Speed of sound

For other uses, see Speed of sound (disambiguation).

The speed of sound is the distance travelled per unit timeby a sound wave propagating through an elastic medium.The SI unit of speed is the metre per second (m/s). Indry air at 20 °C, the speed of sound is 343.2 metres persecond (1,126 ft/s). This is 1,236 kilometres per hour(768 mph; 667 kn), or a kilometre in 2.914 s or a mile in4.689 s.The speed of sound in an ideal gas is independent of fre-quency, but does vary slightly with frequency in a real gas.It is proportional to the square root of the absolute tem-perature, but is independent of pressure or density for agiven ideal gas. The speed of sound in air varies slightlywith pressure only because air is not quite an ideal gas.Although (in the case of gases only) the speed of soundis expressed in terms of a ratio of both density and pres-sure, these quantities cancel in ideal gases at any giventemperature, composition, and heat capacity. This leadsto a velocity formula for ideal gases which includes onlythe latter independent variables.In common everyday speech, speed of sound refers to thespeed of sound waves in air. However, the speed of soundvaries from substance to substance. Sound travels faster inliquids and non-porous solids than it does in air. It travelsabout 4.3 times as fast in water (1,484 m/s), and nearly 15times as fast in iron (5,120 m/s), as in air at 20 °C. Soundwaves in solids are composed of compression waves (justas in gases and liquids), but there is also a different typeof sound wave called a shear wave, which occurs onlyin solids. These different types of waves in solids usu-ally travel at different speeds, as exhibited in seismology.The speed of a compression sound wave in solids is de-termined by themedium’s compressibility, shearmodulusand density. The speed of shear waves is determined onlyby 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 speedof an object moving through the medium. The speedof an object divided by the speed of sound in the fluidis called the Mach number. Objects moving at speedsgreater than Mach1 are travelling at supersonic speeds.

1 History

Sir Isaac Newton computed the speed of sound in air as979 feet per second (298 m/s), which is too low by about

15%,[1] but had neglected the effect of fluctuating tem-perature; that was later rectified by Laplace.[2]

During the 17th century, there were several attemptsto measure the speed of sound accurately, including at-tempts by Marin Mersenne in 1630 (1,380 Parisian feetper second), Pierre Gassendi in 1635 (1,473 Parisian feetper second) and Robert Boyle (1,125 Parisian feet persecond).[3]

In 1709, the Reverend William Derham, Rector of Up-minster, published a more accurate measure of the speedof sound, at 1,072 Parisian feet per second.[3] Derhamused a telescope from the tower of the church of St Lau-rence, Upminster to observe the flash of a distant shotgunbeing fired, and then measured the time until he heardthe gunshot with a half second pendulum. Measurementswere made of gunshots from a number of local land-marks, including North Ockendon church. The distancewas known by triangulation, and thus the speed that thesound had travelled could be calculated.[4]

2 Basic concept

The transmission of sound can be illustrated by using amodel consisting of an array of balls interconnected bysprings. For real material the balls represent moleculesand the springs represent the bonds between them. Soundpasses through the model by compressing and expand-ing the springs, transmitting energy to neighbouring balls,which transmit energy to their springs, and so on. Thespeed of sound through the model depends on the stiff-ness of the springs (stiffer springs transmit energy morequickly). Effects like dispersion and reflection can alsobe understood using this model.In a real material, the stiffness of the springs is called theelastic modulus, and the mass corresponds to the density.All other things being equal (ceteris paribus), sound willtravel more slowly in spongymaterials, and faster in stifferones. For instance, sound will travel 1.59 times fasterin nickel than in bronze, due to the greater stiffness ofnickel at about the same density. Similarly, sound trav-els about 1.41 times faster in light hydrogen (protium)gas than in heavy hydrogen (deuterium) gas, since deu-terium has similar properties but twice the density. At thesame time, “compression-type” sound will travel faster insolids than in liquids, and faster in liquids than in gases,because the solids are more difficult to compress than liq-uids, while liquids in turn are more difficult to compress

1

2 3 EQUATIONS

than gases.Some textbooks mistakenly state that the speed of soundincreases with increasing density. This is usually illus-trated by presenting data for three materials, such as air,water and steel, which also have vastly different com-pressibilities which more than make up for the densitydifferences. An illustrative example of the two effectsis that sound travels only 4.3 times faster in water thanair, despite enormous differences in compressibility ofthe two media. The reason is that the larger density ofwater, which works to slow sound in water relative to air,nearly makes up for the compressibility differences in thetwo media.

2.1 Compression and shear waves

Pressure-pulse or compression-type wave (longitudinal wave)confined to a plane. This is the only type of sound wave thattravels in fluids (gases and liquids)

Transverse wave affecting atoms initially confined to a plane.This additional type of sound wave (additional type of elasticwave) travels only in solids, and the sideways shearing motionmay take place in any direction at right angles to the direction ofwave-travel (only one shear direction is shown here, at right an-gles to the plane). Furthermore, the right-angle shear directionmay change over time and distance, resulting in different typesof polarization of shear-waves

In a gas or liquid, sound consists of compression waves.In solids, waves propagate as two different types. Alongitudinal wave is associated with compression anddecompression in the direction of travel, which is thesame process as all sound waves in gases and liquids.A transverse wave, called a shear wave in solids, isdue to elastic deformation of the medium perpendicularto the direction of wave travel; the direction of shear-deformation is called the "polarization" of this type ofwave. In general, transverse waves occur as a pair oforthogonal polarizations. These different waves (com-pression waves and the different polarizations of shearwaves) may have different speeds at the same frequency.Therefore, they arrive at an observer at different times, anextreme example being an earthquake, where sharp com-pression waves arrive first, and rocking transverse wavesseconds later.The speed of a compression wave in fluid is determinedby the medium’s compressibility and density. In solids,the compression waves are analogous to those in fluids,depending on compressibility, density, and the additionalfactor of shear modulus. The speed of shear waves, whichcan occur only in solids, is determined simply by the solidmaterial’s shear modulus and density.

3 Equations

In general, the speed of sound c is given by the Newton–Laplace equation:

c =

√Ks

ρ,

where

• Ks is a coefficient of stiffness, the isentropic bulkmodulus (or the modulus of bulk elasticity forgases);

• ρ is the density.

Thus the speed of sound increases with the stiffness (theresistance of an elastic body to deformation by an appliedforce) of the material, and decreases with the density. Forideal gases the bulk modulus K is simply the gas pressuremultiplied by the dimensionless adiabatic index, which isabout 1.4 for air under normal conditions of pressure andtemperature.For general equations of state, if classical mechanics isused, the speed of sound c is given by

c =

√(∂p

∂ρ

)s

,

where

3

• p is the pressure;

• ρ is the density and the derivative is taken isentrop-ically, that is, at constant entropy s.

If relativistic effects are important, the speed of sound iscalculated from the relativistic Euler equations.In a non-dispersive medium, the speed of sound is in-dependent of sound frequency, so the speeds of energytransport and sound propagation are the same for all fre-quencies. Air, a mixture of oxygen and nitrogen, consti-tutes a non-dispersive medium. However, air does con-tain a small amount of CO2 which is a dispersivemedium,and causes dispersion to air at ultrasonic frequencies (>28 kHz).[5]

In a dispersive medium, the speed of sound is a functionof sound frequency, through the dispersion relation. Eachfrequency component propagates at its own speed, calledthe phase velocity, while the energy of the disturbancepropagates at the group velocity. The same phenomenonoccurs with light waves; see optical dispersion for a de-scription.

4 Dependence on the properties ofthe medium

The speed of sound is variable and depends on the proper-ties of the substance through which the wave is travelling.In solids, the speed of transverse (or shear) waves dependson the shear deformation under shear stress (called theshear modulus), and the density of the medium. Longi-tudinal (or compression) waves in solids depend on thesame two factors with the addition of a dependence oncompressibility.In fluids, only the medium’s compressibility and densityare the important factors, since fluids do not transmitshear stresses. In heterogeneous fluids, such as a liquidfilled with gas bubbles, the density of the liquid and thecompressibility of the gas affect the speed of sound inan additive manner, as demonstrated in the hot chocolateeffect.In gases, adiabatic compressibility is directly related topressure through the heat capacity ratio (adiabatic in-dex), and pressure and density are inversely related ata given temperature and composition, thus making onlythe latter independent properties (temperature, molecu-lar composition, and heat capacity ratio) important. Inlow molecular weight gases such as helium, sound prop-agates faster compared to heavier gases such as xenon(for monatomic gases the speed of sound is about 75%of the mean speed that molecules move in the gas). For agiven ideal gas the speed of sound depends only on itstemperature. At a constant temperature, the ideal gaspressure has no effect on the speed of sound, becausepressure and density (also proportional to pressure) have

equal but opposite effects on the speed of sound, and thetwo contributions cancel out exactly. In a similar way,compression waves in solids depend both on compress-ibility and density—just as in liquids—but in gases thedensity contributes to the compressibility in such a waythat some part of each attribute factors out, leaving onlya dependence on temperature, molecular weight, and heatcapacity ratio (see derivations below). Thus, for a singlegiven gas (where molecular weight does not change) andover a small temperature range (where heat capacity isrelatively constant), the speed of sound becomes depen-dent on only the temperature of the gas.In non-ideal gases, such as a van der Waals gas, the pro-portionality is not exact, and there is a slight dependenceof sound velocity on the gas pressure.Humidity has a small but measurable effect on speed ofsound (causing it to increase by about 0.1%–0.6%), be-cause oxygen and nitrogen molecules of the air are re-placed by lighter molecules of water. This is a simplemixing effect.

5 Altitude variation and implica-tions for atmospheric acoustics

Troposphere

Tropopause

Stratosphere

Stratopause

Mesosphere

Mesopause

Thermosphere

0

10

20

30

40

50

60

70

80

90

100

Geom

etr

ic a

ltitude (

km)

Temperature

Temperature (K)200 250 300150

Speed of sound

Speed of sound (m/s)250 300 350200

Pressure

Pressure (kN/m²)50 100 1500

Density

Density (kg/m³)0.5 1 1.50

Kármán line

ionosphere D layer

aurora

meteor

weather balloon

ozone layer

NASA X-43A

SR-71 Blackbird

Concorde

typical airliner

Mt Everest

Burj Khalifa

Density and pressure decrease smoothly with altitude, but tem-perature (red) does not. The speed of sound (blue) depends onlyon the complicated temperature variation at altitude and can becalculated from it, since isolated density and pressure effects onthe speed of sound cancel each other. Speed of sound increaseswith 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

4 7 DETAILS

speed of sound is the temperature. For a given ideal gaswith constant heat capacity and composition, the speed ofsound is dependent solely upon temperature; see Detailsbelow. In such an ideal case, the effects of decreased den-sity and decreased pressure of altitude cancel each otherout, save for the residual effect of temperature.Since temperature (and thus the speed of sound) de-creases with increasing altitude up to 11 km, sound isrefracted upward, away from listeners on the ground,creating an acoustic shadow at some distance from thesource.[6] The decrease of the speed of sound with heightis referred to as a negative sound speed gradient.However, there are variations in this trend above 11 km.In particular, in the stratosphere above about 20 km, thespeed of sound increases with height, due to an increasein temperature from heating within the ozone layer. Thisproduces a positive speed of sound gradient in this region.Still another region of positive gradient occurs at veryhigh altitudes, in the aptly-named thermosphere above 90km.

6 Practical formula for dry air

Approximation of the speed of sound in dry air based on the heatcapacity ratio (in green) against the truncated Taylor expansion(in red).

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

cair = (331.3 + 0.606 · ϑ) m/s,

where θ is the temperature in degrees Celsius (°C).This equation is derived from the first two terms of theTaylor expansion of the following more accurate equa-tion:

cair = 331.3 m/s√1 +

ϑ

273.15.

Dividing the first part, andmultiplying the second part, onthe right hand side, by √273.15 gives the exactly equiva-lent form

cair = 20.05 m/s√ϑ+ 273.15.

The value of 331.3 m/s, which represents the speed at0 °C (or 273.15 K), is based on theoretical (and somemeasured) values of the heat capacity ratio, γ, as well ason the fact that at 1 atm real air is very well describedby the ideal gas approximation. Commonly found valuesfor the speed of sound at 0 °C may vary from 331.2 to331.6 due to the assumptions made when it is calculated.If ideal gas γ is assumed to be 7/5 = 1.4 exactly, the 0 °Cspeed is calculated (see section below) to be 331.3 m/s,the coefficient used above.This equation is correct to a much wider temperaturerange, but still depends on the approximation of heat ca-pacity ratio being independent of temperature, and forthis reason will fail, particularly at higher temperatures.It gives good predictions in relatively dry, cold, low pres-sure conditions, such as the Earth’s stratosphere. Theequation fails at extremely low pressures and short wave-lengths, due to dependence on the assumption that thewavelength of the sound in the gas is much longer thanthe average mean free path between gas molecule colli-sions. A derivation of these equations will be given in thefollowing section.A graph comparing results of the two equations is at right,using the slightly different value of 331.5 m/s for thespeed of sound at 0 °C.

7 Details

7.1 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) isgiven by

K = γ · p,

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

c =

√γ · p

ρ,

where

7.1 Speed of sound in ideal gases and air 5

• γ is the adiabatic index also known as the isentropicexpansion factor. It is the ratio of specific heats ofa gas at a constant-pressure to a gas at a constant-volume( Cp/Cv ), and arises because a classicalsound wave induces an adiabatic compression, inwhich the heat of the compression does not haveenough time to escape the pressure pulse, and thuscontributes to the pressure induced by the compres-sion;

• p is the pressure;

• ρ is the density.

Using the ideal gas law to replace p with nRT/V, and re-placing ρ with nM/V, the equation for an ideal gas be-comes

cideal =

√γ · p

ρ=

√γ ·R · T

M=

√γ · k · T

m,

where

• cᵢ ₑₐ is the speed of sound in an ideal gas;

• R (approximately 8.314,5 J · mol−1 · K−1) is themolar gas constant;[7]

• k is the Boltzmann constant;

• γ (gamma) is the adiabatic index (sometimes as-sumed 7/5 = 1.400 for diatomic molecules from ki-netic theory, assuming from quantum theory a tem-perature range at which thermal energy is fully par-titioned into rotation (rotations are fully excited),but none into vibrational modes. Gamma is ac-tually experimentally measured over a range from1.399,1 to 1.403 at 0 °C, for air. Gamma is assumedfrom kinetic theory to be exactly 5/3 = 1.666,7 formonatomic gases such as noble gases);

• T is the absolute temperature;

• M is the molar mass of the gas. The mean molarmass for dry air is about 0.028,964,5 kg/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 smallperturbation on the ambient condition, and the certainother noted conditions are fulfilled, as noted below. Cal-culated values for cₐᵢᵣ have been found to vary slightlyfrom experimentally determined values.[8]

Newton famously considered the speed of sound be-fore most of the development of thermodynamics andso incorrectly used isothermal calculations instead ofadiabatic. His result was missing the factor of γ but wasotherwise correct.

Numerical substitution of the above values gives the idealgas approximation of sound velocity for gases, whichis accurate at relatively low gas pressures and densities(for air, this includes standard Earth sea-level conditions).Also, for diatomic gases the use of γ = 1.400,0 requiresthat the gas exists in a temperature range high enough thatrotational heat capacity is fully excited (i.e., molecularrotation is fully used as a heat energy “partition” or reser-voir); but at the same time the temperature must be lowenough that molecular vibrational modes contribute noheat capacity (i.e., insignificant heat goes into vibration,as all vibrational quantum modes above the minimum-energy-mode, have energies too high to be populated bya significant number of molecules at this temperature).For air, these conditions are fulfilled at room temper-ature, and also temperatures considerably below roomtemperature (see tables below). See the section on gasesin specific heat capacity for a more complete discussionof this phenomenon.For air, we use a simplified symbol

R∗ = R/Mair.

Additionally, if temperatures in degrees Celsius(°C) areto be used to calculate air speed in the region near 273kelvin, then Celsius temperature θ = T − 273.15 may beused. Then

cideal =√γ ·R∗ · T =

√γ ·R∗ · (ϑ+ 273.15),

cideal =√γ ·R∗ · 273.15 ·

√1 +

ϑ

273.15.

For dry air, where θ (theta) is the temperature in degreesCelsius(°C).Making the following numerical substitutions,

R = 8.314, 510 J/(mol · K)

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

Mair = 0.028, 964, 5 kg/mol

is the mean molar mass of air, in kg; and using the idealdiatomic gas value of γ = 1.400,0.Then

cair = 331.3 m/s√1 +

ϑ◦C273.15◦C .

Using the first two terms of the Taylor expansion:

cair = 331.3 m/s(1 + ϑ◦C2 · 273.15◦C ),

6 8 EFFECT OF FREQUENCY AND GAS COMPOSITION

cair = (331.3 + 0.606◦C−1 · ϑ) m/s.The derivation includes the first two equations given inthe Practical formula for dry air section above.

7.2 Effects due to wind shear

The speed of sound varies with temperature. Since tem-perature and sound velocity normally decrease with in-creasing altitude, sound is refracted upward, away fromlisteners on the ground, creating an acoustic shadow atsome distance from the source.[6] Wind shear of 4 m/(s ·km) can produce refraction equal to a typical temperaturelapse rate of 7.5 °C/km.[9] Higher values of wind gradi-ent will refract sound downward toward the surface in thedownwind direction,[10] eliminating the acoustic shadowon the downwind side. This will increase the audibilityof sounds downwind. This downwind refraction effectoccurs because there is a wind gradient; the sound is notbeing carried along by the wind.[11]

For sound propagation, the exponential variation of windspeed with height can be defined as follows:[12]

U(h) = U(0)hζ ,

dUdH (h) = ζ

U(h)

h,

where

• U(h) is the speed of the wind at height h;

• ζ is the exponential coefficient based on ground sur-face 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 acous-tic shadow, believed to have been enhanced by a north-east wind, kept two divisions of Union soldiers out of thebattle,[13] because they could not hear the sounds of battleonly 10 km (six miles) downwind.[14]

7.3 Tables

In the standard atmosphere:

• T0 is 273.15 K (= 0 °C = 32 °F), giving a theoreticalvalue of 331.3 m/s (= 1086.9 ft/s = 1193 km/h =741.1 mph = 644.0 kn). Values ranging from 331.3-331.6 may be found in reference literature, however;

• T20 is 293.15 K (= 20 °C = 68 °F), giving a value of343.2 m/s (= 1126.0 ft/s = 1236 km/h = 767.8 mph= 667.2 kn);

• T25 is 298.15 K (= 25 °C = 77 °F), giving a value of346.1 m/s (= 1135.6 ft/s = 1246 km/h = 774.3 mph= 672.8 kn).

In fact, assuming an ideal gas, the speed of sound c de-pends on temperature only, not on the pressure or den-sity (since these change in lockstep for a given temper-ature and cancel out). Air is almost an ideal gas. Thetemperature of the air varies with altitude, giving the fol-lowing variations in the speed of sound using the standardatmosphere—actual conditions may vary.Given normal atmospheric conditions, the temperature,and thus speed of sound, varies with altitude:

8 Effect of frequency and gas com-position

8.1 General physical considerations

The medium in which a sound wave is travelling does notalways respond adiabatically, and as a result the speed ofsound can vary with frequency.[15]

The limitations of the concept of speed of sound due toextreme attenuation are also of concern. The attenua-tion which exists at sea level for high frequencies appliesto successively lower frequencies as atmospheric pressuredecreases, or as the mean free path increases. For thisreason, the concept of speed of sound (except for fre-quencies approaching zero) progressively loses its rangeof applicability at high altitudes.[8] The standard equa-tions for the speed of sound apply with reasonable ac-curacy only to situations in which the wavelength of thesoundwave is considerably longer than the mean free pathof molecules in a gas.The molecular composition of the gas contributes both asthe mass (M) of the molecules, and their heat capacities,and so both have an influence on speed of sound. In gen-eral, at the same molecular mass, monatomic gases haveslightly higher speed of sound (over 9% higher) becausethey have a higher γ (5/3 = 1.66…) than diatomics do(7/5 = 1.4). Thus, at the same molecular mass, the speedof sound of a monatomic gas goes up by a factor of

cgas,monatomiccgas,diatomic

=

√5/3

7/5=

√25

21= 1.091 . . .

This gives the 9% difference, and would be a typical ra-tio for speeds of sound at room temperature in heliumvs. deuterium, each with a molecular weight of 4. Soundtravels faster in helium than deuterium because adia-batic compression heats helium more, since the heliummolecules can store heat energy from compression onlyin translation, but not rotation. Thus helium molecules(monatomic molecules) travel faster in a sound wave andtransmit sound faster. (Sound generally travels at about70% of the mean molecular speed in gases).Note that in this example we have assumed that tempera-ture is low enough that heat capacities are not influenced

10.1 Single-shot timing methods 7

by molecular vibration (see heat capacity). However, vi-brational modes simply cause gammas which decrease to-ward 1, since vibration modes in a polyatomic gas givesthe gas additional ways to store heat which do not affecttemperature, and thus do not affect molecular velocityand sound velocity. Thus, the effect of higher temper-atures and vibrational heat capacity acts to increase thedifference between the speed of sound in monatomic vs.polyatomic molecules, with the speed remaining greaterin monatomics.

8.2 Practical application to air

By far the most important factor influencing the speed ofsound in air is temperature. The speed is proportionalto the square root of the absolute temperature, giving anincrease of about 0.6 m/s per degree Celsius. For thisreason, the pitch of a musical wind instrument increasesas its temperature increases.The speed of sound is raised by humidity but decreasedby carbon dioxide. The difference between 0% and100% humidity is about 1.5 m/s at standard pressure andtemperature, but the size of the humidity effect increasesdramatically with temperature. The carbon dioxide con-tent of air is not fixed, due to both carbon pollution andhuman breath (e.g., in the air blown through wind instru-ments).The dependence on frequency and pressure are normallyinsignificant in practical applications. In dry air, the speedof sound increases by about 0.1 m/s as the frequency risesfrom 10 Hz to 100 Hz. For audible frequencies above100 Hz it is relatively constant. Standard values of thespeed of sound are quoted in the limit of low frequencies,where the wavelength is large compared to the mean freepath.[16]

9 Mach number

Main article: Mach numberMach number, a useful quantity in aerodynamics, is theratio of air speed to the local speed of sound. At alti-tude, for reasons explained, Mach number is a functionof temperature. Aircraft flight instruments, however, op-erate using pressure differential to compute Mach num-ber, not temperature. The assumption is that a particularpressure represents a particular altitude and, therefore, astandard temperature. Aircraft flight instruments need tooperate this way because the stagnation pressure sensedby a Pitot tube is dependent on altitude as well as speed.

10 Experimental methods

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

U.S. Navy F/A-18 traveling near the speed of sound. The whitehalo consists of condensed water droplets formed by the suddendrop in air pressure behind the shock cone around the aircraft(see Prandtl-Glauert singularity).[17]

The earliest reasonably accurate estimate of the speedof sound in air was made by William Derham, and ac-knowledged by Isaac Newton. Derham had a telescopeat the top of the tower of the Church of St Laurencein Upminster, England. On a calm day, a synchronizedpocket watch would be given to an assistant who wouldfire a shotgun at a pre-determined time from a conspicu-ous point some miles away, across the countryside. Thiscould be confirmed by telescope. He then measuredthe interval between seeing gunsmoke and arrival of thesound using a half-second pendulum. The distance fromwhere the gun was fired was found by triangulation, andsimple division (distance/time) provided velocity. Lastly,by making many observations, using a range of differ-ent distances, the inaccuracy of the half-second pendu-lum could be averaged out, giving his final estimate of thespeed of sound. Modern stopwatches enable this methodto be used today over distances as short as 200–400 me-ters, and not needing something as loud as a shotgun.

10.1 Single-shot timing methods

The simplest concept is the measurement made using twomicrophones and a fast recording device such as a digitalstorage scope. This method uses the following idea.If a sound source and two microphones are arranged in astraight line, with the sound source at one end, then thefollowing can be measured:

1. The distance between the microphones (x), calledmicrophone basis.

2. The time of arrival between the signals (delay)reaching the different microphones (t).

Then v = x/t.

8 11 NON-GASEOUS MEDIA

10.2 Other methods

In these methods the time measurement has beenreplaced by a measurement of the inverse of time(frequency).Kundt’s tube is an example of an experiment which canbe used to measure the speed of sound in a small volume.It has the advantage of being able to measure the speedof sound in any gas. This method uses a powder to makethe nodes and antinodes visible to the human eye. This isan example of a compact experimental setup.A tuning fork can be held near the mouth of a long pipewhich is dipping into a barrel of water. In this system itis the case that the pipe can be brought to resonance ifthe length of the air column in the pipe is equal to (1 +2n)λ/4 where n is an integer. As the antinodal point forthe pipe at the open end is slightly outside the mouth ofthe pipe it is best to find two or more points of resonanceand then measure half a wavelength between these.Here it is the case that v = fλ.

10.3 High-precision measurements in air

The effect from impurities can be significant when mak-ing high-precision measurements. Chemical desiccantscan be used to dry the air, but will in turn contaminate thesample. The air can be dried cryogenically, but this hasthe effect of removing the carbon dioxide as well; there-fore many high-precision measurements are performedwith air free of carbon dioxide rather than with naturalair. A 2002 review[18] found that a 1963 measurementby Smith and Harlow using a cylindrical resonator gave“the most probable value of the standard speed of soundto date.” The experiment was done with air from whichthe carbon dioxide had been removed, but the result wasthen corrected for this effect so as to be applicable to realair. The experiments were done at 30 °C but correctedfor temperature in order to report them at 0 °C. The resultwas 331.45 ± 0.01 m/s for dry air at STP, for frequenciesfrom 93 Hz to 1,500 Hz.

11 Non-gaseous media

11.1 Speed of sound in solids

11.1.1 Three-dimensional solids

In a solid, there is a non-zero stiffness both for volumet-ric deformations and shear deformations. Hence, it ispossible to generate sound waves with different veloci-ties dependent on the deformation mode. Sound wavesgenerating volumetric deformations (compression) andshear deformations (shearing) are called pressure waves(longitudinal waves) and shear waves (transverse waves),respectively. In earthquakes, the corresponding seismic

waves are called P-waves (primary waves) and S-waves(secondary waves), respectively. The sound velocities ofthese two types of waves propagating in a homogeneous3-dimensional solid are respectively given by[19]

csolid,p =

√K + 4

3G

ρ=

√E(1− ν)

ρ(1 + ν)(1− 2ν),

csolid,s =

√G

ρ,

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 E = 3K(1− 2ν). Note that the speed of pressure waves dependsboth on the pressure and shear resistance properties ofthe material, while the speed of shear waves depends onthe shear properties only.Typically, pressure waves travel faster in materials thando shear waves, and in earthquakes this is the reasonthat the onset of an earthquake is often preceded by aquick upward-downward shock, before arrival of wavesthat produce a side-to-side motion. For example, for atypical steel alloy, K = 170 GPa, G = 80 GPa and ρ =7,700 kg/m3, yielding a compressional speed c ₒ ᵢ , of6,000 m/s.[19] This is in reasonable agreement with c ₒ ᵢ ,measured experimentally at 5,930 m/s for a (possibly dif-ferent) type of steel.[20] The shear speed c ₒ ᵢ , is esti-mated at 3,200 m/s using the same numbers.

11.1.2 One-dimensional solids

The speed of sound for pressure waves in stiff materi-als such as metals is sometimes given for “long rods” ofthe material in question, in which the speed is easier tomeasure. In rods where their diameter is shorter than awavelength, the speed of pure pressure wavesmay be sim-plified and is given by:

csolid =

√E

ρ,

where E is the Young’s modulus. This is similar to theexpression for shear waves, save that Young’s modulusreplaces the shear modulus. This speed of sound for pres-sure waves in long rods will always be slightly less than the

11.2 Speed of sound in liquids 9

same speed in homogeneous 3-dimensional solids, andthe ratio of the speeds in the two different types of ob-jects depends on Poisson’s ratio for the material.

11.2 Speed of sound in liquids

Speed of sound in water vs temperature.

In a fluid the only non-zero stiffness is to volumetric de-formation (a fluid does not sustain shear forces).Hence the speed of sound in a fluid is given by

cfluid =

√K

ρ,

where K is the bulk modulus of the fluid.

11.2.1 Water

In fresh water, sound travels at about 1497 m/s at25 °C.[21] See Technical Guides - Speed of Sound inPure Water for an online calculator. Applications ofunderwater sound can be found in sonar, acoustic com-munication and acoustical oceanography. See Discoveryof Sound in the Sea for other examples of the uses ofsound in the ocean (by both man and other animals).

11.2.2 Seawater

In salt water that is free of air bubbles or suspended sed-iment, sound travels at about 1500 m/s. The speed ofsound in seawater depends on pressure (hence depth),temperature (a change of 1 °C ~ 4 m/s), and salinity (achange of 1‰ ~ 1 m/s), and empirical equations havebeen derived to accurately calculate the speed of soundfrom these variables.[22] Other factors affecting the speedof sound are minor. Since temperature decreases withdepth while pressure and generally salinity increase, theprofile of the speed of sound with depth generally showsa characteristic curve which decreases to a minimum at

Depth

(km

)

Sound Speed (m/s)1480 1500 1520 1540

-5.5

-5.0

-4.5

-4.0

-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

Speed of sound as a function of depth at a position north ofHawaii in the Pacific Ocean derived from the 2005 World OceanAtlas. The SOFAR channel is centered on the minimum in thespeed of sound at ca. 750-m depth.

a depth of several hundred meters, then increases againwith increasing depth (right).[23] For more informationsee Dushaw et al.[24]

A simple empirical equation for the speed of sound in seawater with reasonable accuracy for the world’s oceans isdue to Mackenzie:[25]

c(T, S, z) = a1+a2T+a3T2+a4T

3+a5(S−35)+a6z+a7z2+a8T (S−35)+a9Tz

3,

where

• 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

a1 = 1, 448.96, a2 = 4.591, a3 = −5.304× 10−2,

a4 = 2.374× 10−4, a5 = 1.340, a6 = 1.630× 10−2,

a7 = 1.675× 10−7, a8 = −1.025× 10−2, a9 = −7.139× 10−13,

10 14 REFERENCES

with check value 1550.744m/s forT = 25 °C, S = 35 partsper thousand, z = 1,000 m. This equation has a standarderror of 0.070 m/s for salinity between 25 and 40 ppt.See Technical Guides. Speed of Sound in Sea-Water foran online calculator.Other equations for the speed of sound in sea water areaccurate over a wide range of conditions, but are far morecomplicated, e.g., that by V. A. Del Grosso[26] and theChen-Millero-Li Equation.[24][27]

11.3 Speed of sound in plasma

The speed of sound in a plasma for the common case thatthe electrons are hotter than the ions (but not too muchhotter) is given by the formula (see here)

cs = (γZkTe/mi)1/2 = 9.79× 103(γZTe/µ)

1/2 m/s,

where

• mᵢ is the ion mass;

• μ is the ratio of ion mass to proton mass μ = mᵢ/m ;

• Tₑ 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 pro-vided by separate species, the pressure by the electronsand the density by the ions. The two are coupled througha fluctuating electric field.

12 Gradients

Main article: Sound speed gradient

When sound spreads out evenly in all directions in threedimensions, the intensity drops in proportion to the in-verse square of the distance. However, in the ocean thereis a layer called the 'deep sound channel' or SOFAR chan-nel which can confine sound waves at a particular depth.In the SOFAR channel, the speed of sound is lower thanthat in the layers above and below. Just as light waveswill refract towards a region of higher index, sound waveswill refract towards a region where their speed is reduced.The result is that sound gets confined in the layer, muchthe way light can be confined in a sheet of glass or opticalfiber. Thus, the sound is confined in essentially two di-mensions. In two dimensions the intensity drops in pro-portion to only the inverse of the distance. This allows

waves to travel much further before being undetectablyfaint.A similar effect occurs in the atmosphere. Project Mogulsuccessfully used this effect to detect a nuclear explosionat a considerable distance.

13 See also

• Acoustoelastic effect

• Elastic wave

• Second sound

• Sonic boom

• Sound barrier

• Underwater acoustics

• Vibrations

14 References

[1] “The Speed of Sound”. mathpages.com. Retrieved 3May2015.

[2] Bannon, Mike; Kaputa, Frank. “The Newton–LaplaceEquation and Speed of Sound”. Thermal Jackets. Re-trieved 3 May 2015.

[3] Murdin, Paul (Dec 25, 2008). Full Meridian of Glory:Perilous Adventures in the Competition to Measure theEarth. Springer Science & Business Media. p. 35–36.ISBN 9780387755342.

[4] Fox, Tony (2003). Essex Journal. Essex Arch & Hist Soc.pp. 12–16.

[5] Dean, E. A. (August 1979). Atmospheric Effects on theSpeed of Sound, Technical report of Defense TechnicalInformation Center

[6] Everest, F. (2001). The Master Handbook of Acoustics.New York: McGraw-Hill. pp. 262–263. ISBN 0-07-136097-2.

[7] “CODATA Value: molar gas constant”. Physics.nist.gov.Retrieved 24 October 2010.

[8] U.S. Standard Atmosphere, 1976, U.S. GovernmentPrinting Office, Washington, D.C., 1976.

[9] Uman, Martin (1984). Lightning. New York: Dover Pub-lications. ISBN 0-486-64575-4.

[10] Volland, Hans (1995). Handbook of Atmospheric Electro-dynamics. Boca Raton: CRC Press. p. 22. ISBN 0-8493-8647-0.

11

[11] 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 occuronly because there is a wind gradient and it is not due tothe result of sound being convected along by the wind.

[12] Bies, David (2004). Engineering Noise Control, Theoryand 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 re-fract sound waves downwards, resulting in increased noiselevels.

[13] Cornwall, Sir (1996). Grant as Military Commander.New York: Barnes & Noble. p. 92. ISBN 1-56619-913-1.

[14] Cozens, Peter (2006). The Darkest Days of the War: theBattles of Iuka and Corinth. Chapel Hill: The Universityof North Carolina Press. ISBN 0-8078-5783-1.

[15] A B Wood, A Textbook of Sound (Bell, London, 1946)

[16] “Speed of Sound in Air”. Phy.mtu.edu. Retrieved 13 June2014.

[17] “APOD: 19 August 2007, A Sonic Boom”.Antwrp.gsfc.nasa.gov. Retrieved 24 October 2010.

[18] Zuckerwar, Handbook of the speed of sound in real gases,p. 52

[19] L. E. Kinsler et al. (2000), Fundamentals of acoustics, 4thEd., John Wiley and sons Inc., New York, USA

[20] J. Krautkrämer and H. Krautkrämer (1990), Ultrasonictesting of materials, 4th fully revised edition, Springer-Verlag, Berlin, Germany, p. 497

[21] “Speed of Sound in Water at Temperatures between 32–212 oF (0–100 oC) — imperial and SI units”. The Engi-neering Toolbox.

[22] APL-UW TR 9407 High-Frequency Ocean Environmen-tal Acoustic Models Handbook, pp. I1-I2.

[23] “How Fast Does Sound Travel?". Discovery of Soundin the Sea. University of Rhode Island. Retrieved 30November 2010.

[24] Dushaw, Brian D.; Worcester, P. F.; Cornuelle,B. D.; Howe, B. M. (1993). “On Equations forthe Speed of Sound in Seawater”. Journal ofthe Acoustical Society of America 93 (1): 255–275.Bibcode:1993ASAJ…93..255D. doi:10.1121/1.405660.

[25] Kenneth V., Mackenzie (1981). “Discussion ofsea-water sound-speed determinations”. Journal ofthe Acoustical Society of America 70 (3): 801–806.Bibcode:1981ASAJ…70..801M. doi:10.1121/1.386919.

[26] Del Grosso, V. A. (1974). “New equation for speedof sound in natural waters (with comparisons to otherequations)". Journal of the Acoustical Society of Amer-ica 56 (4): 1084–1091. Bibcode:1974ASAJ…56.1084D.doi:10.1121/1.1903388.

[27] Meinen, Christopher S.; Watts, D. Randolph (1997).“Further Evidence that the Sound-Speed Algorithm ofDel Grosso Is More Accurate Than that of Chen andMillero”. Journal of the Acoustical Society of America102 (4): 2058–2062. Bibcode:1997ASAJ..102.2058M.doi:10.1121/1.419655.

15 External links• Calculation: Speed of Sound in Air and the Tem-perature

• Speed of sound: Temperature Matters, Not AirPressure

• Properties of the U.S. Standard Atmosphere 1976

• The Speed of Sound

• How toMeasure the Speed of Sound in a Laboratory

• Teaching Resource for 14-16 Years on Sound In-cluding Speed of Sound

• Technical Guides. Speed of Sound in Pure Water

• Technical Guides. Speed of Sound in Sea-Water

• Did Sound Once Travel at Light Speed?

• Acoustic Properties of Various Materials Includingthe Speed of Sound

12 16 TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES

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