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Resonance is a phenomenon in which an external force causes a system to oscillate at an amplitude that is disproportionately large compared to the amplitude of the external force. In linear systems it occurs when the system has one or more resonant frequencies at which it is prone to resonate and the external force is applied at a frequency at or near one of those resonant frequencies. Resonance is an extremely common phenomenon that has been studied and applied across a wide variety physics and engineering disciplines. Many types of systems that oscillate have the potential to resonate.

Definition

Resonance is a phenomenon in which an external force applied at or near the resonant frequency causes a system to oscillate in steady-state with an amplitude that is disproportionately large compared to the amplitude of the oscillations if the same force were applied at other frequencies.^[1]

Resonance occurs when a system has one or more modes of oscillation that are significantly underdamped, and an external input drives that mode of oscillation at or near the frequency that maximizes its steady-state amplitude. The mode of oscillation has an undamped frequency of oscillation or natural frequency. The resonant frequency or driving frequency that maximizes the amplitude of steady-state oscillations will be near that natural frequency. The exact resonant frequency associated with that mode depends on other dynamics of the system and can be determined by finding maxima of the system's gain as a function of frequency or maxima of the amplitude of oscillations as a function of frequency. In nonlinear systems the resonant frequency can depend on a variety of other factors, including the amplitude of the input driving the system.

Child on a Swing and Other Common 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 of resonance is a playground swing, which acts as a type of pendulum. When first set in motion, the child swings with a frequency that depends on the length of the chain, eventually settling back at the bottom as energy is lost to friction. As the child swings, pushing the child in time with the frequency of swinging makes the swing go higher and higher. The child experiences resonance between the pushing and the height of the swinging. Pushing at other frequencies does not produce the same gains. For example, the child could be pushed at double the frequency of swinging by pushing the child forward each time the child passes through the bottom regardless of whether the child is moving forward or backward. This speeds the child up when the child is moving forward and slows the child down when the child is moving backward, with no net effect over many swings.

Resonance is ubiquitous in the physical and biological sciences as well as engineering, with many natural systems exhibiting resonance and many man-made systems making use of resonance. It occurs at the atomic scale and in planetary motions. It occurs in mechanical devices, circuits, and quantum systems. It is the basis for sound in musical instruments, light from lasers, and certain medical imaging technologies. A few other familiar examples of resonance include:

Timekeeping mechanisms of modern clocks and watches, e.g., the balance wheel in a mechanical watch and the quartz crystal in a quartz watch
Acoustic resonances in musical instruments and the human vocal tract
Shattering of a crystal wineglass when exposed to a musical tone of the right pitch
Friction idiophones, such as making a glass object 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
Magnetic resonance imaging to form pictures of the anatomy and the physiological processes of the body

History

Etymology, who first studied it and named it, etc. Nonlinear resonance book had some info

Resonance in nonlinear systems

Some ideas:

Linear systems as linearizations of nonlinear systems around an equilibrium point, then determine resonance as before
First example from Nonlinear Resonances book with resonant frequency that is a function of amplitude
List of other examples of nonlinear resonance

https://www.google.com/books/edition/Nonlinear_Resonances/3fQUCwAAQBAJ?hl=en&gbpv=1&dq=isbn:9783319248868&printsec=frontcover

Mathematical description of standing waves

Existing content on the standing wave page needs better and more illustrations. Go find them!

2D standing wave in a circular bowl

3D standing wave

Expressing arbitrary waves as a linear combination of standing waves

In a system that allows standing waves, an arbitrary waveform traveling through the system can be expressed as a sum or linear combination of the various standing wave modes. In this formulation, the standing wave normal modes form an eigenbasis for the system.

To see this, first return to the general solution of the wave equation. One way to express the general solution to the wave equation is to assume a solution

Wave equation#Plane wave eigenmode should be modified to have a time Fourier transform instead of assuming eigenmode solution.

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Resonance and Musical Instruments

Extend standing wave resonance to musical instruments. Intro physics textbooks give some good examples and connections. Check acoustic resonance page for relevant info.

Waves on a string --> String instruments, pianos
Pressure waves in a pipe --> Woodwind instruments, pipe organs
Drum pitch
Vibrations --> Bells, xylophones

Wind Instrument Frequencies

Normal modes of coupled oscillators

Sympathetic resonance

Examples of Resonance

A bunch of material currently in the existing article gets pushed here with light cleanup.

Notes

^ Physics (3rd ed.). John Wiley & Sons. 1977. 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. {{cite book}}: Unknown parameter |authors= ignored (help)

References

Halliday, David; Resnick, Robert; Walker, Jearl (2005). Fundamentals of Physics (7th ed.). John Wiley & Sons. ISBN 0-471-42959-7.
Serway, Raymond A.; Faughn, Jerry S. (1992). College Physics (3rd ed.). Saunders College Publishing. ISBN 0-03-076377-0.
Streets, J. (2010). "Chapter 16 - Superposition and Standing Waves" (PDF). Department of Physics. PHYS122 Fundamentals of Physics II. University of Maryland. Retrieved August 23, 2020.

[Resnick_and_Halliday-1] Physics (3rd ed.). John Wiley & Sons. 1977. 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. {{cite book}}: Unknown parameter |authors= ignored (help)

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