Outside clock
Forced response
A sinusoidal force supplies a frequency.
m x″ + c x′ + k x = F cos(ωt)
Steady amplitudecomputing
Exact peakcomputing
Computing the bounded steady response.
A room full of possible notes
The Note the Object Chooses
An object can hold many natural modes while its driver selects which ones receive energy. Pick an ideal string, bar, or tube, hear its live-computed spectrum under three model drives, then compare bounded forced response with the negative-damping model that captures the essential distinction behind Tacoma Narrows flutter.
The object does not keep one secret note. It keeps a ladder. What reaches your ear begins with the rung that the driver can reach, then passes through radiation and hearing. This first instrument isolates that first step.
First instrument
Give the same object a different hand
Choose an ideal object, choose a model drive, then play the computed modes.
Model drive, same object
Ready. Sound starts only when you press Play.
line = natural mode
height = model excitation
height = model excitation
Ideal object
computing modes
Strongest model mode
computing selection
| mode | frequency | relative excitation |
|---|
The vertical lines stay put when you change the driver. Those are the object's possible modes under the stated ideal boundary conditions. Only their heights move. The synthesizer sends those same computed frequencies to Web Audio, weighted by the selected drive, so the dominant pitch can change without changing the object.
StrikeA brief contact has a broad spectrum. Where it lands controls its projection onto each mode.
BowStick-slip feedback can maintain motion after a strike would have decayed.
AirflowA flow can feed selected acoustic or structural motion. The energy source can be steady even when the response oscillates.
computing the selected object's equation
These frequencies are clean because the objects are not real objects. The string is perfectly flexible, the bar obeys Euler-Bernoulli beam theory, and the tube has no end correction. Material loss, stiffness, end geometry, mounting, and coupling move real frequencies. The excitation weights are deliberate teaching choices, exposed in the table rather than disguised as measurements. Radiation efficiency and the ear still stand between vibration and heard loudness.
The frequency match is not the whole story
A smart objection says that every case is just resonance: find a natural frequency, drive at that frequency, and the object responds. That describes one mechanism. It does not describe a system whose motion changes the force so that the force feeds the motion.
The tempting shortcut
“An oscillator grows because an outside force keeps arriving at its natural frequency.
That is resonance, and that is why Tacoma fell.”
Use the same mass, stiffness, and structural damping in both panels below. On the left, change the frequency of an imposed sinusoidal force. On the right, there is no imposed frequency at all. Change only the speed of a steady flow.
Feedback from motion
Minimal flutter model
Steady flow changes the effective damping.
m x″ + (c − aU) x′ + k x = 0
Effective dampingcomputing
Critical speedcomputing
Computing stability from the damping sign.
The left curve is a frequency response. Damping makes every steady response finite, and the largest displacement occurs near the natural frequency. The right curve is a time history from a tiny disturbance. Below the threshold, its envelope shrinks. Above the threshold, motion-dependent aerodynamic work overwhelms structural damping and the envelope grows. No matching external frequency appears in that equation.
Tacoma belongs on the right, with a warning label
Billah and Scanlan examined the documented Tacoma Narrows behaviour and the scaled-model evidence, and rejected the common textbook account of externally forced resonance. The Billah and Scanlan's account identifies the destructive torsional motion as aeroelastic flutter, a self-excited instability in which the moving deck and the airflow formed a feedback system. Externally forced linear resonance is not supported, but the precise onset and transition mechanism remains debated.
The right panel is not a model of that bridge. Tacoma required torsional degrees of freedom, unsteady aerodynamic forces, mode shapes, and nonlinear behaviour. This one-coordinate equation keeps only the distinction that matters here: a steady flow can make effective damping negative, so oscillation can grow without an outside clock matching a natural frequency.
The check
The browser recomputes every quantitative result in the instruments from the equations and stated inputs. A separately maintained, dependency-free Node verifier mirrors those calculations, and its current output has been compared with the page.
Live mode check
computing all three spectra
Live oscillator check
computing peak response and stability threshold
What is measured, derived, chosen, and open
- Derived inside the page: all mode frequencies, the forced-response curve and exact peak, the critical flow speed, the effective-damping sign, and each growth or decay rate.
- Reference constants: the aluminium elastic modulus and density are representative handbook values. The speed of sound is a stated model value. The cantilever eigenvalue roots come from the standard characteristic equation.
- Free model choices: every object dimension, string tension and density, oscillator coefficient, six-mode truncation, audio duration, and drive-weight profile. The profiles demonstrate selection. They do not predict a particular violin, beam, flute, bow position, strike point, or wind speed.
- Idealizations: linear motion, small amplitude, ideal boundaries, no beam shear or rotary inertia, no string stiffness, no tube end correction, and no acoustic radiation model.
- Not claimed: that the loudest vibrating mode must be the pitch every listener reports, or that the one-coordinate flutter equation reconstructs Tacoma.
- Historical inference: the Tacoma correction rests on the cited engineering analysis and evidence, not on this toy model.
Reproduce the numerical spine with
node research/the-note-the-object-chooses/verify-the-note-the-object-chooses.mjs.
The exact displacement peak is not exactly at ωₙ
For the forced equation, the steady displacement amplitude is A(ω) = F / √((k − mω²)² + (cω)²). With nonzero damping, differentiating that denominator puts the displacement peak slightly below the undamped natural frequency. At the natural frequency, the normalized model gives A = 1/c, but the exact peak is slightly larger. Velocity response, not displacement response, peaks exactly at the natural frequency. The live graph and verifier keep this distinction rather than repeating the common shortcut.
Where the object ends and the ear begins
A normal mode is a pattern of motion with its own frequency. Driving determines how much energy enters each pattern. Sound radiation then depends on how that motion couples to the surrounding air, and hearing adds frequency sensitivity, masking, and pitch inference. This page stops before those latter stages. “Strongest model mode” means strongest synthesized excitation weight, not a universal psychoacoustic verdict.
Sources and the record correction
For the Tacoma correction, see K. Yusuf Billah and Robert H. Scanlan, “Resonance, Tacoma Narrows bridge failure, and undergraduate physics textbooks”. A 2022 fluid-structure simulation supports aeroelastic flutter, while a competing 2022 structural analysis records the continuing dispute over the onset and transition. The fuller citations and model references are recorded in research/the-note-the-object-chooses/README.md.
Three neighbouring questions
This page asks which available modes a driver feeds. Its neighbours ask how a mode draws a shape, whether a whole spectrum reveals a shape, and what motion does to a tone already in flight.