Speed of Sound
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Speed of Sound
Sound measurements
 Sound pressure  p, SPL,LPA
 Particle velocity  v, SVL
 Particle displacement  ?
 Sound intensity  I, SIL
 Sound power  P, SWL, LWA
 Sound energy  W
 Sound energy density  w
 Sound exposure  E, SEL
 Acoustic impedance  Z
 Speed of sound  c
 Audio frequency  AF
 Transmission loss  TL

The speed of sound is the distance travelled per unit time by a sound wave as it propagates through an elastic medium. In dry air at 0 °C (32 °F), the speed of sound is 331.2 metres per second (1,087 ft/s; 1,192 km/h; 741 mph; 644 kn). At 20 °C (68 °F), the speed of sound is 343 metres per second (1,125 ft/s; 1,235 km/h; 767 mph; 667 kn), or a kilometre in or a mile in .

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 ordinary air, deviating slightly from ideal behavior.

In common everyday speech, speed of sound refers to the speed of sound waves in air. However, the speed of sound varies from substance to substance: sound travels most slowly in gases; it travels faster in liquids; and faster still in solids. For example, (as noted above), sound travels at in air; it travels at in water (4.3 times as fast as in air); and at in iron (about 15 times as fast as in air). In an exceptionally stiff material such as diamond, sound travels at 12,000 metres per second (26,843 mph);[1] (about 35 times as fast as in air) which is around the maximum speed that sound will travel under normal conditions.

Sound waves in solids are composed of compression waves (just as in gases and liquids), and a different type of sound wave called a shear wave, which occurs only in solids. Shear waves in solids usually travel at different speeds, 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.



Sir Isaac Newton computed the speed of sound in air as 979 feet per second (298 m/s), which is too low by about 15%,[2]. Newton's analysis was good save for 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). This error was later rectified by Laplace.[3]

During the 17th century, there were several attempts to measure the speed of sound accurately, including attempts by Marin Mersenne in 1630 (1,380 Parisian feet per second), Pierre Gassendi in 1635 (1,473 Parisian feet per second) and Robert Boyle (1,125 Parisian feet per second).[4]

In 1709, the Reverend William Derham, Rector of Upminster, published a more accurate measure of the speed of sound, at 1,072 Parisian feet per second.[4] Derham used a telescope from the tower of the church of St 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.[5]

Basic concepts

The transmission of sound can be illustrated by using a model consisting of an array of spherical objects interconnected by springs.

In real material terms, the spheres represent the material's molecules and the springs represent the bonds between them. Sound passes through the system by compressing and expanding the springs, transmitting the acoustic energy to neighboring spheres. This helps transmit the energy in-turn to the neighboring sphere's springs (bonds), and so on.

The speed of sound through the model depends on the stiffness/rigidity of the springs, and the mass of the spheres. As long as the spacing of the spheres remains constant, stiffer springs/bonds transmit energy quicker, while larger spheres transmit the energy slower.

In a real material, the stiffness of the springs is known as the "elastic modulus", and the mass corresponds to the material density. Given that all other things being equal (ceteris paribus), sound will travel slower in spongy materials, and faster in stiffer ones. Effects like dispersion and reflection can also be understood using this model.[]



A practical example can be observed in Edinburgh, when the "One o' Clock Gun" is fired at the eastern end of Edinburgh Castle. Standing at the base of the western end of the Castle Rock, the sound of the Gun can be heard through the rock, slightly before it arrives by the air route, partly delayed by the slightly longer route. It is particularly effective if a multi-gun salute such as for "The Queen's Birthday" is being fired.

Compression and shear waves

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 waves, see below)
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 at right-angle 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 of shear-waves

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 supported in 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" of this type of wave. In general, transverse waves occur as a pair of orthogonal polarizations.


The speed of a compression wave in 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.


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

For fluids in general, the speed of sound c is given by the Newton-Laplace equation:


  • Ks is a coefficient of stiffness, the isentropic bulk modulus (or the modulus of bulk elasticity for gases);
  • ? is the density.

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 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.



  • p is the pressure;
  • ? is the density and the derivative is taken isentropically, that is, at constant entropy s.


In a 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 CO2 which is a dispersive medium, and causes dispersion to air at ultrasonic frequencies .[6]

In a 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 optical dispersion for a description.

Dependence on the properties of the medium


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).

In low molecular weight gases such as helium, sound propagates faster as compared to 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.


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.

Altitude variation and implications for atmospheric acoustics

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. 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 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.

Since temperature (and thus the speed of sound) decreases with increasing altitude up to , sound is refracted upward, away from listeners on the ground, creating an acoustic shadow at some distance from the source.[7] 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 . In particular, in the stratosphere above about , 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 aptly-named thermosphere above .

Practical formula for dry air


The approximate speed of sound in dry (0% humidity) air, in meters per second, at temperatures near , can be calculated from

where is the temperature in degrees Celsius (°C).


Dividing the first part, and multiplying the second part, on the right hand side, by gives the exactly equivalent form

which can also be written as

where T denotes the thermodynamic temperature.

The value of , which represents the speed at (or ), is based on theoretical (and some measured) values of the heat capacity ratio, ?, as well as on the fact that at 1 atm real air is very well described by the ideal gas approximation. Commonly found values for the speed of sound at may vary from 331.2 to 331.6 due to the assumptions made when it is calculated. If ideal gas ? is assumed to be exactly, the speed is calculated (see section below) to be , the coefficient used above.


A graph comparing results of the two equations is at right, using the slightly different value of for the speed of sound at .


Speed of sound in ideal gases and air

For an ideal gas, K (the bulk modulus in equations above, equivalent to C, the coefficient of stiffness in solids) is given by

thus, from the Newton-Laplace equation above, the speed of sound in an ideal gas is given by


  • ? is the adiabatic index also known as the isentropic expansion factor. It is the ratio of specific heats of a gas at a constant-pressure to a gas at a constant-volume(), 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;
  • ? is the density.

Using the ideal gas law to replace p with nRT/V, and replacing ? with nM/V, the equation for an ideal gas becomes


  • cideal is the speed of sound in an ideal gas;
  • R (approximately ) is the molar gas constant(universal gas constant);[8]
  • k is the Boltzmann constant;
  • ? (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 for diatomic molecules, according to kinetic theory. Gamma is actually experimentally measured over a range from 1.399,1 to 1.403 at , for air. Gamma is exactly for monatomic gases such as noble gases;
  • T is the absolute temperature;
  • M is the molar mass of the gas. The mean molar mass for dry air is about ;[]
  • 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 cair have been found to vary slightly from experimentally determined values.[9]

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 ? but was otherwise correct.


For air, we introduce the shorthand

In addition, we switch to the Celsius temperature , which is useful to calculate air speed in the region near 0°C (about 273 kelvin). Then, for dry air,

where (theta) is the temperature in degrees Celsius(°C).

Substituting numerical values

for the molar gas constant in J/mole/Kelvin, and

for the mean molar mass of air, in kg; and using the ideal diatomic gas value of , we have

Finally, Taylor expansion of the remaining square root in yields

The above derivation includes the first two equations given in the "Practical formula for dry air" section above.

Effects due to wind shear

The speed of sound varies with temperature. Since temperature and sound velocity normally decrease with increasing altitude, sound is refracted upward, away from listeners on the ground, creating an acoustic shadow at some distance from the source.[7] Wind shear of 4 m/(s · km) can produce refraction equal to a typical temperature lapse rate of .[10] Higher values of wind gradient will refract sound downward toward the surface in the downwind direction,[11] 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 sound is not being carried along by the wind.[12]

For sound propagation, the exponential variation of wind speed with height can be defined as follows:[13]


  • U(h) is the speed of the wind at height h;
  • ? is the exponential coefficient based on ground surface roughness, typically between 0.08 and 0.52;
  • dU/dH(h) is the expected wind gradient at 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,[14] because they could not hear the sounds of battle only (six miles) downwind.[15]


In the standard atmosphere:

  • T0 is (= ), giving a theoretical value of (= = = = ). Values ranging from 331.3 to 331.6 m/s may be found in reference literature, however;
  • T20 is (= = ), giving a value of (= = = = );
  • T25 is (= = ), giving a value of (= = = = ).


Effect of temperature on properties of air
T (°C)
Speed of sound
c (m/s)
Density of air
? (kg/m3)
Characteristic specific acoustic impedance
z0 (Pa·s/m)
35 351.88 1.1455 403.2
30 349.02 1.1644 406.5
25 346.13 1.1839 409.4
20 343.21 1.2041 413.3
15 340.27 1.2250 416.9
10 337.31 1.2466 420.5
5 334.32 1.2690 424.3
0 331.30 1.2922 428.0
-5 328.25 1.3163 432.1
-10 325.18 1.3413 436.1
-15 322.07 1.3673 440.3
-20 318.94 1.3943 444.6
-25 315.77 1.4224 449.1


Altitude Temperature m/s km/h mph kn
Sea level 340 1,225 761 661
(Cruising altitude of commercial jets,
and first supersonic flight)
295 1,062 660 573
29,000 m (Flight of X-43A) 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.[16]


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 ? (...) than diatomics do . Thus, at the same molecular mass, the speed of sound of a monatomic gas goes up by a factor of

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).


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 per degree Celsius. For this reason, the pitch of a musical wind instrument increases as its temperature increases.


The dependence on frequency and pressure are normally insignificant in practical applications. In dry air, the speed of sound increases by about as the frequency rises from to . For audible frequencies above 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.[17]

Mach number



Experimental methods


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 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 meters, and not needing something as loud as a shotgun.

Single-shot timing methods




Then v = x/t.

Other methods



A tuning fork can be held near the mouth of a long 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 (1 + 2n)?/4 where n is an integer. As the 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 = f?.

High-precision measurements in air

The effect from impurities can be significant when making high-precision measurements. Chemical desiccants 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[19] 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 but corrected for temperature in order to report them at . The result was for dry air at STP, for frequencies from to .

Non-gaseous media

Speed of sound in solids

Three-dimensional solids



The last quantity is not an independent one, as . Note that 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, , and , yielding a compressional speed csolid,p of .[20] This is in reasonable agreement with csolid,p measured experimentally at for a (possibly different) type of steel.[21] The shear speed csolid,s is estimated at using the same numbers.

One-dimensional solids



Speed of sound in liquids

Speed of sound in water vs temperature.


Hence the speed of sound in a fluid is given by

where K is the bulk modulus of the fluid.


In fresh water, sound travels at about at (see the External Links section below for online calculators).[22] Applications of underwater sound can be found in sonar, acoustic communication and acoustical oceanography.


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 is centred on the minimum in the speed of sound at ca. 750-m depth.

In salt water that is free of air bubbles or suspended sediment, sound travels at about ( at , 10°C and 3% salinity by one method).[23] The speed of sound in seawater depends on pressure (hence depth), temperature (a change of ~ ), and salinity (a change of 1? ~ ), and empirical equations have been derived to accurately calculate the speed of sound from these variables.[24][25] Other factors affecting the speed of sound are minor. Since temperature decreases with depth while pressure and generally salinity increase, the profile of the speed of sound with depth generally shows a characteristic curve which decreases to a minimum at a depth of several hundred meters, then increases again with increasing depth (right).[26] For more information see Dushaw et al.[27]



  • T is the temperature in degrees Celsius;
  • S is the salinity in parts per thousand;
  • z is the depth in meters.

The constants a1, a2, ..., a9 are

with check value for , , . This equation has a standard error of for salinity between 25 and 40 ppt. See Technical Guides. Speed of Sound in Sea-Water for an online calculator.

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[29] and the Chen-Millero-Li Equation.[27][30]

Speed of sound in plasma








See also


  1. ^ Speed of Sound
  2. ^ "The Speed of Sound". mathpages.com. Retrieved 2015. 
  3. ^ Bannon, Mike; Kaputa, Frank. "The Newton-Laplace Equation and Speed of Sound". Thermal Jackets. Retrieved 2015. 
  4. ^ a b Murdin, Paul (25 December 2008). Full Meridian of Glory: Perilous Adventures in the Competition to Measure the Earth. Springer Science & Business Media. pp. 35-36. ISBN 9780387755342. 
  5. ^ Fox, Tony (2003). Essex Journal. Essex Arch & Hist Soc. pp. 12-16. 
  7. ^ a b Everest, F. (2001). The Master Handbook of Acoustics. New York: McGraw-Hill. pp. 262-263. ISBN 0-07-136097-2. 
  8. ^ "CODATA Value: molar gas constant". Physics.nist.gov. Retrieved 2010. 
  12. ^ Singal, S. (2005). Noise Pollution and Control Strategy. Oxford: Alpha Science International. p. 7. ISBN 1-84265-237-0. 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. 
  13. ^ Bies, David (2004). Engineering Noise Control, Theory and Practice. London: Spon Press. p. 235. ISBN 0-415-26713-7. 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. 
  14. ^ Cornwall, Sir (1996). Grant as Military Commander. New York: Barnes & Noble. p. 92. ISBN 1-56619-913-1. 
  15. ^ Cozens, Peter (2006). The Darkest Days of the War: the Battles of Iuka and Corinth. Chapel Hill: The University of North Carolina Press. ISBN 0-8078-5783-1. 
  17. ^ "Speed of Sound in Air". Phy.mtu.edu. Retrieved 2014. 
  22. ^ "Speed of Sound in Water at Temperatures between 32-212 oF (0-100 oC) -- imperial and SI units". The Engineering Toolbox. 
  23. ^ Wong, George S. K.; Zhu, Shi-ming (1995). "Speed of sound in seawater as a function of salinity, temperature, and pressure". The Journal of the Acoustical Society of America. 97 (3): 1732. Bibcode:1995ASAJ...97.1732W. doi:10.1121/1.413048. 
  25. ^ Robinson, Stephen (22 Sep 2005). "Technical Guides - Speed of Sound in Sea-Water". National Physical Laboratory. Retrieved 2016. 
  26. ^ "How Fast Does Sound Travel?". Discovery of Sound in the Sea. University of Rhode Island. Retrieved 2010. 
  27. ^ a b Dushaw, Brian D.; Worcester, P. F.; Cornuelle, B. D.; Howe, B. M. (1993). "On Equations for the Speed of Sound in Seawater". Journal of the Acoustical Society of America. 93 (1): 255-275. Bibcode:1993ASAJ...93..255D. doi:10.1121/1.405660. 
  28. ^ Kenneth V., Mackenzie (1981). "Discussion of sea-water sound-speed determinations". Journal of the Acoustical Society of America. 70 (3): 801-806. Bibcode:1981ASAJ...70..801M. doi:10.1121/1.386919. 
  29. ^ Del Grosso, V. A. (1974). "New equation for speed of sound in natural waters (with comparisons to other equations)". Journal of the Acoustical Society of America. 56 (4): 1084-1091. Bibcode:1974ASAJ...56.1084D. doi:10.1121/1.1903388. 
  30. ^ Meinen, Christopher S.; Watts, D. Randolph (1997). "Further Evidence that the Sound-Speed Algorithm of Del Grosso Is More Accurate Than that of Chen and Millero". Journal of the Acoustical Society of America. 102 (4): 2058-2062. Bibcode:1997ASAJ..102.2058M. doi:10.1121/1.419655. 

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