A note at your fingertips: sources & model

Pluck a guitar, move a fret and hear the model. Pull a note apart into harmonics, investigate resonance, and open the instrument beside real research photographs.

What supports the explanation?

What this model assumes

1. Original explanatory guitar geometry, not a scanned instrument or a finite-element body model.
2. The uniform flexible fixed-end string assumes small motion and constant tension; it omits stiffness dispersion, torsion, finger contact and bridge loading.
3. The declared mode loss rates are illustrative and not assigned to measured woods or brands.
4. Thirty-two modes approximate the initial triangle; the energy deficit is retained and disclosed.
5. Mode solos inspect one component without rewriting the full-pluck energy account.
6. The single-mode driven test has its own fixed mass, natural frequency and damping, independent of the pluck’s fret and tension.
7. The exploded guitar is disconnected for inspection; strings are hidden while separated.
8. Model sound is a fixed-gain sonification, not a real guitar recording, hearing test or calibrated loudness.
9. Research spectra and photographs keep their actual unstrung-body or suspended-plate conditions. No sound-quality ranking follows from matched modal frequencies.
10. 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.
11. 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.
12. 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.
13. 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.
14. 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.
15. 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.
16. 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.
17. 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.
18. 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.
19. 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.
20. 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.
21. 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.

What has been checked

Analytical reference cases, conservation or transition invariants, finite drawing commands, bounded setup parsing, discovery and route integrity are checked automatically. These checks do not establish anatomical fidelity, learner outcomes or browser/device compatibility. Independent subject review, learner trials, comprehensive accessibility review and browser video encoding checks remain pending.

Each source supports the associated claim. Sources do not certify this implementation or its visuals.

About the cover illustration

Original acoustic guitar with computed selected-string motion and movable inspection assembly. Not a manufacturer scan or a calculated soundboard mode.