String Vibration Modes (2D): Harmonics, Pluck Position, Inharmonicity and Melde Resonance

Presets

Live readouts

Graphs

Spectrum: amplitude of each harmonic mode relative to the strongest (dB)

Pickup signal over three periods of the fundamental (mm)

Resonance curve of the driven string: pickup amplitude per coil force (mm) against f / f1

Decay of harmonics 1, 2, 3, 5, 8 (dB against time): high modes die first

Inharmonicity: partial frequency minus n times the ideal fundamental (cents)

Numerical checks

running...

How to use

Pick a preset or drag the string with the mouse and release to pluck it. The picture is slowed down (time-scale slider) and the displacement is exaggerated. Try: (1) pluck at the midpoint and watch the even harmonics disappear from the spectrum; (2) Piano wire to see the harmonics drift sharp (inharmonicity graph); (3) Melde resonance: sweep the drive frequency slider and watch the pickup amplitude jump near f / f1 = 1, 2, 3 and so on; (4) raise the tension and use Tune the string to note; (5) Play sound to hear the modal synthesis of what you see. Space pauses, R re-plucks.

The science

The wave equation. A string of linear density mu under tension T obeys mu y_tt = T y_xx - E I y_xxxx - 2 mu s y_t The first term is the restoring force of tension (waves at speed c = sqrt(T/mu)), the second is the bending stiffness of the wire (I = pi d^4 / 64), the last is damping.

Standing waves. With fixed ends the solutions are modes sin(n pi x / L) with angular frequencies w_n^2 = (T/mu)(n pi / L)^2 + (E I/mu)(n pi / L)^4 = n^2 w_1^2 (1 + B n^2), B = pi^2 E I / (T L^2) For a perfectly flexible string f_n = n c / (2 L): the harmonic series, so the string sounds a clean pitch. A real wire has B > 0, so the partials are stretched sharp, f_n ~ n f_1 (1 + B n^2 / 2). This is why piano tuners "stretch" the octaves.

How hard each mode is excited. Pulling the string into a triangle of height h at x0 and releasing it gives the mode amplitudes a_n = 2 h L^2 sin(n pi x0 / L) / (n^2 pi^2 x0 (L - x0)) so the harmonics fall off as 1/n^2, and any harmonic with a node at the pluck point (n x0 / L an integer) is absent: pluck at the middle and the even harmonics vanish; pluck near the bridge and you get a bright, nasal tone. A hammer strike (initial velocity) gives amplitudes v_n / w_n, falling off more slowly.

Damping. Each mode decays as exp(-s_n t) with s_n = s_0 (1 + g (n^2 - 1)): air drag and internal friction hit short wavelengths harder, so the tone gets mellower as it fades.

Melde's experiment (driven string). An electromagnet driving the string at xd with force F cos(W t) gives each mode the steady amplitude y_n = (2F / mu L) sin(n pi xd / L) / sqrt((w_n^2 - W^2)^2 + 4 s_n^2 W^2) which peaks when W = w_n; the string then shows n loops. A coil placed at a node of a mode cannot excite it.

Honest simplifications. (1) Small displacements only: the tension is constant, so there is no pitch glide after a hard pluck. (2) One polarisation (vertical); a real string also whirls and has two slightly split polarisations. (3) The bridge and body are perfectly rigid, so there is no coupling to a soundboard and no beating between strings. (4) Damping is a simple two-parameter model, not a fit to a particular instrument. (5) The picture is slowed down (time scale slider) and displacements are exaggerated by the stated factor. (6) The simulation keeps 48 modes; the tail above that is negligible for the amplitudes shown.

Checks run in the page. (a) An independent finite-difference solution of the full PDE (240 cells, leapfrog) against the modal sum at four instants and two points; (b) the energy stored in the plucked triangle against T h^2 L / (2 x0 (L - x0)); (c) the missing harmonics when plucked at L/3; (d) the frequency of mode 6 of a stiff steel wire measured in the finite-difference run against the formula with inharmonicity; (e) the live integrator driven on resonance against the analytic steady-state amplitude.