Resonance

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Increase of amplitude according to the reduction of damping and the frequency approaches the resonant frequency of a simple dampened harmonic oscillator driven.

In physics, resonance describes the phenomenon of amplitude increase that occurs when the frequency of a periodically applied force (or a Fourier component thereof) is equal to or close to a natural frequency of the system in which it works. When an oscillating force is applied at a resonant frequency to a dynamic system, the system oscillates at a higher amplitude than when the same force is applied at another, non-resonant frequency.

Frequencies at which the response amplitude is a relative maximum are also known as the resonant frequencies or resonant frequencies of the system. Small periodic forces that are close to a resonant frequency of the system have the ability to produce oscillations of large amplitudes in it due to the storage of vibratory energy.

Resonance phenomena occur with all types of vibrations or waves: there are mechanical resonance, acoustic resonance, electromagnetic resonance, nuclear magnetic resonance (NMR), electron spin resonance (ESR) and quantum wave function resonance. Resonant systems can be used to generate vibrations of a specific frequency (for example, musical instruments), or to choose specific frequencies from a complex vibration that contains many frequencies (for example, filters).

The term resonance (from the Latin resonantia, 'echo', from resonare, "to resonate") originates from the field of acoustics, particularly the & #34;sympathetic resonance" observed in musical instruments, for example, when a string begins to vibrate and produces a sound after a different string has been struck. Another example can be electrical resonance, it occurs in a circuit with capacitors and inductors because the collapsing magnetic field of the inductor generates an electric current in its windings that charges the capacitor, and then the discharge of the capacitor provides an electric current that generates a magnetic field. in the inducer. Once the circuit is charged, the oscillation is self-sustaining, and there is no external periodic driving action. This is analogous to a mechanical pendulum, where mechanical energy is converted back and forth from kinetic to potential, and both systems are forms of simple harmonic oscillators.

Overview

Resonance occurs when a system is capable of easily storing and transferring energy between two or more different forms of storage (such as kinetic energy and potential energy in the case of a simple pendulum). However, there are some losses from cycle to cycle, called damping. When the damping is small, the resonant frequency is approximately equal to the natural frequency of the system, which is a frequency of unforced vibrations. Some systems have multiple resonant frequencies.

Examples

Pushing a person on a swing is a common example of resonance. The charged swing, is a pendulum, has a natural frequency of oscillation, resonance frequency, and resists being pushed into a faster or slower index.

A common example is a playground swing, which acts like a pendulum. By pushing a person on a swing in sync with the swing's natural interval (its resonance frequency), the swing gets higher and higher (maximum amplitude), while attempts to push the swing at a faster or faster “tempo” slow cause the arcs to be smaller. This is because the energy absorbed by the swing is maximized when the pushes are matched with the natural oscillations of the swing.

Resonance occurs widely in nature and is exploited in many man-made devices. It is the mechanism by which virtually all sine waves and vibrations are generated. 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 of resonance:

The timing mechanisms of modern watches, for example the steering wheel within a mechanical clock and quartz on a quartz clock.
The resonance of the tides in Fundy Bay.
Acoustic resonance of musical instruments and the human vocal tract.
The breaking of a glass cup exposed to the correct musical tone (their resonance frequency).
Frictional idiosphones, as well as glass objects (glass, bottle, glass), vibrate by rubbing their edge with the tip of the finger.
The electrical resonance of tuned circuits on radios and televisions that allows to receive radio frequencies.
The creation of light coherent by optical resonance in a laser cavity.
The orbital resonance that exemplify some moons of the gas giants of the solar system.
The material resonances in atomic scale are the basis of several spectroscopic techniques used in the physics of the condensed matter.
Electron spin resonance.
Mössbauer effect.
Nuclear magnetic resonance.

The Tacoma Narrows Bridge

Main article: Tacoma Bridge (1940)

The dramatically visible and rhythmic torsion that resulted in the collapse of the Tacoma Narrows Bridge in 1940 is mischaracterized as an example of resonance phenomena in certain texts. The catastrophic vibrations that destroyed the bridge were not due to mechanical resonance simple, but to the interaction between the bridge and the wind that crossed it, producing a phenomenon known as fluttering or flameo, which periodically pushed the bridge, causing movement like a "self-supported oscillation". Robert H. Scanlan, the father of bridge aerodynamics, wrote an article about this misunderstanding.

The International Space Station

The rocket engines for the International Space Station (ISS) are controlled by autopilot. Normally, the parameters installed to control the engine control system for the Zvezda module cause the rocket engines to propel the International Space Station into a higher orbit. The rocket motors are mounted on hinges, the crew are usually unaware of the operation. However on January 14, 2009, the uploaded parameters caused the autopilot to oscillate the rocket motors in ever-widening oscillations, at a frequency of 0.5 Hz. These oscillations were captured on video and lasted 142 seconds.

Resonance in linear systems

Resonance manifests itself in various linear and non-linear systems as oscillations around an equilibrium point. When the system is driven by an external sinusoidal input, an output of the system can oscillate in response. The ratio between the amplitude of the stable output oscillations and the input oscillations is called the gain, and the gain can be a function of the frequency of the external sinusoidal input. Peaks in gain at certain frequencies correspond to resonances, where the amplitude of the output oscillations are disproportionately long.

Since most linear and nonlinear systems are modeled as harmonic oscillators near their equilibrium, this section begins with the derivative of the resonant frequency of a damped harmonic oscillator. The section continues with a use of an RLC circuit to illustrate the connections between resonance and the transfer function of a system, frequency response, poles, and zeros. Starting with the example of the RLC circuit, the section generalizes this relationship for higher-order linear systems with multiple inputs and outputs.

Resonance of a damped and driven oscillator

Main article: Resonance#Simple Harmonic Movement

Considering a mass attached to a spring that is driven by an external applied sinusoidal force in which the system damps. Newton's second law takes the form of:

${displaystyle m{frac {mathrm {d} ^{2}x}{mathrm {d} t^{2}}}=F_{0}sin(omega t)-kx-c{frac {mathrm {d} x}{mathrm {d} t}},}$

(1)

Where m is the mass, x the displacement of the mass from the equilibrium point, F₀ the amplitude ω is the angular frequency, k is the spring constant and c is the coefficient of viscosity. This can take the form of:

${displaystyle {frac {mathrm {d} ^{2}x}{mathrm {d} t^{2}}}+2zeta omega _{0}{frac {mathrm {d} x}{mathrm {d} t}}+omega _{0}^{2}x={frac {F_{0}}{m}}sin(omega t),}$

(2)

Where

{displaystyle omega _{0}={sqrt {frac {k}{m}}}}

is called Angular Speed or Angular Frequency.

{displaystyle zeta ={frac {c}{2{sqrt {mk}}}}}

is called a buffer factor.

Elsewhere ω₀ is also known as resonance frequency. However, as shown below, when we analyze the oscillations of the displacement x(t), the resonant frequency is close to but not equal to ω₀. The example of an RLC circuit in the next section provides examples of different resonant frequencies for the same system.

The general solution of the equation (2) is the sum of a transient solution that depends on the initial conditions and a Steady State steady-state solution that is independent of the initial conditions and depends only of the amplitude F₀, external angular frequency ω, the natural frequency ω₀, and the damping factor ζ. The stationary state decays in a relatively short period of time, so to study the resonance it is enough to consider the solution in stationary state. It is possible to write the steady-state solution for x(t) as a function proportional to the driving force with a change of Phase (wave) φ,

${displaystyle x(t)={frac {F_{0}}{m{sqrt {left(2omega omega _{0}zeta right)^{2}+(omega _{0}^{2}-omega ^{2})^{2}}}}}sin(omega t+varphi),}$

(3)

Where

{displaystyle varphi =arctan left({frac {2omega omega _{0}zeta }{omega ^{2}-omega _{0}^{2}}}right)+npi.}

The value of the phase is normally between -180° and 0°, which is why it represents a delay or offset for the positive and negative values of the arctan argument.

Variation in stationary status of amplitude and frequency

{displaystyle omega /omega _{0}}

and damping

zeta

or buffer factor of a Simple Arminc Oscillator

Resonance occurs when, at certain frequencies of conduction, the steady-state amplitude of x(t) is large compared to its amplitude at other frequencies. For the mass in a spring, resonance physically corresponds to oscillations of the mass that have large displacements from the equilibrium position of the spring at certain firing frequencies. Observing the amplitude of x(t) as a function of the frequency ω, the amplitude is maximum at the natural frequency.

${displaystyle omega _{r}=omega _{0}{sqrt {1-2zeta ^{2}}}.}$

ω_r is the resonant frequency for this system. Again, note that the resonant frequency is not equal to the natural angular frequency ω₀ of the oscillator. They are proportional, and if the damping factor reaches zero they are equal, but if it is not zero they are not the same frequency. As shown in the figure, resonance can also occur at other frequencies close to the resonance frequency, including ω₀, but the maximum response is the resonance frequency.

Note that ω_r is real and different from zero if $<math alttext="{displaystyle zeta γ γ .1/2{displaystyle zeta \leq1/{sqrt {2}}}} <img alt="{displaystyle zeta$ , therefore, this system can only resonate when the harmonic oscillator is significantly damped. For systems with a very small buffer relationship and a driving frequency close to the resonance frequency, stable state oscillations can become very large.

Resonance of a pendulum

For other damped controlled harmonic oscillators whose equations of motion do not look exactly like the mass in a spring example, the resonant frequency remains

{displaystyle omega _{r}=omega _{0}{sqrt {1-2zeta ^{2}}},}

But the definitions of ω₀ and ζ change depending on the physics of the system. For a pendulum of length l and a small angular displacement θ <15° Equation (1) becomes:

{displaystyle ml{frac {mathrm {d} ^{2}theta }{mathrm {d} t^{2}}}=F_{0}sin(omega t)-mgtheta -cl{frac {mathrm {d} theta }{mathrm {d} t}}}

and therefore

{displaystyle omega _{0}={sqrt {frac {g}{l}}},}

{displaystyle zeta ={frac {c}{2m}}{sqrt {frac {l}{g}}}.}

Transfer function, frequency response and resonance for a series RLC circuit

Voltage resonance across an inductor

The resonant frequency does not always have to take the form given in the previous examples. Suppose that for the RLC circuit, the output voltage of interest is the voltage across the inductor. As shown above, in the Laplace domain, the voltage across the inductor is

{displaystyle V_{salida}(s)=sLI(s),}

{displaystyle V_{salida}(s)={frac {s^{2}}{s^{2}+{frac {R}{L}}s+{frac {1}{LC}}}}V_{in}(s),}

{displaystyle V_{salida}(s)={frac {s^{2}}{s^{2}+2zeta omega _{0}s+omega _{0}^{2}}}V_{entrada}(s),}

using the same definitions for ω ₀ and ζ as in the previous example. The transfer function between V _input (s) and this new V _output (s) through the inductor is

{displaystyle H(s)={frac {s^{2}}{s^{2}+2zeta omega _{0}s+omega _{0}^{2}}}.}

Notice that this transfer function has the same poles as the transfer function in the previous example, but also has two zeros in the numerator at s = 0. Evaluating H (s) along the imaginary axis, its gain becomes

{displaystyle G(omega)={frac {omega ^{2}}{sqrt {left(2omega omega _{0}zeta right)^{2}+(omega _{0}^{2}-omega ^{2})^{2}}}}.}

Compared to the gain in equation (6) which uses the capacitor voltage as output, this gain has a factor of ω ² in the numerator and will therefore have a different resonant frequency that maximizes gain. That frequency is

{displaystyle omega _{r}={frac {omega _{0}}{sqrt {1-2zeta ^{2}}}},}

So, for the same RLC circuit but with the voltage across the inductor as the output, the resonant frequency is now greater than the natural frequency, although it still tends toward the natural frequency as the damping ratio goes to zero. That the same circuit can have different resonant frequencies for different output options is not contradictory. As shown in equation (4), the voltage drop in the circuit is divided among the three elements of the circuit, with each element having a different dynamic. Capacitor voltage grows slowly by integrating current over time and is therefore more sensitive at lower frequencies, whereas inductor voltage grows when current changes rapidly and is therefore more sensitive to higher frequencies. While the circuit as a whole has a natural frequency at which it tends to oscillate, the different dynamics of each element in the circuit cause each element to resonate at a slightly different frequency.

Voltage resonance across a resistor

Assume the output voltage of interest is the voltage across the resistor. In Laplace's domain, the voltage across the resistor is

{displaystyle V_{salida}(s)=RI(s),}

{displaystyle V_{salida}(s)={frac {Rs}{L(s^{2}+{frac {R}{L}}s+{frac {1}{LC}})}}V_{entrada}(s),}

and using the same natural frequency and damping ratio as in the capacitor example, the transfer function is

{displaystyle H(s)={frac {2zeta omega _{0}s}{s^{2}+2zeta omega _{0}s+omega _{0}^{2}}}.}

Notice that this transfer function also has the same poles as the RLC circuit examples above, but only has a zero in the numerator at s = 0. For this transfer function, its gain is

{displaystyle G(omega)={frac {2zeta omega _{0}omega }{sqrt {left(2omega omega _{0}zeta right)^{2}+(omega _{0}^{2}-omega ^{2})^{2}}}}.}

The resonant frequency that maximizes this gain is

{displaystyle omega _{r}=omega _{0},}

and the gain is one at this frequency, so the voltage across the resistor resonates at the natural frequency of the circuit and at this frequency the amplitude of the voltage across the resistor is equal to the amplitude of the input voltage.

Types of resonance

Mechanical and acoustic resonance

Main article: Mechanical resonance

School resonating mass experiment

Mechanical resonance is the tendency of a mechanical system to absorb more energy, when the frequency of its oscillations coincides with the natural frequency of vibration of the system than it does at other frequencies. It can cause violent rocking movements and even catastrophic failures in improperly built structures, including bridges, buildings, trains, and airplanes. When designing objects, engineers must ensure that the mechanical resonance frequencies of component parts do not match the vibrational frequencies of motors or other oscillating parts, a phenomenon known as a resonance disaster.

Avoiding resonance disasters is a major concern in every construction project, tower and bridge construction. As a countermeasure, a floating mount can be installed to absorb resonant frequencies and thus dissipate the absorbed energy. The Taipei 101 building relies on a 660 tonne (727.5 ST; 660,000 kg)—a mass damper—to cancel resonance. Furthermore, the structure is designed to resonate at a frequency that would not normally occur. Buildings in seismic zones are often built to account for the oscillating frequencies of expected ground motion. In addition, engineering design objects that have motors must ensure that the mechanical resonant frequencies of component parts do not coincide with the vibrational frequencies of motors or other strongly oscillating parts.

Clock keeps time by mechanical resonance in a balance wheel, pendulum, or quartz crystal.

It has been hypothesized that the cadence of runners is energetically favorable due to the resonance between the elastic energy stored in the lower extremity and the mass of the runner.

Acoustic resonance is a branch of mechanical resonance that deals with mechanical vibrations across the frequency range of the human ear, in other words sound. For humans, hearing is normally limited to frequencies between approximately 20 Hz and 20,000 Hz (20 kHz). Many objects and materials act as resonators with resonant frequencies within this range, and when struck they vibrate mechanically, pushing up the surrounding air. to create sound. This is the source of many drum sounds we hear.

Acoustic resonance is an important consideration for instrument makers, since most instruments use a resonator, such as the strings and body of a violin, the length of the tube in a flute, and the shape and tension of a flute. drum membrane.

Like mechanical resonance, acoustic resonance can cause catastrophic failure of the resonant object. The classic example of this is breaking a wine glass with sound at the precise resonant frequency of the glass, although this is difficult in practice.

Electrical Resonance

Main article: Electrical resonance

Animation that illustrates the electrical resonance in a LC circuit, which consists of a condenser (C) and an inducer (L) connected to each other. The load flows from one side to the other between the capacitor plates through the inducer. The energy oscillates from one side to the other between the electric field capacitor (

E

) and the inducing magnetic field (

B

Electrical resonance occurs in an electrical circuit at a particular resonant frequency when the impedance of the circuit is minimal in a series circuit or in a parallel circuit (usually when the transfer function reaches its absolute maximum value). Resonance in circuits is used to transmit and receive wireless communications, such as television, cell phones, and radio.

Optical Resonance

Main article: Optical cavity

An optical cavity, also called an "optical resonator," is a mirror arrangement that forms a standing wave resonator for light. Optical cavities are a major component of the active medium, they surround the gain medium and provide feedback of the laser light. They are also used in optical parametric oscillators and some interferometers. The light confined in the cavity is reflected several times producing standing waves for certain resonant frequencies. The standing wave patterns produced are called “modes”. Longitudinal modes differ only in frequency while transverse modes differ for different frequencies and have different intensity patterns in the cross section of the beam. The Ring and Whispering Gallery Resonators are examples of optical resonators that do not form standing waves.

The different types of resonators are distinguished by the focal lengths of the two mirrors and the distance between them; Flat mirrors are not often used due to the difficulty in aligning them precisely. The geometry (type of resonator) must be chosen so that the beam remains stable, that is, the size of the beam does not continue to grow with each reflection. Resonator types are also designed to meet other criteria, such as minimal beam waist, or having no focal point (and therefore intense light at that point) within the cavity.

Optical cavities are designed to have a Q factor. A beam reflects a large number of times with little attenuation - therefore the frequency linewidth of the beam is small in comparison with the laser frequency.

Additional optical resonances are guided mode resonances and surface plasmon resonances, which result in anomalous reflection and high evanescent fields at the resonance. In this case, the resonant modes are guided modes of a waveguide or surface plasmon modes of a dielectric-metal interface. These modes are generally driven by a lower wavelength grating.

Orbital Resonance

Main article: Orbital resonance

In celestial mechanics, orbital resonance occurs when two orbiting bodies exert a periodic gravitational influence on each other, usually because their orbital period is related by a ratio of two small integers. Orbital resonances greatly enhance the mutual gravitational influence of bodies. In most cases, this results in an "unsettled" interaction, in which the bodies exchange momentum and change orbits until the resonance no longer exists. In some circumstances, a resonant system can be stable and self-correcting, so that the bodies remain in resonance. Examples are the 1:2:4 resonance of Jupiter's moons Ganymede, Europa, and Io, and the 2:3 resonance between Pluto and Neptune. Unstable resonances with Saturn's inner moons give rise to holes in Saturn's rings. The special case of 1:1 resonance (between bodies with similar orbital radii) causes large Solar System bodies to clear the neighborhood around their orbits by ejecting almost everything else around them. This effect is used in the current definition of a planet.

Atomic, particle and molecular resonance

Main articles: Nuclear MRI and Resonance (particle physics).

NMR Magnet at HWB-NMR, Birmingham, UK. In its strong field 21.2 - tesla, the proton resonance is in 900 MHz.

Nuclear magnetic resonance (NMR) is the name given to a physical resonance phenomenon that involves the observation of quantum mechanical magnetic properties of an atom ic nucleus in the presence of an applied external magnetic field. Many scientific techniques exploit NMR phenomena to study molecular physics, crystal, and non-crystalline materials through nuclear magnetic resonance spectroscopy. NMR is also commonly used in advanced medical imaging techniques, such as magnetic resonance imaging (MRI).

All nuclei containing odd numbers of nucleons have intrinsic magnetic momentum and angular momentum. A key feature of NMR is that the resonant frequency of a particular substance is directly proportional to the strength of the applied magnetic field. It is this characteristic that is exploited in imaging techniques; If a sample is placed in a non-uniform magnetic field, the resonant frequencies of the nuclei in the sample depend on where in the field they are. Therefore, the particle can be located quite accurately by its resonant frequency.

Electronic paramagnetic resonance, also known as "electron spin resonance" (ESR), is a spectroscopic technique similar to NMR, but it uses unpaired electrons. The materials for which it can be applied are much more limited since the material must have unpaired spin and be paramagnetic.

The Mössbauer effect is the resonant and recoil-free emission and absorption of gamma-ray photons by solidly bound atoms.

Resonance in particle physics appears in circumstances similar to classical physics at the level of quantum mechanics and quantum field theory. However, they can also be considered unstable particles, with the above formula valid if Γ is decay rate and Ω replaced by the mass of the particle M . In that case, the formula comes from the ] propagator of the particle, with its mass replaced by the complex number M + iΓ. The formula is more related to the disintegration of particles by the optical theorem.

Resonators

A physical system can have as many resonant frequencies as 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, balance wheels, pendulum, and LC tuned circuits have a resonant frequency. Systems with two degrees of freedom, such as coupled pendulums and resonant transformers, can have two resonant frequencies. As the number of coupled harmonic oscillators grows, the time it takes to transfer power from one to the other 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 can experience resonance due to vibrations within them are called a resonator, such as organ pipe, vibrating string, quartz, microwave, and laser. Since you can see that they are made of many moving, coupled parts (like atoms), they can have many resonant frequencies. The vibrations within them travel like waves, at a roughly constant speed, 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 . To cause resonance, the phase of a sine wave after a round trip must equal the initial phase, so the waves reinforce the oscillation. So, the condition for resonance in a resonator is that the round-trip distance, 2d , is equal to an integer number of wavelengths λ of the wave:

$2d=Nlambdaqquad qquad Nin {1,2,3,dots }$

If the speed of a wave is v , the frequency is: f = v /λ then the resonant frequencies are:

$f={frac {Nv}{2d}}qquad qquad Nin {1,2,3,dots }$ So, the resonant frequencies of the resonators,

called normal modes, they are equally spaced multiples of a lower frequency called the fundamental frequency. Multiples are often called harmonics. There can be several series of resonant frequencies, corresponding to different modes of oscillation.

Q-factor

The quality factor is a dimensionless quantity (see dimensionless magnitude), a parameter that describes a damped motion such as an oscillator or resonator. A higher Q indicates a lower rate of energy loss relative to the oscillator's stored energy, so oscillations will slow to a halt. A high quality ball bearing pendulum suspended oscillating in air has a higher Q, whereas a pendulum immersed in oil would have a lower Q. To keep a system in resonance with constant amplitude it must be supplied with energy externally, the energy supplied in each cycle must be less than the energy stored in the system (i.e. the sum of potential and kinetics) by a factor of Q/2π.Oscillators with high quality factors have low damping, which tends to make them resonate more weather

Resonators with higher Q factors resonate with larger amplitudes (at the resonant frequency) but have a smaller range of frequencies around the frequency at which they resonate. The range of frequencies in which the oscillator resonates is called the bandwidth. Therefore, a high Q tuned circuit in a radio receiver would be more difficult to tune, but would have higher selectivity, and would do a better job of filtering out signals from other stations near the spectrum. high "Q" they operate in a smaller range of frequencies and are more stable. (See phase noise.)

The quality factor of oscillators varies substantially from system to system. In systems for which damping is important (such as dampers that prevent a door from slamming) have Q = 1/2, on the other hand clocks, lasers and other systems that need strong resonance or high frequency stability need high quality factors, for example tuning forks have a quality factor of around Q = 1000, while the quality factor of an atomic clock and a high laser can reach Q =10¹¹ and even they can become larger. Physicists and engineers use many alternative quantities to describe how damped an oscillator is that are closely related to its quality factor.

Universal resonance curve

"Universal resonance curve", is a symmetrical approach to the standardized response of a resonating circuit; Abcisa are deviations from the central frequency, in central frequency units divided by 2Q; ordered is relative amplitude, and the phase in cycles; discontinuous curves compare the range of bipolar circuit responses for a value Q 5; Q higher, there is less deviation from the universal curve. crosses mark the edges of the bandwidth of 3 dB(g. 0.707, phase change 45° or 0.125 cycle).

The exact response of a resonance, especially for frequencies far from the resonant frequency, depends on the details of the physical system and is usually not exactly symmetric with respect to the resonant frequency, as illustrated for the harmonic oscillator. For a slightly damped linear oscillator with resonant frequency Ω, the intensity of the oscillations I when the system operates with a trigger frequency ω is typically approximated by a formula that is symmetric with respect to the resonant frequency:

{displaystyle I(omega)equiv |chi |^{2}propto {frac {1}{(omega -Omega)^{2}+left({frac {Gamma }{2}}right)^{2}}}.}

Where susceptibility ${displaystyle chi (omega)}$ links the amplitude of the oscillator to the driving force in the frequency space:

${displaystyle x(omega)=chi (omega)F(omega)}$

Intensity is defined as the square of the amplitude of the oscillations. This is a Lorentzian function, or Cauchy distribution, and this response is found in many physical situations involving resonant systems. Γ is a parameter that depends on the damping of the oscillator, and is known as the line width of the resonance. Heavily damped oscillators tend to have wide line widths, and respond to a wider range of driving frequencies around the resonant frequency. The line width is inversely proportional to the quality factor, which is a measure of the sharpness of the resonance.

In broadcast engineering and electronic engineering, this approximate symmetric response is known as the universal resonance curve, a concept introduced by Frederick Terman in 1932 to simplify the approximate analysis of radio circuits with a range of center frequencies and Q values.

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