Speed of sound

For other uses, seeSpeed of sound (disambiguation).

AnF/A-18 Hornetdisplaying rare localized condensation breaking the speed of sound

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

Thespeed of soundis the distance travelled per unit of time by asound waveas it propagates through anelasticmedium. At 20 °C (68 °F), the speed of sound in air is about 343m/s(1,125ft/s;1,235km/h;767mph;667kn), or1kmin2.91 sor onemilein4.69 s.It depends strongly on temperature as well as the medium through which asound waveis propagating. At 0 °C (32 °F), the speed of sound in air is about 331 m/s (1,086 ft/s; 1,192 km/h; 740 mph; 643 kn).More simply, the speed of sound is how fast vibrations travel.

The speed of sound in anideal gasdepends 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 colloquial speech,speed of soundrefers to the speed of sound waves inair.However, the speed of sound varies from substance to substance: typically, sound travels most slowly ingases,faster inliquids,and fastest insolids.For example, while sound travels at343 m/sin air, it travels at1481 m/sinwater(almost 4.3 times as fast) and at5120 m/siniron(almost 15 times as fast). In an exceptionally stiff material such as diamond, sound travels at 12,000 m/s (39,000 ft/s),– 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 ashear wave,which occurs only in solids. Shear waves in solids usually travel at different speeds than compression waves, as exhibited inseismology.The speed of compression waves in solids is determined by the medium'scompressibility,shear modulus,and density. The speed of shear waves is determined only by the solid material's shear modulus and density.

Influid 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'sMach number.Objects moving at speeds greater than the speed of sound (Mach1) are said to be traveling atsupersonicspeeds.

Earth

In Earth's atmosphere, the speed of sound varies greatly from about 295 m/s (1,060 km/h; 660 mph) at high altitudes to about 355 m/s (1,280 km/h; 790 mph) at high temperatures.

History

Sir Isaac Newton's 1687Principiaincludes a computation of the speed of sound in air as 979 feet per second (298 m/s). This is too low by about 15%.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 anadiabatic process,not anisothermal process). This error was later rectified byLaplace.

During the 17th century there were several attempts to measure the speed of sound accurately, including attempts byMarin Mersennein 1630 (1,380 Parisian feet per second),Pierre Gassendiin 1635 (1,473 Parisian feet per second) andRobert Boyle(1,125 Parisian feet per second).In 1709, the ReverendWilliam Derham,Rectorof Upminster, published a more accurate measure of the speed of sound, at 1,072Parisian feetper second.(TheParisian footwas325 mm.This is longer than the standard "international foot" in common use today, which was officially defined in 1959 as304.8 mm,making the speed of sound at 20 °C (68 °F) 1,055 Parisian feet per second).

Derham used a telescope from the tower of thechurch of St. Laurence, Upminsterto 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, includingNorth Ockendonchurch. The distance was known bytriangulation,and thus the speed that the sound had travelled was calculated.

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 thebondsbetween 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 thestiffness/rigidity of thesprings,and the mass of the spheres. As long as the spacing of the spheres remains constant, stiffer springs/bonds transmit energy more quickly, while more massive spheres transmit energy more slowly.

In a real material, the stiffness of the springs is known as the "elastic modulus",and the mass corresponds to the materialdensity.Sound will travel more slowly inspongy materialsand faster in stiffer ones. Effects like dispersion and reflection can also be understood using this model.

Some textbooks mistakenly state that the speed of soundincreaseswith density. This notion is illustrated by presenting data for three materials, such as air, water, and steel and noting that the speed of sound is higher in the denser materials. But the example fails to take into account that the materials have vastly different compressibility, which more than makes up for the differences in density, which wouldslowwave speeds in the denser materials. An illustrative example of the two effects is that sound travels only 4.3 times faster in water than air, despite enormous differences in compressibility of the two media. The reason is that the greater density of water, which works toslowsound in water relative to the air, nearly makes up for the compressibility differences in the two media.

For instance, sound will travel 1.59 times faster in nickel than in bronze, due to the greater stiffness of nickel at about the same density. Similarly, sound travels about 1.41 times faster in light hydrogen (protium) gas than in heavy hydrogen (deuterium) gas, since deuterium has similar properties but twice the density. At the same time, "compression-type" sound will travel faster in solids than in liquids, and faster in liquids than in gases, because the solids are more difficult to compress than liquids, while liquids, in turn, are more difficult to compress than gases.

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 waveaffecting 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 inanydirection 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 ofpolarizationof shear waves.

In a gas or liquid, sound consists of compression waves. In solids, waves propagate as two different types. Alongitudinal waveis 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, thetransverse wave,also called ashear 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 oforthogonalpolarizations.

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 anearthquake,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'scompressibilityanddensity.In solids, the compression waves are analogous to those in fluids, depending on compressibility and density, but with the additional factor ofshear moduluswhich 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.

Equations

The speed of sound in mathematical notation is conventionally represented byc,from the Latinceleritasmeaning "swiftness".

For fluids in general, the speed of soundcis given by the Newton–Laplace equation: $$c={\sqrt {\frac {K_{s}}{\rho }}},$$ where

• $K_{s}$is a coefficient of stiffness, the isentropicbulk modulus(or the modulus of bulk elasticity for gases);
• $\rho$is thedensity.

$K_{s}=\rho \left({\frac {\partial P}{\partial \rho }}\right)_{s}$,where$P$is the pressure and thederivativeis taken isentropically, that is, at constantentropys.This is because a sound wave travels so fast that its propagation can be approximated as anadiabatic 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 modulusKis simply the gas pressure multiplied by the dimensionlessadiabatic index,which is about 1.4 for air under normal conditions of pressure and temperature.

For generalequations of state,ifclassical mechanicsis used, the speed of soundccan be derivedas follows:

Consider the sound wave propagating at speed$v$through a pipe aligned with the$x$axis and with a cross-sectional area of$A$.In time interval$dt$it moves length$dx=v\,dt$.Insteady state,themass flow rate${\dot {m}}=\rho vA$must be the same at the two ends of the tube, therefore themass flux$j=\rho v$is constant and$v\,d\rho =-\rho \,dv$.PerNewton's second law,thepressure-gradient forceprovides the acceleration: $${\begin{aligned}{\frac {dv}{dt}}&=-{\frac {1}{\rho }}{\frac {dP}{dx}}\\[1ex]\rightarrow dP&=(-\rho \,dv){\frac {dx}{dt}}=(v\,d\rho )v\\[1ex]\rightarrow v^{2}&\equiv c^{2}={\frac {dP}{d\rho }}\end{aligned}}$$

And therefore:

$$c={\sqrt {\left({\frac {\partial P}{\partial \rho }}\right)_{s}}}={\sqrt {\frac {K_{s}}{\rho }}},$$

Ifrelativisticeffects are important, the speed of sound is calculated from therelativistic Euler equations.

In anon-dispersive medium,the speed of sound is independent ofsound 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 CO₂whichisa dispersive medium, and causes dispersion to air atultrasonicfrequencies (greater than28kHz).

In adispersive medium,the speed of sound is a function of sound frequency, through thedispersion relation.Each frequency component propagates at its own speed, called thephase velocity,while the energy of the disturbance propagates at thegroup velocity.The same phenomenon occurs with light waves; seeoptical dispersionfor 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 undershear stress(called theshear 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 oncompressibility.

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 thehot chocolate effect.

In gases, adiabatic compressibility is directly related to pressure through theheat capacity ratio(adiabatic index), while pressure and density are inversely related to the temperature and molecular weight, thus making only the completely independent properties oftemperature and molecular structureimportant (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 lowmolecular weightgases such asheliumthan it does in heavier gases such asxenon.For monatomic gases, the speed of sound is about 75% of the mean speed that the atoms move in that gas.

For a givenideal gasthe molecular composition is fixed, and thus the speed of sound depends only on itstemperature.At a constant temperature, the gaspressurehas no effect on the speed of sound, since the density will increase, and since pressure anddensity(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 theVan der Waals gasequation 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%), becauseoxygenandnitrogenmolecules of the air are replaced by lighter molecules ofwater.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. The speed of sound increases with height in two regions of the stratosphere and thermosphere, due to heating effects in these regions.

In theEarth's atmosphere,the chief factor affecting the speed of sound is thetemperature.For a given ideal gas with constant heat capacity and composition, the speed of sound is dependentsolelyupon temperature; see§ Detailsbelow. 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 to11 km,sound isrefractedupward, away from listeners on the ground, creating anacoustic shadowat some distance from the source.The decrease of the speed of sound with height is referred to as a negativesound speed gradient.

However, there are variations in this trend above11 km.In particular, in thestratosphereabove about20 km,the speed of sound increases with height, due to an increase in temperature from heating within theozone layer.This produces a positive speed of sound gradient in this region. Still another region of positive gradient occurs at very high altitudes, in thethermosphereabove90 km.

Details

Speed of sound in ideal gases and air

For an ideal gas,K(thebulk modulusin equations above, equivalent toC,the coefficient of stiffness in solids) is given by $$K=\gamma \cdot p.$$ Thus, from the Newton–Laplace equation above, the speed of sound in an ideal gas is given by $$c={\sqrt {\gamma \cdot {p \over \rho }}},$$ where

• γis theadiabatic indexalso known as theisentropic expansion factor.It is the ratio of the specific heat of a gas at constant pressure to that of a gas at constant volume ($C_{p}/C_{v}$) 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;
• pis thepressure;
• ρis thedensity.

Using theideal gaslaw to replacepwithnRT/V,and replacingρwithnM/V,the equation for an ideal gas becomes $$c_{\mathrm {ideal} }={\sqrt {\gamma \cdot {p \over \rho }}}={\sqrt {\gamma \cdot R\cdot T \over M}}={\sqrt {\gamma \cdot k\cdot T \over m}},$$ where

• c_idealis the speed of sound in anideal gas;
• Ris themolar gas constant;
• kis theBoltzmann constant;
• γ(gamma) is theadiabatic 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 is7/5 = 1.400for diatomic gases (such asoxygenandnitrogen), according to kinetic theory. Gamma is actually experimentally measured over a range from 1.3991 to 1.403 at0 °C,for air. Gamma is exactly5/3 = 1.667for monatomic gases (such asargon) and it is4/3 = 1.333for triatomic molecule gases that, likeH
  ₂O,are not co-linear (a co-linear triatomic gas such asCO
  ₂is equivalent to a diatomic gas for our purposes here);
• Tis the absolute temperature;
• Mis the molar mass of the gas. The mean molar mass for dry air is about 0.02897 kg/mol (28.97 g/mol);
• nis the number of moles;
• mis 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 forc_airhave been found to vary slightly from experimentally determined values.

Newtonfamously considered the speed of sound before most of the development ofthermodynamicsand so incorrectly usedisothermalcalculations instead ofadiabatic.His result was missing the factor ofγbut 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 ofγ= 1.4000requires 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 inspecific heat capacityfor a more complete discussion of this phenomenon.

For air, we introduce the shorthand $$R_{*}=R/M_{\mathrm {air} }.$$

Approximation of the speed of sound in dry air based on theheat capacity ratio(in green) against the truncatedTaylor expansion(in red)

In addition, we switch to theCelsiustemperatureθ=T−273.15 K,which is useful to calculate air speed in the region near0 °C(273 K). Then, for dry air, $${\begin{aligned}c_{\mathrm {air} }&={\sqrt {\gamma \cdot R_{*}\cdot T}}={\sqrt {\gamma \cdot R_{*}\cdot (\theta +273.15\,\mathrm {K} )}},\\c_{\mathrm {air} }&={\sqrt {\gamma \cdot R_{*}\cdot 273.15\,\mathrm {K} }}\cdot {\sqrt {1+{\frac {\theta }{273.15\,\mathrm {K} }}}}.\end{aligned}}$$

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

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

A graph comparing results of the two equations is to the right, using the slightly more accurate value of 331.5 m/s (1,088 ft/s) for the speed of sound at0 °C.

Effects due to wind shear

The speed of sound varies with temperature. Since temperature and sound velocity normally decrease with increasing altitude, sound isrefractedupward, away from listeners on the ground, creating anacoustic shadowat some distance from the source.Wind shear of 4 m/(s · km) can produce refraction equal to a typical temperaturelapse rateof7.5 °C/km.Higher values of wind gradient will refract sound downward toward the surface in the downwind direction,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.

For sound propagation, the exponential variation of wind speed with height can be defined as follows: $${\begin{aligned}U(h)&=U(0)h^{\zeta },\\{\frac {\mathrm {d} U}{\mathrm {d} H}}(h)&=\zeta {\frac {U(h)}{h}},\end{aligned}}$$ where

• U(h) is the speed of the wind at heighth;
• ζis the exponential coefficient based on ground surface roughness, typically between 0.08 and 0.52;
• dU/dH(h) is the expected wind gradient at heighth.

In the 1862American Civil WarBattle of Iuka,an acoustic shadow, believed to have been enhanced by a northeast wind, kept two divisions of Union soldiers out of the battle,because they could not hear the sounds of battle only10 km(six miles) downwind.

Tables

In thestandard atmosphere:

• T₀is273.15 K(=0 °C=32 °F), giving a theoretical value of331.3 m/s(=1086.9 ft/s=1193 km/h=741.1 mph=644.0kn). Values ranging from 331.3 to331.6 m/smay be found in reference literature, however;
• T₂₀is293.15 K(=20 °C=68 °F), giving a value of343.2 m/s(=1126.0 ft/s=1236 km/h=767.8 mph=667.2kn);
• T₂₅is298.15 K(=25 °C=77 °F), giving a value of346.1 m/s(=1135.6 ft/s=1246 km/h=774.3 mph=672.8kn).

In fact, assuming anideal gas,the speed of soundcdepends on temperature and composition only,not on the pressureordensity(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.

Effect of temperature on properties of air

Celsius tempe­rature θ[°C]	Speed of sound c[m/s]	Density of air ρ[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

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

Altitude	Temperature	m/s	km/h	mph	kn
Sea level	15 °C(59 °F)	340	1,225	761	661
11,000 mto20,000 m (cruising altitude of commercial jets, andfirst supersonic flight)	−57 °C(−70 °F)	295	1,062	660	573
29,000 m (flight ofX-43A)	−48 °C(−53 °F)	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.

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 themean free pathincreases. For this reason, the concept of speed of sound (except for frequencies approaching zero) progressively loses its range of applicability at high altitudes.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 higherγ(5/3 = 1.66...) than diatomics do (7/5 = 1.4). Thus, at the same molecular mass, the speed of sound of a monatomic gas goes up by a factor of $${c_{\mathrm {gas,monatomic} } \over c_{\mathrm {gas,diatomic} }}={\sqrt {{5/3} \over {7/5}}}={\sqrt {25 \over 21}}=1.091\ldots$$

This gives the 9% difference, and would be a typical ratio for speeds of sound at room temperature inheliumvs.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 (seeheat 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 about0.6 m/sper 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 about1.5 m/sat 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 about0.1 m/sas the frequency rises from10 Hzto100 Hz.For audible frequencies above100 Hzit 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.

As shown above, the approximate value 1000/3 = 333.33... m/s is exact a little below5 °Cand 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.

Mach number

Main article:Mach number

Mach number, a useful quantity in aerodynamics, is the ratio of airspeedto the local speed of sound. At altitude, for reasons explained, Mach number is a function of temperature. Aircraftflight 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 aPitot tubeis dependent on altitude as well as speed.

Experimental methods

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

The earliest reasonably accurate estimate of the speed of sound in air was made byWilliam Derhamand acknowledged byIsaac Newton.Derham had a telescope at the top of the tower of theChurch of St LaurenceinUpminster,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 twomicrophonesand a fast recording device such as adigitalstorage 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:

1. The distance between the microphones (x), called microphone basis.
2. The time of arrival between the signals (delay) reaching the different microphones (t).

Thenv=x/t.

Other methods

In these methods, thetimemeasurement has been replaced by a measurement of the inverse of time (frequency).

Kundt's tubeis 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 thenodesandantinodesvisible to the human eye. This is an example of a compact experimental setup.

Atuning forkcan be held near the mouth of a longpipewhich is dipping into a barrel ofwater.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)λ/4wherenis an integer. As theantinodalpoint 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 thatv=fλ.

High-precision measurements in air

The effect of impurities can be significant when making high-precision measurements. Chemicaldesiccantscan 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 reviewfound 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 at30 °Cbut corrected for temperature in order to report them at0 °C.The result was331.45 ± 0.01 m/sfor dry air at STP, for frequencies from93 Hzto1,500 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. Inearthquakes,the corresponding seismic waves are calledP-waves(primary waves) andS-waves(secondary waves), respectively. The sound velocities of these two types of waves propagating in a homogeneous 3-dimensional solid are respectively given by $$c_{\mathrm {solid,p} }={\sqrt {\frac {K+{\frac {4}{3}}G}{\rho }}}={\sqrt {\frac {E(1-\nu )}{\rho (1+\nu )(1-2\nu )}}},$$ $$c_{\mathrm {solid,s} }={\sqrt {\frac {G}{\rho }}},$$ where

• Kis thebulk modulusof the elastic materials;
• Gis theshear modulusof the elastic materials;
• Eis theYoung's modulus;
• ρis the density;
• νisPoisson's ratio.

The last quantity is not an independent one, asE = 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,K= 170 GPa,G= 80 GPaandp=7700 kg/m³,yielding a compressional speedc_solid,pof6,000 m/s.This is in reasonable agreement withc_solid,pmeasured experimentally at5,930 m/sfor a (possibly different) type of steel.The shear speedc_solid,sis estimated at3,200 m/susing the same numbers.

Speed of sound in semiconductor solids can be very sensitive to the amount of electronic dopant in them.

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: $$c_{\mathrm {solid} }={\sqrt {\frac {E}{\rho }}},$$ whereEisYoung's modulus.This is similar to the expression for shear waves, save thatYoung's modulusreplaces theshear 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 onPoisson's ratiofor the material.

Speed of sound in liquids

Speed of sound in water vs temperature

In a fluid, the only non-zerostiffnessis to volumetric deformation (a fluid does not sustain shear forces).

Hence the speed of sound in a fluid is given by $$c_{\mathrm {fluid} }={\sqrt {\frac {K}{\rho }}},$$ whereKis thebulk modulusof the fluid.

Water

In fresh water, sound travels at about1481 m/sat20 °C(see the External Links section below for online calculators).Applications ofunderwater soundcan be found insonar,acoustic communicationandacoustical oceanography.

Seawater

See also:Sound speed profile

Speed of sound as a function of depth at a position north of Hawaii in thePacific Oceanderived from the 2005World Ocean Atlas.TheSOFAR channelspans the minimum in the speed of sound at about 750 m depth.

In salt water that is free of air bubbles or suspended sediment, sound travels at about1500 m/s(1500.235 m/sat1000kilopascals,10 °Cand 3%salinityby one method).The speed of sound in seawater depends on pressure (hence depth), temperature (a change of1 °C~4 m/s), andsalinity(a change of 1‰~1 m/s), and empirical equations have been derived to accurately calculate the speed of sound from these variables.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).For more information see Dushaw et al.

An empirical equation for the speed of sound in sea water is provided by Mackenzie: $$c(T,S,z)=a_{1}+a_{2}T+a_{3}T^{2}+a_{4}T^{3}+a_{5}(S-35)+a_{6}z+a_{7}z^{2}+a_{8}T(S-35)+a_{9}Tz^{3},$$ where

• Tis the temperature in degrees Celsius;
• Sis the salinity in parts per thousand;
• zis the depth in metres.

The constantsa₁,a₂,...,a₉are $${\begin{aligned}a_{1}&=1,448.96,&a_{2}&=4.591,&a_{3}&=-5.304\times 10^{-2},\\a_{4}&=2.374\times 10^{-4},&a_{5}&=1.340,&a_{6}&=1.630\times 10^{-2},\\a_{7}&=1.675\times 10^{-7},&a_{8}&=-1.025\times 10^{-2},&a_{9}&=-7.139\times 10^{-13},\end{aligned}}$$ with check value1550.744 m/sforT=25 °C,S= 35 parts per thousand,z= 1,000 m.This equation has a standard error of0.070 m/sfor salinity between 25 and 40ppt.See[1]for an online calculator.

(The Sound Speed vs. Depth graph doesnotcorrelate directly to the MacKenzie formula. This is due to the fact that the temperature and salinity varies at different depths. WhenTandSare 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 Grossoand the Chen-Millero-Li Equation.

Speed of sound in plasma

The speed of sound in aplasmafor the common case that the electrons are hotter than the ions (but not too much hotter) is given by the formula (seehere) $$c_{s}=\left({\frac {\gamma ZkT_{\mathrm {e} }}{m_{\mathrm {i} }}}\right)^{1/2}=\left({\frac {\gamma ZT_{e}}{\mu }}\right)^{1/2}\times 90.85~\mathrm {m/s},$$ where

• m_iis theionmass;
• μis the ratio of ion mass toprotonmassμ=m_i/m_p;
• T_eis theelectrontemperature;
• Zis the charge state;
• kisBoltzmann constant;
• γis theadiabatic 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 onMarsvaries as a function of frequency. Higher frequencies travel faster than lower frequencies. Higher frequency sound from lasers travels at 250 m/s (820 ft/s), while low frequency sound topped out at 240 m/s (790 ft/s).

Gradients

Main article: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' orSOFAR channelwhich 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 higherrefractive index,sound waves willrefracttowards 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 oroptical 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 Mogulsuccessfully used this effect to detect anuclear explosionat 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

External links

• Speed of Sound Calculator
• Calculation: Speed of Sound in Air and the Temperature
• Speed of sound: Temperature Matters, Not Air Pressure
• Properties of the U.S. Standard Atmosphere 1976
• The Speed of Sound
• How to Measure the Speed of Sound in a Laboratory
• Did Sound Once Travel at Light Speed?
• Acoustic Properties of Various Materials Including the Speed of SoundArchived16 February 2014 at theWayback Machine
• Discovery of Sound in the Sea(uses of sound by humans and other animals)