Math Is Fun Forum

Jai Ganesh

Administrator

Registered: 2005-06-28

Posts: 46,317

Resonance

Resonance

Gist

Resonance is the state of a system in which an abnormally large vibration is produced in response to an external stimulus, occurring when the frequency of the stimulus is the same, or nearly the same, as the natural vibration frequency of the system.

Summary

Resonance, in physics, is a relatively large selective response of an object or a system that vibrates in step or phase, with an externally applied oscillatory force. Resonance was first investigated in acoustical systems such as musical instruments and the human voice. An example of acoustical resonance is the vibration induced in a violin or piano string of a given pitch when a musical note of the same pitch is sung or played nearby.

The concept of resonance has been extended by analogy to certain mechanical and electrical phenomena. Mechanical resonance, such as that produced in bridges by wind or by marching soldiers, is known to have built up to proportions large enough to be destructive, as in the case of the destruction of the Tacoma Narrows Bridge (q.v.) in 1940. Spacecraft, aircraft, and surface vehicles must be designed so that the vibrations caused by their engines or by their movement through air are kept to a safe minimum.

Resonance in electrical systems is of a somewhat different nature. Its occurrence in frequency-sensitive (alternating-current) circuits makes it possible for communication devices equipped with such circuits to accept signals of certain frequencies while rejecting others. In a television receiver, for example, resonance occurs when the frequency of one of the incoming signals reaching the circuit is near the natural frequency of the circuit, which then responds by absorbing maximum energy from the signal as the current within the circuit surges back and forth in step with the very weak current in the antenna.

A form of resonance somewhat analogous to a certain kind of mechanical resonance has been detected on the nuclear scale. This phenomenon, called magnetic resonance, occurs when atoms or their nuclei respond to the application of various magnetic fields by emitting or absorbing electromagnetic radiation of radio and microwave frequencies. See also magnetic resonance.

Details

Resonance is a phenomenon that occurs when an object or system is subjected to an external force or vibration that matches its natural frequency. When this happens, the object or system absorbs energy from the external force and starts vibrating with a larger amplitude. Resonance can occur in various systems, such as mechanical, electrical, or acoustic systems, and it is often desirable in certain applications, such as musical instruments or radio receivers. However, resonance can also be detrimental, leading to excessive vibrations or even structural failure in some cases.

All systems, including molecular systems and particles, tend to vibrate at a natural frequency depending upon their structure; this frequency is known as a resonant frequency or resonance frequency. When an oscillating force, an external vibration, is applied at a resonant frequency of a dynamic system, object, or particle, the outside vibration will cause the system to oscillate at a higher amplitude (with more force) than when the same force is applied at other, non-resonant frequencies.

The resonant frequencies of a system can be identified when the response to an external vibration creates an amplitude that is a relative maximum within the system. Small periodic forces that are near a resonant frequency of the system have the ability to produce large amplitude oscillations in the system due to the storage of vibrational energy.

Resonance phenomena occur with all types of vibrations or waves: there is mechanical resonance, orbital 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') originated from the field of acoustics, particularly the sympathetic resonance observed in musical instruments, e.g., when one string starts to vibrate and produce sound after a different one is struck.

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.

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 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:

* Timekeeping mechanisms of modern clocks and watches, e.g., the balance wheel in a mechanical watch and the quartz crystal in a quartz watch
* Tidal resonance of the Bay of Fundy
* Acoustic resonances of musical instruments and the human vocal tract
* Shattering of a crystal wineglass when exposed to a musical tone of the right pitch (its resonant frequency)
* Friction idiophones, such as making a glass object (glass, bottle, vase) vibrate by rubbing around its rim with a fingertip
* Electrical resonance of tuned circuits in radios and TVs that allow radio frequencies to be selectively received
* Creation of coherent light by optical resonance in a laser cavity
* Orbital resonance as exemplified by some moons of the Solar System's gas giants
* Material resonances in atomic scale are the basis of several spectroscopic techniques that are used in condensed matter physics
** Electron spin resonance
** Mössbauer effect
** Nuclear magnetic resonance

Linear systems

Resonance manifests itself in many linear and nonlinear systems as oscillations around an equilibrium point. When the system is driven by a sinusoidal external input, a measured output of the system may oscillate in response. The ratio of the amplitude of the output's steady-state oscillations to the input's oscillations is called the gain, and the gain can be a function of the frequency of the sinusoidal external input. Peaks in the gain at certain frequencies correspond to resonances, where the amplitude of the measured output's oscillations are disproportionately large.

Since many linear and nonlinear systems that oscillate are modeled as harmonic oscillators near their equilibria, this section begins with a derivation of the resonant frequency for a driven, damped harmonic oscillator. The section then uses an RLC circuit to illustrate connections between resonance and a system's transfer function, frequency response, poles, and zeroes. Building off the RLC circuit example, the section then generalizes these relationships for higher-order linear systems with multiple inputs and outputs.

---

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.