Bowed String: Helmholtz Motion, Stick-Slip and the Schelleng Map (2D)

Setting up the lab bench...

Presets

Live readouts

Graphs

String velocity at the bow (last two cycles)

Force on the bridge (last two cycles)

Harmonic content of the bridge force against an ideal 1/n sawtooth

Regime map (beta against bow force) with the ideal F_max line

What is happening now

Press Play or choose a preset.

Numerical checks

running...

How to use

The default preset gives clean Helmholtz motion. Time runs in slow motion (1/100) so you can follow the corner round the string; choose 1x real time and press Sound: on to hear the bridge force. Space pauses, R resets.

Lower the bow force F_N: the string slips several times per cycle and the tone turns scratchy.
Raise it too far: the tone turns raucous.
Move the bow towards the bridge (small beta): you need much more force.
The last graph is a regime map computed from this simulator; the dashed blue line is the ideal limit F_max.

The science

The string is modelled with d'Alembert's solution: transverse velocity splits into a right-going and a left-going wave, each travelling at c = sqrt(T/mu), so the fundamental is f1 = c/(2L). The simulation is a digital waveguide: four delay lines (nut to bow, bow to nut, bow to bridge, bridge to bow), inverting reflections at the rigid ends, a small loss per round trip and a one-pole low-pass at the bridge that damps high harmonics.

At the bow the waves meet a friction junction. The string velocity there is v_s = v_in + f/(2 Z0) with Z0 = mu c. If the bow can hold the string (|f| <= mu_s F_N) it sticks, v_s = v_b; otherwise it slips and the friction force follows mu(v) = mu_d + (mu_s - mu_d) exp(-|v|/v_s), which is solved by bisection every sample.

Helmholtz motion: stick for (1 - beta) of the period, slip for betav_slip = -v_b (1 - beta)/beta , mean string velocity at the bow = 0Raman limit: F_max = 2 Z0 v_b / ((mu_s - mu_d) beta)

The bridge force is close to a sawtooth with harmonics falling as 1/n, except that harmonics near multiples of 1/beta are suppressed (the bow sits near a node of those modes). The map in the last graph classifies short simulations into Helmholtz motion, several slips per cycle, raucous and no stick-slip. Schelleng's classical diagram has the same shape: a window between a minimum and a maximum bow force that narrows and rises as the bow moves toward the bridge.

Honest simplifications. One polarisation, perfectly flexible string (no stiffness or dispersion), frequency-independent loss except the bridge filter, a rigid bridge (no body feedback), a point bow with an instantaneous Stribeck-type friction law with no temperature or hair-width effects, and a pitch quantised to a whole number of grid steps (the pitch shown is the exact one used). The simulated F_max is lower than the ideal Raman limit because of losses and the finite Stribeck slope. Audio is the bridge force passed through three generic body-like resonances; it is not a model of a real violin.

In-page checks (reference run at 440 Hz, beta 0.12, v_b 0.2 m/s, F_N 0.6 N): pitch from stick-slip events against the grid frequency, stuck fraction against 1 - beta, mean string velocity against 0, mean slip velocity against -v_b(1 - beta)/beta.