Acoustic_resonance - Wikiparency

# Acoustic resonance > *For resonance in general, see [[Resonance]]. For the bottle-and-neck resonator, see [[Helmholtz_resonance]].* **Acoustic resonance** is the tendency of an acoustic system to take up and give back far more [[Energy|energy]] at certain frequencies than at others: the frequencies at which a [[Sound|sound]] wave, reflected back and forth inside the system, returns in step with itself. Those frequencies are the system's [[Normal_mode|normal modes]]. A stretched string, the air in a pipe, the air in a closed room and the air in a vented bottle each have their own set, fixed by their size, by the [[Speed_of_sound|speed of sound]] in them and by what happens at their boundaries. Every such system is a [[Resonator|resonator]], and nearly every musical instrument is built from one or two of them. The mechanism is the same as for any driven [[Oscillation|oscillation]]. A wave that fits its container a whole number of half-wavelengths reinforces itself on every round trip and builds a [[Standing_wave|standing wave]]; a wave that does not fit cancels itself out. How sharp each peak is depends on [[Damping|damping]], summarized by the [[Q_factor|Q factor]]. The same physics that makes an organ pipe sing makes a small room boom in the bass and, at high enough sound levels, lets a pure tone shatter a wine glass. The article's microsim is *Acoustic resonance: the modes of a room*, a closed box of air in which the reader picks the mode numbers (l, m, n) and the floor plan and watches the standing pressure pattern and the room's lowest resonances on a 0–200 Hz axis. On the Minnesota side, the article follows the 32-foot pipes of the Northrop organ in Minneapolis and a physics class that sang a glass to pieces. ## Vibrating string A string fixed at both ends is the simplest acoustic resonator. A transverse wave runs along it at speed `v = √(T/μ)`, set by the tension T and the mass per unit length μ, and reflects at each fixed end with its displacement inverted. Only waves with a node at both ends survive the round trip, so the allowed wavelengths are `λn = 2L/n` and the resonant frequencies are `fn = (n/2L)·√(T/μ)`, with n = 1, 2, 3, ….[^up16-6] The lowest is the [[Fundamental_frequency|fundamental frequency]]; the rest are whole-number multiples, the [[Harmonic|harmonics]]. This is the same equation as for a pipe open at both ends, because the boundary conditions are symmetric in both cases.[^up17-4] A worked number shows the scale. A guitar string with a vibrating length of 0.648 m tuned to E4 (329.6 Hz) must carry transverse waves at `v = 2Lf` ≈ 427 m/s. Doubling the tension raises the pitch by a factor of √2, about six semitones; halving the length raises it by an octave. The [[String_vibration|string vibration]] article develops the full theory. ### String resonance in music instruments A bare string is a poor radiator: it is too thin to push much air. Instruments couple it through a bridge to a soundboard and an enclosed air cavity, which have resonances of their own. The textbook description of the violin and the guitar is that the vibrating string's sound "resonates in the sounding box, greatly amplifying the sound and creating overtones that give the instrument its characteristic timbre," and that the more complex the box's shape, the wider the range of frequencies over which it resonates.[^up17-5] The shape of the body thus decides which [[Overtone|overtones]] are strengthened, and so the [[Timbre|timbre]]. Instruments also exploit [[Sympathetic_resonance|sympathetic resonance]]: undamped strings tuned to a played note's harmonics pick up energy from it and ring on their own, as the undamped strings of a piano do when the sustain pedal is held. ## Resonance of a tube of air Air in a tube carries a longitudinal pressure wave at the speed of sound, about 343 m/s at 20 °C. At a closed end the air cannot move, so the end is a displacement node and a pressure antinode. At an open end the pressure is held close to atmospheric, so the end is a pressure node and a displacement antinode. The pattern of ends decides which modes fit.[^up17-4] ### Cylinders A cylinder open at both ends has antinodes of displacement at both ends, and resonates at `fn = n·c/2L` for every whole number n. A cylinder closed at one end has a node at the closed end and an antinode at the open one; only odd quarter-wavelengths fit, so it resonates at `fn = (2n − 1)·c/4L`, the odd harmonics alone. The closed tube therefore sounds an octave lower than an open tube of the same length and has a hollower tone.[^up17-4] For a 0.60 m tube the numbers are 286 Hz, 572 Hz, 858 Hz … open, and 143 Hz, 429 Hz, 715 Hz … closed. The open end is not exactly at the pipe's mouth. The air just outside moves with the air inside, so the tube acts as if it were longer by an end correction of about 0.61 times the radius for an unflanged pipe, the value found by Harold Levine and Julian Schwinger in 1948.[^levine1948] For a flute-like tube 0.60 m long and 0.95 cm in radius, the two open ends add about 1.2 cm and lower the fundamental from 286 Hz to about 280 Hz. Because every frequency scales with c, and c rises by about 0.6 m/s per degree Celsius, wind instruments go sharp as they warm, which is why players warm them before tuning.[^up17-4] ### Cones A cone closed at its narrow tip behaves, in the ideal case, not like a closed cylinder but like an open one: its resonances form the complete harmonic series `fn = n·c/2L`. The pressure waves inside a cone are spherical rather than plane, and the geometry puts the missing even harmonics back. That is why the oboe and the saxophone, with conical bores, overblow at the octave, while the clarinet, a near-cylinder closed by the reed, overblows at the twelfth and is dominated by odd harmonics in its low register. The subject belongs to [[Musical_acoustics|musical acoustics]]; real bores are truncated cones with tone holes and flared bells, and their modes are only approximately harmonic. ### Closed rectangular box A box closed on all six sides is a three-dimensional tube with hard walls everywhere. Every wall is a pressure antinode, and the modes are labeled by three whole numbers (l, m, n), the number of half-wavelengths along each side. Their frequencies, first worked out by Lord Rayleigh, are `f(l,m,n) = (c/2)·√((l/Lx)² + (m/Ly)² + (n/Lz)²)`.[^rayleigh1878] A mode with one non-zero index is axial (it bounces between one pair of walls), two is tangential, and three is oblique. The lowest mode of a room 5 m long is `c/2Lx` = 34.3 Hz; its 2.7 m ceiling height gives an axial mode at 63.5 Hz; the tangential mode (2, 1, 0) of a 5 m × 4 m floor lies at 80.9 Hz with a wavelength of 4.24 m. In a small room these low modes are few and far apart, so some bass notes are boosted and others lost depending on where the listener sits. As frequency rises the modes crowd together; above the Schroeder frequency, `fS ≈ 2000·√(T60/V)`, they overlap so densely that the room behaves as a diffuse field governed by [[Reverberation|reverberation]] rather than by separate resonances.[^schroeder1962] For the 54 m³ room above with a reverberation time of 0.6 s, fS is about 211 Hz. The [[Room_acoustics|room acoustics]] article carries this into studio and home-theater design. *Try: drag room length Lx and room width Ly, and step mode l, mode m and mode n (the height index enters the frequency only) — the surface redraws the standing pressure pattern over the floor, the orange bar marks the mode among the room's lowest twelve, and the readout gives f, λ and whether the mode is axial, tangential or oblique; set all three to zero to see why (0, 0, 0) is not a mode.* ## Resonance of a sphere of air (vented) A cavity of air with a small opening has one resonance far below any of its standing-wave modes. The plug of air in the neck acts as a mass; the air shut inside acts as a spring that is squeezed when the plug moves in and stretched when it moves out. This mass-on-a-spring is the [[Helmholtz_resonance|Helmholtz resonator]], named for Hermann von Helmholtz, who used tuned glass spheres to pick single harmonics out of complex sounds.[^helmholtz1863] Its frequency is `fH = (c/2π)·√(A/(V·Leff))`, where A is the neck's cross-section, V the cavity volume and Leff the neck length plus its end corrections. The shape of the cavity hardly matters, only its volume, which is why a sphere, a bottle and a loudspeaker cabinet obey the same law. A 750 mL bottle with an 8 cm neck of radius 0.95 cm resonates at about 108 Hz; four times the volume halves the pitch to 54 Hz. The same calculation tunes the port of a bass-reflex [[Loudspeaker|loudspeaker]] and the side-branch chambers of an automotive [[Muffler|muffler]]. Because the whole cavity moves in one phase, the Helmholtz resonance is the one acoustic resonance that can be treated as a single [[Lumped-element_model|lumped element]]. ## Breaking glass with sound via resonance A wine glass is a thin shell with its own flexural modes. Tapped, it rings at its lowest one; driven by sound at exactly that frequency, it takes up energy on every cycle, and because glass is lightly damped the motion can grow until the rim cracks. The University Physics text presents the effect as its opening example of sound inducing resonance and describes a filmed demonstration in which a glass driven by a loudspeaker shatters when the tone reaches its resonant frequency.[^up17-1] It is the driven [[Harmonic_oscillator|harmonic oscillator]] at its most dramatic: the less the damping, the taller and narrower the resonance peak.[^up15-6] The practical difficulty is precision and level. UCLA's physics lecture-demonstration manual tunes a signal generator to the glass one hertz and then a tenth of a hertz at a time while watching a microphone signal on an oscilloscope, and reports a level of approximately 140 dB at the glass.[^ucla-glass] That is an RMS [[Sound_pressure|sound pressure]] of about 200 Pa, ten million times the threshold of hearing, far beyond a safe exposure for anyone nearby. Trained singers have broken glasses with the voice alone, but it takes a steady, loud, exactly pitched tone and a glass with a sharp resonance. ## In musical composition Composers have used the resonances of a room as material. In Alvin Lucier's *I Am Sitting in a Room*, first performed in 1969, a spoken text is recorded, played back into the same room and re-recorded, over and over. With each generation the room reinforces its own resonant frequencies and suppresses the rest, until the words dissolve into tones "entirely specific to the architectural particularity of a given space," as the Museum of Modern Art put it when it acquired a 2014 performance.[^moma-lucier] The piece is a room-mode measurement made audible: the tones that survive are the room's (l, m, n) modes, weighted by where the microphone and loudspeaker stand. Instrument builders exploit the same principle in reverse, choosing body and bore shapes so that the resonances they cannot avoid fall where they help the tone. ## Minnesota *This section is specific to Wikitube.* Northrop Memorial Auditorium at the University of Minnesota, Minneapolis, houses Aeolian-Skinner Opus 892, a four-manual organ of 108 ranks installed in four stages between 1932 and 1935 and first dedicated on December 12, 1932. Its Pedal division carries four 32-foot stops, among them a Double Open Diapason.[^diapason2003] An open pipe 32 ft (9.75 m) long has, by the open–open formula, a fundamental of about 17.6 Hz at 20 °C, close to the 16.4 Hz of the organ's lowest C; the pipe-length names are nominal, and real pipes are shortened or lengthened by end corrections and voicing. Such notes lie at the bottom edge of hearing and are felt as much as heard, and they are the clearest large-scale case of cylinder resonance in the state. The glass-breaking demonstration has a Minnesota record too. In May 2009, NPR's *Talk of the Nation* reported on Sam Terfa, a science teacher at Minnehaha Academy in Minneapolis, who had the school's strongest singer serenade a wine glass to demonstrate resonant frequency; the glass shattered on camera.[^npr2009] ## See also - [[Helmholtz_resonance]] - [[Resonator]] - [[Harmonic_oscillator]] — damped and driven (section 2) - [[Resonance]] (section 3) - [[Standing_wave]] (section 11) - [[Reverberation]] (section 17) - [[Room_acoustics]] ## References [^up16-6]: OpenStax (2016). *University Physics Volume 1*, §16.6 "Standing Waves and Resonance," pp. 781–789. Portal Book 077. https://openstax.org/details/books/university-physics-volume-1 [^up17-4]: OpenStax (2016). *University Physics Volume 1*, §17.4 "Normal Modes of a Standing Sound Wave," Eqs. 17.13–17.16, pp. 831–833 (tube closed at one end, tube open at both ends, temperature dependence of wind instruments). Portal Book 077. [^up17-5]: OpenStax (2016). *University Physics Volume 1*, §17.5 "Sources of Musical Sound," p. 837 and Fig. 17.27 (violin and guitar sounding boxes). Portal Book 077. [^up17-1]: OpenStax (2016). *University Physics Volume 1*, §17.1 "Sound Waves," pp. 808–809 and Fig. 17.2 (glass shattered by a sound wave at its resonant frequency). Portal Book 077. [^up15-6]: OpenStax (2016). *University Physics Volume 1*, §15.6 "Forced Oscillations," pp. 739–741 (resonance curves narrow as damping falls; Check Your Understanding 15.6 on the singer and the crystal glass). Portal Book 077. [^levine1948]: Levine, Harold; Schwinger, Julian (1948). "On the radiation of sound from an unflanged circular pipe." *Physical Review* 73 (4): 383–406. https://doi.org/10.1103/PhysRev.73.383 (end correction ≈ 0.61 r at low frequency). [^rayleigh1878]: Strutt, John William, Lord Rayleigh (1877–1878). *The Theory of Sound*, 2 vols. London: Macmillan. Rectangular-enclosure modes are treated in vol. 2; exact section number not re-checked for this article. [^schroeder1962]: Schroeder, M. R. (1962). "Frequency-correlation functions of frequency responses in rooms." *Journal of the Acoustical Society of America* 34 (12): 1819–1823. The constant 2000 in SI units follows the form used in the portal's acoustic pack (`acoustic.room.schroeder`); DOI not re-checked for this article. [^helmholtz1863]: Helmholtz, Hermann von (1863). *Die Lehre von den Tonempfindungen als physiologische Grundlage für die Theorie der Musik*. Braunschweig: Vieweg. English translation by A. J. Ellis, *On the Sensations of Tone* (1875). [^ucla-glass]: UCLA Department of Physics and Astronomy, Instructional Research Lab. "Breaking Glass with Sound." *Lecture Demonstration Manual*, Acoustics. https://www.physics.ucla.edu/demoweb/demomanual/acoustics/effects_of_sound/breaking_glass_with_sound.html [^moma-lucier]: Joseph, Martha (January 20, 2015). "Collecting Alvin Lucier's *I Am Sitting in a Room*." *Inside/Out*, Museum of Modern Art. https://www.moma.org/explore/inside_out/2015/01/20/collecting-alvin-luciers-i-am-sitting-in-a-room/ [^diapason2003]: Hendrickson, Charles (June 9, 2003). "Northrop Auditorium, University of Minnesota, Aeolian-Skinner Restoration." *The Diapason*. https://www.thediapason.com/content/northrop-auditorium-university-minnesota-aeolian-skinner-restoration [^npr2009]: NPR (May 22, 2009). "Belting Out A Musical Physics Lesson." *Talk of the Nation*. https://www.npr.org/2009/05/22/104447724/belting-out-a-musical-physics-lesson — the school's Minneapolis location: Minnehaha Academy, "Contact." https://www.minnehahaacademy.net/about/contact  **Microsim — three.js (Wikitube framework):** *Acoustic resonance* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/acoustics/Acoustic_resonance.html" data-title="Acoustic resonance"></div> *Built from `MICROSIM_GUIDE/specs/acoustics/sims/Acoustic_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/Acoustic_resonance) : [Wikitube](https://en.wikitube.io/wiki/Acoustic_resonance) - skeleton pinned to revision 1350107359 (2026-09-11).