A bow moves at a perfectly steady speed. Yet the string under it vibrates in an exact rhythm, hundreds of times a second. Nobody is tapping out that rhythm. So where does it come from?
Bow hair is rubbed with rosin, and rosin is odd stuff: it grips hard when two surfaces are stuck together, and grips far less once they start sliding. That gap between "static" and "sliding" friction is the whole trick.
The bow grabs the string and drags it sideways. The string's tension pulls back harder and harder. Eventually the pull beats the grip, and the string slips back past the bow. Then the bow catches it again and the stick phase starts over. Drag, stretch, snap back, repeat.
The speed of that cycle is set by the string alone, not by how fast you bow. An open G on a violin vibrates at about 196 times a second, so the bow sticks and slips about 196 times every second. A steady push is turned into a steady beat.
Slow motion, with the swing exaggerated. Drag the bow along the string or use the sliders. The dot under the bow is green while it sticks and red while it slips. The sound button plays a 196 Hz note (violin open G).
In 1862 Hermann von Helmholtz watched a bowed string through his "vibration microscope" and found it does something unexpected. The string is not a smooth arc. It is two straight lines meeting at a sharp corner, a kink, and the kink runs round and round the string.
The kink travels at 2×(string length)×(frequency), so it completes exactly one round trip, bridge to far end and back, per cycle. Its tip traces two parabolic arcs, which you can see as the faint curves in the demo. The string shape you see is just the two straight lines tied to that moving corner.
Here is the neat part. The string slips only when the kink comes back to the bow. Then the kink runs from the bow to the bridge, bounces, and returns to the bow, and that short trip is the whole slip. Right after it, the bow catches the string again.
The kink's trip from the bow to the bridge and back is a fraction of a full lap, and that fraction equals the bow's distance from the bridge as a fraction of the string. Bow one tenth of the way along, and the string slips for one tenth of each cycle and sticks for nine tenths. The "Slipping" readout in the demo is just your slider value.
The sideways swing follows too. In the idealised picture, the largest swing of the string is proportional to the bow speed divided by the bow's distance from the bridge. At the same speed, bowing at a tenth of the way along swings the string five times as far as bowing at the middle. Close to the bridge you get a big sound from a modest stroke.
It comes at a price. The bow has to press harder there to keep the stick-slip going, and the range of force that works gets narrower. Too light and the note turns thin and whistly. Too heavy and it turns to a scratch. A violinist spends years learning that window.
The bridge carries the vibration to the body of the violin, and what the bridge feels is the string's sideways pull, set by the slope of the string where it meets the bridge. Follow the slope as the kink goes round and you get a slow ramp up, then a sudden snap back. That shape is a sawtooth, and the lower graph in the demo draws it.
A sawtooth is rich. It holds the fundamental plus every harmonic above it, each one weaker than the last, which is why a bowed string sounds bright and a little buzzy. Press the sound button and flip between the sawtooth and a pure sine of the same pitch. Same note, very different character. Pressing the bow harder makes the kink sharper and adds still more high harmonics, so the tone brightens.
So the next time you hear a violin hold a long note, remember what is happening: a corner running up and down the string, a sticky bow letting go once per lap, and a steady drag turned into a perfect rhythm. The string is its own metronome.