Presets
Playback
String
Tension, mass per length and vibrating length set the pitch.
Pluck and loss
Where and how hard you pluck, and how fast the energy leaks away.
Layers
Model checks
The science
Wave equation and normal modes
A string of length L, tension T and linear density mu obeys y_tt = c^2 y_xx with wave speed c = sqrt(T/mu). Fixed at both ends it supports standing waves sin(n pi x / L) with frequencies f_n = n c / (2L), so the fundamental is f_1 = (1/2L) sqrt(T/mu).
A plucked string
Pulling the string aside by h at a fraction p of its length gives a triangular shape. Its sine series has amplitudes a_n = 2h sin(n pi p) / (n^2 pi^2 p (1 - p)), so the n-th harmonic falls as 1/n^2 and vanishes when the pluck sits on one of its nodes (p = k/n). Plucking near the bridge (small p) gives a bright, thin tone; plucking at the middle gives a mellow one with no even harmonics.
Loss and stiffness
Each mode decays as exp(-gamma_n t) with gamma_n = d0 + d2 n^2: a constant loss plus a loss that grows with harmonic number, which is why a note mellows as it fades. A real steel string is slightly stiff, which stretches the partials to f_n = n f_1 sqrt(1 + B n^2) (inharmonicity B, typically 1e-4 to 1e-3 for guitars).
Simplifications
One polarisation only, rigid fixed ends, no body or soundboard coupling, no coupling between strings, no fret buzz or finger damping. The tone button is a simple additive synthesiser of the same decaying partials. The motion is shown in slow motion with the vertical displacement exaggerated, because a real 3 mm pluck on a 648 mm string would be almost invisible.
What to try
- Click the string at its middle, then near the bridge: compare the harmonic bars in the spectrum graph.
- Set the pluck position to 1/3: the 3rd harmonic disappears.
- Double the tension: the fundamental rises by a factor of about 1.41 (square root of 2).
- Raise the stiffness B and watch the upper partials drift above the exact harmonic positions.