Plucked Guitar String in 3D: Wave Model, Modes and Inharmonicity

What is modelled

One stiff, lossy guitar string between a nut and a saddle. You pluck it with a triangular initial shape and watch the finite-difference wave model run in slow motion above a Blender-modelled guitar. This is the 3D companion of the 2D page "Plucked Guitar String"; both pages share the same solver, presets and numerical checks.

Equations

• Transverse displacement y(x,t) with fixed ends: μ ytt = T yxx − EI yxxxx − 2 μ σ yt, solved on 100 intervals with an explicit second-order scheme at its stability limit.
• Ideal string: modes fn = n f1, f1 = √(T/μ)/(2L). Bending stiffness raises the partials: fn = n f1 √(1 + B n²), B = π² EI/(T L²) (inharmonicity, the piano effect).
• A triangular pluck at p excites mode n with amplitude 2h sin(nπp)/(n²π² p(1−p)): plucking at the middle kills the even modes, plucking near the bridge brightens the sound.
• Frets: the Blender neck is built with xn = L(1 − 2−n/12); the page checks this against the model for the current scale length.

Checks run in the page

• Period of the time-stepped string against f₁.
• Total energy of a lossless stiff string stays constant (time-centred leapfrog invariant).
• The numerical pluck projection matches the closed-form Fourier series.
• Fret and saddle positions measured on the Blender mesh agree with 12-tone equal temperament and the scale length.

Honest simplifications

• One polarisation, small amplitude (linear). The bridge and nut are rigid, so nothing is radiated: the guitar body, string-to-string coupling and frequency-dependent damping are not modelled. The loss rate is one constant per string.
• Displacement is drawn exaggerated (slider) and in slow motion; the real motion is hundreds of times faster and a fraction of a millimetre.
• Only one string is simulated and drawn; the neck and body are a scale model for context. Changing the scale length rescales the drawn neck.
• "Hear it" renders the pickup signal of this same model offline and plays it; it has no body resonance, so it sounds like a bare string on a pickup.

What to try

• Pluck at the middle (p = 0.5): only odd modes remain in graph 1. Pluck near the saddle (p = 0.05): many strong modes.
• Raise the bending stiffness EI and watch the orange partials in graph 3 drift above the ideal grey line.
• Choose "Slack rope": slow, low-frequency, heavily damped motion.