PlanetPhysics/Resonance

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One of the first (mechanical) resonance phenomena to have been reported was that discovered by Galileo Galilei in his investigations of (coupled through support) pendulums beginning in 1602.

The very sharp (or marked) increase in amplitude of oscillation of a mechanical, electrical or nuclear [[../SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence/|system]] $O_1$ exposed to a periodic force $F_r(t)$, or another oscillator $O_2$ whose frequency is equal to, or very close to, the natural undamped frequency of the oscillating system $O_1$. The intensity of oscillation is defined as the [[../PiecewiseLinear/|square]] of the amplitude of the oscillations. More precisely, in the case of a linear oscillator with a resonance frequency ${\omega }_0$, the intensity of oscillations $I(\omega )$ when the system $O_1$ is driven with a driving frequency ${\omega }_2$ is given by:

$I({\omega }_2)=const.\frac{\gamma /2}{({\omega }_2-{\omega }_0{)}^2+{\left(\frac{\gamma }{2}\right)}^2}.$

This oscillator intensity is a [[../LebesgueMeasure/|Lorentzian]] [[../Bijective/|function]] which is characteristic of many resonant systems, with $\gamma$ being known as the linewidth of the resonance , or resonance linewidth at half-heigth of the resonance peak which depends on the degree of damping of the oscillator.

Heavily damped oscillators have broad linewidths, and will respond to a wider range of driving frequencies near the resonance frequency. The resonance linewidth $\gamma$ is inversely proportional to the quality factor , or $Q$-factor of the oscillating systems, which is a measure of the sharpness of their resonance. As an example, for [[../SpectralImaging/|NMR]] and rf electrical circuit probes, a high Q-factor is essential for high sensitivity [[../MolecularOrbitals/|detection]] of the signal by the rf probe.

A system is called resonant if it has the property of oscillating by itself (without external coupling) with a maximum amplitude only at a certain frequency or frequencies, known as the resonance frequencies of the system.