Resonance

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This article is about resonance in physics. For other senses of this term, see resonance (disambiguation).

In physics, resonance is the tendency of a system to absorb more energy when the frequency of its oscillations matches the system's natural frequency of vibration (its resonant frequency) than it does at other frequencies. Examples are the acoustic resonances of musical instruments, 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, and resonance in electrical circuits.

A resonant object, whether mechanical, acoustic, or electrical, will probably have more than one resonant 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 resonant 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 resonant frequency Ω, the intensity of oscillations I when the system is driven with a driving frequency ω is given by:

<math>I(\omega) \propto \frac{\frac{\Gamma}{2}}{(\omega - \Omega)^2 + \left( \frac{\Gamma}{2} \right)^2 }</math>.

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 resonant frequency. The linewidth is inversely proportional to the Q factor, which is a measure of the sharpness of the resonance.

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Quantum mechanics

A resonance is a quantum state whose mean energy lies above the fragmentation threshold of a system and is associated with:

a pronounced variation of the cross sections if the fragmentation energy lies in the neighbourhood of the energy of the resonance (energy-dependent definition) - The width of this neighbourhood is called the width of the resonance.

an exponential decay of the system when the system has a mean energy close to the resonance energy (time-dependent definition, i.e. in time-resolved spectroscopy) - The lifetime (or inverse of the exponent of the exponential signal) of the resonance is proportional to the inverse of its width. Resonances are usually classified into shape and Feshbach resonances or into Breit-Wigner and Fano resonances.

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Quantum field theory

In quantum field theory, resonance is an unstable particle/bound state. It is characterized by a complex pole off the real line in the S-matrix (which happens to be analytic). A sharp resonance is a resonance with a sharp peak in the S-matrix (which corresponds to a long lifetime compared to the reciprocal of its mass) while a broad resonance is a resonance with a spread out peak (which corresponds to a short lifetime relative to the reciprocal of its mass). If a resonance is too broad, it might not be considered as a particle at all even if it has a complex pole (far from the real line).

If the resonance happens to be a "fundamental particle" (i.e. described by a "fundamental field" of its own), it shows up as a complex pole off the real line in the 2-point connected correlation function (i.e. the propagator).

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Old Tacoma Narrows Bridge failure

The Old Tacoma Narrows Bridge has been popularized in physics text books as a classical example of resonance, but this description is misleading. It is more correct to say that it failed due to the action of self-excited forces, by an aeroelastic phenomenon known as flutter. Robert H. Scanlan, father of the field of bridge aerodynamics, wrote an article about this misunderstanding<ref>K. Billah and R. Scanlan (1991), Resonance, Tacoma Narrows Bridge Failure, and Undergraduate Physics Textbooks, American Journal of Physics, 59(2), 118--124 (PDF)</ref>.

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