String_vibration - Wikiparency

# String vibration **String vibration** is the motion of a taut, flexible string, most familiar from musical instruments, in which a transverse [[Wave|wave]] runs along the string, reflects from its fixed ends and settles into [[Standing_wave|standing waves]]. Only waves that fit a whole number of half-wavelengths between the ends survive, so a string of length `L` sounds a [[Fundamental_frequency|fundamental frequency]] `f1 = v/(2L)` and a series of higher modes at whole-number multiples of it, the [[Harmonic|harmonics]]. The wave speed `v = √(T/μ)` is set by the tension `T` and the mass per unit length `μ`, which is why a player tunes a string by turning a peg and changes notes by stopping the string with a finger. The same three quantities, length, tension and density, were laid out as rules for musicians in the seventeenth century and derived from [[Newton's_laws_of_motion|Newton's laws]] in the eighteenth. On section 5 of the Acoustics portal, the article's microsim, *String vibration: modes of a string fixed at both ends*, lets the reader pick a mode number, a tension, a length and a string, and read off the wave speed, the fundamental and the frequency of the chosen mode against the harmonic series on an audio-frequency axis. Its default is a violin A string, which carries waves at 282 m/s and sounds 441 Hz. ## Wave A string under tension carries transverse waves: each small piece of string moves sideways, at right angles to the string, while the disturbance travels along it. The restoring force is the tension itself. Where the string is curved, the pulls on the two ends of a short piece no longer cancel, and the leftover sideways force accelerates the piece back toward the straight line.[^up16-3] The string is therefore a chain of coupled oscillators, and its motion is that of a [[Mechanical_wave|mechanical wave]] rather than of a single [[Harmonic_oscillator|harmonic oscillator]]. A disturbance of any shape travels along an ideal string without changing shape, in both directions at once. Jean le Rond d'Alembert showed in 1747 that the general solution of the string's equation is the sum of two such traveling shapes, `y = f(x − vt) + g(x + vt)`, one moving right and one moving left.[^dalembert] A fixed end [[Reflection_(physics)|reflects]] a wave upside down, because the string there cannot move and the reflected wave must cancel the incoming one at that point. A wave running back and forth between two fixed ends therefore meets its own reflection, and the two counter-running waves [[Superposition_principle|add]] to a pattern that does not travel: `y = 2A sin(kx) cos(ωt)`, the standing wave the microsim draws. The factor `sin(kx)` fixes the shape and places nodes where it is zero; the factor `cos(ωt)` makes every point swing in step, all reaching their extremes at the same moment.[^up16-6] The nodes at the ends are the boundary conditions, and they are what turn a continuum of possible wavelengths into a discrete set of [[Normal_mode|normal modes]]. Real strings lose energy to the air, to internal friction and, above all, through the bridge into the instrument body, which is how the vibration becomes audible [[Sound|sound]]. A plucked guitar string stops within a few seconds.[^up15-5] The ideal string of the equations, and of the microsim, has no losses and vibrates forever. ### Derivation Take a short piece of string of length `Δx` and mass `μΔx`, stretched with tension `T`, and displaced sideways by a small amount `y(x, t)`. If the slope of the string stays small, the tension is the same everywhere and its components along the string cancel. The sideways components do not: the pull at each end is `T` times the local slope, so the net sideways force is `T(∂y/∂x at x + Δx − ∂y/∂x at x)`. Newton's second law sets that force equal to mass times acceleration, `μΔx ∂²y/∂t²`. Dividing by `Δx` and letting it shrink to zero gives `∂²y/∂t² = (T/μ) ∂²y/∂x²`, the one-dimensional [[Wave_equation|wave equation]], a [[Partial_differential_equation|partial differential equation]] whose solutions travel at `v = √(T/μ)`.[^up16-3] Doubling the tension raises the speed by `√2`; a string four times heavier per meter carries waves at half the speed. The derivation rests on four assumptions: the string is perfectly flexible, so it resists stretching but not bending; its slope is small everywhere; its tension does not change as it moves; and it has no losses. Each fails in some real instrument. A steel [[Piano_acoustics|piano]] string is stiff, which raises its upper modes above exact harmonics; a hard pluck stretches the string and raises its tension, so the pitch starts slightly sharp and falls; and every string is damped. The ideal model is still close enough that a string's frequencies can be predicted from a ruler, a scale and a tension gauge. Brook Taylor derived the fundamental frequency of a vibrating string in 1713 by treating the string's shape as a curve and the restoring force as proportional to its curvature, and d'Alembert's 1747 paper wrote down the wave equation above and its traveling-wave solution.[^taylor][^dalembert] ## Frequency of the wave Mode `n` fits `n` half-wavelengths between the ends, so its wavelength is `λn = 2L/n` and its frequency is `fn = n v/(2L) = (n/(2L)) √(T/μ)`, with `n = 1, 2, 3, …`[^up16-6] The first mode, with one antinode in the middle, is the fundamental or first harmonic. All higher modes are [[Overtone|overtones]], and because each is an exact whole-number multiple of the fundamental, the overtones of an ideal string are also harmonics: the second harmonic is the first overtone, the third harmonic the second overtone, and so on.[^up16-6] Mode `n` has `n + 1` nodes, counting the two ends, which the microsim marks with orange dots. The three factors in the formula are Mersenne's laws, published by Marin Mersenne in 1636: for a string of given material, frequency is inversely proportional to length, proportional to the square root of tension, and inversely proportional to the square root of mass per unit length.[^mersenne] Each maps onto a player's action. Halving the vibrating length, as a guitarist does by stopping the string at the twelfth fret, doubles the frequency and raises the note an octave. Tightening the peg raises the pitch; doubling the tension raises it by `√2`, the equal-tempered interval of a tritone. Heavier strings, often wound with wire to add mass without stiffness, carry the low notes. #### A violin string, worked through A violin string sounds over a length of about 32 cm from the bridge to the nut. Tuned to concert A at 440 Hz, its fundamental requires a wave speed `v = 2Lf1 = 2 × 0.32 m × 440 Hz = 282 m/s`. With a linear density of 0.66 g/m, the tension that gives that speed is `T = v²μ = 282² × 6.6 × 10⁻⁴ = 52.5 N`, about the weight of a 5.4 kg mass.[^gea-12] The string's vibration passes through the bridge into the violin's plates, which vibrate at the same 440 Hz and radiate sound of wavelength `343/440 ≈ 0.78 m` in air, more than twice the length of the string.[^gea-12] The microsim starts from exactly these values; its readout gives 441 Hz, because 282 m/s is itself a rounded number. Holding the tension at 52.5 N and the length at 0.32 m, the microsim's four strings span more than two octaves: | String (microsim choice) | μ (g/m) | Wave speed v (m/s) | Fundamental f1 (Hz) | |---|---|---|---| | nylon | 0.5 | 324 | 506 | | violin A | 0.66 | 282 | 441 | | steel | 6.6 | 89 | 139 | | bass | 20 | 51 | 80 | A guitar faces the same trade-off with six strings of one length. Its open strings are tuned from 329.63 Hz (high E) down to 82.41 Hz (low E), a factor of four, and the low E has roughly 20 times the linear density of the high E.[^up16-3] A fourfold drop in frequency on the same length needs a fourfold drop in wave speed, which with twenty times the density takes about `16/20 = 0.8` times the tension, so all six strings pull on the neck with comparable force. #### The harmonic series and the sound of a string A real string vibrates in many modes at once, a mixture that [[Fourier_analysis|Fourier analysis]] separates into its harmonics, and the ear hears the mix as one note with the [[Pitch_(music)|pitch]] of the fundamental and a tone color, or [[Timbre|timbre]], set by the relative strengths of the harmonics.[^schmidt-3] The frequency ratios of the harmonics are the ratios of the musical intervals: 2:1 between the first and second harmonics is an octave, 3:2 between the second and third a perfect fifth, and 4:3 between the third and fourth a perfect fourth.[^schmidt-4] The microsim's lower panel draws the series `n f1` for `n = 1` to 8 on a logarithmic axis from 20 Hz to 20 kHz, the [[Hearing_range|range of human hearing]], where equal frequency ratios take equal space: the bars crowd together as `n` grows because each step up the series is a smaller musical interval. Where the string is set in motion decides which harmonics it carries. For an ideal string plucked at a fraction `β` of its length, the amplitude of mode `n` is proportional to `sin(nπβ)/n²`. A string plucked at its center has no even harmonics at all, since every even mode has a node there, and a string plucked at one-fifth of its length lacks the fifth, tenth and fifteenth. Plucking near the bridge gives a brighter, thinner tone rich in upper harmonics; plucking near the middle gives a rounder one.[^fletcher] The harmonics of real strings are not quite exact. A stiff string resists bending as well as stretching, and bending matters more for the short wavelengths of high modes, so mode `n` sits at `n f1 √(1 + Bn²)`, where the inharmonicity coefficient `B` grows with the fourth power of the wire's diameter and falls with tension and the square of length. The effect is small on gut and nylon but audible on the short, thick bass strings of a piano, and it is one reason pianos are tuned with octaves slightly stretched; the [[Piano_acoustics|piano acoustics]] article covers the consequences.[^fletcher] *Try: choose the bass 20 g/m string, set tension T to 200 N and length L to 1.2 m and read f1 ≈ 42 Hz, near a bass guitar's low E; then step mode n from 1 to 8 and watch the nodes multiply and the orange bar climb the harmonic series while f_n and lambda_n update.* ## Observing string vibrations A plucked string is hard to watch because it is too fast. A violin A string completes 440 cycles a second, far beyond what the eye can follow, so a vibrating string appears as a blurred, lens-shaped band whose outline is the envelope of its motion. For a string in its fundamental mode that band is widest in the middle and pinched at the ends; a string driven in a higher mode shows several lobes separated by still points, the nodes. The microsim draws that envelope as a faint curve and slows the string itself to `0.6 n` Hz, an ILLUSTRATIVE rate, so that mode `n` can be seen swinging inside it; the readout always gives the real frequency. The classroom method is to drive the string rather than pluck it. A string tied to a mechanical vibrator at one end, passed over a pulley and loaded with a hanging mass at the other, has a known tension and a clean node at each support. Sweeping the vibrator's frequency upward, the string lies nearly still until the drive reaches `f1`, when a single large loop appears, then two loops at `2f1`, three at `3f1`, and so on, each a [[Resonance|resonance]] of the string, the same modes of a [[Standing_wave|standing wave]] that the lead describes.[^up16-6] Changing the hanging mass at a fixed frequency selects the modes the other way, by changing the wave speed. To freeze the motion itself, a stroboscope that flashes once per cycle makes the string appear to stand still, and one flashing slightly slower than the string makes it appear to move in slow motion, the view the microsim imitates. A digital camera with a rolling shutter, which reads its sensor rows one after another rather than all at once, captures different parts of the string at different moments and turns a vibrating guitar string into a wavy line in a still photo; the waves in the image are an artifact of the scan as well as a record of the motion. Hermann von Helmholtz, observing a bowed violin string through a vibration microscope in the 1860s, found that it did not move in a smooth sinusoid at all: a sharp corner travels around the string's lens-shaped envelope, and the point under the bow sticks to the bow hair and then slips back quickly, once per cycle. That stick–slip pattern, now called Helmholtz motion, is why a bowed string's spectrum is so rich in harmonics.[^helmholtz][^fletcher] Electric instruments observe the string continuously: a magnetic pickup under a steel string produces a voltage from the string's motion at that one point, and so hears the harmonics that have antinodes near it more strongly than those with nodes there. The same idea of making modes visible extends to plates, where [[Ernst_Chladni|Ernst Chladni]] scattered sand that collected along the nodal lines of a vibrating plate.[^chladni] ## See also - [[Harmonic]] - [[Fundamental_frequency]] - [[Overtone]] - [[Oscillation]] (Acoustics portal section 1) - [[Resonance]] (section 3) - [[Standing_wave]] (section 11) - [[Musical_acoustics]] (section 22) ## References [^up16-3]: Ling, Samuel J.; Sanny, Jeff; Moebs, William (2016). *University Physics Volume 1*. OpenStax, Rice University. §16.3 "Wave Speed on a Stretched String," pp. 766–769: linear density (Eq. 16.7), derivation of the wave speed on a string (Eq. 16.8), and Example 16.5 on the six guitar strings (low E about 20 times the density of high E; open-string frequencies 329.63–82.41 Hz). 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 wave from two counter-running waves, string modes λn = 2L/n and fn = nv/2L (Eq. 16.15–16.16), fundamental, harmonics and overtones (p. 786), and the string-vibrator apparatus (Fig. 16.28–16.29, Example 16.7). [^up15-5]: Ling, Sanny and Moebs (2016), *University Physics Volume 1*, §15.5 "Damped Oscillations," p. 734 (a plucked guitar string stops oscillating a few seconds after being plucked). [^gea-12]: Gea-Banacloche, Julio (2019). *University Physics I: Classical Mechanics*. University of Arkansas Open Educational Resources. §12.5.2 "Violin sounds," p. 298 (L = 32 cm, f1 = 440 Hz gives c = 282 m/s; μ = 0.66 g/m gives Ft = 52.5 N; the plates vibrate at 440 Hz). The book uses 340 m/s for sound in air and obtains 0.77 m; this article uses 343 m/s. https://scholarworks.uark.edu/oer/3. Book 076 on the [[PORTAL_Acoustics]] shelf. [^schmidt-3]: Schmidt-Jones, Catherine; Jones, Russell (2007). *Understanding Basic Music Theory*. Connexions, Rice University. §3.2 "Standing Waves and Musical Instruments," pp. 100–104 (nodes at both ends of a string; the harmonics give the string its timbre). http://cnx.org/content/col10363/1.3/. Book 092 on the [[PORTAL_Acoustics]] shelf. [^schmidt-4]: Schmidt-Jones and Jones (2007), *Understanding Basic Music Theory*, §4.6 "Harmonic Series II: Harmonics, Intervals, and Instruments," pp. 146–147 (octave 2:1, perfect fifth 3:2, fourth from the harmonic series). [^fletcher]: Fletcher, Neville H.; Rossing, Thomas D. (1998). *The Physics of Musical Instruments*, 2nd ed. Springer. Ch. 2, "Continuous systems in one dimension: strings and bars" (plucked-string spectra, stiffness and inharmonicity) and Ch. 10 on bowed strings (Helmholtz motion). https://doi.org/10.1007/978-0-387-21603-4 [^mersenne]: Mersenne, Marin (1636). *Harmonie universelle, contenant la théorie et la pratique de la musique*. Paris: Sébastien Cramoisy. [^taylor]: Taylor, Brook (1713). "De motu nervi tensi." *Philosophical Transactions of the Royal Society of London* 28 (337): 26–32. (Issue and pages as commonly cited; not checked against a scan in this run.) [^dalembert]: d'Alembert, Jean le Rond (1749). "Recherches sur la courbe que forme une corde tenduë mise en vibration." *Histoire de l'Académie Royale des Sciences et Belles Lettres de Berlin* 3 (for 1747): 214–219. (Volume and pages as commonly cited; not checked against a scan in this run.) [^helmholtz]: Helmholtz, Hermann von (1863). *Die Lehre von den Tonempfindungen als physiologische Grundlage für die Theorie der Musik*. Braunschweig: Vieweg. English translation by Alexander J. Ellis, *On the Sensations of Tone* (1875). [^chladni]: Chladni, Ernst Florens Friedrich (1787). *Entdeckungen über die Theorie des Klanges*. Leipzig: Weidmanns Erben und Reich.  **Microsim — three.js (Wikitube framework):** *String vibration* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/acoustics/String_vibration.html" data-title="String vibration"></div> *Built from `MICROSIM_GUIDE/specs/acoustics/sims/String_vibration.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/String_vibration) : [Wikitube](https://en.wikitube.io/wiki/String_vibration) - skeleton pinned to revision 1306385524 (2026-09-11).