Resonance

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

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In physics, resonance is the tendency of a system to oscillate with high amplitude when excited by energy at a certain frequency. This frequency is known as the system's natural frequency of vibration, resonant frequency, or eigenfrequency.

[edit] Examples

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, resonance in electrical circuits and the shattering of crystal glasses when exposed to an acoustic note of appropriate pitch and strength.

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.

See also: center frequency

[edit] 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:

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

[edit] 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.

[edit] Quantum field theory

In quantum field theory, most particles are unstable particles, i.e., they decay into sets of lighter particles. If the decay is fast, the mass of the particle is not sharply defined. Such particles are usually called resonances. Typical of a resonance is the decay into a continuum of states, i.e., the center-of-mass energy of the decay products, or daughter particles, vary. The energy dependence of such a resonance is described by the relativistic Breit-Wigner distribution, in the simplest case. In this case the "driving frequency" corresponds to the energy with which the resonance is produced, the "resonant frequency" corresponds to the unstable particle's mass, and the linewidth Γ of the resonance corresponds to the inverse of the lifetime τ of the particle, Γ = 1 / τ.

[edit] 'Old Tacoma Narrows' bridge failure

The Old Tacoma Narrows Bridge has been popularized in physics textbooks 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[1].

[edit] See also

[edit] Reference

  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)

[edit] External links

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