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

# Extensive Definition

In physics, resonance is the
tendency of a system to oscillate at maximum amplitude at certain frequencies,
known as the system's resonance frequencies (or resonant
frequencies). At these frequencies, even small periodic
driving forces can produce large amplitude vibrations, because the
system stores vibrational energy. When damping is small, the resonance
frequency is approximately equal to the natural
frequency of the system, which is the frequency of free
vibrations. Resonant phenomena occur with all type of vibrations or
waves; mechanical (acoustic), electromagnetic,
and quantum wave
functions. Resonant systems can be used to generate vibrations
of a specific frequency, or pick out specific frequencies from a
complex vibration containing many frequencies.

## Examples

One familiar example is a playground swing, which acts as a pendulum. Pushing a person in a swing in time with the natural interval of the swing (its resonance frequency) will make the swing go higher and higher (maximum amplitude), while attempts to push the swing at a faster or slower tempo will result in smaller arcs. This is because the energy the swing absorbs is maximized when the pushes are at the resonance frequency, while some of this energy is canceled out by the inertial energy of the swing when they are not.Resonance occurs widely in nature, and is
exploited in many man-made devices. Many sounds we hear, such as
when hard objects of metal, glass, or wood are struck, are caused
by brief resonant vibrations in the object. Light and other short
wavelength electromagnetic
radiation is produced by resonance on an atomic scale, such as
electrons in atoms. Other examples are:

- acoustic resonances of musical instruments and our vocal cords
- 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 cochlea of the ear, which enables people to distinguish different frequencies or tones in the sounds they hear.
- 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 musical tone of the right pitch (its resonance frequency).

## 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.

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.

## Resonators

A physical system can have as many resonance
frequencies as it has
degrees of freedom; each degree of freedom can vibrate as a
harmonic
oscillator. Systems with one degree of freedom, such as a mass
on a spring, pendulums,
balance
wheels, and LC tuned
circuits have one resonance frequency. Systems with two degrees
of freedom, such as coupled
pendulums and resonant
transformers can have two resonance frequencies. As the number
of coupled harmonic oscillators grows, the time it takes to
transfer energy from one to the next becomes significant. The
vibrations in them begin to travel through the coupled harmonic
oscillators in waves, from one oscillator to the next.

Extended objects that experience resonance due to
vibrations inside them are called resonators, such as organ pipes,
vibrating
strings, quartz
crystals, microwave cavities, and
laser rods. Since these
can be viewed as being made of millions of coupled moving parts
(such as atoms), they can have millions of resonance frequencies.
The vibrations inside them travel as waves, at an approximately
constant velocity, bouncing back and forth between the sides of the
resonator. If the distance between the sides is d\,, the length of
a round trip is 2d\,. In order to cause resonance, the phase of a
sinusoidal wave after
a round trip has to be equal to the initial phase, so the waves
will reinforce. So the condition for resonance in a resonator is
that the round trip distance, 2d\,, be equal to an integral number
of wavelengths of the wave:

- 2d = N\lambda,\qquad\qquad N \in \

If the velocity of a wave is v\,, the frequency
is f = v / \lambda\, so the resonant frequencies are:

- f = \frac\qquad\qquad N \in \

So the resonance frequencies of resonators,
called normal
modes, are equally spaced multiples of a lowest frequency
called the fundamental
frequency. The multiples are often called overtones. There may be several
such series of resonant frequencies, corresponding to different
modes of vibration.

## 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. The catastrophic vibrations that destroyed the bridge were not due to simple mechanical resonance, but to a more complicated oscillation between the bridge and winds passing through it, known as aeroelastic flutter. Robert H. Scanlan, father of the field of bridge aerodynamics, wrote an article about this misunderstanding.## 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 \Gamma is the decay rate and \Omega 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+i\Gamma. The formula is further related to the particle's decay rate by the optical theorem.## String resonance in music instruments

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).## See also

## References

## External links

- Resonance - a chapter from an online textbook
- Greene, Brian, "Resonance in strings". The Elegant Universe, NOVA (PBS)
- Hyperphysics section on resonance concepts
- A short FAQ on quantum resonances
- Resonance versus resonant (usage of terms)
- Wood and Air Resonance in a Harpsichord
- Java applet demonstrating resonances on a string when the frequency of the driving force is varied
- Breaking glass with sound, including high-speed footage of glass breaking

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