The speed of sound is the distance travelled per unit of time by a

_{s}'' is a coefficient of stiffness, the isentropic _{2} which ''is'' a dispersive medium, and causes dispersion to air at frequencies ().
In a

density
The density (more precisely, the volumetric mass density; also known as specific mass), of a substance is its per unit . The symbol most often used for density is ''ρ'' (the lower case Greek letter ), although the Latin letter ''D'' can also ...

(also proportional to pressure) have equal but opposite effects on the speed of sound, and the two contributions cancel out exactly. In a similar way, compression waves in solids depend both on compressibility and density—just as in liquids—but in gases the density contributes to the compressibility in such a way that some part of each attribute factors out, leaving only a dependence on temperature, molecular weight, and heat capacity ratio which can be independently derived from temperature and molecular composition (see derivations below). Thus, for a single given gas (assuming the molecular weight does not change) and over a small temperature range (for which the heat capacity is relatively constant), the speed of sound becomes dependent on only the temperature of the gas.
In non-ideal gas behavior regimen, for which the Van der Waals gas equation would be used, the proportionality is not exact, and there is a slight dependence of sound velocity on the gas pressure.
Humidity has a small but measurable effect on the speed of sound (causing it to increase by about 0.1%–0.6%), because

density
The density (more precisely, the volumetric mass density; also known as specific mass), of a substance is its per unit . The symbol most often used for density is ''ρ'' (the lower case Greek letter ), although the Latin letter ''D'' can also ...

.
Using the _{ideal} is the speed of sound in an ideal gas
An ideal gas is a theoretical composed of many randomly moving s that are not subject to . The ideal gas concept is useful because it obeys the , a simplified , and is amenable to analysis under . The requirement of zero interaction can often be ...

;
* ''R'' (approximately ) is the _{air} have been found to vary slightly from experimentally determined values.U.S. Standard Atmosphere, 1976, U.S. Government Printing Office, Washington, D.C., 1976.

_{0} is (= ), giving a theoretical value of (= = = = ). Values ranging from 331.3 to 331.6 m/s may be found in reference literature, however;
* ''T''_{20} is (= = ), giving a value of (= = = = );
* ''T''_{25} is (= = ), giving a value of (= = = = ).
In fact, assuming an ideal gas
An ideal gas is a theoretical composed of many randomly moving s that are not subject to . The ideal gas concept is useful because it obeys the , a simplified , and is amenable to analysis under . The requirement of zero interaction can often be ...

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

shear modulus
In materials science
The interdisciplinary field of materials science, also commonly termed materials science and engineering, covers the design and discovery of new materials, particularly solids. The intellectual origins of materials scienc ...

of the elastic materials;
* ''E'' is the _{solid,p} of . This is in reasonable agreement with ''c''_{solid,p} measured experimentally at for a (possibly different) type of steel. The shear speed ''c''_{solid,s} is estimated at using the same numbers.

shear modulus
In materials science
The interdisciplinary field of materials science, also commonly termed materials science and engineering, covers the design and discovery of new materials, particularly solids. The intellectual origins of materials scienc ...

. This speed of sound for pressure waves in long rods will always be slightly less than the same speed in homogeneous 3-dimensional solids, and the ratio of the speeds in the two different types of objects depends on

bulk modulus
The bulk modulus (K or B) of a substance is a measure of how resistant to compression that substance is. It is defined as the ratio of the infinitesimal
In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to ze ...

of the fluid.

_{1}, ''a''_{2}, ..., ''a''_{9} are
: $\backslash begin\; a\_1\; \&=\; 1,448.96,\; \&a\_2\; \&=\; 4.591,\; \&a\_3\; \&=\; -5.304\; \backslash times\; 10^,\backslash \backslash \; a\_4\; \&=\; 2.374\; \backslash times\; 10^,\; \&a\_5\; \&=\; 1.340,\; \&a\_6\; \&=\; 1.630\; \backslash times\; 10^,\backslash \backslash \; a\_7\; \&=\; 1.675\; \backslash times\; 10^,\; \&a\_8\; \&=\; -1.025\; \backslash times\; 10^,\; \&a\_9\; \&=\; -7.139\; \backslash times\; 10^,\; \backslash end$
with check value for , , . This equation has a standard error of for salinity between 25 and 40 Parts per thousand, ppt. Se

Technical Guides. Speed of Sound in Sea-Water

for an online calculator. (Note: The Sound Speed vs. Depth graph does ''not'' correlate directly to the MacKenzie formula. This is due to the fact that the temperature and salinity varies at different depths. When ''T'' and ''S'' are held constant, the formula itself is always increasing with depth.) Other equations for the speed of sound in sea water are accurate over a wide range of conditions, but are far more complicated, e.g., that by V. A. Del Grosso and the Chen-Millero-Li Equation.

_{i} is the ion mass;
* ''μ'' is the ratio of ion mass to proton mass ;
* ''T''_{e} is the electron temperature;
* ''Z'' is the charge state;
* ''k'' is adiabatic index
In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volume ...

.
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.

Speed of Sound Calculator

Speed of sound: Temperature Matters, Not Air Pressure

How to Measure the Speed of Sound in a Laboratory

* [http://www.dosits.org/ Discovery of Sound in the Sea] (uses of sound by humans and other animals) {{Authority control Fluid dynamics Aerodynamics Acoustics Sound Sound measurements Physical quantities Chemical properties Velocity Temporal rates

sound wave
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular ...

as it propagates through an elastic medium. At , the speed of sound in air is about , or one kilometre
The kilometre (SI symbol: km; or ), spelt kilometer in American English, is a unit of length in the metric system, equal to one thousand metres (kilo- being the SI prefix for ). It is now the measurement unit used for expressing distances betw ...

in or one mile
The mile, sometimes the international mile or statute mile to distinguish it from other miles, is a British imperial unit and US customary unit
United States customary units (U.S. customary units) are a system of measurements commonly u ...

in . It depends strongly on temperature as well as the medium through which a sound wave
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular ...

is propagating. At , the speed of sound is about .
The speed of sound in an ideal gas
An ideal gas is a theoretical composed of many randomly moving s that are not subject to . The ideal gas concept is useful because it obeys the , a simplified , and is amenable to analysis under . The requirement of zero interaction can often be ...

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 colloquial speech, ''speed of sound'' refers to the speed of sound waves in air
File:Atmosphere gas proportions.svg, Composition of Earth's atmosphere by volume, excluding water vapor. Lower pie represents trace gases that together compose about 0.043391% of the atmosphere (0.04402961% at April 2019 concentration ). Number ...

. However, the speed of sound varies from substance to substance: typically, sound travels most slowly in gas
Gas is one of the four fundamental states of matter
In physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space ...

es, faster in liquid
A liquid is a nearly incompressible
In fluid mechanics
Fluid mechanics is the branch of physics concerned with the mechanics
Mechanics (Ancient Greek, Greek: ) is the area of physics concerned with the motions of physical objects, ...

s, and fastest in solids
Solid is one of the four fundamental states of matter (the others being liquid
A liquid is a nearly incompressible
In fluid mechanics or more generally continuum mechanics, incompressible flow (isochoric process, isochoric flow) re ...

. For example, while sound travels at in air, it travels at in water
Water (chemical formula H2O) is an , transparent, tasteless, odorless, and , which is the main constituent of 's and the s of all known living organisms (in which it acts as a ). It is vital for all known forms of , even though it provide ...

(almost 4.3 times as fast) and at in iron (almost 15 times as fast). In an exceptionally stiff material such as diamond, sound travels at , about 35 times its speed in air and about the fastest it can 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
__NOTOC__
In seismology
Seismology (; from Ancient Greek
Ancient Greek includes the forms of the Greek language used in ancient Greece and the classical antiquity, ancient world from around 1500 BC to 300 BC. It is often roughly divided ...

, which occurs only in solids. Shear waves in solids usually travel at different speeds than compression waves, as exhibited in seismology
Seismology (; from Ancient Greek σεισμός (''seismós'') meaning "Earthquake, earthquake" and -λογία (''-logía'') meaning "study of") is the scientific study of earthquakes and the propagation of Linear elasticity#Elastic wave, elast ...

. The speed of compression waves in solids is determined by the medium's compressibility
In thermodynamics
Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quan ...

, shear modulus
In materials science
The interdisciplinary field of materials science, also commonly termed materials science and engineering, covers the design and discovery of new materials, particularly solids. The intellectual origins of materials scienc ...

and density. The speed of shear waves is determined only by the solid material's shear modulus and density.
In fluid dynamics
In and , fluid dynamics is a subdiscipline of that describes the flow of s—s and es. It has several subdisciplines, including ' (the study of air and other gases in motion) and hydrodynamics (the study of liquids in motion). Fluid dynamic ...

, the speed of sound in a fluid medium (gas or liquid) is used as a relative measure for the speed of an object moving through the medium. The ratio of the speed of an object to the speed of sound (in the same medium) is called the object's Mach number#REDIRECT Mach number
300px, An F/A-18 Hornet creating a vapor cone at transonic speed">vapor_cone.html" ;"title="F/A-18 Hornet creating a vapor cone">F/A-18 Hornet creating a vapor cone at transonic speed just before reaching the speed of sound
...

. Objects moving at speeds greater than the speed of sound (') are said to be traveling at supersonic
Supersonic speed is the speed of an object that exceeds the speed of sound
The speed of sound is the distance travelled per unit of time by a sound wave as it propagates through an elasticity (solid mechanics), elastic medium. At , the spe ...

speeds.
History

Sir Isaac Newton
Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of s ...

's 1687 '' Principia'' includes a computation of the speed of sound in air as . 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 an adiabatic process
In thermodynamics
Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these qu ...

, not an isothermal process
In thermodynamics, an isothermal process is a type of thermodynamic process in which the temperature ''T'' of a Thermodynamic system, system remains constant: Δ''T'' = 0. This typically occurs when a system is in contact with an outside the ...

). This error was later rectified by Laplace
Pierre-Simon, marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French scholar and polymath
A polymath ( el, πολυμαθής, ', "having learned much"; Latin
Latin (, or , ) is a classical language belonging to the I ...

.
During the 17th century there were several attempts to measure the speed of sound accurately, including attempts by Marin Mersenne
Marin Mersenne (also known as Marinus Mersennus or ''le Père'' Mersenne; ; 8 September 1588 – 1 September 1648) was a French whose works touched a wide variety of fields. He is perhaps best known today among mathematicians for numbers, thos ...

in 1630 (1,380 Parisian feet per second), Pierre Gassendi
Pierre Gassendi (; also Pierre Gassend, Petrus Gassendi; 22 January 1592 – 24 October 1655) was a French
French (french: français(e), link=no) may refer to:
* Something of, from, or related to France
France (), officially the French R ...

in 1635 (1,473 Parisian feet per second) and Robert Boyle
Robert Boyle (; 25 January 1627 – 31 December 1691) was an Anglo-Irish natural philosopher, chemist, physicist, and inventor. Boyle is largely regarded today as the first modern chemist, and therefore one of the founders of modern che ...

(1,125 Parisian feet per second). In 1709, the Reverend William Derham
William Derham FRS (26 November 16575 April 1735)Smolenaars, Marja.Derham, William (1657–1735), ''Oxford Dictionary of National Biography
The ''Dictionary of National Biography'' (''DNB'') is a standard work of reference on notable figures ...

, Rector
Rector (Latin for the member of a vessel's crew who steers) may refer to:
Style or title
*Rector (ecclesiastical), a cleric who functions as an administrative leader in some Christian denominations
*Rector (academia), a senior official in an educ ...

of Upminster, published a more accurate measure of the speed of sound, at 1,072 Parisian feet per second. (The Parisian foot was 325 mm. This is longer than the standard "international foot" in common use today, which was officially defined in 1959 as 304.8 mm, making the speed of sound at 1,055 Parisian feet per second).
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
North Ockendon is the easternmost and most outlying settlement of Greater London
Greater London is a Ceremonial counties of England, ceremonial county of England that makes up the majority of the London region. This Regions of England, regio ...

church. The distance was known by triangulation
In trigonometry
Trigonometry (from Greek '' trigōnon'', "triangle" and '' metron'', "measure") is a branch of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathe ...

, 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 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 thestiffness
Stiffness is the extent to which an object resists Deformation (mechanics), deformation in response to an applied force.
The complementary concept is flexibility or pliability: the more flexible an object is, the less stiff it is.
Calculations
...

/rigidity of the spring
Spring(s) may refer to:
Common uses
* Spring (season), a season of the year
* Spring (device), a mechanical device that stores energy
* Spring (hydrology), a natural source of water
* Spring (mathematics), a geometric surface in the shape of a heli ...

s, 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
An elastic modulus (also known as modulus of elasticity) is a quantity that measures an object or substance's resistance to being deformed elastically (i.e., non-permanently) when a stress is applied to it. The elastic modulus of an object is defi ...

", and the mass corresponds to the material density
The density (more precisely, the volumetric mass density; also known as specific mass), of a substance is its per unit . The symbol most often used for density is ''ρ'' (the lower case Greek letter ), although the Latin letter ''D'' can also ...

. Given that all other things being equal (ceteris paribus
' or ' () is a Latin phrase meaning "other things equal"; English translations of the phrase include "all other things being equal" or "other things held constant" or "all else unchanged". A prediction or a statement about a ontic, causal, epist ...

), sound will travel slower in spongy materials, and faster in stiffer ones. Effects like dispersion and reflection can also be understood using this model.
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 () gas than in heavy hydrogen (deuterium
Deuterium (or hydrogen-2, symbol or deuterium, also known as heavy hydrogen) is one of two stable isotopes
The term stable isotope has a meaning similar to stable nuclide, but is preferably used when speaking of nuclides of a specific elemen ...

) 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.
Some textbooks mistakenly state that the speed of sound increases with density. This notion is illustrated by presenting data for three materials, such as air, water, and steel; they each have vastly different compressibility, which more than makes up for the density differences. 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 larger density of water, which works to ''slow'' sound in water relative to air, nearly makes up for the compressibility differences in the two media.
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

In a gas or liquid, sound consists of compression waves. In solids, waves propagate as two different types. Alongitudinal wave
Longitudinal waves are waves in which the vibration of the medium is parallel to the direction the wave travels and displacement of the medium is in the same (or opposite) direction of the wave propagation. Mechanical wave, Mechanical longitudinal ...

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 , also called a shear wave
__NOTOC__
In seismology
Seismology (; from Ancient Greek
Ancient Greek includes the forms of the Greek language used in ancient Greece and the classical antiquity, ancient world from around 1500 BC to 300 BC. It is often roughly divided ...

, 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
Polarization or polarisation may refer to:
In the physical sciences
*Polarization (waves), the ability of waves to oscillate in more than one direction, in particular polarization of light, responsible for example for the glare-reducing effect of ...

" of this type of wave. In general, transverse waves occur as a pair of orthogonal
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

polarizations.
These different waves (compression waves and the different polarizations of shear waves) may have different speeds at the same frequency. Therefore, they arrive at an observer at different times, an extreme example being an earthquake
An earthquake (also known as a quake, tremor or temblor) is the shaking of the surface of the Earth resulting from a sudden release of energy in the Earth
Earth is the third planet from the Sun and the only astronomical object known ...

, where sharp compression waves arrive first and rocking transverse waves seconds later.
The speed of a compression wave in a fluid is determined by the medium's compressibility
In thermodynamics
Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quan ...

and density
The density (more precisely, the volumetric mass density; also known as specific mass), of a substance is its per unit . The symbol most often used for density is ''ρ'' (the lower case Greek letter ), although the Latin letter ''D'' can also ...

. In solids, the compression waves are analogous to those in fluids, depending on compressibility and density, but with the additional factor of shear modulus
In materials science
The interdisciplinary field of materials science, also commonly termed materials science and engineering, covers the design and discovery of new materials, particularly solids. The intellectual origins of materials scienc ...

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.
Equations

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: : $c\; =\; \backslash sqrt,$ where * ''Kbulk modulus
The bulk modulus (K or B) of a substance is a measure of how resistant to compression that substance is. It is defined as the ratio of the infinitesimal
In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to ze ...

(or the modulus of bulk elasticity for gases);
* $\backslash rho$ is the density
The density (more precisely, the volumetric mass density; also known as specific mass), of a substance is its per unit . The symbol most often used for density is ''ρ'' (the lower case Greek letter ), although the Latin letter ''D'' can also ...

.
Thus, the speed of sound increases with the stiffness (the resistance of an elastic body to deformation by an applied force) of the material and decreases with an increase in density. For ideal gases, the bulk modulus ''K'' is simply the gas pressure multiplied by the dimensionless adiabatic index
In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volume ...

, which is about 1.4 for air under normal conditions of pressure and temperature.
For general equations of state
In physics
Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through S ...

, if classical mechanics is used, the speed of sound ''c'' can be derived as 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 $dx\; =\; v\; \backslash ,\; dt$. In steady state
In systems theory
Systems theory is the interdisciplinary study of system
A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole.
A system, surrounded and influen ...

, the mass flow rate
In physics and engineering, mass flow rate is the mass of a substance which passes per unit of time. Its unit of measurement, unit is kilogram per second in SI units, and Slug (unit), slug per second or pound (mass), pound per second in US custo ...

$\backslash dot\; m\; =\; \backslash rho\; v\; A$ must be the same at the two ends of the tube, therefore the mass fluxIn physics
Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through Spac ...

$j=\backslash rho\; v$ is constant and $v\; \backslash ,\; d\backslash rho\; =\; -\backslash rho\; \backslash ,\; dv$. Per Newton's second law
Newton's laws of motion are three Scientific law, laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows:
''Law 1''. A body continues ...

, the pressure-gradient forceThe pressure-gradient force is the force that results when there is a difference in pressure across a surface. In general, a pressure
Pressure (symbol: ''p'' or ''P'') is the force
In physics
Physics (from grc, φυσική ( ...

provides the acceleration:
:$\backslash begin\; \backslash frac\; \&=-\backslash frac\backslash frac\backslash \backslash \; \backslash rightarrow\; dP\&=(-\backslash rho\; \backslash ,dv)\backslash frac=(v\; \backslash ,\; d\backslash rho)v\backslash \backslash \; \backslash rightarrow\; v^2\&\; \backslash equiv\; c^2=\backslash frac\; \backslash end$
And therefore:
:$c\; =\; \backslash sqrt,$
where
* ''P'' is the pressure;
* $\backslash rho$ is the density and the derivative
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

is taken isentropically, that is, at constant entropy
Entropy is a scientific concept as well as a measurable physical property that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynamic ...

''s''. This is because a sound wave travels so fast that its propagation can be approximated as an adiabatic process
In thermodynamics
Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these qu ...

.
If relativistic effects are important, the speed of sound is calculated from the relativistic Euler equations
In fluid mechanics
Fluid mechanics is the branch of physics concerned with the mechanics
Mechanics (Ancient Greek, Greek: ) is the area of physics concerned with the motions of physical objects, more specifically the relationships among forc ...

.
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 COdispersive medium
In optics
Optics is the branch of physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, ...

, the speed of sound is a function of sound frequency, through the dispersion relation
In the physical sciences and electrical engineering, dispersion relations describe the effect of #Dispersion, dispersion on the properties of waves in a medium. A dispersion relation relates the wavelength or wavenumber of a wave to its frequency ...

. Each frequency component propagates at its own speed, called the phase velocity
The phase velocity of a wave
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or ...

, while the energy of the disturbance propagates at the group velocity
in groups of gravity waves on the surface of deep water. The red square moves with the phase velocity, and the green circles propagate with the group velocity. In this deep-water case, ''the phase velocity is twice ...

. The same phenomenon occurs with light waves; see optical dispersion
A compact fluorescent lamp seen through an Amici prism ">Amici_prism.html" ;"title="compact fluorescent lamp seen through an Amici prism">compact fluorescent lamp seen through an Amici prism
In optics, dispersion is the phenomenon in which the ...

for a description.
Dependence on the properties of the medium

The speed of sound is variable and depends on the properties of the substance through which the wave is travelling. In solids, the speed of transverse (or shear) waves depends on the shear deformation under shear stress (called theshear modulus
In materials science
The interdisciplinary field of materials science, also commonly termed materials science and engineering, covers the design and discovery of new materials, particularly solids. The intellectual origins of materials scienc ...

), and the density of the medium. Longitudinal (or compression) waves in solids depend on the same two factors with the addition of a dependence on compressibility
In thermodynamics
Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quan ...

.
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
In thermal physics
Example of a thermal column between the ground and a cumulus
A thermal column (or thermal) is a column of rising air in the lower altitudes of Earth's atmosphere
File:Atmosphere gas proportions.svg, Composition of Ear ...

(adiabatic index), while pressure and density are inversely related to the temperature and molecular weight, thus making only the completely independent properties of ''temperature and molecular structure'' important (heat capacity ratio may be determined by temperature and molecular structure, but simple molecular weight is not sufficient to determine it).
Sound propagates faster in low molecular weight
A molecule is an electrically
Electricity is the set of physical phenomena associated with the presence and motion
Image:Leaving Yongsan Station.jpg, 300px, Motion involves a change in position
In physics, motion is the phenomenon ...

gases such as helium
Helium (from el, ἥλιος, helios
Helios; Homeric Greek: ), Latinized as Helius; Hyperion and Phaethon are also the names of his father and son respectively. often given the epithets Hyperion ("the one above") and Phaethon ("the shining" ...

than it does in heavier gases such as xenon
Xenon is a chemical element
In chemistry, an element is a pure Chemical substance, substance consisting only of atoms that all have the same numbers of protons in their atomic nucleus, nuclei. Unlike chemical compounds, chemical elemen ...

. 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
An ideal gas is a theoretical composed of many randomly moving s that are not subject to . The ideal gas concept is useful because it obeys the , a simplified , and is amenable to analysis under . The requirement of zero interaction can often be ...

the molecular composition is fixed, and thus the speed of sound depends only on its temperature
Temperature ( ) is a physical quantity that expresses hot and cold. It is the manifestation of thermal energy
Thermal radiation in visible light can be seen on this hot metalwork.
Thermal energy refers to several distinct physical concept ...

. At a constant temperature, the gas pressure
Pressure (symbol: ''p'' or ''P'') is the force
In physics, a force is an influence that can change the motion (physics), motion of an Physical object, object. A force can cause an object with mass to change its velocity (e.g. moving fr ...

has no effect on the speed of sound, since the density will increase, and since pressure and oxygen
Oxygen is the chemical element
Image:Simple Periodic Table Chart-blocks.svg, 400px, Periodic table, The periodic table of the chemical elements
In chemistry, an element is a pure substance consisting only of atoms that all have the same ...

and nitrogen
Nitrogen is the chemical element
upright=1.0, 500px, The chemical elements ordered by link=Periodic table
In chemistry
Chemistry is the science, scientific study of the properties and behavior of matter. It is a natural science ...

molecules of the air are replaced by lighter molecules of water
Water (chemical formula H2O) is an , transparent, tasteless, odorless, and , which is the main constituent of 's and the s of all known living organisms (in which it acts as a ). It is vital for all known forms of , even though it provide ...

. This is a simple mixing effect.
Altitude variation and implications for atmospheric acoustics

In theEarth's atmosphere
The atmosphere of Earth is the layer of gas
Gas is one of the four fundamental states of matter (the others being solid
Solid is one of the four fundamental states of matter (the others being liquid, gas and plasma). The mo ...

, the chief factor affecting the speed of sound is the temperature
Temperature ( ) is a physical quantity that expresses hot and cold. It is the manifestation of thermal energy
Thermal radiation in visible light can be seen on this hot metalwork.
Thermal energy refers to several distinct physical concept ...

. For a given ideal gas with constant heat capacity and composition, the speed of sound is dependent ''solely'' upon temperature; see 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
In physics
Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior throu ...

upward, away from listeners on the ground, creating an acoustic shadowAn acoustic shadow or sound shadow is an area through which sound waves fail to propagate, due to topographical
Topography is the study of the forms and features of land surfaces. The topography of an area could refer to the surface forms and ...

at some distance from the source. The decrease of the speed of sound with height is referred to as a negative sound speed gradient
In physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. ...

.
However, there are variations in this trend above . In particular, in the stratosphere
The stratosphere () is the second layer of the atmosphere of the Earth, located above the troposphere
The troposphere is the first and lowest layer of the atmosphere of the Earth, and contains 75% of the total mass of the planetary atmosphe ...

above about , the speed of sound increases with height, due to an increase in temperature from heating within the . 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
The thermosphere is the layer in the Earth's atmosphere directly above the mesosphere and below the exosphere. Within this layer of the atmosphere, ultraviolet radiation causes photoionization/photodissociation of molecules, creating ions; the th ...

above .
Practical formula for dry air

The approximate speed of sound in dry (0% humidity) air, in metres per second, at temperatures near , can be calculated from : $c\_\; =\; (331.3\; +\; 0.606\; \backslash cdot\; \backslash vartheta)~~~\backslash mathrm,$ where ''$\backslash vartheta$'' is the temperature in degreesCelsius
The degree Celsius is a unit of temperature on the Celsius scale, a temperature scale
Scale of temperature is a methodology of calibrating the physical quantity temperature in metrology. Empirical scales measure temperature in relation to conv ...

(°C).
This equation is derived from the first two terms of the Taylor expansion
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

of the following more accurate equation:
: $c\_\; =\; 331.3~\; \backslash sqrt~~~~\backslash mathrm.$
Dividing the first part, and multiplying the second part, on the right hand side, by gives the exactly equivalent form
: $c\_\; =\; 20.05~\; \backslash sqrt~~~~\backslash mathrm.$
which can also be written as
: $c\_\; =\; 20.05~\; \backslash sqrt~~~~\backslash mathrm$
where ''T'' denotes the thermodynamic temperature
Thermodynamic temperature is a quantity defined in thermodynamics as distinct from Kinetic theory of gases, kinetic theory or statistical mechanics. A thermodynamic temperature reading of zero is of particular importance for the third law of therm ...

.
The value of , which represents the speed at (or ), is based on theoretical (and some measured) values of the heat capacity ratio
In thermal physics
Example of a thermal column between the ground and a cumulus
A thermal column (or thermal) is a column of rising air in the lower altitudes of Earth's atmosphere
File:Atmosphere gas proportions.svg, Composition of Ear ...

, ''γ'', as well as on the fact that at 1 atm
ATM or atm often refers to:
* Atmosphere (unit) or atm, a unit of atmospheric pressure
* Automated teller machine, a cash dispenser or cash machine
ATM or atm may also refer to:
Computing
* ATM (computer), a ZX Spectrum clone developed in Mos ...

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 ''γ'' is assumed to be exactly, the speed is calculated (see section below) to be , the coefficient used above.
This equation is correct to a much wider temperature range, but still depends on the approximation of heat capacity ratio being independent of temperature, and for this reason will fail, particularly at higher temperatures. It gives good predictions in relatively dry, cold, low-pressure conditions, such as the Earth's stratosphere
The stratosphere () is the second layer of the atmosphere of the Earth, located above the troposphere
The troposphere is the first and lowest layer of the atmosphere of the Earth, and contains 75% of the total mass of the planetary atmosphe ...

. The equation fails at extremely low pressures and short wavelengths, due to dependence on the assumption that the wavelength of the sound in the gas is much longer than the average mean free path
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succ ...

between gas molecule collisions. A derivation of these equations will be given in the following section.
A graph comparing results of the two equations is at right, using the slightly different value of for the speed of sound at .
Details

Speed of sound in ideal gases and air

For an ideal gas, ''K'' (thebulk modulus
The bulk modulus (K or B) of a substance is a measure of how resistant to compression that substance is. It is defined as the ratio of the infinitesimal
In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to ze ...

in equations above, equivalent to C, the coefficient of stiffness in solids) is given by
: $K\; =\; \backslash gamma\; \backslash cdot\; p,$
thus, from the Newton–Laplace equation above, the speed of sound in an ideal gas is given by
: $c\; =\; \backslash sqrt,$
where
* ''γ'' is the adiabatic index
In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volume ...

also known as the ''isentropic expansion factor''. It is the ratio of the specific heat of a gas at constant pressure to that of a gas at constant volume ($C\_p/C\_v$) 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;
* ''p'' is the pressure
Pressure (symbol: ''p'' or ''P'') is the force
In physics, a force is an influence that can change the motion (physics), motion of an Physical object, object. A force can cause an object with mass to change its velocity (e.g. moving fr ...

;
* ''ρ'' is the ideal gas
An ideal gas is a theoretical composed of many randomly moving s that are not subject to . The ideal gas concept is useful because it obeys the , a simplified , and is amenable to analysis under . The requirement of zero interaction can often be ...

law to replace ''p'' with ''nRT''/''V'', and replacing ''ρ'' with ''nM''/''V'', the equation for an ideal gas becomes
: $c\_\; =\; \backslash sqrt\; =\; \backslash sqrt\; =\; \backslash sqrt,$
where
* ''c''molar gas constant
The gas constant (also known as the molar gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . It is equivalent to the Boltzmann constant, but expressed in units of energy per temperature increment per '' m ...

(universal gas constant);
* ''k'' is the Boltzmann constant
The Boltzmann constant ( or ) is the proportionality factor
In mathematics, two varying quantities are said to be in a Binary relation, relation of proportionality,
Multiplication, multiplicatively connected to a Constant (mathematics), c ...

;
* ''γ'' (gamma) is the adiabatic index
In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volume ...

. 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 1.3991 to 1.403 at , for air. Gamma is exactly for monatomic gases such as noble gas
The noble gases (historically also the inert gases; sometimes referred to as aerogens) make up a class of chemical element
In chemistry
Chemistry is the study of the properties and behavior of . It is a that covers the that m ...

es and it is for triatomic molecule gases that, like H2O, are not co-linear (a co-linear triatomic gas such as CO2 is equivalent to a diatomic gas for our purposes here);
* ''T'' is the absolute temperature;
* ''M'' is the molar mass of the gas. The mean molar mass for dry air is about ;
* ''n'' is the number of moles;
* ''m'' is the mass of a single molecule.
This equation applies only when the sound wave is a small perturbation on the ambient condition, and the certain other noted conditions are fulfilled, as noted below. Calculated values for ''c''Newton
Newton most commonly refers to:
* Isaac Newton (1642–1726/1727), English scientist
* Newton (unit), SI unit of force named after Isaac Newton
Newton may also refer to:
Arts and entertainment
* Newton (film), ''Newton'' (film), a 2017 Indian fil ...

famously considered the speed of sound before most of the development of thermodynamics
Thermodynamics is a branch of physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in ot ...

and so incorrectly used isothermal
In thermodynamics, an isothermal process is a type of thermodynamic process in which the temperature ''T'' of a Thermodynamic system, system remains constant: Δ''T'' = 0. This typically occurs when a system is in contact with an outside the ...

calculations instead of . His result was missing the factor of ''γ'' but was otherwise correct.
Numerical substitution of the above values gives the ideal gas approximation of sound velocity for gases, which is accurate at relatively low gas pressures and densities (for air, this includes standard Earth sea-level conditions). Also, for diatomic gases the use of requires that the gas exists in a temperature range high enough that rotational heat capacity is fully excited (i.e., molecular rotation is fully used as a heat energy "partition" or reservoir); but at the same time the temperature must be low enough that molecular vibrational modes contribute no heat capacity (i.e., insignificant heat goes into vibration, as all vibrational quantum modes above the minimum-energy-mode, have energies too high to be populated by a significant number of molecules at this temperature). For air, these conditions are fulfilled at room temperature, and also temperatures considerably below room temperature (see tables below). See the section on gases in specific heat capacity
In thermodynamics
Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these qua ...

for a more complete discussion of this phenomenon.
For air, we introduce the shorthand
: $R\_*\; =\; R/M\_.$
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,
: $c\_\; =\; \backslash sqrt\; =\; \backslash sqrt,$
: $c\_\; =\; \backslash sqrt\; \backslash cdot\; \backslash sqrt,$
where ''$\backslash vartheta$'' (theta) is the temperature in degrees Celsius
The degree Celsius is a unit of temperature on the Celsius scale, a temperature scale
Scale of temperature is a methodology of calibrating the physical quantity temperature in metrology. Empirical scales measure temperature in relation to conv ...

(°C).
Substituting numerical values
: $R\; =\; 8.314\backslash ,463~\backslash mathrm$
for the molar gas constant
The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . It is the molar equivalent to the Boltzmann constant
The Boltzmann constant ( or ) is the proportionality f ...

in J/mole/Kelvin, and
: $M\_\; =\; 0.028\backslash ,964\backslash ,5~\backslash mathrm$
for the mean molar mass of air, in kg; and using the ideal diatomic gas value of , we have
: $c\_\; =\; 331.3~~\; \backslash sqrt~~~\backslash mathrm.$
Finally, Taylor expansion of the remaining square root in $\backslash vartheta$ yields
: $c\_\; =\; 331.3~(1\; +\; \backslash frac)~~~\backslash mathrm,$
: $c\_\; =\; (331.3\; +\; 0.606\; \backslash cdot\; \backslash vartheta)~~~\backslash mathrm.$
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 isrefracted
In physics
Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior throu ...

upward, away from listeners on the ground, creating an acoustic shadowAn acoustic shadow or sound shadow is an area through which sound waves fail to propagate, due to topographical
Topography is the study of the forms and features of land surfaces. The topography of an area could refer to the surface forms and ...

at some distance from the source. Wind shear of 4 m/(s · km) can produce refraction equal to a typical temperature lapse rate
The lapse rate is the rate at which an atmospheric variable, normally temperature in Earth's atmosphere, falls with altitude. ''Lapse rate'' arises from the word ''lapse'', in the sense of a gradual fall. In dry air, the adiabatic lapse rate is 9 ...

of . 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 sound is not being carried along by the wind.
For sound propagation, the exponential variation of wind speed with height can be defined as follows:
: $U(h)\; =\; U(0)\; h^\backslash zeta,$
: $\backslash frac(h)\; =\; \backslash zeta\; \backslash frac,$
where
* ''U''(''h'') is the speed of the wind at height ''h'';
* ''ζ'' is the exponential coefficient based on ground surface roughness, typically between 0.08 and 0.52;
* ''dU''/''dH''(''h'') is the expected wind gradient at height ''h''.
In the 1862 American Civil War
The American Civil War (also known by other names
Other most often refers to:
* Other (philosophy), a concept in psychology and philosophy
Other or The Other may also refer to:
Books
* The Other (Tryon novel), ''The Other'' (Tryon nove ...

Battle of Iuka
The Battle of Iuka was fought on September 19, 1862, in Iuka, Mississippi, during the American Civil War. In the opening battle of the Iuka-Corinth Campaign, Union Army, Union Major general (United States), Maj. Gen. William Rosecrans stopped the ...

, 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 only (six miles) downwind.
Tables

In the standard atmosphere: * ''T''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 path
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succ ...

increases. 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 higher ''γ'' (...) than diatomics do (). Thus, at the same molecular mass, the speed of sound of a monatomic gas goes up by a factor of
: $=\; \backslash sqrt\; =\; \backslash sqrt\; =\; 1.091\backslash ldots$
This gives the 9% difference, and would be a typical ratio for speeds of sound at room temperature in helium
Helium (from el, ἥλιος, helios
Helios; Homeric Greek: ), Latinized as Helius; Hyperion and Phaethon are also the names of his father and son respectively. often given the epithets Hyperion ("the one above") and Phaethon ("the shining" ...

vs. deuterium
Deuterium (or hydrogen-2, symbol or deuterium, also known as heavy hydrogen) is one of two stable isotopes
The term stable isotope has a meaning similar to stable nuclide, but is preferably used when speaking of nuclides of a specific elemen ...

, 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).
Note that in this example we have assumed that temperature is low enough that heat capacities are not influenced by molecular vibration (see heat capacity
Heat capacity or thermal capacity is a physical property
A physical property is any property
Property is a system of rights that gives people legal control of valuable things, and also refers to the valuable things themselves. Depending on t ...

). However, vibrational modes simply cause gammas which decrease toward 1, since vibration modes in a polyatomic gas give the gas additional ways to store heat which do not affect temperature, and thus do not affect molecular velocity and sound velocity. Thus, the effect of higher temperatures and vibrational heat capacity acts to increase the difference between the speed of sound in monatomic vs. polyatomic molecules, with the speed remaining greater in monatomics.
Practical application to air

By far, the most important factor influencing the speed of sound in air is temperature. The speed is proportional to the square root of the absolute temperature, giving an increase of about per degree Celsius. For this reason, the pitch of a musical wind instrument increases as its temperature increases. The speed of sound is raised by humidity. The difference between 0% and 100% humidity is about at standard pressure and temperature, but the size of the humidity effect increases dramatically with temperature. The dependence on frequency and pressure are normally insignificant in practical applications. In dry air, the speed of sound increases by about 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. As shown above, the approximate value 1000/3 = 333.33... m/s is exact a little below 5 °C and is a good approximation for all "usual" outside temperatures (in temperate climates, at least), hence the usual rule of thumb to determine how far lightning has struck: count the seconds from the start of the lightning flash to the start of the corresponding roll of thunder and divide by 3: the result is the distance in kilometers to the nearest point of the lightning bolt.Mach number

Mach number, a useful quantity in aerodynamics, is the ratio of airspeed
In everyday use and in kinematics
Kinematics is a subfield of physics, developed in classical mechanics, that describes the Motion (physics), motion of points, bodies (objects), and systems of bodies (groups of objects) without considerin ...

to the local speed of sound. At altitude, for reasons explained, Mach number is a function of temperature.
Aircraft flight instruments
Flight instruments are the instruments in the cockpit of an aircraft that provide the pilot with data about the flight situation of that aircraft, such as altitude, airspeed, Variometer, vertical speed, heading and much more other crucial inform ...

, however, operate using pressure differential to compute Mach number, not temperature. The assumption is that a particular pressure represents a particular altitude and, therefore, a standard temperature. Aircraft flight instruments need to operate this way because the stagnation pressure sensed by a Pitot tube
, combines a pitot tube (right) with a static port and an angle-of-attack vane (left). Air-flow is right to left.
helicopter
A pitot ( ) tube, also known as pitot probe, is a flow measurement
Flow measurement is the quantification of bulk ...

is 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 Derham
William Derham FRS (26 November 16575 April 1735)Smolenaars, Marja.Derham, William (1657–1735), ''Oxford Dictionary of National Biography
The ''Dictionary of National Biography'' (''DNB'') is a standard work of reference on notable figures ...

and acknowledged by Isaac Newton
Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Greek: ) includes the study of such topics a ...

. Derham had a telescope at the top of the tower of the Church of St Laurence in Upminster
Upminster is a suburb
The Swedish suburbs of Husby/Kista/Akalla are built according to the typical city planning of the Million Programme.
A suburb (or suburban area or suburbia) is a mixed-use or residential area, existing either as pa ...

, 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 twomicrophone
A microphone, colloquially called a mic or mike (), is a device – a transducer
A transducer is a device that energy from one form to another. Usually a transducer converts a in one form of energy to a signal in another.
Transducers ar ...

s and a fast recording device such as a digital
Digital usually refers to something using discrete digits, often binary digits.
Technology and computing Hardware
*Digital electronics
Digital electronics is a field of electronics
The field of electronics is a branch of physics and electr ...

storage scope. This method uses the following idea.
If a sound source and two microphones are arranged in a straight line, with the sound source at one end, then the following can be measured:
# The distance between the microphones (''x''), called microphone basis.
# The time of arrival between the signals (delay) reaching the different microphones (''t'').
Then ''v'' = ''x''/''t''.
Other methods

In these methods, thetime
Time is the continued sequence of existence and event (philosophy), events that occurs in an apparently irreversible process, irreversible succession from the past, through the present, into the future. It is a component quantity of various me ...

measurement has been replaced by a measurement of the inverse of time (frequency
Frequency is the number of occurrences of a repeating event per unit of time
A unit of time is any particular time
Time is the indefinite continued sequence, progress of existence and event (philosophy), events that occur in an apparen ...

).
Kundt's tube
Kundt's tube is an experimental acoustical apparatus invented in 1866 by German physicist August Kundt for the measurement of the speed of sound in a gas or a solid rod. The experiment is still taught today due to its ability to demonstrate longit ...

is an example of an experiment which can be used to measure the speed of sound in a small volume. It has the advantage of being able to measure the speed of sound in any gas. This method uses a powder to make the nodes
In general, a node is a localized swelling (a "knot") or a point of intersection (a Vertex (graph theory), vertex).
Node may refer to:
In mathematics
*Vertex (graph theory), a vertex in a mathematical graph
*Node (autonomous system), behaviour fo ...

and s visible to the human eye. This is an example of a compact experimental setup.
A tuning fork
A tuning fork is an acoustic
Acoustic may refer to:
Music Albums
* Acoustic (Bayside EP), ''Acoustic'' (Bayside EP)
* Acoustic (Britt Nicole EP), ''Acoustic'' (Britt Nicole EP)
* Acoustic (Joey Cape and Tony Sly album), ''Acoustic'' (Joey Cap ...

can be held near the mouth of a long pipe
Pipe(s) or PIPE(S) may refer to:
Common uses
* Pipe (fluid conveyance)
A pipe is a tubular section or hollow Cylinder (geometry), cylinder, usually but not necessarily of circle, circular cross section (geometry), cross-section, used m ...

which is dipping into a barrel of water
Water (chemical formula H2O) is an , transparent, tasteless, odorless, and , which is the main constituent of 's and the s of all known living organisms (in which it acts as a ). It is vital for all known forms of , even though it provide ...

. 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 + 2''n'')λ/4'' where ''n'' is an integer. As the point for the pipe at the open end is slightly outside the mouth of the pipe it is best to find two or more points of resonance and then measure half a wavelength between these.
Here it is the case that ''v'' = ''fλ''.
High-precision measurements in air

The effect of impurities can be significant when making high-precision measurements. Chemicaldesiccant
A desiccant is a hygroscopic
Hygroscopy is the phenomenon of attracting and holding water molecules via either absorption (chemistry), absorption or adsorption from the surrounding Natural environment, environment, which is usually at norma ...

s can be used to dry the air, but will, in turn, contaminate the sample. The air can be dried cryogenically, but this has the effect of removing the carbon dioxide as well; therefore many high-precision measurements are performed with air free of carbon dioxide rather than with natural air. A 2002 review 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

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. Inearthquake
An earthquake (also known as a quake, tremor or temblor) is the shaking of the surface of the Earth resulting from a sudden release of energy in the Earth
Earth is the third planet from the Sun and the only astronomical object known ...

s, the corresponding seismic waves are called P-wave
A P wave (primary wave or pressure wave) is one of the two main types of elastic body waves, called seismic waves
Seismic waves are waves of energy
In physics
Physics (from grc, φυσική (ἐπιστήμη), physik ...

s (primary waves) and S-wave
__NOTOC__
In seismology
Seismology (; from Ancient Greek σεισμός (''seismós'') meaning "Earthquake, earthquake" and -λογία (''-logía'') meaning "study of") is the scientific study of earthquakes and the propagation of Linear el ...

s (secondary waves), respectively. The sound velocities of these two types of waves propagating in a homogeneous 3-dimensional solid are respectively given byL. E. Kinsler et al. (2000), ''Fundamentals of acoustics'', 4th Ed., John Wiley and sons Inc., New York, USA.
: $c\_\; =\; \backslash sqrt\; =\; \backslash sqrt,$
: $c\_\; =\; \backslash sqrt,$
where
* ''K'' is the bulk modulus
The bulk modulus (K or B) of a substance is a measure of how resistant to compression that substance is. It is defined as the ratio of the infinitesimal
In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to ze ...

of the elastic materials;
* ''G'' is the Young's modulus
Young's modulus E, the Young modulus, or the modulus of elasticity
An elastic modulus (also known as modulus of elasticity) is a quantity that measures an object or substance's resistance to being deformed elastically (i.e., non-permanently) wh ...

;
* ''ρ'' is the density;
* ''ν'' is Poisson's ratio
In materials science and solid mechanics, Poisson's ratio \nu (Nu (letter), nu) is a measure of the Poisson effect, the Deformation (engineering), deformation (expansion or contraction) of a material in directions perpendicular to the specific dir ...

.
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 ''c''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: : $c\_\; =\; \backslash sqrt,$ where ''E'' isYoung's modulus
Young's modulus E, the Young modulus, or the modulus of elasticity
An elastic modulus (also known as modulus of elasticity) is a quantity that measures an object or substance's resistance to being deformed elastically (i.e., non-permanently) wh ...

. This is similar to the expression for shear waves, save that Young's modulus
Young's modulus E, the Young modulus, or the modulus of elasticity
An elastic modulus (also known as modulus of elasticity) is a quantity that measures an object or substance's resistance to being deformed elastically (i.e., non-permanently) wh ...

replaces the Poisson's ratio
In materials science and solid mechanics, Poisson's ratio \nu (Nu (letter), nu) is a measure of the Poisson effect, the Deformation (engineering), deformation (expansion or contraction) of a material in directions perpendicular to the specific dir ...

for the material.
Speed of sound in liquids

In a fluid, the only non-zerostiffness
Stiffness is the extent to which an object resists Deformation (mechanics), deformation in response to an applied force.
The complementary concept is flexibility or pliability: the more flexible an object is, the less stiff it is.
Calculations
...

is to volumetric deformation (a fluid does not sustain shear forces).
Hence the speed of sound in a fluid is given by
: $c\_\; =\; \backslash sqrt,$
where ''K'' is the Water

In fresh water, sound travels at about at (see the External Links section below for online calculators). Applications of underwater sound can be found insonar
Sonar (sound navigation and ranging) is a technique that uses sound propagation (usually underwater, as in submarine navigation) to navigation, navigate, measure distances (ranging), communicate with or detect objects on or under the surface o ...

, acoustic communication and acoustical oceanography.
Seawater

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). The speed of sound in seawater depends on pressure (hence depth), temperature (a change of ~ ), and salinity (a change of 1Per mil, ‰ ~ ), 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: : $c(T,\; S,\; z)\; =\; a\_1\; +\; a\_2\; T\; +\; a\_3\; T^2\; +\; a\_4\; T^3\; +\; a\_5\; (S\; -\; 35)\; +\; a\_6\; z\; +\; a\_7\; z^2\; +\; a\_8\; T(S\; -\; 35)\; +\; a\_9\; T\; z^3,$ where * ''T'' is the temperature in degrees Celsius; * ''S'' is the salinity in parts per thousand; * ''z'' is the depth in metres. The constants ''a''Technical Guides. Speed of Sound in Sea-Water

for an online calculator. (Note: The Sound Speed vs. Depth graph does ''not'' correlate directly to the MacKenzie formula. This is due to the fact that the temperature and salinity varies at different depths. When ''T'' and ''S'' are held constant, the formula itself is always increasing with depth.) Other equations for the speed of sound in sea water are accurate over a wide range of conditions, but are far more complicated, e.g., that by V. A. Del Grosso and the Chen-Millero-Li Equation.

Speed of sound in plasma

The speed of sound in a Plasma (physics), plasma for the common case that the electrons are hotter than the ions (but not too much hotter) is given by the formula (see Plasma parameters#Velocities, here) : $c\_s\; =\; (\backslash gamma\; ZkT\_\backslash mathrm/m\_\backslash mathrm)^\; =\; 90.85\; (\backslash gamma\; ZT\_e/\backslash mu)^~\backslash mathrm,$ where * ''m''Boltzmann constant
The Boltzmann constant ( or ) is the proportionality factor
In mathematics, two varying quantities are said to be in a Binary relation, relation of proportionality,
Multiplication, multiplicatively connected to a Constant (mathematics), c ...

;
* ''γ'' is the Gradients

When sound spreads out evenly in all directions in three dimensions, the intensity drops in proportion to the inverse square of the distance. However, in the ocean, there is a layer called the 'deep sound channel' or SOFAR channel which can confine sound waves at a particular depth. In the SOFAR channel, the speed of sound is lower than that in the layers above and below. Just as light waves will refract towards a region of higher refractive index, index, sound waves will refraction, refract towards a region where their speed is reduced. The result is that sound gets confined in the layer, much the way light can be confined to a sheet of glass or optical fiber. Thus, the sound is confined in essentially two dimensions. In two dimensions the intensity drops in proportion to only the inverse of the distance. This allows waves to travel much further before being undetectably faint. A similar effect occurs in the atmosphere. Project Mogul successfully used this effect to detect a nuclear explosion at a considerable distance.See also

* Acoustoelastic effect * Elastic wave * Second sound * Sonic boom * Sound barrier * Speeds of sound of the elements * Underwater acoustics * VibrationsReferences

External links

Speed of Sound Calculator

Speed of sound: Temperature Matters, Not Air Pressure

How to Measure the Speed of Sound in a Laboratory

* [http://www.dosits.org/ Discovery of Sound in the Sea] (uses of sound by humans and other animals) {{Authority control Fluid dynamics Aerodynamics Acoustics Sound Sound measurements Physical quantities Chemical properties Velocity Temporal rates