Resonance - Wikiparency

# Resonance > *For resonance of air in tubes, rooms and cavities, see [[Acoustic_resonance]].* **Resonance** is the large response of a system that is pushed periodically at, or close to, one of its own natural [[Frequency|frequencies]]. A small push repeated in step with the motion adds energy on every cycle, so the swing grows until the energy lost to [[Damping|damping]] each cycle equals the energy fed in. Children on swings, a piano string answering a sung note, a radio tuned to one station and the protons in a hospital scanner all resonate in this sense; so, less happily, do bridges and buildings shaken by feet or earthquakes. How strongly an [[Oscillation|oscillator]] resonates, and how choosy it is about the frequency, comes down to one number, the [[Q_factor|quality factor]] `Q`. For a lightly damped system driven exactly at its natural frequency, the response is `Q` times what a slow push of the same size would produce, and the band of frequencies that excites it well is `f0/Q` wide. The section 3 microsim of the Acoustics portal, *Resonance: the universal response curve*, puts a wine glass beside a loudspeaker: the reader chooses `Q` and the drive level and sweeps the tone through the glass's 660 Hz resonance, watching the gain climb the curve toward an illustrative breaking line. ## Overview Every system that can [[Vibration|vibrate]] has natural frequencies at which it oscillates when disturbed and left alone: one for a mass on a spring, a whole series for a string or an air column. When such a system is driven by a periodic force, it follows the drive frequency, but the amplitude it reaches depends on how close that frequency is to a natural one. Far below, the system moves with the force as if the force were steady; far above, it cannot keep up and barely moves; near the natural frequency, the response rises to a peak. The phenomenon of driving a system at its natural frequency is what a standard textbook calls resonance, and a system being driven there is said to resonate.[^up15-6] Two things set the peak. The first is timing: at resonance the force pushes in step with the velocity, so it does positive work through the whole cycle instead of alternately giving and taking energy back. The second is loss: with nothing to remove energy, the amplitude would grow without limit, so the height of the peak is set by the damping. Less damping gives a taller and narrower peak.[^up15-6] Resonance is therefore both an amplifier and a filter, and engineering exploits either role or fights it. ## Examples The textbook example is a parent pushing a child on a swing. The parent does not run up and shove; small pushes timed to the swing's own period build a large swing over many cycles.[^up15-6] A paddle ball on a rubber band does the same when the hand moves at the ball's natural frequency, and follows the hand without bouncing when the hand moves slowly.[^up15-6] Sound supplies the acoustic cases. With the dampers lifted, the strings of a piano pick out and sing back a note sung loudly at them, because the strings tuned to the voice's frequencies are driven at their natural frequencies; this is the [[Sympathetic_resonance|sympathetic resonance]] that the portal treats as a See-also variant.[^up15-6] A struck [[Tuning_fork|tuning fork]] held over a tube of the right length makes the air column sound, and a singer's tone can set a wine glass ringing, the demonstration the microsim is built around.[^up15-6][^rossing] Hospital scanners use resonance of another kind: in [[Magnetic_resonance_imaging|magnetic resonance imaging]], hydrogen nuclei in a strong magnetic field absorb energy from radio waves tuned to their precession frequency, on the order of 100 MHz.[^up15-6] ## Linear systems In a linear system the response to a sum of forces is the sum of the responses, so the steady response to a sinusoidal drive at each frequency describes the system completely. That frequency-by-frequency description is its [[Frequency_response|frequency response]], and resonances are the frequencies where it peaks. ### The driven, damped harmonic oscillator The model is the [[Harmonic_oscillator|harmonic oscillator]] of section 2, `m x'' + c x' + k x = F0 cos(ωt)`. Written with the frequency ratio `r = f/f0` and `Q = √(mk)/c`, its steady amplitude relative to the static deflection is `A/A_static = 1/√((1 − r²)² + (r/Q)²)`, the equation the microsim computes and prints in its header. At `r = 1` the gain is exactly `Q`; far below, it approaches 1; far above, it falls as `1/r²`. At `r = 2` the glass moves only a third as far as a slow push would move it, a gain of −9.5 dB, whatever `Q` is. The phase of the response swings from in step with the force, through a 90° lag at resonance, to nearly opposite it well above.[^up15-6] For light damping the amplitude peak sits within a fraction of a percent of `f0`; the microsim reports the exact peak, 659.82 Hz for `Q = 30`, a shift of −0.18 Hz. ### RLC series circuits A resistor, inductor and capacitor in series obey the same equation with charge in place of displacement, inductance in place of mass, resistance in place of the damping coefficient and inverse capacitance in place of stiffness. The circuit resonates at `f0 = 1/(2π√(LC))` with `Q = (1/R)√(L/C)`. With `L = 10 mH`, `C = 100 nF` and `R = 10 Ω`, the resonance falls at 5.03 kHz with `Q = 31.6` and a bandwidth of 159 Hz. A radio tuner is such a circuit with an adjustable capacitor; the narrower its resonance, the better it picks one station out of its neighbors.[^up15-6] The same circuit is the core of most analog filters in [[Electrical_engineering|electrical engineering]]. ### Generalizing resonance and antiresonance for linear systems Any linear system with inertia and stiffness has a [[Transfer_function|transfer function]], and its natural frequencies appear as poles, values of the complex frequency where the response becomes infinite. For the damped oscillator the poles lie at `s = −ζω0 ± iω0√(1 − ζ²)`; a pole close to the imaginary axis, meaning light damping, produces a tall, narrow resonance, and the distance from the axis sets the width. Systems with more than one degree of freedom also have zeros, frequencies at which the response measured at a particular point drops toward nothing. That dip is an antiresonance. A parallel inductor–capacitor circuit presents a very large impedance at its antiresonance, and the dynamic vibration absorber, a small mass on a spring tuned to the troublesome frequency and bolted to a machine, places an antiresonance of the machine where the unwanted resonance used to be.[^denhartog] [[Vibration_isolation|Vibration isolation]] works on the other side of the peak, putting the forcing frequency well above the mount's resonance. ## Standing waves Extended systems such as strings, air columns, plates and buildings have not one natural frequency but many, one for each pattern of [[Standing_wave|standing wave]] that fits their boundaries. A standing wave forms when two waves of equal amplitude travel in opposite directions and alternately reinforce and cancel; the result vibrates in place, with fixed nodes and moving antinodes. A textbook treats resonance and standing waves together because each normal mode behaves like its own oscillator: driven at the mode's frequency, it builds a large amplitude from small inputs.[^up16-6] ### Standing waves on a string A string fixed at both ends can hold only waves that fit a whole number of half-wavelengths between the ends, so its resonant frequencies are `f_n = n v/(2L)`, where `v = √(T/μ)` is the wave speed set by the tension and the mass per unit length. A string vibrator driving a string over a pulley shows the modes appearing one after another as the drive frequency passes each `f_n`, with only small, irregular motion in between.[^up16-6] The [[String_vibration|string vibration]] article (section 5) and its microsim follow the modes of a violin string. ## Types ### Mechanical Mechanical resonance matters wherever machines turn or structures carry moving loads. An [[Automotive_engineering|automobile]] part whose natural frequency matches the engine's running frequency can vibrate enough to cause premature failure. Engineers record the sound of the engine, use [[Fourier_analysis|Fourier analysis]] to find the loudest frequencies, and look for a part that resonates there; the cure may be a change of material or length.[^up16-6] ### Acoustic Air in a tube or cavity resonates at frequencies set by its dimensions and the speed of sound, which is how wind instruments turn the buzz of lips or a reed into a pitched tone. String instruments use the air and wood of their bodies as resonators to amplify and color the string's vibration, and the marimba hangs gourds or tubes under its bars as resonance chambers.[^up17-5] The [[Acoustic_resonance|acoustic resonance]] article (section 14) covers tubes, rooms and the [[Helmholtz_resonance|Helmholtz resonator]]. ### Electrical Electrical resonance is the RLC circuit above, together with its many relatives. Resonant circuits select radio channels, set the frequency of oscillators, and match antennas to transmitters across [[Radio-frequency_engineering|radio-frequency engineering]]. Quartz crystals, which convert mechanical vibration to voltage through [[Piezoelectricity|piezoelectricity]], behave electrically as very high-Q resonant circuits and keep time in clocks and computers. ### Optical An optical cavity, two mirrors facing each other, resonates when a whole number of half-wavelengths of light fits between them. Its resonant frequencies are spaced by `c/(2L)`, where `c` is the speed of light: 500 MHz for mirrors 30 cm apart. A laser is built on such a cavity, which feeds back only the frequencies that fit. ### Orbital In celestial mechanics, an orbital resonance occurs when bodies exert regular, periodic gravitational tugs on each other because their orbital periods stand in a ratio of small whole numbers. Jupiter's moons Io, Europa and Ganymede orbit in the ratio 1:2:4, and the gaps Daniel Kirkwood identified in the asteroid belt fall where an asteroid's period would be a simple fraction of Jupiter's.[^murray] ### Atomic, particle, and molecular Atoms, nuclei and molecules absorb and emit energy most strongly at sharply defined frequencies, and these resonances are among the most precisely measured quantities in physics. Since 1967 the second has been defined by a resonance of the cesium-133 atom, whose ground-state hyperfine transition is fixed at exactly 9,192,631,770 Hz.[^bipm] [[Nuclear_magnetic_resonance|Nuclear magnetic resonance]] drives the [[Spin_(physics)|spin]] of protons, which precess at about 42.58 MHz per tesla of field, or 63.9 MHz in a 1.5 T scanner.[^codata] Molecular vibrations, studied numerically in [[Molecular_dynamics|molecular dynamics]] and each close to a small [[Harmonic_oscillator|harmonic oscillator]], resonate at infrared frequencies and give every molecule its absorption fingerprint. ## Disadvantages Resonance can destroy what it amplifies. During an earthquake a building whose natural frequency matches the ground shaking can collapse while its neighbors stand, and wide roofs supported only at their edges, over gymnasiums, supermarkets and churches, resonate at earthquake frequencies more readily than houses do; [[Earthquake_engineering|earthquake engineering]] treats the natural frequency of a structure as a design quantity for that reason.[^up16-6] London's Millennium Footbridge swayed sideways under crowds soon after opening in 2000 and was closed for about two years for damping to be added.[^up15-6][^strogatz] Its motion grew because walkers, feeling the deck move, fell into step with it, a feedback that fed energy in at the bridge's lateral frequency.[^strogatz] The best-known example is also the most often misstated. The Tacoma Narrows Bridge, which tore itself apart in a moderate wind on November 7, 1940, is widely described as a case of forced resonance. Billah and Scanlan showed that the wind supplied no periodic force at the bridge's frequency; the failure was aeroelastic flutter, a self-excited oscillation in which the twisting deck extracted energy from a steady wind.[^billah] The distinction matters for design: a forced resonance is cured by moving the natural frequency or adding damping, while flutter requires changing the deck's aerodynamic shape. ## Q factor The [[Q_factor|Q factor]] measures how sharp a resonance is. Two definitions agree for light damping. By bandwidth, `Q = f0/Δf`, where `Δf` is the width of the peak between the frequencies at which the power response falls to half its maximum, the points where the amplitude is `1/√2` of the peak. By energy, `Q = 2π` times the energy stored divided by the energy lost per cycle. For the harmonic oscillator both give `Q = 1/(2ζ)` in terms of the damping ratio.[^up15-6] Some introductory texts measure the width at half the maximum amplitude instead; that width is `√3` times the half-power width, so the two conventions give values of `Q` that differ by the same factor.[^up15-6] The two faces of `Q` are linked. A high-Q system is sharply tuned when driven and rings for a long time when struck, since its amplitude takes `Q/π` cycles to fall by a factor of `e`. At the microsim's default `Q = 30`, the 660 Hz glass rings for about 9.5 cycles, 14 ms, before its amplitude falls to 37 percent; at `Q = 1,000`, its tuning band narrows to 0.66 Hz and the ring lasts about half a second. The gain is the other face: at resonance the response is `Q` times the static response, or `20 log10 Q` [[Decibel|decibels]], which is +29.5 dB at `Q = 30` and +40 dB at `Q = 100`. ## Universal resonance curve Near a resonance every lightly damped system has the same shape of response. Measuring the detuning in units of the half-power half-width, `x = 2Q(f − f0)/f0`, the amplitude relative to the peak is `1/√(1 + x²)` and the phase shifts by `−arctan x`. At `x = ±1` the amplitude is down 3 dB and the phase has moved 45°. This single curve, often tabulated in radio engineering as the universal resonance curve, describes a tuned circuit, a tuning fork and a wine glass alike once the detuning is scaled by `Q`. The microsim draws the full curves rather than the near-resonance approximation: gray curves for `Q = 1, 3, 10, 30` and `100` on a decibel scale against `f/f0`, with the chosen `Q` in orange and a white marker riding the curve as the sweep moves the tone. The red line marks where the drive level plus the gain reaches 145 dB, an ILLUSTRATIVE breaking threshold chosen for the demonstration, not a measured property of glass.[^rossing] At a 105 dB drive, a [[Sound_pressure|sound pressure]] of about 3.6 Pa, the glass needs a gain of 40 dB to reach it, so no glass with `Q` below 100 breaks at any frequency. At `Q = 300`, with a peak gain of 49.5 dB, the tone must still sit within about ±3 Hz of 660 Hz, since the gain falls to 40 dB when `x ≈ ±2.8`. The rim is drawn swinging at 2.5 Hz and exaggerated so that it can be seen. *Try: untick sweep, set the quality factor Q to about 300 and the drive level to 105 dB SPL, then nudge drive f / f0 through 1.000 and watch the rim / breaking strain readout pass 100 % only in a band a few hertz wide; drop Q to 3 and the peak flattens to about +10 dB.* ## See also - [[Sympathetic_resonance]] - [[Tuning_fork]] - [[Harmonic_oscillator]] (Acoustics portal section 2) - [[String_vibration]] (section 5) - [[Acoustic_resonance]] (section 14) - [[Thermoacoustics]] (section 28) ## References [^up15-6]: Ling, Samuel J.; Sanny, Jeff; Moebs, William (2016). *University Physics Volume 1*. OpenStax, Rice University. §15.6 "Forced Oscillations," pp. 737–741: definition of resonance, Eq. 15.27–15.29, Fig. 15.28 (piano), 15.29 (paddle ball), 15.31 (amplitude against driving frequency), 15.32 (quality defined from the half-amplitude width), the swing and radio-tuning examples, MRI at about 100 MHz, the London Millennium Footbridge closed for roughly two years (Fig. 15.33), and Check Your Understanding 15.6 on the singer and the crystal glass. https://openstax.org/details/books/university-physics-volume-1. Book 077 on the [[PORTAL_Acoustics]] shelf. [^up16-6]: Ling, Sanny and Moebs (2016), *University Physics Volume 1*, §16.6 "Standing Waves and Resonance," pp. 781–790: standing waves, string modes Eq. 16.15–16.16 and Fig. 16.28–16.29, the automobile-part and Fourier-analysis example, and earthquake resonance of buildings and wide roofs. [^up17-5]: Ling, Sanny and Moebs (2016), *University Physics Volume 1*, §17.5 "Sources of Musical Sound," pp. 834–838, Fig. 17.27 (violin and guitar sounding boxes) and Fig. 17.28 (marimba resonance chambers). [^rossing]: Rossing, Thomas D.; Moore, F. Richard; Wheeler, Paul A. (2002). *The Science of Sound*, 3rd ed. Addison-Wesley. Ch. 2 and 4 (quality factor; resonance of a wine glass). The microsim's f0 = 660 Hz is a typical value and its 145 dB breaking threshold is ILLUSTRATIVE; neither is taken from a measurement in this book. [^denhartog]: Den Hartog, J. P. (1956). *Mechanical Vibrations*, 4th ed. McGraw-Hill (Dover reprint 1985). Ch. 3, the dynamic vibration absorber. [^strogatz]: Strogatz, Steven H.; Abrams, Daniel M.; McRobie, Allan; Eckhardt, Bruno; Ott, Edward (2005). "Crowd synchrony on the Millennium Bridge." *Nature* 438: 43–44. https://doi.org/10.1038/438043a [^billah]: Billah, K. Yusuf; Scanlan, Robert H. (1991). "Resonance, Tacoma Narrows bridge failure, and undergraduate physics textbooks." *American Journal of Physics* 59 (2): 118–124. https://doi.org/10.1119/1.16590 [^murray]: Murray, Carl D.; Dermott, Stanley F. (1999). *Solar System Dynamics*. Cambridge University Press. Ch. 8, "Resonant perturbations" (the Laplace resonance of the Galilean satellites and the Kirkwood gaps). https://doi.org/10.1017/CBO9781139174817 [^bipm]: Bureau International des Poids et Mesures (2019). *The International System of Units (SI)*, 9th ed. §2.3.1, definition of the second (Δν_Cs = 9 192 631 770 Hz); the cesium definition dates from the 13th CGPM, 1967. https://www.bipm.org/en/publications/si-brochure [^codata]: National Institute of Standards and Technology. "Proton gyromagnetic ratio over 2 pi," CODATA recommended values of the fundamental physical constants (≈ 42.577 MHz/T). https://physics.nist.gov/cgi-bin/cuu/Value?gammapbar  **Microsim — three.js (Wikitube framework):** *Resonance* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/acoustics/Resonance.html" data-title="Resonance"></div> *Built from `MICROSIM_GUIDE/specs/acoustics/sims/Resonance.json`; part of the [[PORTAL_Acoustics|Acoustics portal]] spine (section sims and See-also variants).*  ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Resonance) : [Wikitube](https://en.wikitube.io/wiki/Resonance) - skeleton pinned to revision 1370697120 (2026-09-11).