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Oscillayne

The Harmonic Pulse / Waveform Resonance Physics

\[\ddot x + \gamma\dot x + \omega_0^{2}x = \frac{F_0}{m}\cos\omega t\]
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Size guide S–5XL · inches

Push a swing at just the right moment and it climbs higher each time. Resonance: drive a system at its natural frequency and small pushes build into large oscillations. Oscillayne personifies it. Oscillayne wears it in pixel art on a unisex tee. The physics panel below has the equation, every symbol and the sources.

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Details

Size & fitS–5XL

Unisex heavy cotton (Gildan 5000), classic fit.

SizeWidthLengthSleeve
S182815.1
M202916.5
L223018
XL243119.5
2XL263221
3XL283322.4
4XL303423.7
5XL323525

Dla skali: rozmiar M ma 20 in (50.80 cm) szerokości, czyli około 3.143 × 1034 długości Plancka (lP = 1.616255 × 10−35 m). Miarka krawiecka jest szybsza.

Measurements in inches, ±1.5 in tolerance.

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The physics

A damped oscillator driven by a periodic force responds most strongly when the drive is close to its natural frequency. The less damping there is, the taller and narrower that resonance peak becomes, which is how small, well-timed pushes build up large swings.

Oscillayne does not push hard. She pushes on time.

Anything that can wobble has a natural frequency: a swing, a wine glass, a bridge deck, the quartz crystal in a watch. The first formula on this page is the standard model of driving one. A mass on a spring, a friction term γ that drains energy, and an outside force pushing back and forth at frequency ω. Wait for the starting jitters to die away and the system settles into moving at the driving frequency, whatever its own preference.

How far it moves is the second formula. Push far below the natural frequency \(\omega_0\) and it simply follows the force, as if only the spring mattered. Push far above and it barely responds, because the mass cannot keep up. Push near \(\omega_0\) and the first term under the square root nearly vanishes, leaving only friction to limit the motion. There the amplitude exceeds the slow-push response by a factor \(\omega_0/\gamma\), called the quality factor Q. Low-friction oscillators can have a Q in the thousands or far more; a watch's quartz crystal is one, which is why it keeps such steady time.

Resonance also gets blamed for things it did not do. The Tacoma Narrows Bridge tore itself apart in a steady wind on 7 November 1940, four months after it opened, and textbooks long called that resonance. It was not, as K. Yusuf Billah and Robert Scanlan argued in 1991: a steady wind is not a rhythmic push at the bridge's natural frequency. The twisting deck shaped the airflow around it, and the airflow fed the twisting, a self-excited instability engineers call flutter. Resonance needs a rhythm supplied from outside. Flutter makes its own.

Oscillayne waits for the beat. Then she adds a little to it, every time.

Equations

\[\ddot x + \gamma\dot x + \omega_0^{2}x = \frac{F_0}{m}\cos\omega t\]
\[A(\omega) = \frac{F_0/m}{\sqrt{(\omega_0^{2}-\omega^{2})^{2} + \gamma^{2}\omega^{2}}}\]
SymbolMeaningUnit
\(x\) displacement \(\mathrm{m}\)
\(\gamma\) damping rate \(\mathrm{s⁻¹}\)
\(\omega_0\) natural angular frequency \(\mathrm{rad/s}\)
\(F_0\) driving force amplitude \(\mathrm{N}\)
\(m\) mass \(\mathrm{kg}\)
\(\omega\) driving angular frequency \(\mathrm{rad/s}\)
\(A\) steady-state amplitude \(\mathrm{m}\)

Sources

  1. Feynman Lectures on Physics Vol. I Ch. 21: The Harmonic Oscillator
  2. Feynman Lectures on Physics Vol. I Ch. 23: Resonance
  3. Billah & Scanlan (1991) Resonance, Tacoma Narrows bridge failure, and undergraduate physics textbooks, Am. J. Phys. 59, 118
  4. Encyclopaedia Britannica — Tacoma Narrows Bridge

Source Strutt, J. W. (Lord Rayleigh) (1877). The Theory of Sound, Vol. I, Ch. III (vibrating systems with one degree of freedom; forced vibrations). London: Macmillan.

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