How does a guitar string make a musical note?

Pull, then let go

Pulling a string stores elastic energy. After release, deformation and motion trade energy. The model begins with zero velocity, just as a held pluck should. Its declared damping gradually removes energy from the retained motion.

Make the vibrating part shorter

An active fret changes the boundary. At the same tension and linear density, halving the vibrating length doubles the ideal fundamental frequency. The section behind that boundary remains part of the real instrument but is outside this single-span calculation.

Turn the tension control

Greater tension gives a higher wave speed in this approximation. Raising tension by a factor of four would double frequency at fixed length and density; the interactive range is smaller. The displayed tension is a model input, not advice for retuning an actual instrument.

One string contains a mixture

The fundamental is harmonic 1. Harmonic 2 is the first overtone, not the second overtone. Each component has its own pattern of nodes and moving regions. A pluck generally excites many components at once.

Position changes the mixture

A midpoint triangular pluck omits even ideal modes. One-fifth omits modes 5, 10, 15 and so on. The same fundamental can remain while the mixture changes, helping explain timbre: the character of a sound.

A repeating force is a new experiment

Ordinary picking gives an initial displacement and then stops forcing. The laboratory driver keeps pushing. Its timing determines how it exchanges work with the moving string. Damping eventually balances average input power.

Switch off and keep the state

Removing the force does not instantly remove the motion. Position and velocity continue, while stored energy dissipates. This decaying continuation is ring-down.

The body is part of the application

In a real acoustic guitar, strings, bridge, plates, cavity and surrounding air interact. A broad moving surface can couple to air differently from a thin string alone. The passive body does not create extra energy; its modes and losses shape the response.

What defines this teaching string?

The open length is 0.65 m and linear density is 0.004 kg/m. Tension is 81.796 N, giving an ideal open fundamental of 110 Hz, the A2 reference under A440 tuning. These authored parameters do not describe a measured brand or the specific source instruments.

Frequency from boundaries

For a uniform flexible string under constant tension, c = √(T/μ), and fₙ = nc/(2L). The fretted span uses L = L₀2^(−fret/12). The model omits stiffness, real fret compensation, and the additional tension caused by pressing a real string down.

A pluck is a projection

For a triangular displacement h at fraction a of length, Bₙ = 2h sin(nπa)/[π²n²a(1−a)]. Negative Bₙ describes relative phase, not negative energy. The complete shape is a sum of Bₙ sin(nπx/L), with each component then following its damped time solution.

Release with the correct initial velocity

The model solves qₙ(t) = Bₙe^(−σₙt)[cos(ωdₙt) + (σₙ/ωdₙ)sin(ωdₙt)], where ωdₙ² = ωₙ² − σₙ². The additional sine term makes the initial velocity zero. Using only an exponentially shrinking cosine would violate that held-release condition.

Loss and truncation are explicit

The free-pluck loss rate is σₙ = 0.8 + 0.02n² per second, an authored profile rather than a fit to wood or string measurements. Thirty-two modes round the held triangle’s corner. At default settings they retain about 98.05% of the full triangle’s initial energy; the curve is not rescaled to hide that difference.

Energy depends on more than displacement

Each mode has generalized mass μL/2 and energy ½M(q̇² + ω²q²). Its loss rate is 2Mσq̇². At equilibrium displacement, kinetic energy can still be large. For the full triangular pluck, E = Th²/[2La(1−a)]. Doubling h quadruples E under the small-motion assumptions.

Work is an integral

Holding force grows with displacement in the linear approximation. Work from rest is ½Fholdh, not the endpoint product Fholdh. For a moving driver, instantaneous power is force times velocity. Source work values with an unresolved force-displacement convention were not imported as the model’s energy.

The separate resonance bench

M = 0.0013 kg, f₀ = 110 Hz and force amplitude 0.01 N define a fundamental-only test. The damping ratio is adjustable from 0.02 to 0.15. This stronger test damping makes settling visible; it is neither the free-pluck loss profile nor a fitted guitar-body Q.

Which response reaches its maximum?

For a constant-amplitude sinusoidal force, steady displacement peaks at f₀√(1−2ζ²) in this underdamped range. Velocity amplitude and average absorbed power peak at f₀. At ζ = 0.05, the displacement peak is 109.725 Hz. A response curve must say which quantity it shows.

A fixed bridge cannot also receive calculated work

The ideal string has a perfectly fixed endpoint, so its endpoint velocity is zero. The same solver cannot claim to transfer a quantified stream of work into a moving soundboard. The 3D construction route is qualitative; predicting the coupled instrument requires the body’s mobility and reciprocal loading.

Different sensors answer different questions

A line-scan kymograph records one location across time. A laser measures movement; a microphone measures pressure; a pickup has its own response. Their waveforms need not match. The real source evidence preserves distinctions between observed, filtered and simulated traces.

Listening is an explicitly chosen signal

The optional sound uses the model’s displacement at x = 0.41L, 48 kHz mono sampling, a fixed gain and short edge fades. The 32 modes remain below the sample band in the selected range. It is not calibrated acoustic pressure or a prediction of this wood’s exact timbre. Its real-time note speed differs from the slowed animation.

Playing with tone

Players change excitation position as well as pitch. The ideal mode tray explains one mechanism; a real hand also changes several contact and timing variables.

Building and studying instruments

Makers work with connected structures, materials and air. Actual mobility and mode measurements can test a model, while listening preferences require their own evidence.

Understanding resonance elsewhere

The same energy questions help explain a swing or a vibrating structure: which mode is excited, what force supplies work, and where does energy leave? Specific systems need their own parameters.

Pull the same ideal string twice as far. What changes?

• Its fundamental must double.
• Its initial energy becomes four times as large.
• It gains four times as many frets.

Halve the vibrating length at the same tension and density.

• The fundamental doubles.
• The fundamental halves.
• Only the color changes.

Which modes are absent from an ideal midpoint triangular pluck?

• Every mode.
• Only the fundamental.
• Even-numbered modes.

Can the midpoint, a node of mode 2, move in the full pluck?

• Yes: other modes can move it.
• No: one mode’s node freezes every component.
• Yes: all nodes must travel down the string.

Why does the continuously driven damped mode settle?

• The driver disappears at a frequency match.
• Average input power balances loss.
• Frequency matching creates free energy.

At steady 110 Hz, force and velocity align. What does that mean?

• The driver can transfer positive work efficiently.
• The string has no acceleration.
• Zero displacement means zero energy.

What does an acoustic guitar’s body do?

• Create energy without receiving any.
• Repeat every string component equally.
• Help couple vibration into air, with its own response.

A microphone trace differs from a laser trace. Must one be wrong?

• Yes: all sensors must show the same waveform.
• No: pressure and string motion are different observables.
• No: a microphone photographs the inside of the wood.

Wave speed, fixed-end modes and frequency controls

Research-lab educational derivation. Its existing drawings and recordings are not copied as product assets.

The acoustic instrument’s bridge, body and radiation

Supports physical context; the local model does not solve the full coupled instrument.

Actual string kymograph and measured departure from ideal motion

Original experiment with 44,100-line/s camera. The 196.36 Hz source case is separate from the authored 110 Hz reference. Figure 3a extracted raster, CC BY 3.0.

Body mobility and the need for coupled modeling

Original measurements and simulation of an unstrung travel-guitar body as a step toward coupling. No figures from this paper are redistributed.

Real bracing, geometry changes and measured top-plate modes

Freely suspended top plates, not complete played guitars. Figure 5 combines actual construction photographs and mobility measurements, CC BY 4.0. Mode agreement is not a perceptual quality score.

Actual measurement method and unstrung-body response

Figures 1–3, CC BY 4.0. Body peaks 104.7 and 111.1 Hz are source observations; unresolved work and Q conventions are not used to calibrate this lesson.

Different measurement observables and sensor loading

Actual laser/pickup signals and separately labeled simulation. Source photographs are a simplified electric-guitar research platform, not the original 3D acoustic guitar. CC BY 4.0.

Quiet optional listening and control over volume

No loudness challenge or instruction to turn sound up until a signal is detectable. Digital volume cannot establish sound pressure at a user’s ear.