Speed_of_sound - Wikiparency

# Speed of sound The **speed of sound** is the distance a [[Sound|sound]] wave travels per unit time through an elastic medium. In dry air at 20 °C it is about 343 m/s (1,125 ft/s), roughly a kilometer in three seconds or a mile in five. It is about four times faster in fresh [[Water|water]] (about 1,480 m/s) and about seventeen times faster in [[Steel|steel]] (about 5,960 m/s), because what sets the speed is the ratio of a medium's stiffness to its [[Density|density]], and liquids and solids are far stiffer than gases.[^openstax17-2] In a gas the speed depends on temperature and molecular mass but not on pressure, following `c = sqrt(gamma R T / M)`; for air this is `c ≈ 331.3 sqrt(1 + T/273.15)` m/s with `T` in degrees Celsius. The speed is the constant `c` of the [[Acoustic_wave_equation|acoustic wave equation]], it fixes the wavelength of every tone through `c = f lambda`, it defines the Mach number of an aircraft, and its variation with height and depth bends sound through the [[Atmosphere_of_Earth|atmosphere]] and ocean. The Wikitube microsim *Speed of sound: a race through four media* starts the same tone at the left end of four tubes of gas, air, water and steel; the reader changes the gas and its temperature and the tone frequency, and watches both the fronts and the wavelengths separate. ## History Early estimates came from timing the delay between a distant flash and its report. Marin Mersenne described such measurements in *Harmonie universelle* (1636), and later timings of cannon fire over measured baselines brought the figure for air close to its modern value.[^mersenne1636] [[Isaac_Newton|Isaac Newton]] gave the first theory in the *Principia* (1687), treating air as a spring compressed at constant temperature; his result was about 15 percent below the measured values.[^laplace1816] Pierre-Simon Laplace resolved the discrepancy in 1816. The compressions in a sound wave happen too quickly for heat to leave them, so the process is adiabatic, the effective stiffness of air is larger by the ratio of specific heats `gamma ≈ 1.4` (see [[Heat_capacity|heat capacity]]), and the speed rises by `sqrt(1.4) ≈ 1.18`.[^laplace1816] The first measurement in water followed in 1826, when Jean-Daniel Colladon and Charles Sturm struck an underwater bell on Lake Geneva while igniting gunpowder and timed the flash-to-sound delay at a boat about 16 km (10 mi) away: 1,435 m/s in water at 8 °C.[^dosits-colladon] ## Compression and shear waves Fluids resist only a change of volume, so they carry only compression waves, the longitudinal waves of [[Longitudinal_wave|sound]] proper. Solids also resist a change of shape, so they carry two kinds of bulk wave: compression (longitudinal) waves, in which the material moves along the direction of travel, and shear (transverse) waves, in which it moves across it. The compression wave is always faster. In an isotropic solid with bulk modulus `K`, shear modulus `G` and density `rho`, the two speeds are `c_p = sqrt((K + 4G/3)/rho)` and `c_s = sqrt(G/rho)`.[^kinsler2000] Earthquakes show the difference directly. The primary or [[P_wave|P wave]] of seismology is the compression wave and arrives first; the secondary or [[S_wave|S wave]] is the shear wave and arrives later, and the delay between them at a seismometer grows with the distance to the source. The P and S waves are both [[Seismic_wave|seismic waves]]. A thin rod adds a third speed, `sqrt(Y/rho)` with `Y` the Young's modulus (OpenStax Eq. 17.5), because the rod can bulge sideways.[^openstax17-2] For steel (`Y ≈ 200 GPa`, `rho ≈ 7,850 kg/m³`) that is about 5,050 m/s, against 5,960 m/s for a longitudinal wave in bulk steel, the Table 17.1 value the microsim uses. ## Equations Every formula for the speed of a mechanical wave has the same shape, the square root of a restoring (elastic) property over an inertial property; OpenStax writes it as `v = sqrt(elastic property / inertial property)`.[^openstax17-2] For a string the elastic property is the tension and the inertial one the mass per length; for a [[Fluid_dynamics|fluid]] it is the adiabatic bulk modulus `B` over the density, `c = sqrt(B/rho)` (Eq. 17.4), called the Newton–Laplace equation; for a solid rod it is `sqrt(Y/rho)` (Eq. 17.5). For an ideal gas, `B = gamma p` and `p/rho = R T/M`, so `c = sqrt(gamma R T / M)` (Eq. 17.6) with `gamma` the ratio of specific heats, `R = 8.314 J/(mol·K)`, `T` the absolute temperature and `M` the molar mass. For dry air (`gamma = 1.40`, `M = 0.02897 kg/mol`) at 293.15 K this gives `sqrt(1.40 × 8.314 × 293.15 / 0.02897) ≈ 343 m/s`. The pressure has dropped out: at a fixed temperature, compressing a gas raises its stiffness and its density in the same proportion.[^openstax17-2] The speed links [[Frequency|frequency]] and wavelength by `c = f lambda` (Eq. 17.3). A 250 Hz tone, the microsim's default, is 1.37 m long in air at 20 °C, 5.93 m in water and 23.8 m in steel; the source fixes the frequency, and each medium stretches the wavelength in proportion to its speed. ## Dependence on the properties of the medium Stiffness and density pull in opposite directions, and stiffness usually wins. Water is about 830 times denser than air, which alone would make sound 29 times slower; but its bulk modulus, about 2.2 GPa, is some 15,000 times the adiabatic bulk modulus of air (1.4 × 101 kPa ≈ 142 kPa), and `sqrt(2.2×10⁹/998) ≈ 1,485 m/s`. The same comparison explains solids: metals are denser than water but stiffer still.[^gea12] Among gases, molar mass dominates: [[Helium|helium]] (4 g/mol) carries sound about three times faster than air, and sulfur hexafluoride (146 g/mol) about 2.5 times slower, which is why inhaling them shifts the resonances of the vocal tract.[^gea12] In a gas, `c` grows as the square root of the absolute temperature, a consequence of the link between sound and the random molecular motion of the [[Kinetic_theory_of_gases|kinetic theory of gases]]. OpenStax notes that this "is not a strong dependence": from 0 °C to 20 °C the speed rises from 331 m/s to 343 m/s, under 4 percent.[^openstax17-2] Liquids and solids depend on temperature through their elastic moduli, which must be measured rather than derived. The microsim's four lanes share one clock and one tone; the gas lane follows `acoustic.waves.cGas(gamma, T, M)` and the chart plots `c` against temperature, while water (1,482 m/s) and steel (5,960 m/s) are held at 20 °C. At −40 °C the air lane slows to about 306 m/s, 11 percent below its 20 °C value, and a kilometer of air then takes 3.27 s instead of 2.91 s. *Try: drag **gas temperature** from −40 °C to 60 °C and watch the air front and the orange dot on the chart move together while water and steel do not; set **gas in tube 1** to helium, then SF6, to see molar mass win; raise **tone frequency** and watch every wavelength shrink by the same factor; press **restart** to rerun the race.* ## Altitude variation and implications for atmospheric acoustics Because the speed in air depends on temperature and not on pressure, it changes with altitude only as the temperature does. In the U.S. Standard Atmosphere the temperature falls by 6.5 °C per kilometer from 15 °C at sea level to −56.5 °C at 11 km, and then stays constant to 20 km.[^ussa1976] The speed of sound falls from about 340 m/s at sea level to about 295 m/s at the tropopause; an [[Aircraft|airliner]] at Mach 0.85 near 11 km is making about 251 m/s. A vertical change in sound speed is a [[Sound_speed_gradient|sound speed gradient]], and it bends sound toward the slower layer by [[Refraction|refraction]]. On a sunny afternoon, air near the ground is warmest, sound curves upward, and an [[Acoustic_shadow|acoustic shadow]] can form a few hundred meters from a source. On a still night or over cold water or snow, a temperature inversion puts the faster air aloft, sound curves back down, and distant traffic and voices carry unusually far.[^pierce2019] ## Details ### Speed of sound in ideal gases and air For dry air the ideal-gas formula reduces to `c = 331.3 sqrt(1 + T/273.15)` m/s, with `T` in degrees Celsius, OpenStax Eq. 17.7; near room temperature a linear form, `c ≈ 331.3 + 0.606 T` m/s, is accurate to a few tenths of a meter per second.[^openstax17-2] A re-evaluation of the published measurements put the zero-frequency speed in standard dry air at 0 °C at 331.29 m/s.[^wong1986] Water vapor lowers the mean molar mass of moist air and so raises the speed, by less than one percent at ordinary temperatures; [[Carbon|carbon]] dioxide lowers it slightly.[^cramer1993] ### Effects due to wind shear Sound travels at `c` relative to the air, so relative to the ground its speed is `c` plus the component of the wind along the path. Wind usually increases with height, so downwind the effective [[Velocity|velocity]] of sound grows upward and rays bend back toward the ground, while upwind they bend away. A source is therefore heard farther downwind, although the wind is only a few percent of `c`.[^pierce2019] ### Tables | Medium | Speed (m/s) | Condition | |---|---|---| | Air | 331 | 0 °C | | Carbon dioxide | 259 | 0 °C | | Helium | 965 | 0 °C | | Hydrogen | 1,290 | 0 °C | | Water, fresh | 1,480 | 20 °C | | Sea water | 1,540 | 20 °C | | Human tissue | 1,540 | 20 °C | | Vulcanized rubber | 54 | longitudinal | | Lead | 1,960 | longitudinal | | Aluminum | 5,120 | longitudinal | | Steel | 5,960 | longitudinal | Values from OpenStax Table 17.1.[^openstax17-2] The temperature column for dry air, from `331.3 sqrt(1 + T/273.15)`: | Temperature (°C) | −40 | −20 | 0 | 20 | 40 | 60 | |---|---|---|---|---|---|---| | Speed in dry air (m/s) | 306.1 | 318.9 | 331.3 | 343.2 | 354.7 | 365.9 | ## Effect of frequency and gas composition ### General physical considerations In an ideal gas the speed of sound does not depend on frequency. OpenStax points to a marching band: if high notes traveled faster than low ones, a listener far away would hear the instruments drift out of step, and they do not.[^openstax17-2] The approximation fails when a wave's period approaches the time molecules need to share [[Energy|energy]] between translation and internal rotation and vibration. Above that relaxation frequency the internal modes lag, the gas behaves as if stiffer, and sound is faster; the medium becomes dispersive, and [[Acoustic_attenuation|attenuation]] peaks near the relaxation frequency. ### Practical application to air In Earth's air the relaxation frequencies of [[Oxygen|oxygen]] and [[Nitrogen|nitrogen]] depend on humidity, and the dispersion is a small fraction of a meter per second across the audible range, though the absorption is large enough that high frequencies die out first over distance. Mars is the exception, described below.[^maurice2022] ## Mach number The Mach number is the ratio of an object's speed through a fluid to the local speed of sound, `Ma = v/c`. It is named for Ernst Mach, who with Peter Salcher photographed the shock waves around supersonic bullets in 1887.[^mach1887] Below Ma ≈ 0.3 air behaves as nearly incompressible; from about 0.8 to 1.2 the [[Compressible_flow|compressible flow]] is transonic; above 1 an object outruns its own pressure waves, which pile into a [[Shock_wave|shock wave]] heard on the ground as a [[Sonic_boom|sonic boom]]. The half-angle of the Mach cone is `arcsin(1/Ma)`, the limiting case of the [[Doppler_effect|Doppler effect]] for a source moving faster than its waves. On October 14, 1947, U.S. Air Force Captain Chuck Yeager flew the Bell XS-1 faster than sound, in what NASA describes as history's first supersonic flight.[^nasa-xs1] ## Experimental methods ### Single-shot timing methods The simplest measurement times a pulse over a known distance. The flash-to-bang rule for lightning is its everyday form: OpenStax gives "every five seconds converts to about one mile," and indeed 343 m/s × 5 s ≈ 1,715 m ≈ 1.07 mi.[^openstax17-2] Modern versions time a spark between two [[Microphone|microphones]] a measured distance apart, removing the observer's reaction time. ### Other methods Resonance methods measure a wavelength instead of a time. In a tube closed at one end, successive resonances of a [[Tuning_fork|tuning fork]] appear at air-column lengths half a wavelength apart, and `c = f lambda` follows; [[Kundt's_tube|Kundt's tube]] makes the standing-wave nodes visible as heaps of powder. Sonic anemometers use the difference between upwind and downwind transit times to measure wind. ### High-precision measurements in air The most precise acoustic measurements use a gas-filled resonator whose modes are measured to parts per million. Moldover and colleagues at the National Bureau of Standards used the radial modes of a spherical resonator filled with argon to determine the universal gas constant `R` from `c = sqrt(gamma R T/M)`, turning the speed of sound into a primary thermometer; acoustic gas thermometry of this kind contributed to fixing the Boltzmann constant in the 2019 redefinition of the SI.[^moldover1988][^moldover2014] ## Non-gaseous media ### Speed of sound in solids In [[Solid_mechanics|solids]] the speed depends on direction in crystals and on wave type in all materials. The practical consequence is [[Ultrasonic_testing|ultrasonic testing]]: a transducer sends a pulse into a steel part and times the echo from a flaw or the back wall. An echo returning after 10 µs in steel places the reflector 5,960 × 10×10⁻⁶ / 2 ≈ 30 mm deep, which is why an inspector calibrates the instrument on a block of the same alloy. ### Speed of sound in liquids Fresh water carries sound at about 1,480 m/s at 20 °C, and sea water at about 1,500–1,540 m/s depending on temperature, salinity and depth; oceanographers use empirical formulas such as Mackenzie's nine-term equation to compute it from those three.[^mackenzie1981] [[Medical_ultrasound|Medical ultrasound]] scanners assume that sound travels through all soft tissue at exactly 1,540 m/s and convert echo time to depth with that value; where the real speed is higher, reflectors are drawn too shallow, and where it is lower, too deep, the propagation speed error artifact.[^thapaliya2024] ### Speed of sound in plasma In a [[Plasma_(physics)|plasma]], the compression wave is the ion acoustic wave. The ions supply the inertia and the hotter electrons most of the pressure, so the speed is approximately `sqrt((gamma_e k T_e + gamma_i k T_i)/m_i)`, set mainly by the electron temperature and the ion mass.[^chen2016] ## Mars The atmosphere of [[Mars]] is mostly carbon dioxide at a surface pressure below one percent of Earth's, and near the ground its temperature is around 240 K. The ideal-gas formula, with `gamma ≈ 1.3` and `M = 0.044 kg/mol`, gives about 240 m/s, some 30 percent slower than on Earth. In the thin carbon-dioxide atmosphere the relaxation frequency falls inside the audible band, and the SuperCam microphone on the [[NASA]] Perseverance rover measured two distinct speeds about 10 m/s apart, below and above about 240 Hz, so the high notes of a sharp sound arrive slightly before the low ones.[^maurice2022] ## Gradients Wherever the speed of sound changes from place to place, sound refracts toward the region of lower speed. In the atmosphere the gradient comes from temperature and wind; in the ocean it comes from temperature, which falls with depth, and pressure, which rises with depth and speeds sound up. The two effects make a minimum in the sound-speed profile, typically near 1,000 m at mid-latitudes, and sound launched near that depth is bent back toward it from above and below. This deep sound channel lets low-frequency [[Sonar|sonar]] signals cross ocean basins, and Walter Munk's canonical profile is the standard idealized model of it in [[Underwater_acoustics|underwater acoustics]].[^munk1974] ## Minnesota *This section is specific to Wikitube.* Minnesota's recorded temperature extremes span a range that changes the speed of sound in its air by about 20 percent. The state's record low, −60 °F (−51.1 °C), was set near Tower in St. Louis County on February 2, 1996, and its record high, 115 °F (46.1 °C), at Beardsley in Big Stone County on July 29, 1917.[^mndnr-extremes] From `331.3 sqrt(1 + T/273.15)`, dry air carried sound at about 299 m/s on the coldest day and about 358 m/s on the hottest, so a thunderclap a kilometer away would have reached a listener near Tower in 3.35 s and one in Beardsley in 2.79 s. ## See also - [[Sound_speed_gradient]] - [[Refraction]] - [[Acoustic_wave_equation]] - [[Sound]] - [[Doppler_effect]] - [[Underwater_acoustics]] ## References [^openstax17-2]: Ling, Samuel J.; Sanny, Jeff; Moebs, William (2016). *University Physics Volume 1*. OpenStax. §17.2 "Speed of Sound," pp. 810–817: Eqs. 17.3–17.7, Table 17.1 (gases at 0 °C; liquids at 20 °C; solids longitudinal or bulk), the flash-to-thunder rule and the marching-band argument. https://openstax.org/details/books/university-physics-volume-1 (Portal Books 077.) [^gea12]: Gea-Banacloche, Julio (2019). *University Physics I: Classical Mechanics*. University of Arkansas Open Educational Resources. §12.1.3 "The wave velocity," pp. 285–286 (c = sqrt(B/ρ₀) and sqrt(Y/ρ₀); helium faster and sulfur hexafluoride slower than air; water stiffer than air). https://scholarworks.uark.edu/oer/3 (Portal Books 076.) [^mersenne1636]: Mersenne, Marin (1636). *Harmonie universelle*. Paris: Sébastien Cramoisy. *Citation needed* for the later cannon-fire timings: that they approached the modern value is standard in histories of acoustics, but no primary figure was checked for this article; a scan of *Harmonie universelle* and of William Derham's 1708 paper in *Philosophical Transactions* would settle it. [^laplace1816]: Laplace, Pierre-Simon (1816). "Sur la vitesse du son dans l'air et dans l'eau." *Annales de chimie et de physique* 3: 238–241. Volume and page as commonly cited; not checked against a scan for this article. Newton's isothermal theory: *Philosophiæ Naturalis Principia Mathematica* (1687), Book II, Section VIII. The 15 percent figure is the ratio 1/sqrt(1.4). [^dosits-colladon]: Discovery of Sound in the Sea (University of Rhode Island Graduate School of Oceanography). "The First Studies of Underwater Acoustics: The 1800s." https://dosits.org/people-and-sound/history-of-underwater-acoustics/the-first-studies-of-underwater-acoustics-the-1800s/ Primary report: Colladon, J.-D.; Sturm, C. (1827). "Mémoire sur la compression des liquides." *Annales de chimie et de physique* 36 (volume as commonly cited; not checked against a scan). [^kinsler2000]: Kinsler, Lawrence E.; Frey, Austin R.; Coppens, Alan B.; Sanders, James V. (2000). *Fundamentals of Acoustics* (4th ed.). Wiley. ISBN 9780471847892. Ch. 3 "Vibrations of bars" (the thin-rod speed sqrt(Y/ρ)); the isotropic-solid speeds c_p and c_s are the standard results of linear elasticity treated there and in Pierce (2019). [^ussa1976]: NOAA, NASA and U.S. Air Force (1976). *U.S. Standard Atmosphere, 1976*. NOAA-S/T 76-1562. https://ntrs.nasa.gov/citations/19770009539 [^pierce2019]: Pierce, Allan D. (2019). *Acoustics: An Introduction to Its Physical Principles and Applications* (3rd ed.). Springer / ASA Press. Chapters on sound in inhomogeneous and moving media (ray acoustics, refraction by temperature and wind gradients). https://doi.org/10.1007/978-3-030-11214-1 [^wong1986]: Wong, George S. K. (1986). "Speed of sound in standard air." *Journal of the Acoustical Society of America* 79 (5): 1359–1366. https://doi.org/10.1121/1.393664 [^cramer1993]: Cramer, Owen (1993). "The variation of the specific heat ratio and the speed of sound in air with temperature, pressure, humidity, and CO2 concentration." *Journal of the Acoustical Society of America* 93 (5): 2510–2516. https://doi.org/10.1121/1.405827 [^maurice2022]: Maurice, S.; Chide, B.; Murdoch, N.; et al. (2022). "In situ recording of Mars soundscape." *Nature* 605: 653–658. https://doi.org/10.1038/s41586-022-04679-0 [^mach1887]: Mach, Ernst; Salcher, Peter (1887). "Photographische Fixirung der durch Projectile in der Luft eingeleiteten Vorgänge." *Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften in Wien, Mathematisch-Naturwissenschaftliche Classe* 95: 764–780. Volume and pages as commonly cited; not checked against a scan for this article. [^nasa-xs1]: NASA (image article). "October 14, 1947." https://www.nasa.gov/image-article/october-14-1947/ [^moldover1988]: Moldover, M. R.; Trusler, J. P. M.; Edwards, T. J.; Mehl, J. B.; Davis, R. S. (1988). "Measurement of the universal gas constant R using a spherical acoustic resonator." *Journal of Research of the National Bureau of Standards* 93 (2): 85–144. https://doi.org/10.6028/jres.093.010 [^moldover2014]: Moldover, M. R.; Gavioso, R. M.; Mehl, J. B.; Pitre, L.; de Podesta, M.; Zhang, J. T. (2014). "Acoustic gas thermometry." *Metrologia* 51 (1): R1–R19. https://doi.org/10.1088/0026-1394/51/1/R1 [^mackenzie1981]: Mackenzie, K. V. (1981). "Nine-term equation for sound speed in the oceans." *Journal of the Acoustical Society of America* 70 (3): 807–812. https://doi.org/10.1121/1.386920 [^thapaliya2024]: Thapaliya, Arbin; Sithole, Alec; Welsh, Michael; Dana, Gaston (2024). *Ultrasound Physics and its Application in Medicine*. PALNI Open Press. §1.8 "Propagation of Ultrasound Through Tissues," p. 9, and §2.9.10 "Propagation Speed Error Artifact." (Portal Books 091.) [^chen2016]: Chen, Francis F. (2016). *Introduction to Plasma Physics and Controlled Fusion* (3rd ed.). Springer. Ch. 4, ion acoustic waves. https://doi.org/10.1007/978-3-319-22309-4 [^munk1974]: Munk, Walter H. (1974). "Sound channel in an exponentially stratified ocean, with application to SOFAR." *Journal of the Acoustical Society of America* 55 (2): 220–226. https://doi.org/10.1121/1.1914492 [^mndnr-extremes]: Minnesota Department of Natural Resources, State Climatology Office. "Minnesota Climate Extremes." https://www.dnr.state.mn.us/climate/summaries_and_publications/extremes.html (the 115 °F Beardsley record as amended by NOAA's National Centers for Environmental Information from 114.5 °F). See also "Minnesota's All-Time Record Low," https://www.dnr.state.mn.us/climate/journal/960202_60_below.html  **Microsim — three.js (Wikitube framework):** *Speed of sound* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/acoustics/Speed_of_sound.html" data-title="Speed of sound"></div> *Built from `MICROSIM_GUIDE/specs/acoustics/sims/Speed_of_sound.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/Speed_of_sound) : [Wikitube](https://en.wikitube.io/wiki/Speed_of_sound) - skeleton pinned to revision 1373106219 (2026-09-11).