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# Resonance

In physics, resonance is the phenomenon of producing large amplitude of vibrations by a small periodic driving force. It is the tendency of a system to oscillate at maximum amplitude at a certain frequency. This frequency is known as the system's resonance frequency (or resonant frequency). When damping is small, the resonance frequency is approximately equal to the natural frequency of the system, which is the frequency of free vibrations. Under resonance condition the energy supplied by the driving force is sufficient enough to overcome friction.

## Examples

One familiar example is a playground swing, which is a crude pendulum. When pushing someone in a swing, pushes that are timed with the correct interval between them (the resonant frequency), will make the swing go higher and higher (maximum amplitude), while attempting to push the swing at a faster or slower rate will result in much smaller arcs.

Resonance occurs in nature, and is exploited in many man-made devices. Some examples:

• acoustic resonances of musical instruments
• the oscillations of the balance wheel in a mechanical watch
• the tidal resonance of the Bay of Fundy
• orbital resonance as exemplified by some moons of the solar system's gas giants
• the resonance of the basilar membrane in the biological transduction of auditory input
• electrical resonance of tuned circuits in radios that allow individual stations to be picked up
• creation of coherent light by optical resonance in a laser cavity
• the shattering of crystal glasses when exposed to a strong enough sound that causes the glass to resonate.

A resonator, whether mechanical, acoustic, or electrical, will probably have more than one resonance frequency (especially harmonics of the strongest resonance). It will be easy to vibrate at those frequencies, and more difficult to vibrate at other frequencies. It will "pick out" its resonance frequency from a complex excitation, such as an impulse or a wideband noise excitation. In effect, it is filtering out all frequencies other than its resonance.

## Theory

For a linear oscillator with a resonance frequency Ω, the intensity of oscillations I when the system is driven with a driving frequency ω is given by:

$I(\omega) \propto \frac{\frac{\Gamma}{2}}{(\omega - \Omega)^2 + \left( \frac{\Gamma}{2} \right)^2 }.$

The intensity is defined as the square of the amplitude of the oscillations. This is a Lorentzian function, and this response is found in many physical situations involving resonant systems. Γ is a parameter dependent on the damping of the oscillator, and is known as the linewidth of the resonance. Heavily damped oscillators tend to have broad linewidths, and respond to a wider range of driving frequencies around the resonance frequency. The linewidth is inversely proportional to the Q factor, which is a measure of the sharpness of the resonance.

## Old Tacoma Narrows bridge failure

The collapse of the Old Tacoma Narrows Bridge, nicknamed Galloping Gertie, in 1940 is sometimes characterized in physics textbooks as a classical example of resonance. This description is misleading, however. It would be more correct to say that the bridge failed due to the action of self-excited forces upon it, largely through a phenomenon known as aeroelastic flutter. Robert H. Scanlan, father of the field of bridge aerodynamics, wrote an article about this misunderstanding[1].

## Resonances in quantum mechanics

In quantum mechanics and quantum field theory resonances may appear in similar circumstances to classical physics. However, they can also be thought of as unstable particles, with the formula above still valid if the Γ is the decay rate and Ω replaced by the particle's mass M. In that case, the formula just comes from the particle's propagator, with its mass replaced by the complex number M + iΓ. The formula is further related to the particle's decay rate by the optical theorem.

## String resonance in music instruments

Main article: String resonance (music)

String resonance occurs on string instruments. Strings or parts of strings may resonate at their fundamental or overtone frequencies when other strings are sounded. For example, an A string at 440 Hz will cause an E string at 330 Hz to resonate, because they share an overtone of 1320 Hz (the third overtone of A and fourth overtone of E).

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• Center frequency
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• Schumann resonance
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• Tuned circuit
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• Sympathetic string

## References

1. ^ K. Billah and R. Scanlan (1991), Resonance, Tacoma Narrows Bridge Failure, and Undergraduate Physics Textbooks, American Journal of Physics, 59(2), 118--124 (PDF)