# Resonance

Increase of amplitude as damping decreases and frequency approaches resonant frequency of a driven damped simple harmonic oscillator. [1] [2]

In mechanical systems, resonance is a phenomenon that only occurs when the frequency at which a force is periodically applied is equal or nearly equal to one of the natural frequencies of the system on which it acts. This causes the system to oscillate with larger amplitude than when the force is applied at other frequencies.[3]

Frequencies at which the response amplitude is a relative maximum are also known as resonant frequencies or resonance frequencies of the system.[3] Near resonant frequencies, small periodic forces have the ability to produce large amplitude oscillations, due to the storage of vibrational energy.

In other systems, such as electrical or optical, phenomena occur which are described as resonance but depend on the interaction between different aspects of the system, not on an external driver.

For example, electrical resonance 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 discharging capacitor provides an electric current that builds the magnetic field in the inductor. 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 between kinetic and potential, and both systems are forms of simple harmonic oscillators.

In optical cavities, light confined in the cavity reflects back and forth multiple times. This produces standing waves, and only certain patterns and frequencies of radiation are sustained, due to the effects of interference, while the others are suppressed by destructive interference. Once the light enters the cavity, the oscillation is self-sustaining, and there is no external periodic driving action.

Some behavior is mistaken for resonance but instead is a form of self-oscillation, such as aeroelastic flutter, speed wobble, or Hunting oscillation. In these cases, the external energy source does not oscillate, but the components of the system interact with each other in a periodic fashion.[4]

### Overview

Resonance occurs when a system is able to store and easily transfer energy between two or more different storage modes (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 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, distinct, resonant frequencies.

Resonance phenomena occur with all types of vibrations or waves: there is mechanical resonance, acoustic resonance, electromagnetic resonance, nuclear magnetic resonance (NMR), electron spin resonance (ESR) and resonance of quantum wave functions. Resonant systems can be used to generate vibrations of a specific frequency (e.g., musical instruments), or pick out specific frequencies from a complex vibration containing many frequencies (e.g., filters).

The term resonance (from Latin resonantia, 'echo', from resonare, 'resound') originates from the field of acoustics, particularly observed in musical instruments, e.g., when strings started to vibrate and to produce sound without direct excitation by the player.

### Examples

Pushing a person in a swing is a common example of resonance. The loaded swing, a pendulum, has a natural frequency of oscillation, its resonant frequency, and resists being pushed at a faster or slower rate.

A 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 resonant frequency) makes the swing go higher and higher (maximum amplitude), while attempts to push the swing at a faster or slower tempo produce smaller arcs. This is because the energy the swing absorbs is maximized when the pushes match the swing's natural oscillations.

Resonance occurs widely in nature, and is exploited in many manmade devices. It is the mechanism by which virtually all sinusoidal 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:

#### Tacoma Narrows Bridge

The dramatically visible, rhythmic twisting that resulted in the 1940 collapse of "Galloping Gertie", the original Tacoma Narrows Bridge, is mistakenly characterized as an example of resonance phenomenon in certain textbooks.[3] The catastrophic vibrations that destroyed the bridge were not due to simple mechanical resonance, but to a more complicated interaction between the bridge and the winds passing through it—a phenomenon known as aeroelastic flutter, which is a kind of "self-sustaining vibration" as referred to in the nonlinear theory of vibrations. Robert H. Scanlan, father of bridge aerodynamics, has written an article about this misunderstanding.[4]

#### International Space Station

The rocket engines for the International Space Station (ISS) are controlled by an autopilot. Ordinarily, uploaded parameters for controlling the engine control system for the Zvezda module make the rocket engines boost the International Space Station to a higher orbit. The rocket engines are hinge-mounted, and ordinarily the crew doesn't notice the operation. On January 14, 2009, however, the uploaded parameters made the autopilot swing the rocket engines in larger and larger oscillations, at a frequency of 0.5 Hz. These oscillations were captured on video, and lasted for 142 seconds.[5]

### Types of resonance

#### Mechanical and acoustic resonance

School resonating mass experiment

Mechanical resonance is the tendency of a mechanical system to absorb more energy when the frequency of its oscillations matches the system's natural frequency of vibration than it does at other frequencies. It may cause violent swaying motions and even catastrophic failure in improperly constructed structures including bridges, buildings, trains, and aircraft. When designing objects, engineers must ensure the mechanical resonance frequencies of the component parts do not match driving vibrational frequencies of motors or other oscillating parts, a phenomenon known as resonance disaster.

Avoiding resonance disasters is a major concern in every building, tower, and bridge construction project. As a countermeasure, shock mounts can be installed to absorb resonant frequencies and thus dissipate the absorbed energy. The Taipei 101 building relies on a 660-tonne pendulum (730-short-ton)—a tuned mass damper—to cancel resonance. Furthermore, the structure is designed to resonate at a frequency that does not typically occur. Buildings in seismic zones are often constructed to take into account the oscillating frequencies of expected ground motion. In addition, engineers designing objects having engines must ensure that the mechanical resonant frequencies of the component parts do not match driving vibrational frequencies of the motors or other strongly oscillating parts.

Clocks keep time by mechanical resonance in a balance wheel, pendulum, or quartz crystal.

The cadence of runners has been hypothesized to be energetically favorable due to resonance between the elastic energy stored in the lower limb and the mass of the runner.[6]

Acoustic resonance is a branch of mechanical resonance that is concerned with the mechanical vibrations across the frequency range of human hearing, in other words sound. For humans, hearing is normally limited to frequencies between about 20 Hz and 20,000 Hz (20 kHz),[7] Many objects and materials act as resonators with resonant frequencies within this range, and when struck vibrate mechanically, pushing on the surrounding air to create sound waves. This is the source of many percussive sounds we hear.

Acoustic 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, and tension on, a drum membrane.

Like mechanical resonance, acoustic resonance can result in catastrophic failure of the object at resonance. 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.[8]

#### Electrical resonance

Animation illustrating electrical resonance in a tuned circuit, consisting of a capacitor (C) and an inductor (L) connected together. Charge flows back and forth between the capacitor plates through the inductor. Energy oscillates back and forth between the capacitor's electric field ( E) and the inductor's magnetic field ( B).

Electrical resonance occurs in an electric circuit at a particular resonant frequency when the impedance of the circuit is at a minimum in a series circuit or at maximum in a parallel circuit (usually when the transfer function peaks in absolute value). Resonance in circuits are used for both transmitting and receiving wireless communications such as television, cell phones and radio.[9]

#### Optical resonance

An optical cavity, also called an optical resonator, is an arrangement of mirrors that forms a standing wave cavity resonator for light waves. Optical cavities are a major component of lasers, surrounding the gain medium and providing feedback of the laser light. They are also used in optical parametric oscillators and some interferometers. Light confined in the cavity reflects multiple 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 across the cross-section of the beam. Ring resonators and whispering galleries are examples of optical resonators that do not form standing waves.

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

Optical cavities are designed to have a very large Q factor.[10] A beam reflects a large number of times with little attenuation—therefore the frequency line width of the beam is small compared to the frequency of the laser.

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

#### Orbital resonance

In celestial mechanics, an orbital resonance occurs when two orbiting bodies exert a regular, periodic gravitational influence on each other, usually due to their orbital periods being related by a ratio of two small integers. Orbital resonances greatly enhance the mutual gravitational influence of the bodies. In most cases, this results in an unstable interaction, in which the bodies exchange momentum and shift orbits until the resonance no longer exists. Under 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 gaps in the rings of Saturn. 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 nearly everything else around them; this effect is used in the current definition of a planet.

#### Atomic, particle, and molecular resonance

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

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

All nuclei containing odd numbers of nucleons have an intrinsic magnetic moment 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 feature that is exploited in imaging techniques; if a sample is placed in a non-uniform magnetic field then the resonant frequencies of the sample's nuclei depend on where in the field they are located. Therefore, the particle can be located quite precisely by its resonant frequency.

Electron paramagnetic resonance, otherwise known as electron spin resonance (ESR), is a spectroscopic technique similar to NMR, but uses unpaired electrons instead. Materials for which this can be applied are much more limited since the material needs to both have an unpaired spin and be paramagnetic.

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

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

### Theory

"Universal Resonance Curve", a symmetric approximation to the normalized response of a resonant circuit; abscissa values are deviation from center frequency, in units of center frequency divided by 2Q; ordinate is relative amplitude, and phase in cycles; dashed curves compare the range of responses of real two-pole circuits for a Q value of 5; for higher Q values, there is less deviation from the universal curve. Crosses mark the edges of the 3 dB bandwidth (gain 0.707, phase shift 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 about the resonant frequency, as illustrated for the simple harmonic oscillator above. For a lightly damped linear oscillator with a resonance frequency Ω, the intensity of oscillations I when the system is driven with a driving frequency ω is typically approximated by a formula that is symmetric about the resonance frequency:[11]

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

Where the susceptibility ${\displaystyle \chi (\omega )}$ links the amplitude of the oscillator to the driving force in frequency space:[12]

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

The 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 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.

In radio engineering and electronics engineering, this approximate symmetric response is known as the universal resonance curve, a concept introduced by Frederick E. Terman in 1932 to simplify the approximate analysis of radio circuits with a range of center frequencies and Q values.[13][14]

### Resonators

A physical system can have as many resonant 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 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 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 can experience resonance due to vibrations inside them are called resonators, such as organ pipes, vibrating strings, quartz crystals, microwave and laser cavities. Since these can be viewed as being made of many coupled moving parts (such as atoms), they can have correspondingly many resonant 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 roundtrip is 2d. To cause resonance, the phase of a sinusoidal wave after a roundtrip must be equal to the initial phase, so the waves reinforce the oscillation. So the condition for resonance in a resonator is that the roundtrip distance, 2d, be equal to an integer number of wavelengths λ of the wave:

${\displaystyle 2d=N\lambda ,\qquad \qquad N\in \{1,2,3,\dots \}}$

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

${\displaystyle f={\frac {Nv}{2d}}\qquad \qquad N\in \{1,2,3,\dots \}}$

So the resonant 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 oscillation.

### Q factor

The Q factor or quality factor is a dimensionless parameter that describes how under-damped an oscillator or resonator is,[15] or equivalently, characterizes a resonator's bandwidth relative to its center frequency.[16] Higher Q indicates a lower rate of energy loss relative to the stored energy of the oscillator, i.e., the oscillations die out more slowly. A pendulum suspended from a high-quality bearing, oscillating in air, has a high Q, while a pendulum immersed in oil has a low Q. To sustain a system in resonance in constant amplitude by providing power externally, the energy provided in each cycle must be less than the energy stored in the system (i.e., the sum of the potential and kinetic) by a factor of Q/2π. Oscillators with high-quality factors have low damping, which tends to make them ring longer.

Sinusoidally driven resonators having higher Q factors resonate with greater amplitudes (at the resonant frequency) but have a smaller range of frequencies around the frequency at which they resonate. The range of frequencies at which the oscillator resonates is called the bandwidth. Thus, a high-Q tuned circuit in a radio receiver would be more difficult to tune, but would have greater selectivity, it would do a better job of filtering out signals from other stations that lie nearby on the spectrum. High Q oscillators operate over a smaller range of frequencies and are more stable. (See oscillator phase noise.)

The quality factor of oscillators varies substantially from system to system. Systems for which damping is important (such as dampers keeping a door from slamming shut) have Q = 1/2. Clocks, lasers, and other systems that need either strong resonance or high frequency stability need high-quality factors. Tuning forks have quality factors around Q = 1000. The quality factor of atomic clocks and some high-Q lasers can reach as high as 1011[17] and higher.[18]

There are many alternate quantities used by physicists and engineers to describe how damped an oscillator is that are closely related to its quality factor. Important examples include: the damping ratio, relative bandwidth, linewidth, and bandwidth measured in octaves.

### References

1. ^ Katsuhiko Ogata (2005). System Dynamics (4th ed.). University of Minnesota. p. 617.
2. ^ Ajoy Ghatak (2005). Optics, 3E (3rd ed.). Tata McGraw-Hill. p. 6.10. ISBN 978-0-07-058583-6.
3. ^ a b c Resnick and Halliday (1977). Physics (3rd ed.). John Wiley & Sons. p. 324. ISBN 9780471717164. There is a characteristic value of the driving frequency ω" at which the amplitude of oscillation is a maximum. This condition is called resonance and the value of ω" at which resonance occurs is called the resonant frequency.CS1 maint: Uses authors parameter (link)
4. ^ a b K. Yusuf Billah and Robert H. Scanlan (1991). "Resonance, Tacoma Narrows Bridge Failure, and Undergraduate Physics Textbooks" (PDF). American Journal of Physics. 59 (2): 118–124. Bibcode:1991AmJPh..59..118B. doi:10.1119/1.16590. Retrieved 2011-05-29.
5. ^ Oberg, James (4 February 2009). "Shaking on Space Station Rattles NASA". NBC News.
6. ^ Snyder; Farley (2011). "Energetically optimal stride frequency in running: the effects of incline and decline". The Journal of Experimental Biology. 214 (12): 2089–95. doi:10.1242/jeb.053157. PMID 21613526.
7. ^ Harry F. Olson Music, Physics and Engineering. Dover Publications, 1967, pp. 248–249. "Under very favorable conditions most individuals can obtain tonal characteristics as low as 12 Hz."
8. ^ "Breaking Glass with Sound". Instructional Resource Lab. UCLA Physics & Astronomy.
9. ^ "The Physics Of Resonance". Intuitor. Retrieved 10 July 2017.
10. ^ Encyclopedia of Laser Physics and Technology - Q factor, quality factor, cavity, resonator, oscillator, frequency standards
11. ^ A. E. Siegman (1986). Lasers. University Science Books. pp. 105–108. ISBN 978-0-935702-11-8.
12. ^ Aspelmeyer M.; et al. (2014). "Cavity optomechanics". Review of modern physics. p. 1397.
13. ^ Frederick Emmons Terman (1932). Radio Engineering. McGraw-Hill Book Company.
14. ^ William McC. Siebert (1986). Circuits, Signals, and Systems. MIT Press. p. 113. ISBN 978-0-262-19229-3.
15. ^ James H. Harlow (2004). Electric Power Transformer Engineering. CRC Press. pp. 2–216. ISBN 978-0-8493-1704-0.
16. ^ Michael H. Tooley (2006). Electronic Circuits: Fundamentals and Applications. Newnes. pp. 77–78. ISBN 978-0-7506-6923-8.
17. ^ Encyclopedia of Laser Physics and Technology: Q factor
18. ^ Time and Frequency from A to Z: Q to Ra Archived 2008-05-04 at the Wayback Machine