# Resonance

In physics, resonance is the tendency of a system to absorb more energy when the frequency of the 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 electronic circuits.

A resonant object, whether mechanical, acoustic, or electromagnetic, 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.

## Mechanics

A swing set is a simple example of a resonant system that most people have practical experience with. It is a form of pendulum, a type of resonant system. If you excite the system (push the swing) with a period between pushes equal to the inverse of the pendulum's natural frequency, the swing will swing higher and higher, but if you excite it at a different frequency, it will be very difficult. The resonant frequency of a pendulum, the only frequency at which it will vibrate, is given approximately, for small displacements, by the equation

$f = {1 \over 2 \pi} \sqrt {g \over L}$

where g is the acceleration due to gravity (9.8 m/s2 for Earth), and L is the length from the pivot point to the center of mass. (An elliptic integral yields a description for any displacement.) Note that, in this approximation, the frequency does not depend on mass. A swing cannot easily be excited by harmonic frequencies, but can be excited by subharmonics.

Resonance may cause violent swaying motions in improperly constructed bridges. Both the Tacoma Narrows Bridge (nicknamed Galloping Gertie) and the London Millennium Footbridge (nicknamed the Wobbly Bridge) exhibited this problem. A bridge can even be destroyed by its resonance; that is why soldiers are trained not to march in lockstep across a bridge, but rather in breakstep.

Mechanical resonators work by transferring energy repeatedly from kinetic to potential form and back again. In the pendulum, for example, all the energy is stored as gravitational energy (a form of potential energy) when the bob is instantaneously motionless at the top of its swing. This energy is proportional to both the mass of the bob and its height above the lowest point. As the bob descends and picks up speed, its potential energy is gradually converted to kinetic energy (energy of movement), which is proportional to the bob's mass and to the square of its speed. When the bob is at the bottom of its travel, it has maximum kinetic energy and minimum potential energy. The same process then happens in reverse as the bob climbs towards the top of its swing.

Other mechanical systems store potential energy in different forms. For example, a spring/mass system stores energy as tension in the spring, which is ultimately stored as the energy of bonds between atoms.

## Electronic circuits

In an electrical circuit, resonance occurs at a particular frequency when the inductive reactance and the capacitive reactance are of equal magnitude, causing electrical energy to oscillate between the magnetic field of the inductor and the electric field of the capacitor.

Resonance occurs because the collapsing magnetic field of the inductor generates an electric current in its windings that charges the capacitor and the discharging capacitor provides an electric current that builds the magnetic field in the inductor, and the process is repeated. An analogy is a mechanical pendulum.

At resonance, the series impedance of the two elements is at a minimum and the parallel impedance is a maximum. Resonance is used for tuning and filtering, because resonance occurs at a particular frequency for given values of inductance and capacitance. Resonance can be detrimental to the operation of communications circuits by causing unwanted sustained and transient oscillations that may cause noise, signal distortion, and damage to circuit elements.

Since the inductive reactance and the capacitive reactance are of equal magnitude, ωL = 1/ωC, where ω = 2πf, in which f is the resonant frequency in Hertz, L is the inductance in Henry, and C is the capacitance in Farad when standard SI units are used.

Source: Federal Standard 1037C

## Acoustics

Resonance is an important consideration for instrument builders as most acoustic instruments use resonators, such as the strings and body of a violin, the length of tube in a flute, and the shape of a drum membrane.

• Resonance of a string.

Lute (harp, guitar, piano, violin etc.) strings have a fundamental resonant frequency directly related to the length and tension of the string. The wavelength that will create the first resonance on the string is equal to twice the length of the string. This frequency is related to the speed v of a wave traveling down the string by the equation

$f = {v \over 2L}$

where L is the length of the string (for a string fixed at both ends). The speed of a wave through a string or wire is related to its tension T and the mass per unit length ρ:

$v = \sqrt {T \over \rho}$

So the frequency is related to the properties of the string by the equation

$f = {\sqrt {T \over \rho} \over 2 L} = {\sqrt {T \over m / L} \over 2 L}$

where T is the tension, ρ is the mass per unit length, and m is the total mass.

Higher tension and shorter lengths increase the resonant frequency, and vice versa. The string also has a resonance at integer multiples of the fundamental frequency f. It will then also resonate at 2f, 3f, 4f, and so on. When the string is excited with an impulsive function (a finger pluck or a strike by a hammer), the string vibrates at all the frequencies present in the impulse (an impulsive function theoretically contains 'all' frequencies). Those frequencies that are not one of the resonances are quickly filtered out—they are attenuated—and all that is left is the harmonic vibrations that we hear as a musical note.

• Resonance of a tube of air.

The resonance of a tube of air is related to the length of the tube and whether it has closed or open ends. When a wave reaches the end of the tube, part of it will be reflected back into the tube, and part will be transmitted to the outside air. An open end will reflect a wave with no inversion; in other words, a compression wave will be reflected as a compression wave. A closed end will invert the wave that is reflected; in other words, a compression wave will be reflected as a rarefaction wave. Examples of instruments that have both ends open are the flute, saxophone, oboe, and trombone. An example of an instrument that has one closed end and one open end is the clarinet. Vibrating air columns also have resonances at harmonics, like strings. Tubes with both ends open resonate at the frequency

$f = {v \over 2L}$

This is similar to the string formula, except v now becomes the speed of sound in air (which is approximately 340 meters per second at 20 °C and at sea level).

Note however that in practise the exact point at which a sound wave is reflecting at an open end is not perfectly at the end section of the tube. The wave in fact progresses over a small distance outside the tube and the reflection ratio is also not perfectly equal to one. This phenomena is caused by the fact that the open end does not behave like an infinite acoustical impedance. It has a finite value, called radiation impedance, which is dependent on the diameter of the tube, the wavelength and the type of reflection board possibly present around the opening of the tube.

A tube with one end closed will have a resonance of

$f = {v \over 4L}$

This type of tube can only produce odd harmonics, f, 3f, 5f, and so on.

Composers have begun to make resonance the subject of compositions. Alvin Lucier has used acoustic instruments and sine wave generators to explore the resonance of objects large and small in many of his compositions. The complex inharmonic partials of a swell shaped crescendo and decrescendo on a tam tam or other percussion instrument interact with room resonances in James Tenney's Koan: Having Never Written A Note For Percussion. Pauline Oliveros and Stuart Dempster regularly perform in large reverberant spaces such as the two million gallon cistern at Fort Warden, WA, which has a reverb with a 45 second decay.

## Theory

For an 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.

## 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.

## 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).