that line (the spike), and the string breaks loose. While it slips, friction drops toward the lower dashed line, \(\mu_d N\), then climbs back as the string slows down, until the string is moving at the bow's speed again and sticks (the dashed line in the first chart). So a force limit decides when the string starts to slip, and the bow's speed decides when it stops.

When you change a setting, the view skips ahead to the motion the string settles into, which takes dozens of vibrations. To watch it get started instead, click "Start from rest". The vertical scale is exaggerated a great deal. A real string swings by a millimeter or two at most, less than 1% of its length.

Helmholtz motion

With a good balance of bow force, speed, and position, the string settles into a regular pattern, first described by Hermann von Helmholtz in the 1860s. At any moment the string is two straight lines with one sharp bend between them, the Helmholtz corner. The corner races around the string, back and forth between nut and bridge, once per vibration. Turn on "Trail" to see its path: the corner traces out two curved arcs, so the string seems to fill a lens shape, which is what your eye sees on a real bowed string.

The bow only notices the corner when it passes underneath. While the corner makes the long trip to the nut and back, the string under the bow moves steadily with the bow hair (sticking). When the corner passes the bow on its way to the bridge, it releases the string, which slips back the other way until the corner returns from the bridge and the bow catches it again. If the bow is a fraction \(\beta\) of the string's length from the bridge, the string sticks for a fraction \(1 - \beta\) of each vibration and slips for \(\beta\). The first chart above shows this: a long, flat stretch at the bow's speed, then a short, fast swing in the other direction, once per vibration.

A few consequences:

  • The bow doesn't set the pitch. The corner takes the same time \(2L/c\) to go around no matter how the string is bowed, so the frequency is the string's own.
  • Faster bowing and bowing nearer the bridge are louder. The string has to keep up with the bow for the whole sticking phase, so the size of the vibration is proportional to \(v_{\text{bow}}/\beta\).
  • The string drives the bridge with a sawtooth. The force of the string on the bridge rises steadily and then drops suddenly once per vibration. A sawtooth contains every harmonic, which is why a bowed string sounds bright and rich. In ideal Helmholtz motion, the harmonics with a node at the bow are missing (bowing at exactly \(\beta = 1/8\) would remove the 8th, 16th, 24th, …). Real motion isn't quite ideal, and in this simulation those harmonics don't disappear.

The bridge and the sound

The string doesn't make much sound by itself, because it's too thin to push much air. Instead, it rocks the bridge, which shakes the violin's body, and the body makes the sound. Here the bridge is a single mass \(m_b = 1\) g on a spring and a damper. The string pulls it sideways, the spring pulls it back toward its rest position, and the damper resists its motion: \[ m_b \ddot{y}_b = \frac{T}{h}\bigl(y_{N-1} - y_b\bigr) - k\,y_b - c\,\dot{y}_b, \] where \(y_b\) is the bridge's displacement, \(k\) is the stiffness, and \(c\) is the damping. On its own, the bridge would ring at its resonance frequency \(f_b = \frac{1}{2\pi}\sqrt{k/m_b}\), dying away more slowly the larger its quality factor \(Q = \sqrt{k\,m_b}/c\) is. The default bridge resonates near 2.8 kHz, roughly where a real violin has a broad peak in its response called the "bridge hill".

The default bridge is stiff, so it moves only a few micrometers, far too little to see in the animation. The third chart shows its motion. Each slip gives the bridge a sudden kick, which sets it ringing at its resonance. "Listen" plays two seconds of this displacement, starting from rest with the current settings. "Play continuously" plays it without stopping, from a second copy of the string that runs in real time. Changing a setting while it plays changes the bowing of that string without restarting it, as a player would mid-stroke. In both, the volume follows the size of the bridge's motion. A very soft bridge can move a hundred times more than the default one, more than can be played without distortion, so motion beyond about four times the default is turned down to fit. The bridge responds strongly to harmonics near its resonance and weakly to those far above it, so the stiffness and damping change the tone color. Move the resonance and listen to the difference.

The oscilloscope below the charts shows this output signal over one natural period of the string, \(2L/c\), lined up on each vibration's peak so that the trace holds still. (With a moving bridge the actual period is a little different, so the trace doesn't quite join up at its ends.) Next to it is the output's spectrum: how strong each frequency is, measured over the last few dozen vibrations. A periodic signal contains only the frequency it repeats at, the fundamental \(f_0\), and whole-number multiples of it, the harmonics \(2f_0, 3f_0, \ldots\), which appear as the numbered peaks. Move the bridge resonance (the dashed line) and watch the harmonics near it get louder. Or click "Pluck instead": the string is plucked at 0.3 of its length from the bridge, which is a node of the 10th harmonic, so the 10th harmonic is missing.

Turn the stiffness down far enough and the bridge moves visibly with the end of the string (watch the spring in the animation). The end of the string is then no longer fixed, so waves reflect from it late and weakened, and that can break up Helmholtz motion that a stiff bridge would allow. A real violin's body has many resonances, not one, so it sounds richer than this model, but the chain is the same: bow to string to bridge to body.

Too little force, too much force

The violinist's problem is to keep the string in Helmholtz motion. Each cell in the map below is a bowing (a bow force and a distance from the bridge, at the current bow speed and bridge), colored by what the string settles into when bowed that way from rest. Click or drag on the map to try a bowing. The green bands under the bowing sliders show the same thing one setting at a time.

Clean Helmholtz motion (one slip per vibration) Not clean (extra slips, or wobbling during the slip) Raucous (sticks too long) Schelleng's maximum force

  • Too much force. The bow grips so hard that when the corner arrives, it can't pull the string free. The corner gets held up, the motion turns irregular, and the result is a harsh, crunching ("raucous") sound. The physicist John Schelleng worked out that the largest force that still allows Helmholtz motion is \[ N_{\max} = \frac{2 Z_0\, v_{\text{bow}}}{\beta\,(\mu_s - \mu_d)}, \] where \(Z_0 = \sqrt{T\mu}\) is the string's characteristic impedance. This is the dashed line in the map.
  • Too little force. The bow can't hold the string against the ripples that travel along it, so the string slips twice or more per vibration. This gives a thin, airy "surface" sound, often with the octave or higher harmonics louder than they should be. The smallest force that works depends on the string's losses, and it grows like \(v_{\text{bow}}/\beta^2\).

Because the minimum grows faster than the maximum as the bow moves toward the bridge, the window between them gets narrower. Close to the bridge you need more force, and you have to control it more precisely. Far from the bridge (sul tasto), a light bow does fine. A faster bow moves the whole window up, so it needs more force to match. Try the "Bow speed" slider and watch the map change.

This model is much simpler than a real violin. The bow touches one point instead of a ribbon of hair, the friction depends only on the sliding speed (real rosin softens as it heats up during each slip), and the string has no stiffness or twisting motion. Its losses are also larger than a real string's, which makes the playable window here narrower than on a real instrument. The basic picture, though, of a sticking-and-slipping string carrying one traveling corner and playable between two force limits, is the one that physicists have confirmed by experiment.

Further reading

  • J. C. Schelleng, "The bowed string and the player," Journal of the Acoustical Society of America 53, 26–41 (1973).
  • M. E. McIntyre, R. T. Schumacher, and J. Woodhouse, "On the oscillations of musical instruments," Journal of the Acoustical Society of America 74, 1325–1345 (1983).
  • J. Woodhouse and P. M. Galluzzo, "The bowed string as we know it today," Acta Acustica united with Acustica 90, 579–589 (2004).