Acoustic Guitar Physics: Strings, Harmonics and the Body That Sings

A string fixed at both ends

A guitar string is clamped at the nut and the bridge, so only vibration shapes that go to zero at both ends can survive - standing waves. The lowest of these, the fundamental, has one antinode in the middle and sets the note you hear as the pitch. Its frequency depends on exactly three things: the string's length L, its tension T, and its mass per unit length μ (linear density).

f1 = (1 / 2L) * sqrt(T / μ)          fundamental frequency
fn = n * f1                          n-th harmonic, n = 1,2,3…
T  = μ * (2 * L * f1)^2              tension needed for a target pitch

Halve the length (fret the 12th fret) and the frequency doubles - an octave up. This is the same wave equation that governs any string, membrane or air column; a guitar just happens to package it with frets, a body and six of them tuned a fourth or a third apart.

Harmonics, and why pluck position matters

Plucking a string doesn't excite just the fundamental - it excites the fundamental plus a whole ladder of harmonics fn = n·f1 simultaneously, and the initial pluck shape (roughly triangular) determines how much energy lands in each one, via a Fourier series. A harmonic whose node happens to sit exactly at the pluck point gets almost no energy: pluck at exactly 1/5 of the string length and the 5th harmonic, along with its multiples (10th, 15th…), is largely suppressed.

That is the physical reason plucking near the bridge sounds bright and metallic (it excites many high harmonics strongly, since a point near a fixed end is close to a node for very few of the low modes) while plucking over the middle of the string sounds warm and rounded (it favours the fundamental and suppresses odd high harmonics whose nodes cluster there).

Real strings aren't ideal: inharmonicity

The formula above assumes a perfectly flexible string with zero bending stiffness. Real strings resist bending a little, which adds a restoring force on top of tension and makes higher partials ring slightly sharp of the ideal integer multiple:

fn ≈ n * f1 * sqrt(1 + B * n^2)     B = inharmonicity coefficient

B grows with string stiffness and diameter and shrinks with length and tension, which is why thick, short, wound strings (piano bass strings, and to a lesser extent a guitar's wound low strings) show noticeably more inharmonicity than a thin plain nylon treble string.

The body: from a whisper to a room-filling sound

A vibrating string alone moves almost no air - it's too thin to push much of anything. The bridge transmits the string's motion into the much larger soundboard, which couples efficiently to the air and does the actual radiating. The soundboard itself has its own resonant modes (a low "top" resonance and a "back" resonance, typically somewhere around 80-200 Hz depending on the instrument) that reinforce notes near those frequencies and shape the guitar's characteristic timbre.

The sound hole adds one more resonance: the enclosed body of air plus the hole behaves as a Helmholtz resonator, the same physics as blowing across the top of a bottle, typically tuned to boost the guitar's lowest bass notes.

Frequently asked questions

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