Acoustic Guitar Physics

What is modelled

An acoustic steel-string guitar with a 648 mm (25.5 inch) scale length. A string is plucked a chosen distance from the bridge and then vibrates as a sum of standing-wave modes. The string pulls on the bridge, the bridge drives the thin top plate, and the plate together with the air in the box radiates the sound through two coupled body resonances. The picture shows the string motion in slow motion and exaggerated in size, because a real string moves a few millimetres at 82 to 330 cycles per second.

Equations

• Wave equation of an ideal string: μ ∂²y/∂t² = T ∂²y/∂x², wave speed c = √(T/μ).
• Standing-wave modes of a string of sounding length L: y(x,t) = Σ an sin(nπx/L) cos(ωn t) e−t/τn, fn = n c / (2L). With stiffness fn = n f1 √(1 + B n²) (the upper partials go sharp).
• Triangular pluck of height h at distance d from the bridge: an = 2 h L² sin(nπd/L) / (n²π² d (L − d)). A pluck at a node of mode n contains none of that mode: plucking at the middle removes all even partials.
• Frets are placed so that each one shortens the string by the factor 2−1/12: Ln = L0 2−n/12, giving the equal-tempered semitone.
• Natural harmonic: a light touch at a fraction q of the length forces a node there and keeps only the modes with n q an integer (q = 1/2, 1/3, 1/4 at frets 12, 7 and 5).
• Energy of the string: E = Σ μ ωn² an² L / 4; force on the bridge: F = T ∂y/∂x |x=0 = Σ T (nπ/L) an cos(ωn t).
• Helmholtz resonance of the box: fH = (cair / 2π) √(A / (V Leff)), with soundhole area A, volume V, effective neck length Leff ≈ t + 1.7 r. It couples to the top-plate mode fT; the two coupled modes are f±² = ½[fH² + fT² ± √((fH² − fT²)² + 4 κ² fH² fT²)], which is why a real guitar shows body peaks near 100 and 200 Hz.

Checks run in the page

• A finite-difference simulation of the string (400 cells) oscillates with the period 2L/c.
• The Fourier mode sum used for drawing and sound equals the finite-difference string shape at an arbitrary time.
• Two coupled oscillators for air and plate are integrated numerically and their spectrum peaks fall on the analytic eigenfrequencies.

Honest simplifications

• Linear, planar string motion (no tension modulation, no string-fret buzz, no coupling of the two polarisations). The decay law τn = τ1 / (1 + 0.05 (n−1)²) is a convenient fit, not a measurement. The body is two coupled resonances and a gentle high-frequency roll-off, not a full plate and cavity model; wound strings use a fixed linear density and a common stiffness. The sound is synthesised from the same modes with the body gain applied to each partial, so it is only a sketch of a real guitar tone. The string displacement is drawn up to 30 times larger than reality; the slow-motion factor is applied to the picture only (the sound plays at real speed).

What to try

• Pluck 3 cm from the bridge, then at the middle of the string, and compare the spectrum bars: the bright and the round tone.
• Choose the harmonic over fret 12 and see the node marker and the missing odd partials.
• Press fret 12 and check the pitch is exactly one octave above the open string.
• Switch on the mode lab and step through modes 1 to 6, counting nodes.
• Make the soundhole smaller and the box larger: the lower coupled body mode slides down and the red lines move in the spectrum graph.