View source for Speed of sound

{{Short description|Speed of sound wave through elastic medium}}
{{other uses}}
[[File:FA-18 Hornet breaking sound barrier (7 July 1999) - filtered.jpg|thumb|right|An [[McDonnell Douglas F/A-18 Hornet|F/A-18 Hornet]] displaying rare localized condensation at the speed of sound]]
{{Use dmy dates|date=October 2018}}
{{Sound measurements}}

The '''speed of sound''' is the distance travelled per unit of time by a [[sound wave]] as it propagates through an [[elasticity (solid mechanics)|elastic]] medium. More simply, the speed of sound is how fast vibrations travel. At {{cvt|20|C|F}}, the speed of sound in air is about {{cvt|343|m/s|ft/s km/h mph kn|0|lk=on}}, or {{val|1|ul=km}} in {{val|2.92|u=s}} or one [[mile]] in {{val|4.69|u=s}}. It depends strongly on temperature as well as the medium through which a [[sound wave]] is propagating.

At {{cvt|0|C|F}}, the speed of sound in dry [[air]] (sea level 14.7 psi) is about {{cvt|331|m/s|ft/s km/h mph kn|0}}.<ref>{{cite web |url=https://www.weather.gov/epz/wxcalc_speedofsound |title=Speed of Sound Calculator |publisher=[[National Weather Service]] |access-date=23 July 2021 }}</ref>

The speed of sound in an [[ideal gas]] depends only on its temperature and composition. The speed has a weak dependence on frequency and pressure in dry air, deviating slightly from ideal behavior.

In colloquial speech, ''speed of sound'' refers to the speed of sound waves in [[Earth's atmosphere|air]]. However, the speed of sound varies from substance to substance: typically, sound travels most slowly in [[gas]]es, faster in [[liquid]]s, and fastest in [[solids]]. 

For example, while sound travels at {{val|343|u=m/s}} in air, it travels in [[fresh water]] at {{val|1481|u=m/s}} at a temperature of {{cvt|20|C|F}}<ref>{{Cite book |title=McGraw-Hill encyclopedia of science & technology: an international reference work in twenty volumes including an index |date=2012 |publisher=McGraw-Hill |isbn=978-0-07-179273-8 |edition=10. ed. ff. |volume=17 |location=New York, NY |pages=7}}</ref> (slightly more than 4.3 times as fast) and at {{val|5,120|u=m/s}} in [[iron]] (almost 15 times as fast). In an exceptionally [[stiffness|stiff]] material such as [[diamond]], sound travels at 12,000 m/s (39,000 ft/s),<ref>{{Cite web |title=Speed of Sound |url=http://hyperphysics.phy-astr.gsu.edu/hbase/Tables/Soundv.html |access-date=2022-10-24 |author=C. R. Nave |website=HyperPhysics |publisher=Department of Physics and Astronomy, Georgia State University}}</ref>{{snd}} about 35 times its speed in air and about the fastest it can travel under normal conditions.

In theory, the speed of sound is actually the speed of vibrations. Sound waves in solids are composed of compression waves (just as in gases and liquids) and a different type of sound wave called a [[S-wave|shear wave]], which occurs only in solids. Shear waves in solids usually travel at different speeds than compression waves, as exhibited in [[seismology]]. The speed of compression waves in solids is determined by the medium's [[compressibility]], [[shear modulus]], and density. The speed of shear waves is determined only by the solid material's shear modulus and density.

In [[fluid dynamics]], the speed of sound in a fluid medium (gas or liquid) is used as a relative measure for the speed of an object moving through the medium. The ratio of the speed of an object to the speed of sound (in the same medium) is called the object's [[Mach number]]. Objects moving at speeds greater than the speed of sound (''{{gaps|Mach|1}}'') are said to be traveling at [[supersonic speed]]s.

{{Toclimit}}

==Earth==

In Earth's atmosphere, the speed of sound varies greatly from about {{cvt|295|m/s|km/h mph}} at high altitudes to about {{cvt|355|m/s|km/h mph}} at high temperatures.

==History==

The Pythagorean [[Archytas]] taught that higher pitched sound travels faster, an opinion accepted by some subsequent philosophers, such as those of the [[Platonic Academy|Academy]] and the [[Peripatetic school|Peripatos]], including possibly [[Aristotle]].<ref>{{Citation |last=Pöhlmann |first=Egert |title=Acoustics |date=2020 |work=A Companion to Ancient Greek and Roman Music |pages=243–256 |url=https://onlinelibrary.wiley.com/doi/abs/10.1002/9781119275510.ch18 |publisher=John Wiley & Sons, Ltd |language=en |doi=10.1002/9781119275510.ch18 |isbn=978-1-119-27551-0|url-access=subscription }}</ref>

[[Sir Isaac Newton]]'s 1687 ''[[Philosophiæ Naturalis Principia Mathematica|Principia]]'' includes a computation of the speed of sound in air as {{convert|979|ft/s|m/s|0}}. This is too low by about 15%.<ref>{{cite web
| url = http://mathpages.com/home/kmath109/kmath109.htm
| title = The Speed of Sound
| publisher = mathpages.com
| access-date = 3 May 2015
}}</ref> The discrepancy is due primarily to neglecting the (then unknown) effect of rapidly fluctuating temperature in a sound wave (in modern terms, sound wave compression and expansion of air is an [[adiabatic process]], not an [[isothermal process]]). Newton then invented various [[Fudge factor|fudge factors]], such as the "crassitude of the solid particles of the air", until the number agreed with the experimental measurement.<ref>{{Cite journal |last=Westfall |first=Richard S. |date=1973-02-23 |title=Newton and the Fudge Factor |url=https://www.science.org/doi/abs/10.1126/science.179.4075.751 |journal=Science |volume=179 |issue=4075 |pages=751–758 |doi=10.1126/science.179.4075.751|pmid=17806284 |bibcode=1973Sci...179..751W |url-access=subscription }}</ref> [[Joseph-Louis Lagrange|Lagrange]] and [[Leonhard Euler|Euler]] both attempted and failed to explain the discrepancy. This discrepancy was finally correctly explained by [[Pierre-Simon Laplace]].<ref name=":1" /> In ''[[Traité de mécanique céleste]]'', he used the result from the [[Clément-Desormes experiment]] of 1819, which measured the [[heat capacity ratio]] of air to be 1.35. This produced a near agreement between theory and experiment for the speed of sound. The modern value of 1.40 was found some years later, leading to complete agreement.<ref name=":0" />

During the 17th century there were several attempts to measure the speed of sound accurately. [[Marin Mersenne]] in 1630 found two values. When measuring the time (of a [[seconds pendulum]]) between seeing the flash of a gun and hearing its sound over a known distance, he found a value of 1,380 Parisian feet/second (448 m/s). However when he measured the time between firing a gun and hearing its echo from a reflecting surface of a known distance, he found 970 Paris feet per second. This led to some to theorize that echoed sound is slower than unechoed sound. Most subsequent experimenters used only his first method.<ref>{{Cite journal |last=Gouk |first=Penelope M. |date=1982 |title=Acoustics in the Early Royal Society 1660-1680 |url=https://www.jstor.org/stable/531772 |journal=Notes and Records of the Royal Society of London |volume=36 |issue=2 |pages=155–175 |doi=10.1098/rsnr.1982.0009 |jstor=531772 |issn=0035-9149}}</ref>

[[Pierre Gassendi]] in 1635 found 1,473 Parisian feet/second,<ref>{{cite journal |last=Lenihan |first=J.M.A. |date=1951 |title=Mersenne and Gassendi: An early chapter in the history of sound |journal=Acustica |volume=1 |issue=2 |pages=96–99}}</ref> and [[Robert Boyle]] 1,125 Parisian feet/second.<ref name="meridian" /> In 1650, [[Giovanni Alfonso Borelli|G. A. Borelli]] and [[Vincenzo Viviani|V. Viviani]] of the [[Accademia del Cimento]] found 350 m/s.<ref>{{Citation |last=Rossing |first=Thomas |title=A Brief History of Acoustics |date=2007 |work=Springer Handbook of Acoustics |pages=9–24 |url=https://link.springer.com/rwe/10.1007/978-0-387-30425-0_2 |publisher=Springer, New York, NY |language=en |doi=10.1007/978-0-387-30425-0_2 |bibcode=2007shac.book....9R |isbn=978-0-387-30425-0|url-access=subscription }}</ref> In 1709, the Reverend [[William Derham]], [[Rector (ecclesiastical)|Rector]] of Upminster, published a more accurate measure of the speed of sound, at 1,072 [[Paris inch|Parisian feet]] per second.<ref>{{Cite journal |last1=Motu |first1=Soni |last2=Derham |first2=W. |date=1708 |title=Experimenta & Observationes de Soni Motu, Aliisque ad id Attinentibus, Factae a Reverendo D. W. Derham Ecclesiae Upminsteriensis Rectore, & Societatis Regalis Londinensis Socio. |url=https://www.jstor.org/stable/103216 |journal= Philosophical Transactions |volume=26 |pages=2–35 |jstor=103216 |issn=0260-7085}}</ref><ref name="meridian">{{cite book
| title = Full Meridian of Glory: Perilous Adventures in the Competition to Measure the Earth
| last = Murdin
| first = Paul
| date = 25 December 2008
| publisher = Springer Science & Business Media
| isbn = 9780387755342
| pages = 35–36
}}</ref> (The [[Foot (unit)|Parisian foot]] was {{val|325|u=mm}}. This is longer than the standard "international foot" in common use today, which was officially defined in 1959 as {{val|304.8|u=mm}}, making the speed of sound at {{convert|20|C|F}} 1,055 Parisian feet per second). See <ref name=":1">{{Cite journal |last=Finn |first=Bernard S. |date=March 1964 |title=Laplace and the Speed of Sound |url=https://www.journals.uchicago.edu/doi/abs/10.1086/349791 |journal=Isis |volume=55 |issue=1 |pages=7–19 |doi=10.1086/349791 |issn=0021-1753|url-access=subscription }}</ref> for a table of more speeds of sound measured in the 1636 to 1791 period.

Derham used a telescope from the tower of the [[church of St Laurence, Upminster|church of St.&nbsp;Laurence, Upminster]] to observe the flash of a distant shotgun being fired, and then measured the time until he heard the gunshot with a half-second pendulum. Measurements were made of gunshots from a number of local landmarks, including [[North Ockendon]] church. The distance was known by [[triangulation]], and thus the speed that the sound had travelled was calculated.<ref name="Tony Fox">{{cite book
| title = Essex Journal
| last = Fox
| first = Tony
| pages = 12–16
| year = 2003
| publisher = Essex Arch & Hist Soc
}}</ref> He measured this many times under many circumstances, to find the dependence of the speed on wind, barometric pressure, temperature, and humidity. For example, he found that if wind is blowing towards the observer, the speed of sound is faster, and vice versa. However he thought temperature did not affect it, because the speed was the same in summer and winter.<ref>{{cite book |last1=Embleton |first1=Tony FW |chapter-url=https://books.google.com/books?id=bg-PlIQG4DkC&dq=Academyof+Sciences+of+Paris+in+1738%2C+cannon+as+the+source+sound&pg=PA53 |title=90-3052 Aeroacoustics of Flight Vehicles: Theory and Practice |last2=Daigle |first2=Gilles A. |date=1977 |chapter=12 Atmospheric Propagation}}</ref> He was also mistaken in finding that rain and fog reduced the speed, a conclusion that was accepted until [[John Tyndall|Tyndall]] disproved it.<ref name=":2" />{{Reference page|page=6}}

Early measurements found that the speeds of sound did not agree, and it was suspected that the speed of wind and temperature may change the speed of sound. In 1740, [[Giovanni Ludovico Bianconi|G. L. Bianconi]] showed that the speed of sound in air increases with temperature.<ref>{{cite journal |last=Blancani |first=Ioannes |date=1744 |title=Observationes physicae variae |trans-title=Various physical observations |journal=De Bononiensi Scientiarum et Artium Instituto Atque Academia Commentarii |language=la |location=Bologna |volume=2 |pages=365–366}}</ref><ref name=":0">{{Cite journal |last=Lindsay |first=R. Bruce |date=1966-04-01 |title=The Story of Acoustics |url=https://doi.org/10.1121/1.1909936 |journal=The Journal of the Acoustical Society of America |volume=39 |issue=4 |pages=629–644 |doi=10.1121/1.1909936 |pmid=5326409 |bibcode=1966ASAJ...39..629L |issn=0001-4966|url-access=subscription }}</ref> The [[French Academy of Sciences|Academy of Sciences of Paris]] in 1738 used cannon as the source sound, and found that when there is no wind, the speed of sound at 0°C was 332 m/s, which is within 1% of the modern accepted value.<ref name=":0" />

Chladni measured the speed of sound in solids by comparing the pitch of sound in a tube of air and a solid bar, and found that the speed of sound in tin is about 7.5 times greater than in air, while in copper it was about 12 times greater.<ref name=":2">{{Cite book |last=Beyer |first=Robert T. |title=Sounds of our times: two hundred years of acoustics |date=1999 |publisher=AIP Press |isbn=978-0-387-98435-3 |location=New York}}</ref>{{Reference page|page=7}} [[Jean-Baptiste Biot|Biot]] in 1808 measured the speed of sound in an iron pipe about 1000 m long, and found it was 10.5 times that of air, though he thought it was only an order of magnitude estimate, since his time-measurement had an accuracy of 0.5 seconds, longer than the time actually necessary for sound to propagate through the pipe.<ref>{{Cite journal |last=Frankel |first=Eugene |date=1977 |title=J. B. Biot and the Mathematization of Experimental Physics in Napoleonic France |url=https://www.jstor.org/stable/27757367 |journal=Historical Studies in the Physical Sciences |volume=8 |pages=33–72 |doi=10.2307/27757367 |jstor=27757367 |issn=0073-2672}}</ref><ref>{{cite journal |last=Biot |first=J. B. |date=1809 |title=Sur la propagation du son à travers les corps solides |trans-title=On the propagation of sound through solid bodies |journal=Mémoires d'Arcueil |language=fr |volume=2 |pages=405–443}}</ref> 

The first measurement of speed of sound in water was done by [[Jean-Daniel Colladon]] and [[Jacques Charles François Sturm|Charles Sturm]] at [[Lake Geneva]] in 1826. They were on two boats separated by 10 km. Colladon repeatedly pressed a lever that would, simultaneously, both ignite a bit of gunpowder above water and ring a bell in water. Sturm would listen for the bell with an underwater tube and measure the time until the sound is heard. They found a value of 1437.8 m/s in water at 8 C. This differs from the modern value by 1 m/s.<ref name=":2" />{{Reference page|pages=34-36}}<ref name=":0" /> They presented the result in a monograph.<ref>{{Cite book |last1=Colladon |first1=Daniel |url=http://archive.org/details/CHEPFL_LIPR_AXB191 |title=Mémoire sur la compression des liquides et la vitesse du son dans l'eau |last2=Sturm |first2=Charles-François |date=1887 |publisher=Genève : Schuchardt |language=French}}</ref>

[[Samuel Earnshaw]] reported in 1860 that he was at an experiment in 1822, where the sound of cannon fire came ''before'' the officer standing next to it shouting "fire". He hypothesized that this meant a loud enough sound would create discontinuity in the air (a [[shock wave]] in modern language), which propagates faster than normal sound waves.<ref>{{Citation |last1=Döge |first1=Torsten |title=Historical Development of the Knowledge of Shock and Blast Waves |date=2014 |work=The History of Theoretical, Material and Computational Mechanics - Mathematics Meets Mechanics and Engineering |pages=249–265 |editor-last=Stein |editor-first=Erwin |url=https://doi.org/10.1007/978-3-642-39905-3_15 |place=Berlin, Heidelberg |publisher=Springer |language=en |doi=10.1007/978-3-642-39905-3_15 |isbn=978-3-642-39905-3 |last2=Gebbeken |first2=Norbert|url-access=subscription }}</ref>

== Compression and shear waves ==
[[File:Onde compression impulsion 1d 30 petit.gif|thumb|Pressure-pulse, or compression-type, wave ([[longitudinal wave]]) confined to a plane. This is the only type of sound wave that travels in fluids (gases and liquids). A pressure-type wave may also travel in solids, along with other types of waves ([[transverse wave]]s, see below).|alt=|305x305px]]
[[File:Onde  cisaillement impulsion 1d 30 petit.gif|thumb|[[Transverse wave]] affecting atoms initially confined to a plane. This additional type of sound wave (additional type of elastic wave) travels only in solids, for it requires a sideways shearing motion which is supported by the presence of elasticity in the solid. The sideways shearing motion may take place in ''any'' direction which is perpendicular to the direction of wave travel (only one shear direction is shown here, at right angles to the plane). Furthermore, the right-angle shear direction may change over time and distance, resulting in different types of ''[[Polarization (waves)|polarization]]'' of shear waves.|alt=|305x305px]]

In a gas or liquid, sound consists of compression waves. In solids, waves propagate as two different types. A [[longitudinal wave]] is associated with compression and decompression in the direction of travel, and is the same process in gases and liquids, with an analogous compression-type wave in solids. Only compression waves are propagated through fluids (gases and liquids). An additional type of wave, the [[transverse wave]], also called a [[shear wave]], occurs only in solids because only solids support elastic deformations. It is due to elastic deformation of the medium perpendicular to the direction of wave travel; the direction of shear deformation is called the "[[Polarization (waves)|polarization]]" of this type of wave. In general, transverse waves occur as a pair of [[orthogonal]] polarizations{{Source?|date=May 2025}}.

These different waves (compression waves and the different polarizations of shear waves) may have different speeds at the same frequency. Therefore, they arrive at an observer at different times, an extreme example being an [[earthquake]], where sharp compression waves arrive first and rocking transverse waves seconds later.

The speed of a compression wave in a fluid is determined by the medium's [[compressibility]] and [[density]]. In solids, the compression waves are analogous to those in fluids, depending on compressibility and density, but with the additional factor of [[shear modulus]] which affects compression waves due to off-axis elastic energies which are able to influence effective tension and relaxation in a compression. The speed of shear waves, which can occur only in solids, is determined simply by the solid material's shear modulus and density{{Source?|date=May 2025}}.

==Equations==
The speed of sound in mathematical notation is conventionally represented by ''c'', from the Latin ''celeritas'' meaning "swiftness".

For fluids in general, the speed of sound ''c'' is given by the Newton–Laplace equation:
<math display="block">c = \sqrt{\frac{K_s}{\rho}},</math>
where
* <math>K_s</math> is a coefficient of stiffness, the isentropic [[bulk modulus]] (or the modulus of bulk elasticity for gases);
* <math>\rho</math> is the [[density]].

<math>K_s = \rho \left(\frac{\partial P}{\partial\rho}\right)_s</math>, where <math>P</math> is the pressure and the [[derivative]] is taken isentropically, that is, at constant [[entropy]] ''s''. This is because a sound wave travels so fast that its propagation can be approximated as an [[Adiabatic process#Various applications of the adiabatic assumption|adiabatic process]], meaning that there isn't enough time, during a pressure cycle of the sound, for significant heat conduction and radiation to occur.

Thus, the speed of sound increases with the stiffness (the resistance of an elastic body to deformation by an applied force) of the material and decreases with an increase in density. For ideal gases, the bulk modulus ''K'' is simply the gas pressure multiplied by the dimensionless [[adiabatic index]], which is about 1.4 for air under normal conditions of pressure and temperature.

For general [[equations of state]], if [[classical mechanics]] is used, the speed of sound ''c'' can be derived<ref>{{Cite web|url=https://courses.lumenlearning.com/suny-osuniversityphysics/chapter/17-2-speed-of-sound/|title=17.2 Speed of Sound {{!}} University Physics Volume 1|website=courses.lumenlearning.com|access-date=2020-01-24}}</ref> as follows:

Consider the sound wave propagating at speed <math>v</math> through a pipe aligned with the <math>x</math> axis and with a cross-sectional area of <math>A</math>. In time interval <math>dt</math> it moves length <math>dx = v \, dt</math>. In [[steady state]], the [[mass flow rate]] <math>\dot m = \rho v A </math> must be the same at the two ends of the tube, therefore the [[Mass flux#Equations for fluids|mass flux]] <math>j=\rho v </math> is constant and <math>v \, d\rho = -\rho \, dv</math>. Per [[Newton's second law]], the [[pressure-gradient force]] provides the acceleration:
<math display="block">\begin{align}
\frac{dv}{dt}
&=-\frac{1}{\rho}\frac{dP}{dx} \\[1ex]
\rightarrow
dP&=(-\rho \,dv)\frac{dx}{dt}=(v \, d\rho)v \\[1ex]
\rightarrow
v^2& \equiv c^2=\frac{dP}{d\rho}
\end{align}
</math>

And therefore:

<math display="block">c = \sqrt{\left(\frac{\partial P}{\partial\rho}\right)_s} = \sqrt{\frac{K_s}{\rho}},</math>

If [[special relativity|relativistic]] effects are important, the speed of sound is calculated from the [[relativistic Euler equations]].

In a [[Acoustic dispersion|non-dispersive medium]], the speed of sound is independent of [[sound frequency]], so the speeds of energy transport and sound propagation are the same for all frequencies. Air, a mixture of oxygen and nitrogen, constitutes a non-dispersive medium. However, air does contain a small amount of CO<sub>2</sub> which ''is'' a dispersive medium, and causes dispersion to air at [[ultrasound|ultrasonic]] frequencies (greater than {{val|28|ul=kHz}}).<ref>Dean, E. A. (August 1979). [https://web.archive.org/web/20120531095150/http://handle.dtic.mil/100.2/ADA076060 Atmospheric Effects on the Speed of Sound], Technical report of Defense Technical Information Center</ref>

In a [[Acoustic dispersion|dispersive medium]], the speed of sound is a function of sound frequency, through the [[dispersion relation]]. Each frequency component propagates at its own speed, called the [[phase velocity]], while the energy of the disturbance propagates at the [[group velocity]]. The same phenomenon occurs with light waves; see [[Dispersion (optics)#Group velocity dispersion|optical dispersion]] for a description.

==Dependence on the properties of the medium==
The speed of sound is variable and depends on the properties of the substance through which the wave is travelling. In solids, the speed of transverse (or shear) waves depends on the shear deformation under [[shear stress]] (called the [[shear modulus]]), and the density of the medium. Longitudinal (or compression) waves in solids depend on the same two factors with the addition of a dependence on [[compressibility]]{{Source?|date=May 2025}}.

In fluids, only the medium's compressibility and density are the important factors, since fluids do not transmit shear stresses. In heterogeneous fluids, such as a liquid filled with gas bubbles, the density of the liquid and the compressibility of the gas affect the speed of sound in an additive manner, as demonstrated in the [[hot chocolate effect]].

In gases, adiabatic compressibility is directly related to pressure through the [[heat capacity ratio]] (adiabatic index), while pressure and density are inversely related to the temperature and molecular weight, thus making only the completely independent properties of ''temperature and molecular structure'' important (heat capacity ratio may be determined by temperature and molecular structure, but simple molecular weight is not sufficient to determine it).

Sound propagates faster in low [[molecular weight]] gases such as [[helium]] than it does in heavier gases such as [[xenon]]. For monatomic gases, the speed of sound is about 75% of the mean speed that the atoms move in that gas.

For a given [[ideal gas]] the molecular composition is fixed, and thus the speed of sound depends only on its [[temperature]]. At a constant temperature, the gas [[pressure]] has no effect on the speed of sound, since the density will increase, and since pressure and [[density]] (also proportional to pressure) have equal but opposite effects on the speed of sound, and the two contributions cancel out exactly. In a similar way, compression waves in solids depend both on compressibility and density—just as in liquids—but in gases the density contributes to the compressibility in such a way that some part of each attribute factors out, leaving only a dependence on temperature, molecular weight, and heat capacity ratio which can be independently derived from temperature and molecular composition (see derivations below). Thus, for a single given gas (assuming the molecular weight does not change) and over a small temperature range (for which the heat capacity is relatively constant), the speed of sound becomes dependent on only the temperature of the gas.

In non-ideal gas behavior regimen, for which the [[Van der Waals equation|Van der Waals gas]] equation would be used, the proportionality is not exact, and there is a slight dependence of sound velocity on the gas pressure.

Humidity has a small but measurable effect on the speed of sound (causing it to increase by about 0.1%–0.6%), because [[oxygen]] and [[nitrogen]] molecules of the air are replaced by lighter molecules of [[water]]. This is a simple mixing effect{{Source?|date=May 2025}}.

==Altitude variation and implications for atmospheric acoustics==
[[File:Comparison US standard atmosphere 1962.svg|thumb|Density and pressure decrease smoothly with altitude, but temperature (red) does not. The speed of sound (blue) depends only on the complicated temperature variation at altitude and can be calculated from it since isolated density and pressure effects on the speed of sound cancel each other. The speed of sound increases with height in two regions of the stratosphere and thermosphere, due to heating effects in these regions.]]

In the [[Earth's atmosphere]], the chief factor affecting the speed of sound is the [[temperature]]. For a given ideal gas with constant heat capacity and composition, the speed of sound is dependent ''solely'' upon temperature; see ''{{section link|#Details}}'' below. In such an ideal case, the effects of decreased density and decreased pressure of altitude cancel each other out, save for the residual effect of temperature{{Source?|date=May 2025}}.

Since temperature (and thus the speed of sound) decreases with increasing altitude up to {{val|11|u=km}}, sound is [[refraction|refracted]] upward, away from listeners on the ground, creating an [[acoustic shadow]] at some distance from the source.<ref name="Everest2001">{{cite book
| last = Everest
| first = F.
| title = The Master Handbook of Acoustics
| publisher = McGraw-Hill
| location = New York
| year = 2001
| isbn = 978-0-07-136097-5
| pages = 262–263
}}</ref> The decrease of the speed of sound with height is referred to as a negative [[sound speed gradient]].

However, there are variations in this trend above {{val|11|u=km}}. In particular, in the [[stratosphere]] above about {{val|20|u=km}}, the speed of sound increases with height, due to an increase in temperature from heating within the [[ozone layer]]. This produces a positive speed-of-sound gradient in this region. Still another region of positive gradient occurs at very high altitudes, in the [[thermosphere]] above {{val|90|u=km}}{{Source?|date=May 2025}}.

==Details==
===Speed of sound in ideal gases and air===
<!-- This section is linked from [[Supersonic]] -->
For an ideal gas, ''K'' (the [[bulk modulus]] in equations above, equivalent to ''C'', the coefficient of stiffness in solids) is given by
<math display="block">K = \gamma \cdot p .</math>
Thus, from the Newton–Laplace equation above, the speed of sound in an ideal gas is given by
<math display="block">c = \sqrt{\gamma \cdot {p \over \rho}},</math>
where
* ''γ'' is the [[adiabatic index]] also known as the ''isentropic expansion factor''. It is the ratio of the specific heat of a gas at constant pressure to that of a gas at constant volume (<math>C_p/C_v</math>) and arises because a classical sound wave induces an adiabatic compression, in which the heat of the compression does not have enough time to escape the pressure pulse, and thus contributes to the pressure induced by the compression;
* ''p'' is the [[pressure]];
* ''ρ'' is the [[density]].

Using the [[ideal gas]] law to replace ''p'' with ''nRT''/''V'', and replacing ''ρ'' with ''nM''/''V'', the equation for an ideal gas becomes
<math display="block">c_{\mathrm{ideal}} = \sqrt{\gamma \cdot {p \over \rho}} = \sqrt{\gamma \cdot R \cdot T \over M} = \sqrt{\gamma \cdot k \cdot T \over m},</math>
where
* ''c''<sub>ideal</sub> is the speed of sound in an [[ideal gas]];
* ''R'' is the [[molar gas constant]];
* ''k'' is the [[Boltzmann constant]];
* ''γ'' (gamma) is the [[adiabatic index]]. At room temperature, where thermal energy is fully partitioned into rotation (rotations are fully excited) but quantum effects prevent excitation of vibrational modes, the value is {{math|1=7/5 = 1.400}} for diatomic gases (such as [[oxygen]] and [[nitrogen]]), according to kinetic theory. Gamma is actually experimentally measured over a range from 1.3991 to 1.403 at {{val|0|u=degC}}, for air. Gamma is exactly {{math|1=5/3 = 1.667}} for monatomic gases (such as [[argon]]) and it is {{math|1=4/3 = 1.333}} for triatomic molecule gases that, like {{chem|link=water|H|2|O}}, are not co-linear (a co-linear triatomic gas such as {{chem|C|O|2}} is equivalent to a diatomic gas for our purposes here);
* ''T'' is the absolute temperature;
* ''M'' is the molar mass of the gas. The mean molar mass for dry air is about {{cvt|0.02897|kg/mol|g/mol}};
* ''n'' is the number of moles;
* ''m'' is the mass of a single molecule.

This equation applies only when the sound wave is a small perturbation on the ambient condition, and the certain other noted conditions are fulfilled, as noted below. Calculated values for ''c''<sub>air</sub> have been found to vary slightly from experimentally determined values.<ref name=USSA1976>U.S. Standard Atmosphere, 1976, U.S. Government Printing Office, Washington, D.C., 1976.</ref>

[[Isaac Newton|Newton]] famously considered the speed of sound before most of the development of [[thermodynamics]] and so incorrectly used [[isothermal]] calculations instead of [[adiabatic]]. His result was missing the factor of ''γ'' but was otherwise correct.

Numerical substitution of the above values gives the ideal gas approximation of sound velocity for gases, which is accurate at relatively low gas pressures and densities (for air, this includes standard Earth sea-level conditions). Also, for diatomic gases the use of {{math|1=''γ'' = 1.4000}} requires that the gas exists in a temperature range high enough that rotational heat capacity is fully excited (i.e., molecular rotation is fully used as a heat energy "partition" or reservoir); but at the same time the temperature must be low enough that molecular vibrational modes contribute no heat capacity (i.e., insignificant heat goes into vibration, as all vibrational quantum modes above the minimum-energy-mode have energies that are too high to be populated by a significant number of molecules at this temperature). For air, these conditions are fulfilled at room temperature, and also temperatures considerably below room temperature (see tables below). See the section on gases in [[specific heat capacity]] for a more complete discussion of this phenomenon.

For air, we introduce the shorthand
<math display="block">R_* = R/M_{\mathrm{air}}.</math>

[[File:Speed of sound in dry air.svg|thumb|Approximation of the speed of sound in dry air based on the [[heat capacity ratio]] (in green) against the truncated [[Taylor expansion]] (in red)]]

In addition, we switch to the [[Celsius]] temperature {{math|1=''θ'' = ''T'' − {{val|273.15|u=K}}}}, which is useful to calculate air speed in the region near {{val|0|u=degC}} ({{val|273|u=K}}). Then, for dry air,
<math display=block>\begin{align}
c_{\mathrm{air}} &= \sqrt{\gamma \cdot R_* \cdot T} = \sqrt{\gamma \cdot R_* \cdot (\theta + 273.15\,\mathrm{K})},\\
c_{\mathrm{air}} &= \sqrt{\gamma \cdot R_* \cdot 273.15\,\mathrm{K}} \cdot \sqrt{1 + \frac{\theta}{273.15\,\mathrm{K}}} .
\end{align}</math>

Substituting numerical values
<math display=block>R = 8.314\,462\,618\,153\,24~\mathrm{J/(mol{\cdot}K)}</math>
<math display=block>M_{\mathrm{air}} = 0.028\,964\,5~\mathrm{kg/mol}</math>
and using the ideal diatomic gas value of {{math|1=''γ'' = 1.4000}}, we have
<math display=block>c_{\mathrm{air}} \approx 331.3\,\mathrm{m/s} \times \sqrt{1 + \frac{\theta}{273.15\,\mathrm{K}}} .</math>

Finally, Taylor expansion of the remaining square root in <math>\theta</math> yields
<math display=block>\begin{align}
c_{\mathrm{air}} & \approx 331.3\,\mathrm{m/s} \times \left(1 + \frac{\theta}{2 \times 273.15\,\mathrm{K}}\right),\\
                 & \approx 331.3\,\mathrm{m/s} + \theta \times 0.606 \,\mathrm{(m/s)/^\circ C} .
\end{align}</math>

A graph comparing results of the two equations is to the right, using the slightly more accurate value of {{cvt|331.5|m/s}} for the speed of sound at {{val|0|u=degC}}.<ref name="Kinsler2000">{{cite book| last1=Kinsler|first1=L.E.| title=Fundamentals of Acoustics | edition = 4th | url=https://archive.org/details/fundamentalsacou00kins_265| url-access=limited| last2=Frey|first2=A.R.| last3=Coppens|first3=A.B.| last4=Sanders|first4=J.V. | publisher=John Wiley & Sons|year=2000|isbn=0-471-84789-5|location=New York}}</ref>{{rp|pages=[https://archive.org/details/fundamentalsacou00kins_265/page/n135 120]-121}}

===Effects due to wind shear===
The speed of sound varies with temperature. Since temperature and sound velocity normally decrease with increasing altitude, sound is [[refraction|refracted]] upward, away from listeners on the ground, creating an [[acoustic shadow]] at some distance from the source.<ref name="Everest2001" /> Wind shear of 4 m/(s · km) can produce refraction equal to a typical temperature [[lapse rate]] of {{val|7.5|u=°C/km}}.<ref>{{cite book
| last = Uman
| first = Martin
| title = Lightning
| publisher = Dover Publications
| location = New York
| year = 1984
| isbn = 978-0-486-64575-9
| url-access = registration
| url = https://archive.org/details/trent_0116300718198
}}</ref> Higher values of wind gradient will refract sound downward toward the surface in the downwind direction,<ref>{{cite book
| last = Volland
| first = Hans
| title = Handbook of Atmospheric Electrodynamics
| publisher = CRC Press
| location = Boca Raton
| year = 1995
| isbn = 978-0-8493-8647-3
| page = 22
}}</ref> eliminating the acoustic shadow on the downwind side. This will increase the audibility of sounds downwind. This downwind refraction effect occurs because there is a wind gradient; the fact that sound is carried along by the wind is not important.<ref>{{cite book
| last = Singal
| first = S.
| title = Noise Pollution and Control Strategy
| publisher = Alpha Science International
| location = Oxford
| year = 2005
| isbn = 978-1-84265-237-4
| page = 7
| quote = It may be seen that refraction effects occur only because there is a wind gradient and it is not due to the result of sound being convected along by the wind.
}}</ref>

For sound propagation, the exponential variation of wind speed with height can be defined as follows:<ref>{{cite book
| last = Bies
| first = David
| title = Engineering Noise Control, Theory and Practice
| publisher = CRC Press
| location = London
| year = 2009
| isbn = 978-0-415-26713-7
| page = 249 |
quote = As wind speed generally increases with altitude, wind blowing towards the listener from the source will refract sound waves downwards, resulting in increased noise levels.
}}</ref>
<math display="block">\begin{align}
U(h) &= U(0) h^\zeta, \\
\frac{\mathrm{d}U}{\mathrm{d}h}(h) &= \zeta \frac{U(h)}{h},
\end{align}</math>
where
* ''U''(''h'') is the speed of the wind at height ''h'';
* ''ζ'' is the exponential coefficient based on ground surface roughness, typically between 0.08 and 0.52;
* ''dU''/''dh''(''h'') is the rate of change of the wind speed with respect to the height h, which is proportional to the expected wind gradient at fixed height ''h''.

In the 1862 [[American Civil War]] [[Battle of Iuka]], an acoustic shadow, believed to have been enhanced by a northeast wind, kept two divisions of Union soldiers out of the battle,<ref>{{cite book
| last = Cornwall
| first = Sir
| title = Grant as Military Commander
| publisher = Barnes & Noble
| location = New York
| year = 1996
| isbn = 978-1-56619-913-1
| page = 92
}}</ref> because they could not hear the sounds of battle only {{val|10|u=km}} (six miles) downwind.<ref>{{cite book
| last = Cozens
| first = Peter
| title = The Darkest Days of the War: the Battles of Iuka and Corinth
| publisher = The University of North Carolina Press
| location = Chapel Hill
| year = 2006
| isbn = 978-0-8078-5783-0
}}</ref>

===Tables===
In the [[Standard temperature and pressure|standard atmosphere]]:
* ''T''<sub>0</sub> is {{val|273.15|u=K}} (= {{val|0|u=degC}} = {{val|32|u=degF}}), giving a theoretical value of {{val|331.3|u=m/s}} (= {{val|1086.9|u=ft/s}} = {{val|1193|u=km/h}} = {{val|741.1|u=mph}} = {{nobreak|644.0 [[Knot (unit)|kn]]}}). Values ranging from 331.3 to {{val|331.6|u=m/s}} may be found in reference literature, however;
* ''T''<sub>20</sub> is {{val|293.15|u=K}} (= {{val|20|u=degC}} = {{val|68|u=degF}}), giving a value of {{val|343.2|u=m/s}} (= {{val|1126.0|u=ft/s}} = {{val|1236|u=km/h}} = {{val|767.8|u=mph}} = {{nobreak|667.2 [[Knot (unit)|kn]]}});
* ''T''<sub>25</sub> is {{val|298.15|u=K}} (= {{val|25|u=degC}} = {{val|77|u=degF}}), giving a value of {{val|346.1|u=m/s}} (= {{val|1135.6|u=ft/s}} = {{val|1246|u=km/h}} = {{val|774.3|u=mph}} = {{nobreak|672.8 [[Knot (unit)|kn]]}}).

In fact, assuming an [[ideal gas]], the speed of sound ''c'' depends on temperature and composition only, '''not on the pressure''' or '''density''' (since these change in lockstep for a given temperature and cancel out). Air is almost an ideal gas. The temperature of the air varies with altitude, giving the following variations in the speed of sound using the standard atmosphere—''actual conditions may vary''.

{{Temperature effect}}

Given normal atmospheric conditions, the temperature, and thus speed of sound, varies with altitude:

{| class="wikitable"
|- style="background:#f0f0f0;"
!Altitude
!Temperature
!m/s
!km/h
!mph
!kn
|-
|Sea level
|{{val|15|u=degC}} ({{val|59|u=degF}})
|340
|1,225
|761
|661
|-
|{{nobreak|11,000 m}} to {{nobreak|20,000 m}}<br>(cruising altitude of commercial jets,<br>and [[Bell X-1|first supersonic flight]])
| {{val|−57|u=degC}} ({{val|-70|u=degF}})
|295
|1,062
|660
|573
|-
|29,000 m (flight of [[Boeing X-43|X-43A]])
| {{val|-48|u=degC}} ({{val|-53|u=degF}})
|301
|1,083
|673
|585
|}

==Effect of frequency and gas composition==

===General physical considerations===
The medium in which a sound wave is travelling does not always respond adiabatically, and as a result, the speed of sound can vary with frequency.<ref>[[Albert Beaumont Wood|A B Wood]], A Textbook of Sound (Bell, London, 1946)</ref>

The limitations of the concept of speed of sound due to extreme attenuation are also of concern. The attenuation which exists at sea level for high frequencies applies to successively lower frequencies as atmospheric pressure decreases, or as the [[mean free path]] increases. For this reason, the concept of speed of sound (except for frequencies approaching zero) progressively loses its range of applicability at high altitudes.<ref name="USSA1976"/> The standard equations for the speed of sound apply with reasonable accuracy only to situations in which the wavelength of the sound wave is considerably longer than the mean free path of molecules in a gas.

The molecular composition of the gas contributes both as the mass (M) of the molecules, and their heat capacities, and so both have an influence on speed of sound. In general, at the same molecular mass, monatomic gases have slightly higher speed of sound (over 9% higher) because they have a higher ''γ'' ({{nobreak|1=5/3 = 1.66}}...) than diatomics do ({{nobreak|1=7/5 = 1.4}}). Thus, at the same molecular mass, the speed of sound of a monatomic gas goes up by a factor of
<math display="block">{c_{\mathrm{gas,monatomic}} \over c_{\mathrm{gas,diatomic}}} = \sqrt{{{{5/3} \over {7/5}}}} = \sqrt{25 \over 21} = 1.091\ldots</math>

This gives the 9% difference, and would be a typical ratio for speeds of sound at room temperature in [[helium]] vs. [[deuterium]], each with a molecular weight of 4. Sound travels faster in helium than deuterium because adiabatic compression heats helium more since the helium molecules can store heat energy from compression only in translation, but not rotation. Thus helium molecules (monatomic molecules) travel faster in a sound wave and transmit sound faster. (Sound travels at about 70% of the mean molecular speed in gases; the figure is 75% in monatomic gases and 68% in diatomic gases).

In this example we have assumed that temperature is low enough that heat capacities are not influenced by molecular vibration (see [[heat capacity]]). However, vibrational modes simply cause gammas which decrease toward 1, since vibration modes in a polyatomic gas give the gas additional ways to store heat which do not affect temperature, and thus do not affect molecular velocity and sound velocity. Thus, the effect of higher temperatures and vibrational heat capacity acts to increase the difference between the speed of sound in monatomic vs. polyatomic molecules, with the speed remaining greater in monatomics.

===Practical application to air===
By far, the most important factor influencing the speed of sound in air is temperature. The speed is proportional to the square root of the absolute temperature, giving an increase of about {{val|0.6|u=m/s}} per degree Celsius. For this reason, the pitch of a musical wind instrument increases as its temperature increases.

The speed of sound is raised by humidity. The difference between 0% and 100% humidity is about {{val|1.5|u=m/s}} at standard pressure and temperature, but the size of the humidity effect increases dramatically with temperature.

The dependence on frequency and pressure are normally insignificant in practical applications. In dry air, the speed of sound increases by about {{val|0.1|u=m/s}} as the frequency rises from {{val|10|u=Hz}} to {{val|100|u=Hz}}. For audible frequencies above {{val|100|u=Hz}} it is relatively constant. Standard values of the speed of sound are quoted in the limit of low frequencies, where the wavelength is large compared to the mean free path.<ref>{{cite web
| url = http://www.phy.mtu.edu/~suits/SpeedofSound.html
| title = Speed of Sound in Air
| publisher = Phy.mtu.edu
| access-date = 13 June 2014
| archive-date = 23 June 2017
| archive-url = https://web.archive.org/web/20170623092133/http://www.phy.mtu.edu/~suits/SpeedofSound.html
| url-status = dead
}}</ref>

As shown above, the approximate value 1000/3 = 333.33... m/s is exact a little below {{val|5|u=degC}} and is a good approximation for all "usual" outside temperatures (in temperate climates, at least), hence the usual rule of thumb to determine how far lightning has struck: count the seconds from the start of the lightning flash to the start of the corresponding roll of thunder and divide by 3: the result is the distance in kilometers to the nearest point of the lightning bolt.  Or divide the number of seconds by 5 for an approximate distance in miles.

==Mach number==
{{Main|Mach number}}
Mach number, a useful quantity in aerodynamics, is the ratio of air [[speed]] to the local speed of sound. At altitude, for reasons explained, Mach number is a function of temperature.
Aircraft [[flight instruments]], however, operate using pressure differential to compute Mach number, not temperature. The assumption is that a particular pressure represents a particular altitude and, therefore, a standard temperature. Aircraft flight instruments need to operate this way because the stagnation pressure sensed by a [[Pitot tube]] is dependent on altitude as well as speed.

==Experimental methods==
A range of different methods exist for the measurement of the speed of sound in air.

The earliest reasonably accurate estimate of the speed of sound in air was made by [[William Derham]] and acknowledged by [[Isaac Newton]]. Derham had a telescope at the top of the tower of the [[Church of St Laurence, Upminster|Church of St Laurence]] in [[Upminster]], England. On a calm day, a synchronized pocket watch would be given to an assistant who would fire a shotgun at a pre-determined time from a conspicuous point some miles away, across the countryside. This could be confirmed by telescope. He then measured the interval between seeing gunsmoke and arrival of the sound using a half-second pendulum. The distance from where the gun was fired was found by triangulation, and simple division (distance/time) provided velocity. Lastly, by making many observations, using a range of different distances, the inaccuracy of the half-second pendulum could be averaged out, giving his final estimate of the speed of sound. Modern stopwatches enable this method to be used today over distances as short as 200–400 metres, and not needing something as loud as a shotgun.

===Single-shot timing methods===
The simplest concept is the measurement made using two [[microphone]]s and a fast recording device such as a [[Digital data|digital]] storage scope. This method uses the following idea.

If a sound source and two microphones are arranged in a straight line, with the sound source at one end, then the following can be measured:
# The distance between the microphones ({{mvar|x}}), called microphone basis.
# The time of arrival between the signals (delay) reaching the different microphones ({{mvar|t}}).

Then {{math|1=''v'' = ''x''/''t''}}.

===Other methods===
In these methods, the [[time]] measurement has been replaced by a measurement of the inverse of time ([[frequency]]).

[[Kundt's tube]] is an example of an experiment which can be used to measure the speed of sound in a small volume. It has the advantage of being able to measure the speed of sound in any gas. This method uses a powder to make the [[Node (physics)|nodes]] and [[antinode]]s visible to the human eye. This is an example of a compact experimental setup.

A [[tuning fork]] can be held near the mouth of a long [[pipe (material)|pipe]] which is dipping into a barrel of [[water]]. In this system it is the case that the pipe can be brought to resonance if the length of the air column in the pipe is equal to {{math|(1 + 2''n'')''λ''/4}} where ''n'' is an integer. As the [[antinode|antinodal]] point for the pipe at the open end is slightly outside the mouth of the pipe it is best to find two or more points of resonance and then measure half a wavelength between these.

Here it is the case that ''v'' = ''fλ''.

===High-precision measurements in air===
The effect of impurities can be significant when making high-precision measurements. Chemical [[desiccant]]s can be used to dry the air, but will, in turn, contaminate the sample. The air can be dried cryogenically, but this has the effect of removing the carbon dioxide as well; therefore many high-precision measurements are performed with air free of carbon dioxide rather than with natural air. A 2002 review<ref>Zuckerwar, Handbook of the speed of sound in real gases, p. 52</ref> found that a 1963 measurement by Smith and Harlow using a cylindrical resonator gave "the most probable value of the standard speed of sound to date." The experiment was done with air from which the carbon dioxide had been removed, but the result was then corrected for this effect so as to be applicable to real air. The experiments were done at {{val|30|u=degC}} but corrected for temperature in order to report them at {{val|0|u=degC}}. The result was {{nobreak|331.45 ± 0.01 m/s}} for dry air at STP, for frequencies from {{val|93|u=Hz}} to {{nobreak|1,500 Hz}}.

==Non-gaseous media==

===Speed of sound in solids===

====Three-dimensional solids====
In a solid, there is a non-zero stiffness both for volumetric deformations and shear deformations. Hence, it is possible to generate sound waves with different velocities dependent
on the deformation mode. Sound waves generating volumetric deformations (compression) and shear deformations (shearing) are called pressure waves (longitudinal waves) and shear waves (transverse waves), respectively. In [[earthquake]]s, the corresponding seismic waves are called [[P-wave]]s (primary waves) and [[S-wave]]s (secondary waves), respectively. The sound velocities of these two types of waves propagating in a homogeneous 3-dimensional solid are respectively given by<ref name="Kinsler2000"/>
<math display="block">c_{\mathrm{solid,p}} = \sqrt{\frac{K + \frac{4}{3}G}{\rho}} = \sqrt{\frac{E(1 - \nu)}{\rho (1 + \nu)(1 - 2 \nu)}},</math>
<math display="block">c_{\mathrm{solid,s}} = \sqrt{\frac{G}{\rho}},</math>
where
* ''K'' is the [[bulk modulus]] of the elastic materials;
* ''G'' is the [[shear modulus]] of the elastic materials;
* ''E'' is the [[Young's modulus]];
* ''ρ'' is the density;
* ''ν'' is [[Poisson's ratio]].

The last quantity is not an independent one, as {{nobreak|1=E = 3K(1 − 2ν)}}. The speed of pressure waves depends both on the pressure and shear resistance properties of the material, while the speed of shear waves depends on the shear properties only.

Typically, pressure waves travel faster in materials than do shear waves, and in earthquakes this is the reason that the onset of an earthquake is often preceded by a quick upward-downward shock, before arrival of waves that produce a side-to-side motion. For example, for a typical steel alloy, {{nobreak|1=''K'' = 170 GPa}}, {{nobreak|1=''G'' = 80 GPa}} and {{math|1=''p'' = {{val|7700|u=kg/m3}}}}, yielding a compressional speed ''c''<sub>solid,p</sub> of {{nobreak|6,000 m/s}}.<ref name="Kinsler2000"/> This is in reasonable agreement with ''c''<sub>solid,p</sub> measured experimentally at {{nobreak|5,930 m/s}} for a (possibly different) type of steel.<ref>J. Krautkrämer and H. Krautkrämer (1990), ''Ultrasonic testing of materials'', 4th fully revised edition, Springer-Verlag, Berlin, Germany, p. 497</ref> The shear speed ''c''<sub>solid,s</sub> is estimated at {{nobreak|3,200 m/s}} using the same numbers.

Speed of sound in semiconductor solids can be very sensitive to the amount of electronic dopant in them.<ref>{{cite journal| doi=10.1016/j.joule.2021.03.009|title=Charge-carrier-mediated lattice softening contributes to high zT in thermoelectric semiconductors|journal=Joule|volume=5|issue=5|page=1168-1182 | year=2021| last1=Slade|first1=Tyler |last2=Anand|first2=Shashwat|last3=Wood|first3=Max|last4=Male|first4=James|last5=Imasato|first5=Kazuki | last6=Cheikh|first6=Dean | last7=Al Malki|first7=Muath | last8=Agne|first8=Matthias | last9=Griffith|first9=Kent | last10=Bux|first10=Sabah | last11=Wolverton|first11=Chris|last12=Kanatzidis|first12=Mercouri | last13=Snyder|first13=Jeff | s2cid=233598665 |doi-access=free|bibcode=2021Joule...5.1168S }}</ref>

{| class="wikitable"
|+Longitudinal & Shear Sound Speeds in Select Solids
!Material
!Abbrevation
!Density <math>\frac{kg}{m^3}</math> 
!<math>C_L \frac{m}{s}</math>
!<math>C_s \frac{m}{s}</math>
!Ref
|-
|Lucalox
|<chem>Al2O3</chem>
|3958
|10614±79
|6374±228
|<ref name=":3">{{Cite journal |last=Borg |first=Jack Denman |last2=Dolan |first2=Dan |date=2025 |title=Sound Speeds of Solids from Ultrasonic Pulse Receiver Measurements |url=https://rgdoi.net/10.13140/RG.2.2.18031.50080 |journal=Sandia National Laboratories Report |language=en |doi=10.13140/RG.2.2.18031.50080}}</ref>
|-
|Aluminum Oxynitride
|ALON
|3678
|10091±90
|6015
|<ref name=":3" />
|-
|Aluminum, 1100
|Al-1100
|2693
|6313±32
|3115±3
|<ref name=":3" />
|-
|Aluminum, 5052 H32
|AL-5052
|2674
|6146±42
|3122±42
|<ref name=":3" />
|-
|Brass
|Brass
|8520
|4457±18
|2272±141
|<ref name=":3" />
|-
|Copper
|Cu
|8924
|4664±32
|2356±6
|<ref name=":3" />
|-
|Magnesium
|Mg
|1700
|5827±51
|3186
|<ref name=":3" />
|-
|Molybdenum
|Mo
|10196
|6381±21
|3553
|<ref name=":3" />
|-
|Polylactic acid
|PLA
|1191
|1908±6
|970±10
|<ref name=":3" />
|-
|Polymethylpentene
|TPX
|833
|2133±17
|1065±13
|<ref name=":3" />
|-
|Polycast
|PMMA
|1180
|2786±27
|1409±16
|<ref name=":3" />
|-
|Sapphire (z-axis)
|<chem>Al2O3</chem>
|3990
|11100
|6040
|<ref name=":3" />
|-
|Silica, fused Quartz
|<chem>SiO2</chem>
|2203
|5512±25
|3743
|<ref name=":3" />
|-
|Steel, Stainless, 304
|SS-304
|7834
|5912±97
|3420±4
|<ref name=":3" />
|-
|Steel, Stainless, 316
|SS-316
|7821
|5882±20
|3167±28
|<ref name=":3" />
|-
|Steel, QTLCS
|QTLCS
|7817
|5677±12
|3192±35
|<ref name=":3" />
|-
|Steel, CRS A36/1008 Mild
|A36/1008
|7896
|5596
|3154
|<ref name=":3" />
|-
|Steel, AR500
|AR500
|7787±88
|5697±37
|3254±53
|<ref name=":3" />
|-
|Steel, G90 Galvanized
|G90
|7832
|5759
|3229
|<ref name=":3" />
|-
|Steel, 4130 Chromoly
|4130
|7965
|5604±9
|3306±65
|<ref name=":3" />
|-
|Tantalum
|Ta
|16652
|4091±29
|2004±56
|<ref name=":3" />
|-
|Titanium (Nitride Coating)
|Ti
|4505±1
|6114±15
|3205±67
|<ref name=":3" />
|-
|Titanium Grade 5
|Ti-6Al-4V
|4422
|6175±53
|3460±100
|<ref name=":3" />
|-
|Tungsten
|W
|19198
|5090±38
|2852
|<ref name=":3" />
|-
|Zinc
|Zn
|7193±72
|3842±34
|2308±43
|<ref name=":3" />
|-
|Zirconia
|<chem>ZrO2</chem>
|6081±1
|6864±52
|3597±19
|<ref name=":3" />
|}

==== One-dimensional solids ====
The speed of sound for pressure waves in stiff materials such as metals is sometimes given for "long rods" of the material in question, in which the speed is easier to measure. In rods where their diameter is shorter than a wavelength, the speed of pure pressure waves may be simplified and is given by:<ref name="Kinsler2000" />{{rp|pages=[https://archive.org/details/fundamentalsacou00kins_265/page/n85 70]}}
<math display="block">c_{\mathrm{solid}} = \sqrt{\frac{E}{\rho}},</math>
where {{math|''E''}} is [[Young's modulus]]. This is similar to the expression for shear waves, save that [[Young's modulus]] replaces the [[shear modulus]]. This speed of sound for pressure waves in long rods will always be slightly less than the same speed in homogeneous 3-dimensional solids, and the ratio of the speeds in the two different types of objects depends on [[Poisson's ratio]] for the material.

===Speed of sound in liquids===
[[File:Speed of sound in water.svg|thumb|right|Speed of sound in water vs temperature]]
In a fluid, the only non-zero [[stiffness]] is to volumetric deformation (a fluid does not sustain shear forces).

Hence the speed of sound in a fluid is given by
<math display="block">c_{\mathrm{fluid}} = \sqrt{\frac{K}{\rho}},</math>
where {{math|''K''}} is the [[bulk modulus]] of the fluid.

====Water====
In fresh water, sound travels at about {{val|1481|u=m/s}} at {{val|20|u=degC}} (see the External Links section below for online calculators).<ref>{{cite web
| url = http://www.engineeringtoolbox.com/sound-speed-water-d_598.html
| title = Speed of Sound in Water at Temperatures between 32–212 oF (0–100 oC) — imperial and SI units
| website = The Engineering Toolbox
}}</ref> Applications of [[underwater acoustics|underwater sound]] can be found in [[sonar]], [[Underwater acoustics#Underwater communication|acoustic communication]] and [[acoustical oceanography]].

====Seawater====
{{See also|Sound speed profile}}
[[File:Underwater speed of sound.svg|thumb|Speed of sound as a function of depth at a position north of Hawaii in the [[Pacific Ocean]] derived from the 2005 [[World Ocean Atlas]]. The [[SOFAR channel]] spans the minimum in the speed of sound at about 750&nbsp;m depth.]]

In salt water that is free of air bubbles and suspended sediment, sound travels at about {{val|1500|u=m/s}} ({{val|1500.235|u=m/s}} at {{val|1000|ul=kilopascals}}, {{val|10|u=degC}} and 3% [[salinity]] by one method).<ref>{{cite journal|last1=Wong|first1=George S. K.|last2=Zhu|first2=Shi-ming|title=Speed of sound in seawater as a function of salinity, temperature, and pressure|journal=The Journal of the Acoustical Society of America | date=1995 | volume=97 | issue=3|page=1732|doi=10.1121/1.413048|bibcode=1995ASAJ...97.1732W}}</ref> The speed of sound in seawater depends on pressure (hence depth), temperature (a change of {{val|1|u=degC}} ~ {{val|4|u=m/s}}), and [[salinity]] (a change of 1[[Per mil|‰]] ~ {{val|1|u=m/s}}), and empirical equations have been derived to accurately calculate the speed of sound from these variables.<ref>[http://webarchive.loc.gov/all/20030402194852/http://handle.dtic.mil/100.2/ADB199453 APL-UW TR 9407 High-Frequency Ocean Environmental Acoustic Models Handbook], pp. I1-I2.</ref><ref>{{cite web |last1=Robinson|first1=Stephen |title=Technical Guides – Speed of Sound in Sea-Water|url=http://resource.npl.co.uk/acoustics/techguides/soundseawater/content.html |website=National Physical Laboratory|access-date=7 December 2016|date=22 September 2005 | archive-date=29 April 2017|archive-url=https://web.archive.org/web/20170429192345/http://resource.npl.co.uk/acoustics/techguides/soundseawater/content.html|url-status=dead}}</ref> Other factors affecting the speed of sound are minor. Since in most ocean regions temperature decreases with depth, the profile of the speed of sound with depth decreases to a minimum at a depth of several hundred metres. Below the minimum, sound speed increases again, as the effect of increasing pressure overcomes the effect of decreasing temperature (right).<ref>{{cite web
| url = http://www.dosits.org/science/soundmovement/speedofsound/
| title = How Fast Does Sound Travel?
| work = Discovery of Sound in the Sea
| publisher = University of Rhode Island
| access-date = 30 November 2010
| archive-date = 20 May 2017
| archive-url = https://web.archive.org/web/20170520205400/http://www.dosits.org/science/soundmovement/speedofsound/
| url-status = dead
}}</ref> For more information see Dushaw et al.<ref name=Dushaw93/>

An empirical equation for the speed of sound in sea water is provided by Mackenzie:<ref>{{cite journal
| last = Kenneth V.
| first = Mackenzie
| title = Discussion of sea-water sound-speed determinations
| year = 1981
| journal = Journal of the Acoustical Society of America
| volume = 70
| issue = 3
| pages = 801–806
| doi = 10.1121/1.386919
|bibcode = 1981ASAJ...70..801M}}</ref>
<math display="block">c(T, S, z) = a_1 + a_2 T + a_3 T^2 + a_4 T^3 + a_5 (S - 35) + a_6 z + a_7 z^2 + a_8 T(S - 35) + a_9 T z^3,</math>
where
* ''T'' is the temperature in degrees Celsius;
* ''S'' is the salinity in parts per thousand;
* ''z'' is the depth in metres.

The constants ''a''<sub>1</sub>, ''a''<sub>2</sub>, ..., ''a''<sub>9</sub> are
<math display="block">\begin{align}
a_1 &= 1,448.96,             & a_2 &= 4.591,                 & a_3 &= -5.304 \times 10^{-2},\\
a_4 &= 2.374 \times 10^{-4}, & a_5 &= 1.340,                 & a_6 &= 1.630 \times 10^{-2},\\
a_7 &= 1.675 \times 10^{-7}, & a_8 &= -1.025 \times 10^{-2}, & a_9 &= -7.139 \times 10^{-13},
\end{align}</math>
with check value {{val|1550.744|u=m/s}} for {{math|1=''T'' = {{val|25|u=degC}}}}, {{nobreak|1=''S'' = 35 parts per thousand}}, {{nobreak|1=''z'' = 1,000 m}}. This equation has a standard error of {{val|0.070|u=m/s}} for salinity between 25 and 40 [[Parts per thousand|ppt]]. See [http://resource.npl.co.uk/acoustics/techguides/soundseawater/ Technical Guides - Speed of sound in sea water] for an online calculator.

(The Sound Speed vs. Depth graph does ''not'' correlate directly to the MacKenzie formula.
This is due to the fact that the temperature and salinity varies at different depths.
When ''T'' and ''S'' are held constant, the formula itself is always increasing with depth.)

Other equations for the speed of sound in sea water are accurate over a wide range of conditions, but are far more complicated, e.g., that by V. A. Del Grosso<ref>{{cite journal
| last = Del Grosso
| first = V. A.
| title = New equation for speed of sound in natural waters (with comparisons to other equations)
| year = 1974
| journal = Journal of the Acoustical Society of America
| volume = 56
| issue = 4
| pages = 1084–1091
| doi = 10.1121/1.1903388 | bibcode = 1974ASAJ...56.1084D| doi-access = free
}}</ref> and the Chen-Millero-Li Equation.<ref name=Dushaw93>{{cite journal
| last1 = Dushaw
| first1 = Brian D.
| last2 = Worcester
| first2 = P. F.
| last3 = Cornuelle
| first3 = B. D.
| last4 = Howe
| first4 = B. M.
| title = On Equations for the Speed of Sound in Seawater
| year = 1993
| journal = Journal of the Acoustical Society of America
| volume = 93
| issue = 1
| pages = 255–275
| doi = 10.1121/1.405660|bibcode = 1993ASAJ...93..255D}}</ref><ref>{{cite journal
| last1 = Meinen
| first1 = Christopher S.
| last2 = Watts
| first2 = D. Randolph
| title = Further Evidence that the Sound-Speed Algorithm of Del Grosso Is More Accurate Than that of Chen and Millero
| year = 1997
| journal = Journal of the Acoustical Society of America
| volume = 102
| issue = 4
| pages = 2058–2062
| doi = 10.1121/1.419655|bibcode = 1997ASAJ..102.2058M
| s2cid = 38144335
| url = https://digitalcommons.uri.edu/cgi/viewcontent.cgi?article=1249&context=gsofacpubs
| url-access = subscription
}}</ref>

===Speed of sound in plasma===
The speed of sound in a [[Plasma (physics)|plasma]] for the common case that the electrons are hotter than the ions (but not too much hotter) is given by the formula (see [[Plasma parameters#Velocities|here]])
<math display="block">c_s = \left(\frac{\gamma Z k T_\mathrm{e}}{m_\mathrm{i}}\right)^{1/2} = \left(\frac{\gamma Z T_e}{\mu} \right)^{1/2} \times 90.85~\mathrm{m/s},</math>
where
* ''m''<sub>i</sub> is the [[ion]] mass;
* ''μ'' is the ratio of ion mass to [[proton]] mass {{nobreak|1=''μ'' = ''m''<sub>i</sub>/''m''<sub>p</sub>}};
* ''T''<sub>e</sub> is the [[electron]] temperature;
* ''Z'' is the charge state;
* ''k'' is [[Boltzmann constant]];
* ''γ'' is the [[adiabatic index]].

In contrast to a gas, the pressure and the density are provided by separate species: the pressure by the electrons and the density by the ions. The two are coupled through a fluctuating electric field.

== Mars ==
The speed of sound on [[Mars]] varies as a function of frequency. Higher frequencies travel faster than lower frequencies. Higher frequency sound from lasers travels at {{Convert|250|m/s|abbr=on}}, while low frequency sound travels at {{Convert|240|m/s|abbr=on}}.<ref>{{Cite journal |last1=Maurice |first1=S. |last2=Chide |first2=B. |last3=Murdoch |first3=N. |last4=Lorenz |first4=R. D. |last5=Mimoun |first5=D. |last6=Wiens |first6=R. C. |last7=Stott |first7=A. |last8=Jacob |first8=X. |last9=Bertrand |first9=T. |last10=Montmessin |first10=F. |last11=Lanza |first11=N. L. |last12=Alvarez-Llamas |first12=C. |last13=Angel |first13=S. M. |last14=Aung |first14=M. |last15=Balaram |first15=J. |date=2022-04-01 |title=In situ recording of Mars soundscape |journal=Nature |language=en |volume=605 |issue=7911 |pages=653–658 |doi=10.1038/s41586-022-04679-0 |pmid=35364602 |bibcode=2022Natur.605..653M |issn=1476-4687|doi-access=free |pmc=9132769 }}</ref>

==Gradients==
{{Main|Sound speed gradient}}
When sound spreads out evenly in all directions in three dimensions, the intensity drops in proportion to the inverse square of the distance. However, in the ocean, there is a layer called the 'deep sound channel' or [[SOFAR channel]] which can confine sound waves at a particular depth.

In the SOFAR channel, the speed of sound is lower than that in the layers above and below. Just as light waves will refract towards a region of higher [[refractive index]], sound waves will [[refraction|refract]] towards a region where their speed is reduced. The result is that sound gets confined in the layer, much the way light can be confined to a sheet of glass or [[optical fiber]]. Thus, the sound is confined in essentially two dimensions. In two dimensions the intensity drops in proportion to only the inverse of the distance. This allows waves to travel much further before being undetectably faint.

A similar effect occurs in the atmosphere. [[Project Mogul]] successfully used this effect to detect a [[nuclear explosion]] at a considerable distance.

==See also==
* [[Acoustoelastic effect]]
* [[Elastic wave]]
* [[Second sound]]
* [[Sonic boom]]
* [[Sound barrier]]
* [[Speeds of sound of the elements]]
* [[Underwater acoustics]]
* [[Vibrations]]
* [[Bell X-1]]

==References==
{{Reflist|30em}}

==External links==
* [https://e6bx.com/speed-of-sound/ Speed of Sound Calculator]
* [http://www.sengpielaudio.com/calculator-speedsound.htm Calculation: Speed of Sound in Air and the Temperature]
* [http://www.sengpielaudio.com/SpeedOfSoundPressure.pdf Speed of sound: Temperature Matters, Not Air Pressure]
* [http://www.pdas.com/atmos.html Properties of the U.S. Standard Atmosphere 1976]
* [http://www.mathpages.com/home/kmath109/kmath109.htm The Speed of Sound]
* [https://web.archive.org/web/20060429062028/http://www.bustertests.co.uk/answer/how-to-measure-the-speed-of-sound-in-a-laboratory/ How to Measure the Speed of Sound in a Laboratory]
* [https://www.newscientist.com/article/mg19826504.200-did-sound-once-travel-at-light-speed.html Did Sound Once Travel at Light Speed?]
* [http://www.ndt-ed.org/EducationResources/CommunityCollege/Ultrasonics/Reference%20Information/matproperties.htm Acoustic Properties of Various Materials Including the Speed of Sound] {{Webarchive|url=https://web.archive.org/web/20140216102034/http://www.ndt-ed.org/EducationResources/CommunityCollege/Ultrasonics/Reference%20Information/matproperties.htm |date=16 February 2014 }}
* [http://www.dosits.org/ Discovery of Sound in the Sea] (uses of sound by humans and other animals)

{{Authority control}}

[[Category:Fluid dynamics]]
[[Category:Aerodynamics]]
[[Category:Acoustics]]
[[Category:Sound]]
[[Category:Sound measurements]]
[[Category:Physical quantities]]
[[Category:Chemical properties]]
[[Category:Velocity|Sound]]
[[Category:Temporal rates]]