Speed of sound

# Speed of sound

Overview
{{other uses}} The speed of sound is the distance travelled during a unit of time by a sound wave propagating through an elastic medium. In dry air at 20 °C (68 °F), the speed of sound is 343.2 metres per second (1,126 ft/s). This is 1236 kilometres per hour (768 mph), or about one kilometer in three seconds or approximately one mile in five seconds. In fluid dynamics
Fluid dynamics
In physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...

, the speed of sound in a fluid medium (gas or liquid) is used as a relative measure of speed itself.
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Encyclopedia
{{other uses}}
The speed of sound is the distance travelled during a unit of time by a sound wave propagating through an elastic medium. In dry air at 20 °C (68 °F), the speed of sound is 343.2 metres per second (1,126 ft/s). This is 1236 kilometres per hour (768 mph), or about one kilometer in three seconds or approximately one mile in five seconds. In fluid dynamics
Fluid dynamics
In physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...

, the speed of sound in a fluid medium (gas or liquid) is used as a relative measure of speed itself. The speed of an object (in distance per time) divided by the speed of sound in the fluid is called the Mach number
Mach number
Mach number is the speed of an object moving through air, or any other fluid substance, divided by the speed of sound as it is in that substance for its particular physical conditions, including those of temperature and pressure...

. Objects moving at speeds greater than {{gaps|Mach|1}} are traveling at supersonic
Supersonic
Supersonic speed is a rate of travel of an object that exceeds the speed of sound . For objects traveling in dry air of a temperature of 20 °C this speed is approximately 343 m/s, 1,125 ft/s, 768 mph or 1,235 km/h. Speeds greater than five times the speed of sound are often...

speeds. The speed of sound in an ideal gas
Ideal gas
An ideal gas is a theoretical gas composed of a set of randomly-moving, non-interacting point particles. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is amenable to analysis under statistical mechanics.At normal conditions such as...

is independent of frequency, but it weakly depends on frequency for all real physical situations. It is a function of the square root of temperature, but is nearly independent of pressure
Pressure
Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...

or density
Density
The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...

for a given gas. For different gases, the speed of sound is inversely dependent on square root of the mean molecular weight of the gas, and affected to a lesser extent by the number of ways in which the molecules of the gas can store heat
Heat
In physics and thermodynamics, heat is energy transferred from one body, region, or thermodynamic system to another due to thermal contact or thermal radiation when the systems are at different temperatures. It is often described as one of the fundamental processes of energy transfer between...

from compression, since sound in gases is a type of compression. Although, in the case of gases only, the speed of sound may be expressed in terms of a ratio of both density and pressure, these quantities are not fully independent of each other, and canceling their common contributions from physical conditions, leads to a velocity expression using the independent variables of temperature, composition, and heat capacity noted above. In common everyday speech, speed of sound refers to the speed of sound waves in air
Earth's atmosphere
The atmosphere of Earth is a layer of gases surrounding the planet Earth that is retained by Earth's gravity. The atmosphere protects life on Earth by absorbing ultraviolet solar radiation, warming the surface through heat retention , and reducing temperature extremes between day and night...

. However, the speed of sound varies from substance to substance. Sound travels faster in liquids and non-porous solids than it does in air. It travels about 4.3 times faster in water
Water
Water is a chemical substance with the chemical formula H2O. A water molecule contains one oxygen and two hydrogen atoms connected by covalent bonds. Water is a liquid at ambient conditions, but it often co-exists on Earth with its solid state, ice, and gaseous state . Water also exists in a...

(1,484 m/s), and nearly 15 times as fast in iron (5,120 m/s), than in air at 20 degrees Celsius. In solids, sound waves propagate as two different types. A longitudinal wave
Longitudinal wave
Longitudinal waves, as known as "l-waves", are waves that have the same direction of vibration as their direction of travel, which means that the movement of the medium is in the same direction as or the opposite direction to the motion of the wave. Mechanical longitudinal waves have been also...

is associated with compression and decompression in the direction of travel, which is the same process as all sound waves in gases and liquids. A transverse wave
Transverse wave
A transverse wave is a moving wave that consists of oscillations occurring perpendicular to the direction of energy transfer...

, often called shear wave, is due to elastic deformation of the medium perpendicular to the direction of wave travel; the direction of shear-deformation is called the "polarization" of this type of wave. In general, transverse waves occur as a pair of orthogonal polarizations. 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
Earthquake
An earthquake is the result of a sudden release of energy in the Earth's crust that creates seismic waves. The seismicity, seismism or seismic activity of an area refers to the frequency, type and size of earthquakes experienced over a period of time...

, where sharp compression waves arrive first, and rocking transverse waves seconds later. The speed of an elastic wave in any medium is determined by the medium's compressibility and density
Density
The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...

. The speed of shear waves, which can occur only in solids, is determined by the solid material's stiffness
Stiffness
Stiffness is the resistance of an elastic body to deformation by an applied force along a given degree of freedom when a set of loading points and boundary conditions are prescribed on the elastic body.-Calculations:...

, compressibility and density. {{Sound measurements}}

## Basic concept

The transmission of sound can be illustrated by using a toy model
Toy model
In physics, a toy model is a simplified set of objects and equations relating them that can nevertheless be used to understand a mechanism that is also useful in the full, non-simplified theory....

consisting of an array of balls interconnected by springs. For real material the balls represent molecules and the springs represent the bonds between them. Sound passes through the model by compressing and expanding the springs, transmitting energy to neighboring balls, which transmit energy to their springs, and so on. The speed of sound through the model depends on the stiffness of the springs (stiffer springs transmit energy more quickly). Effects like dispersion and reflection can also be understood using this model. In a real material, the stiffness of the springs is called the elastic modulus
Elastic modulus
An elastic modulus, or modulus of elasticity, is the mathematical description of an object or substance's tendency to be deformed elastically when a force is applied to it...

, and the mass corresponds to the density
Density
The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...

. All other things being equal, sound will travel more slowly in spongy materials, and faster in stiffer ones. For instance, sound will travel much faster in steel than soft iron, due to the greater stiffness of steel at about the same density. Similarly, sound travels about 1.41 times faster in light hydrogen (protium) gas than in heavy hydrogen (deuterium
Deuterium
Deuterium, also called heavy hydrogen, is one of two stable isotopes of hydrogen. It has a natural abundance in Earth's oceans of about one atom in of hydrogen . Deuterium accounts for approximately 0.0156% of all naturally occurring hydrogen in Earth's oceans, while the most common isotope ...

) 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 increasing density. This is usually illustrated by presenting data for three materials, such as air, water and steel, which also have vastly different compressibilities which more than make 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.

## Basic formula

In general, the speed of sound c is given by the Newton-Laplace equation:$c = \sqrt\left\{\frac\left\{P\right\}\left\{\rho\right\}\right\}\,$ whereNEWLINE
NEWLINE
P is a coefficient of stiffness, the bulk modulus
Bulk modulus
The bulk modulus of a substance measures the substance's resistance to uniform compression. It is defined as the pressure increase needed to decrease the volume by a factor of 1/e...

(or the modulus of bulk elasticity for gas mediums),$\rho$ is the density
Density
The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...

NEWLINE 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 the density. For general equations of state, if classical mechanics is used, the speed of sound $c$ is given by$c^2=\frac\left\{\partial p\right\}\left\{\partial\rho\right\}$ where differentiation is taken with respect to adiabatic change.NEWLINE
NEWLINE
where $p$ is the pressure and $\rho$ is the density
NEWLINE If relativistic
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...

effects are important, the speed of sound may be calculated from the relativistic Euler equations
Relativistic Euler equations
In fluid mechanics and astrophysics, the relativistic Euler equations are a generalization of the Euler equations that account for the effects of special relativity....

. In a non-dispersive medium sound speed is independent of sound frequency, so the speeds of energy transport and sound propagation are the same. For audible sounds, a mixture of oxygen and nitrogen constitutes a non-dispersive medium. But air does contain a small amount of CO2 which is a dispersive medium, and it introduces dispersion to air at ultrasonic
Ultrasound
Ultrasound is cyclic sound pressure with a frequency greater than the upper limit of human hearing. Ultrasound is thus not separated from "normal" sound based on differences in physical properties, only the fact that humans cannot hear it. Although this limit varies from person to person, it is...

frequencies (> 28 kHz). In a dispersive medium sound speed is a function of sound frequency, through the dispersion relation
Dispersion relation
In physics and electrical engineering, dispersion most often refers to frequency-dependent effects in wave propagation. Note, however, that there are several other uses of the word "dispersion" in the physical sciences....

. The spatial and temporal distribution of a propagating disturbance will continually change. Each frequency component propagates at its own phase velocity
Phase velocity
The phase velocity of a wave is the rate at which the phase of the wave propagates in space. This is the speed at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave will appear to travel at the phase velocity...

, while the energy of the disturbance propagates at the group velocity
Group velocity
The group velocity of a wave is the velocity with which the overall shape of the wave's amplitudes — known as the modulation or envelope of the wave — propagates through space....

. The same phenomenon occurs with light waves; see optical dispersion for a description.

## Dependence on the properties of the medium

The speed of sound is variable and depends on the properties of the substance through of which the wave is travelling. In solids, the speed of longitudinal waves depend on the stiffness to tensile stress, and the density of the medium. In fluids, the medium's compressibility and density are the important factors. In gases, compressibility and density are related, making other compositional effects and properties important, such as temperature and molecular composition. In low molecular weight gases, such as helium
Helium
Helium is the chemical element with atomic number 2 and an atomic weight of 4.002602, which is represented by the symbol He. It is a colorless, odorless, tasteless, non-toxic, inert, monatomic gas that heads the noble gas group in the periodic table...

, sound propagates faster compared to heavier gases, such as xenon
Xenon
Xenon is a chemical element with the symbol Xe and atomic number 54. The element name is pronounced or . A colorless, heavy, odorless noble gas, xenon occurs in the Earth's atmosphere in trace amounts...

(for monatomic gases the speed of sound is about 75% of the mean speed that molecules move in the gas). For a given ideal gas
Ideal gas
An ideal gas is a theoretical gas composed of a set of randomly-moving, non-interacting point particles. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is amenable to analysis under statistical mechanics.At normal conditions such as...

the sound speed depends only on its temperature
Temperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...

. At a constant temperature, the ideal gas pressure
Pressure
Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...

has no effect on the speed of sound, because pressure and density
Density
The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...

(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 (see derivations below). Thus, for a single given gas (where molecular weight does not change) and over a small temperature range (where heat capacity is relatively constant), the speed of sound becomes dependent on only the temperature of the gas. In non-ideal gases, such as a van der Waals gas
Van der Waals equation
The van der Waals equation is an equation of state for a fluid composed of particles that have a non-zero volume and a pairwise attractive inter-particle force It was derived by Johannes Diderik van der Waals in 1873, who received the Nobel prize in 1910 for "his work on the equation of state for...

, 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 sound speed (causing it to increase by about 0.1%-0.6%), because oxygen
Oxygen
Oxygen is the element with atomic number 8 and represented by the symbol O. Its name derives from the Greek roots ὀξύς and -γενής , because at the time of naming, it was mistakenly thought that all acids required oxygen in their composition...

and nitrogen
Nitrogen
Nitrogen is a chemical element that has the symbol N, atomic number of 7 and atomic mass 14.00674 u. Elemental nitrogen is a colorless, odorless, tasteless, and mostly inert diatomic gas at standard conditions, constituting 78.08% by volume of Earth's atmosphere...

molecules of the air are replaced by lighter molecules of water
Water
Water is a chemical substance with the chemical formula H2O. A water molecule contains one oxygen and two hydrogen atoms connected by covalent bonds. Water is a liquid at ambient conditions, but it often co-exists on Earth with its solid state, ice, and gaseous state . Water also exists in a...

. This is a simple mixing effect.

## Implications for atmospheric acoustics

In the Earth's atmosphere
Earth's atmosphere
The atmosphere of Earth is a layer of gases surrounding the planet Earth that is retained by Earth's gravity. The atmosphere protects life on Earth by absorbing ultraviolet solar radiation, warming the surface through heat retention , and reducing temperature extremes between day and night...

, the most important factor affecting the speed of sound is the temperature
Temperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...

(see Details below). Since temperature and thus the speed of sound normally decrease with increasing altitude, sound is refracted
Refraction
Refraction is the change in direction of a wave due to a change in its speed. It is essentially a surface phenomenon . The phenomenon is mainly in governance to the law of conservation of energy. The proper explanation would be that due to change of medium, the phase velocity of the wave is changed...

upward, away from listeners on the ground, creating an acoustic shadow
An acoustic shadow is an area through which sound waves fail to propagate, due to topographical obstructions or disruption of the waves via phenomena such as wind currents. A gobo refers to a movable acoustic isolation panel and that makes an acoustic shadow. As one website refers to it, "an...

at some distance from the source. The decrease of the sound speed with height is referred to as a negative sound speed gradient
In acoustics, the sound speed gradient is the rate of change of the speed of sound with distance, for example with depth in the ocean,or height in the Earth's atmosphere. A sound speed gradient leads to refraction of sound wavefronts in the direction of lower sound speed, causing the sound rays to...

. However, in the stratosphere
Stratosphere
The stratosphere is the second major layer of Earth's atmosphere, just above the troposphere, and below the mesosphere. It is stratified in temperature, with warmer layers higher up and cooler layers farther down. This is in contrast to the troposphere near the Earth's surface, which is cooler...

, the speed of sound increases with height due to heating within the ozone layer
Ozone layer
The ozone layer is a layer in Earth's atmosphere which contains relatively high concentrations of ozone . This layer absorbs 97–99% of the Sun's high frequency ultraviolet light, which is potentially damaging to the life forms on Earth...

, producing a positive sound speed gradient.

## Practical formula for dry air

The approximate speed of sound in dry (0% humidity) air, in meters per second (m·s−1), at temperatures near 0 °C, can be calculated from:$c_\left\{\mathrm\left\{air\right\}\right\} = \left(331\left\{.\right\}3 + 0\left\{.\right\}606 \cdot \vartheta\right) \ \mathrm\left\{m \cdot s^\left\{-1\right\}\right\}\,$ where $\vartheta$ is the temperature in degrees Celsius
Celsius
Celsius is a scale and unit of measurement for temperature. It is named after the Swedish astronomer Anders Celsius , who developed a similar temperature scale two years before his death...

(°C). This equation is derived from the first two terms of the Taylor expansion of the following more accurate equation: $c_\left\{\mathrm\left\{air\right\}\right\} = 331.3\,\mathrm\left\{m \cdot s^\left\{-1\right\}\right\} \sqrt\left\{1+\frac\left\{\vartheta\right\}\left\{273.15\,^\left\{\circ\right\}\mathrm\left\{C\right\}\right\}\right\}$ Dividing the first part, and multiplying the second part, on the right hand side, by $\sqrt\left\{273.15\right\}$ gives the exactly equivalent form: $c_\left\{\mathrm\left\{air\right\}\right\} = 20.0457\,\mathrm\left\{m \cdot s^\left\{-1\right\}\right\} \sqrt\left\{\left\{\vartheta\right\}+ \left\{273.15\;\right\}\right\}$ The value of 331.3 m/s, which represents the 0 °C speed, is based on theoretical (and some measured) values of the heat capacity ratio
Heat capacity ratio
The heat capacity ratio or adiabatic index or ratio of specific heats, is the ratio of the heat capacity at constant pressure to heat capacity at constant volume . It is sometimes also known as the isentropic expansion factor and is denoted by \gamma or \kappa . The latter symbol kappa is...

, $\gamma$, as well as on the fact that at 1 atm
Atmosphere (unit)
The standard atmosphere is an international reference pressure defined as 101325 Pa and formerly used as unit of pressure. For practical purposes it has been replaced by the bar which is 105 Pa...

real air is very well described by the ideal gas approximation. Commonly found values for the speed of sound at 0 °C may vary from 331.2 to 331.6 due to the assumptions made when it is calculated. If ideal gas $\gamma$ is assumed to be 7/5 = 1.4 exactly, the 0 °C speed is calculated (see section below) to be 331.3 m/s, 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
Stratosphere
The stratosphere is the second major layer of Earth's atmosphere, just above the troposphere, and below the mesosphere. It is stratified in temperature, with warmer layers higher up and cooler layers farther down. This is in contrast to the troposphere near the Earth's surface, which is cooler...

. 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
Mean free path
In physics, the mean free path is the average distance covered by a moving particle between successive impacts which modify its direction or energy or other particle properties.-Derivation:...

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 331.5 m/s for the speed of sound °C.

### Speed in ideal gases and in air

For a gas, K (the bulk modulus
Bulk modulus
The bulk modulus of a substance measures the substance's resistance to uniform compression. It is defined as the pressure increase needed to decrease the volume by a factor of 1/e...

in equations above, equivalent to C, the coefficient of stiffness in solids) is approximately given by$K = \gamma \cdot p\,$ thus$c = \sqrt\left\{\gamma \cdot \left\{p \over \rho\right\}\right\}\,$ Where:$\gamma$ is the adiabatic index also known as the isentropic expansion factor. It is the ratio of specific heats of a gas at a constant-pressure to a gas at a constant-volume($C_p/C_v$), 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.NEWLINE
NEWLINE
p is the pressure
Pressure
Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...

.$\rho$ is the density
Density
The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...

NEWLINE Using the ideal gas
Ideal gas
An ideal gas is a theoretical gas composed of a set of randomly-moving, non-interacting point particles. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is amenable to analysis under statistical mechanics.At normal conditions such as...

law to replace $p$ with nRT/V, and replacing ρ with nM/V, the equation for an ideal gas becomes:$c_\left\{\mathrm\left\{ideal\right\}\right\} = \sqrt\left\{\gamma \cdot \left\{p \over \rho\right\}\right\} = \sqrt\left\{\gamma \cdot R \cdot T \over M\right\}= \sqrt\left\{\gamma \cdot k \cdot T \over m\right\}\,$ whereNEWLINE
NEWLINE
• $c_\left\{\mathrm\left\{ideal\right\}\right\}$ is the speed of sound in an ideal gas.
• NEWLINE
• $R$ (approximately 8.3145 J·mol−1·K−1) is the molar gas constant.
• NEWLINE
• $k$ is the Boltzmann constant
• NEWLINE
• $\gamma$ (gamma) is the adiabatic index (sometimes assumed 7/5 = 1.400 for diatomic molecules from kinetic theory, assuming from quantum theory a temperature range at which thermal energy is fully partitioned into rotation (rotations are fully excited), but none into vibrational modes. Gamma is actually experimentally measured over a range from 1.3991 to 1.403 at 0 degrees Celsius, for air. Gamma is assumed from kinetic theory to be exactly 5/3 = 1.6667 for monoatomic molecules such as noble gas
Noble gas
The noble gases are a group of chemical elements with very similar properties: under standard conditions, they are all odorless, colorless, monatomic gases, with very low chemical reactivity...

es).
• NEWLINE
• $T$ is the absolute temperature in kelvin
Kelvin
The kelvin is a unit of measurement for temperature. It is one of the seven base units in the International System of Units and is assigned the unit symbol K. The Kelvin scale is an absolute, thermodynamic temperature scale using as its null point absolute zero, the temperature at which all...

.
• NEWLINE
• $M$ is the molar mass in kilogram
Kilogram
The kilogram or kilogramme , also known as the kilo, is the base unit of mass in the International System of Units and is defined as being equal to the mass of the International Prototype Kilogram , which is almost exactly equal to the mass of one liter of water...

s per mole
Mole (unit)
The mole is a unit of measurement used in chemistry to express amounts of a chemical substance, defined as an amount of a substance that contains as many elementary entities as there are atoms in 12 grams of pure carbon-12 , the isotope of carbon with atomic weight 12. This corresponds to a value...

. The mean molar mass for dry air is about 0.0289645 kg/mol.
• NEWLINE
• $m$ is the mass of a single molecule in kilograms.
NEWLINE 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 $c_\left\{\mathrm\left\{air\right\}\right\}$ have been found to vary slightly from experimentally determined values. Newton
Isaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...

famously considered the speed of sound before most of the development of thermodynamics
Thermodynamics
Thermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...

and so incorrectly used isothermal calculations instead of adiabatic. His result was missing the factor of $\gamma$ 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 $\ \gamma\, = 1.4000$ requires that the gas exist 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., insigificant 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 for a more complete discussion of this phenomenon. For air, we use a simplified symbol $\ R_* = R/M_\left\{\mathrm\left\{air\right\}\right\}$. Additionally, if temperatures in degrees Celsius
Celsius
Celsius is a scale and unit of measurement for temperature. It is named after the Swedish astronomer Anders Celsius , who developed a similar temperature scale two years before his death...

(°C) are to be used to calculate air speed in the region near 273 kelvin, then Celsius temperature $\vartheta = T - 273.15$ may be used. Then:$c_\left\{\mathrm\left\{ideal\right\}\right\} = \sqrt\left\{\gamma \cdot R_* \cdot T\right\} = \sqrt\left\{\gamma \cdot R_* \cdot \left(\vartheta + 273.15\;^\left\{\circ\right\}\mathrm\left\{C\right\}\right)\right\}\,$$c_\left\{\mathrm\left\{ideal\right\}\right\} = \sqrt\left\{\gamma \cdot R_* \cdot 273.15\right\} \cdot \sqrt\left\{1+\frac\left\{\vartheta\right\}\left\{273.15\;^\left\{\circ\right\}\mathrm\left\{C\right\}\right\}\right\}\,$ For dry air, where $\vartheta\,$ (theta) is the temperature in degrees Celsius
Celsius
Celsius is a scale and unit of measurement for temperature. It is named after the Swedish astronomer Anders Celsius , who developed a similar temperature scale two years before his death...

(°C). Making the following numerical substitutions:$\ R = 8.314510 \cdot \mathrm\left\{J \cdot mol^\left\{-1\right\}\right\} \cdot K^\left\{-1\right\}\,$ is the molar gas constant
Gas constant
The gas constant is a physical constant which is featured in many fundamental equations in the physical sciences, such as the ideal gas law and the Nernst equation. It is equivalent to the Boltzmann constant, but expressed in units of energy The gas constant (also known as the molar, universal,...

in J/mole/Kelvin;$\ M_\left\{\mathrm\left\{air\right\}\right\} = 0.0289645 \cdot \mathrm\left\{kg \cdot mol^\left\{-1\right\}\right\}\,$ is the mean molar mass of air, in kg; and using the ideal diatomic gas value of $\ \gamma\, = 1.4000\,$ Then:$c_\left\{\mathrm\left\{air\right\}\right\} = 331.3 \ \mathrm\left\{m \cdot s^\left\{-1\right\}\right\} \sqrt\left\{1+\frac\left\{\vartheta^\left\{\circ\right\}\mathrm\left\{C\right\}\right\}\left\{273.15\;^\left\{\circ\right\}\mathrm\left\{C\right\}\right\}\right\}\,$ Using the first two terms of the Taylor expansion:$c_\left\{\mathrm\left\{air\right\}\right\} = 331.3 \ \mathrm\left\{m \cdot s^\left\{-1\right\}\right\} \left(1 + \frac\left\{\vartheta^\left\{\circ\right\}\mathrm\left\{C\right\}\right\}\left\{2 \cdot 273.15\;^\left\{\circ\right\}\mathrm\left\{C\right\}\right\}\right)\,$ $c_\left\{\mathrm\left\{air\right\}\right\} = \left( 331\left\{.\right\}3 + 0\left\{.\right\}606\;^\left\{\circ\right\}\mathrm\left\{C\right\}^\left\{-1\right\} \cdot \vartheta\right)\ \mathrm\left\{ m \cdot s^\left\{-1\right\}\right\}\,$ The derivation includes the first two equations given in the Practical formula for dry air section above.

### Effects due to wind shear

The speed of sound varies with temperature. Since temperature and sound velocity normally decrease with increasing altitude, sound is refracted
Refraction
Refraction is the change in direction of a wave due to a change in its speed. It is essentially a surface phenomenon . The phenomenon is mainly in governance to the law of conservation of energy. The proper explanation would be that due to change of medium, the phase velocity of the wave is changed...

upward, away from listeners on the ground, creating an acoustic shadow
An acoustic shadow is an area through which sound waves fail to propagate, due to topographical obstructions or disruption of the waves via phenomena such as wind currents. A gobo refers to a movable acoustic isolation panel and that makes an acoustic shadow. As one website refers to it, "an...

at some distance from the source. Wind shear of 4 m·s−1·km−1 can produce refraction equal to a typical temperature lapse rate
Lapse rate
The lapse rate is defined as the rate of decrease with height for an atmospheric variable. The variable involved is temperature unless specified otherwise. The terminology arises from the word lapse in the sense of a decrease or decline; thus, the lapse rate is the rate of decrease with height and...

of 7.5 °C/km. Higher values of wind gradient will refract sound downward toward the surface in the downwind direction, eliminating the acoustic shadow on the downwind side. This will increase the audibility of sounds downwind. This downwind refraction effect occurs because there is a wind gradient; the 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\left(h\right) = U\left(0\right) h ^ \zeta\,$$\ \frac \left\{dU\right\} \left\{dH\right\} = \zeta \frac \left\{U\left(h\right)\right\} \left\{h\right\}\,$ where: $\ U\left(h\right)$ = speed of the wind at height $\ h$, and $\ U\left(0\right)$ is a constant$\ \zeta$ = exponential coefficient based on ground surface roughness, typically between 0.08 and 0.52$\ \frac \left\{dU\right\} \left\{dH\right\}$ = expected wind gradient at height $h$ In the 1862 American Civil War
American Civil War
The American Civil War was a civil war fought in the United States of America. In response to the election of Abraham Lincoln as President of the United States, 11 southern slave states declared their secession from the United States and formed the Confederate States of America ; the other 25...

Battle of Iuka
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 Maj. Gen. William S. Rosecrans stopped the advance of the army of Confederate Maj. Gen. Sterling Price.Maj. Gen. Ulysses S...

, 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 10 km (six miles) downwind.

### Tables

In the standard atmosphere:NEWLINE
NEWLINE
• T0 is 273.15 K (= 0 °C = 32 °F), giving a theoretical value of 331.3 m·s−1 (= 1086.9 ft/s = 1193 km·h−1 = 741.1 mph = 644.0 knots). Values ranging from 331.3-331.6 may be found in reference literature, however.
• NEWLINE
• T20 is 293.15 K (= 20 °C = 68 °F), giving a value of 343.2 m·s−1 (= 1126.0 ft/s = 1236 km·h−1 = 767.8 mph = 667.2 knots).
• NEWLINE
• T25 is 298.15 K (= 25 °C = 77 °F), giving a value of 346.1 m·s−1 (= 1135.6 ft/s = 1246 km·h−1 = 774.3 mph = 672.8 knots).
NEWLINE In fact, assuming an ideal gas
Ideal gas
An ideal gas is a theoretical gas composed of a set of randomly-moving, non-interacting point particles. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is amenable to analysis under statistical mechanics.At normal conditions such as...

, 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. {{Temperature_effect}} Given normal atmospheric conditions, the temperature, and thus speed of sound, varies with altitude: NEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINE
 Altitude Temperature m·s−1 km·h−1 mph knots Sea level 15 °C (59 °F) 340 1225 761 661 11 000 m−20 000 m(Cruising altitude of commercial jets,and first supersonic flightBell X-1The Bell X-1, originally designated XS-1, was a joint NACA-U.S. Army/US Air Force supersonic research project built by Bell Aircraft. Conceived in 1944 and designed and built over 1945, it eventually reached nearly 1,000 mph in 1948...) −57 °C (−70 °F) 295 1062 660 573 29 000 m (Flight of X-43ABoeing X-43The X-43 is an unmanned experimental hypersonic aircraft with multiple planned scale variations meant to test various aspects of hypersonic flight. It was part of NASA's Hyper-X program. It has set several airspeed records for jet-propelled aircraft....) −48 °C (−53 °F) 301 1083 673 585
NEWLINENEWLINE

## Effect of frequency and gas composition

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 the mean free path
Mean free path
In physics, the mean free path is the average distance covered by a moving particle between successive impacts which modify its direction or energy or other particle properties.-Derivation:...

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 soundwave 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 sound speeds (over 9% higher) because they have a higher $\gamma$ (5/3 = 1.66...) than diatomics do (7/5 = 1.4). Thus, at the same molecular mass, the sound speed of a monatomic gas goes up by a factor of $\left\{ c_\left\{\mathrm\left\{gas: monatomic\right\}\right\} \over c_\left\{\mathrm\left\{gas: diatomic\right\}\right\} \right\} = \sqrt \left\{\left\{other uses\right\}\right\}$
The speed of sound is the distance travelled during a unit of time by a sound wave propagating through an elastic medium. In dry air at 20 °C (68 °F), the speed of sound is 343.2 metres per second (1,126 ft/s). This is 1236 kilometres per hour (768 mph), or about one kilometer in three seconds or approximately one mile in five seconds. In fluid dynamics
Fluid dynamics
In physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...

, the speed of sound in a fluid medium (gas or liquid) is used as a relative measure of speed itself. The speed of an object (in distance per time) divided by the speed of sound in the fluid is called the Mach number
Mach number
Mach number is the speed of an object moving through air, or any other fluid substance, divided by the speed of sound as it is in that substance for its particular physical conditions, including those of temperature and pressure...

. Objects moving at speeds greater than {{gaps|Mach|1}} are traveling at supersonic
Supersonic
Supersonic speed is a rate of travel of an object that exceeds the speed of sound . For objects traveling in dry air of a temperature of 20 °C this speed is approximately 343 m/s, 1,125 ft/s, 768 mph or 1,235 km/h. Speeds greater than five times the speed of sound are often...

speeds. The speed of sound in an ideal gas
Ideal gas
An ideal gas is a theoretical gas composed of a set of randomly-moving, non-interacting point particles. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is amenable to analysis under statistical mechanics.At normal conditions such as...

is independent of frequency, but it weakly depends on frequency for all real physical situations. It is a function of the square root of temperature, but is nearly independent of pressure
Pressure
Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...

or density
Density
The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...

for a given gas. For different gases, the speed of sound is inversely dependent on square root of the mean molecular weight of the gas, and affected to a lesser extent by the number of ways in which the molecules of the gas can store heat
Heat
In physics and thermodynamics, heat is energy transferred from one body, region, or thermodynamic system to another due to thermal contact or thermal radiation when the systems are at different temperatures. It is often described as one of the fundamental processes of energy transfer between...

from compression, since sound in gases is a type of compression. Although, in the case of gases only, the speed of sound may be expressed in terms of a ratio of both density and pressure, these quantities are not fully independent of each other, and canceling their common contributions from physical conditions, leads to a velocity expression using the independent variables of temperature, composition, and heat capacity noted above. In common everyday speech, speed of sound refers to the speed of sound waves in air
Earth's atmosphere
The atmosphere of Earth is a layer of gases surrounding the planet Earth that is retained by Earth's gravity. The atmosphere protects life on Earth by absorbing ultraviolet solar radiation, warming the surface through heat retention , and reducing temperature extremes between day and night...

. However, the speed of sound varies from substance to substance. Sound travels faster in liquids and non-porous solids than it does in air. It travels about 4.3 times faster in water
Water
Water is a chemical substance with the chemical formula H2O. A water molecule contains one oxygen and two hydrogen atoms connected by covalent bonds. Water is a liquid at ambient conditions, but it often co-exists on Earth with its solid state, ice, and gaseous state . Water also exists in a...

(1,484 m/s), and nearly 15 times as fast in iron (5,120 m/s), than in air at 20 degrees Celsius. In solids, sound waves propagate as two different types. A longitudinal wave
Longitudinal wave
Longitudinal waves, as known as "l-waves", are waves that have the same direction of vibration as their direction of travel, which means that the movement of the medium is in the same direction as or the opposite direction to the motion of the wave. Mechanical longitudinal waves have been also...

is associated with compression and decompression in the direction of travel, which is the same process as all sound waves in gases and liquids. A transverse wave
Transverse wave
A transverse wave is a moving wave that consists of oscillations occurring perpendicular to the direction of energy transfer...

, often called shear wave, is due to elastic deformation of the medium perpendicular to the direction of wave travel; the direction of shear-deformation is called the "polarization" of this type of wave. In general, transverse waves occur as a pair of orthogonal polarizations. 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
Earthquake
An earthquake is the result of a sudden release of energy in the Earth's crust that creates seismic waves. The seismicity, seismism or seismic activity of an area refers to the frequency, type and size of earthquakes experienced over a period of time...

, where sharp compression waves arrive first, and rocking transverse waves seconds later. The speed of an elastic wave in any medium is determined by the medium's compressibility and density
Density
The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...

. The speed of shear waves, which can occur only in solids, is determined by the solid material's stiffness
Stiffness
Stiffness is the resistance of an elastic body to deformation by an applied force along a given degree of freedom when a set of loading points and boundary conditions are prescribed on the elastic body.-Calculations:...

, compressibility and density. {{Sound measurements}}

## Basic concept

The transmission of sound can be illustrated by using a toy model
Toy model
In physics, a toy model is a simplified set of objects and equations relating them that can nevertheless be used to understand a mechanism that is also useful in the full, non-simplified theory....

consisting of an array of balls interconnected by springs. For real material the balls represent molecules and the springs represent the bonds between them. Sound passes through the model by compressing and expanding the springs, transmitting energy to neighboring balls, which transmit energy to their springs, and so on. The speed of sound through the model depends on the stiffness of the springs (stiffer springs transmit energy more quickly). Effects like dispersion and reflection can also be understood using this model. In a real material, the stiffness of the springs is called the elastic modulus
Elastic modulus
An elastic modulus, or modulus of elasticity, is the mathematical description of an object or substance's tendency to be deformed elastically when a force is applied to it...

, and the mass corresponds to the density
Density
The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...

. All other things being equal, sound will travel more slowly in spongy materials, and faster in stiffer ones. For instance, sound will travel much faster in steel than soft iron, due to the greater stiffness of steel at about the same density. Similarly, sound travels about 1.41 times faster in light hydrogen (protium) gas than in heavy hydrogen (deuterium
Deuterium
Deuterium, also called heavy hydrogen, is one of two stable isotopes of hydrogen. It has a natural abundance in Earth's oceans of about one atom in of hydrogen . Deuterium accounts for approximately 0.0156% of all naturally occurring hydrogen in Earth's oceans, while the most common isotope ...

) 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 increasing density. This is usually illustrated by presenting data for three materials, such as air, water and steel, which also have vastly different compressibilities which more than make 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.

## Basic formula

In general, the speed of sound c is given by the Newton-Laplace equation:$c = \sqrt\left\{\frac\left\{P\right\}\left\{\rho\right\}\right\}\,$ whereNEWLINE
NEWLINE
P is a coefficient of stiffness, the bulk modulus
Bulk modulus
The bulk modulus of a substance measures the substance's resistance to uniform compression. It is defined as the pressure increase needed to decrease the volume by a factor of 1/e...

(or the modulus of bulk elasticity for gas mediums),$\rho$ is the density
Density
The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...

NEWLINE 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 the density. For general equations of state, if classical mechanics is used, the speed of sound $c$ is given by$c^2=\frac\left\{\partial p\right\}\left\{\partial\rho\right\}$ where differentiation is taken with respect to adiabatic change.NEWLINE
NEWLINE
where $p$ is the pressure and $\rho$ is the density
NEWLINE If relativistic
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...

effects are important, the speed of sound may be calculated from the relativistic Euler equations
Relativistic Euler equations
In fluid mechanics and astrophysics, the relativistic Euler equations are a generalization of the Euler equations that account for the effects of special relativity....

. In a non-dispersive medium sound speed is independent of sound frequency, so the speeds of energy transport and sound propagation are the same. For audible sounds, a mixture of oxygen and nitrogen constitutes a non-dispersive medium. But air does contain a small amount of CO2 which is a dispersive medium, and it introduces dispersion to air at ultrasonic
Ultrasound
Ultrasound is cyclic sound pressure with a frequency greater than the upper limit of human hearing. Ultrasound is thus not separated from "normal" sound based on differences in physical properties, only the fact that humans cannot hear it. Although this limit varies from person to person, it is...

frequencies (> 28 kHz). In a dispersive medium sound speed is a function of sound frequency, through the dispersion relation
Dispersion relation
In physics and electrical engineering, dispersion most often refers to frequency-dependent effects in wave propagation. Note, however, that there are several other uses of the word "dispersion" in the physical sciences....

. The spatial and temporal distribution of a propagating disturbance will continually change. Each frequency component propagates at its own phase velocity
Phase velocity
The phase velocity of a wave is the rate at which the phase of the wave propagates in space. This is the speed at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave will appear to travel at the phase velocity...

, while the energy of the disturbance propagates at the group velocity
Group velocity
The group velocity of a wave is the velocity with which the overall shape of the wave's amplitudes — known as the modulation or envelope of the wave — propagates through space....

. The same phenomenon occurs with light waves; see optical dispersion for a description.

## Dependence on the properties of the medium

The speed of sound is variable and depends on the properties of the substance through of which the wave is travelling. In solids, the speed of longitudinal waves depend on the stiffness to tensile stress, and the density of the medium. In fluids, the medium's compressibility and density are the important factors. In gases, compressibility and density are related, making other compositional effects and properties important, such as temperature and molecular composition. In low molecular weight gases, such as helium
Helium
Helium is the chemical element with atomic number 2 and an atomic weight of 4.002602, which is represented by the symbol He. It is a colorless, odorless, tasteless, non-toxic, inert, monatomic gas that heads the noble gas group in the periodic table...

, sound propagates faster compared to heavier gases, such as xenon
Xenon
Xenon is a chemical element with the symbol Xe and atomic number 54. The element name is pronounced or . A colorless, heavy, odorless noble gas, xenon occurs in the Earth's atmosphere in trace amounts...

(for monatomic gases the speed of sound is about 75% of the mean speed that molecules move in the gas). For a given ideal gas
Ideal gas
An ideal gas is a theoretical gas composed of a set of randomly-moving, non-interacting point particles. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is amenable to analysis under statistical mechanics.At normal conditions such as...

the sound speed depends only on its temperature
Temperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...

. At a constant temperature, the ideal gas pressure
Pressure
Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...

has no effect on the speed of sound, because pressure and density
Density
The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...

(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 (see derivations below). Thus, for a single given gas (where molecular weight does not change) and over a small temperature range (where heat capacity is relatively constant), the speed of sound becomes dependent on only the temperature of the gas. In non-ideal gases, such as a van der Waals gas
Van der Waals equation
The van der Waals equation is an equation of state for a fluid composed of particles that have a non-zero volume and a pairwise attractive inter-particle force It was derived by Johannes Diderik van der Waals in 1873, who received the Nobel prize in 1910 for "his work on the equation of state for...

, 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 sound speed (causing it to increase by about 0.1%-0.6%), because oxygen
Oxygen
Oxygen is the element with atomic number 8 and represented by the symbol O. Its name derives from the Greek roots ὀξύς and -γενής , because at the time of naming, it was mistakenly thought that all acids required oxygen in their composition...

and nitrogen
Nitrogen
Nitrogen is a chemical element that has the symbol N, atomic number of 7 and atomic mass 14.00674 u. Elemental nitrogen is a colorless, odorless, tasteless, and mostly inert diatomic gas at standard conditions, constituting 78.08% by volume of Earth's atmosphere...

molecules of the air are replaced by lighter molecules of water
Water
Water is a chemical substance with the chemical formula H2O. A water molecule contains one oxygen and two hydrogen atoms connected by covalent bonds. Water is a liquid at ambient conditions, but it often co-exists on Earth with its solid state, ice, and gaseous state . Water also exists in a...

. This is a simple mixing effect.

## Implications for atmospheric acoustics

In the Earth's atmosphere
Earth's atmosphere
The atmosphere of Earth is a layer of gases surrounding the planet Earth that is retained by Earth's gravity. The atmosphere protects life on Earth by absorbing ultraviolet solar radiation, warming the surface through heat retention , and reducing temperature extremes between day and night...

, the most important factor affecting the speed of sound is the temperature
Temperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...

(see Details below). Since temperature and thus the speed of sound normally decrease with increasing altitude, sound is refracted
Refraction
Refraction is the change in direction of a wave due to a change in its speed. It is essentially a surface phenomenon . The phenomenon is mainly in governance to the law of conservation of energy. The proper explanation would be that due to change of medium, the phase velocity of the wave is changed...

upward, away from listeners on the ground, creating an acoustic shadow
An acoustic shadow is an area through which sound waves fail to propagate, due to topographical obstructions or disruption of the waves via phenomena such as wind currents. A gobo refers to a movable acoustic isolation panel and that makes an acoustic shadow. As one website refers to it, "an...

at some distance from the source. The decrease of the sound speed with height is referred to as a negative sound speed gradient
In acoustics, the sound speed gradient is the rate of change of the speed of sound with distance, for example with depth in the ocean,or height in the Earth's atmosphere. A sound speed gradient leads to refraction of sound wavefronts in the direction of lower sound speed, causing the sound rays to...

. However, in the stratosphere
Stratosphere
The stratosphere is the second major layer of Earth's atmosphere, just above the troposphere, and below the mesosphere. It is stratified in temperature, with warmer layers higher up and cooler layers farther down. This is in contrast to the troposphere near the Earth's surface, which is cooler...

, the speed of sound increases with height due to heating within the ozone layer
Ozone layer
The ozone layer is a layer in Earth's atmosphere which contains relatively high concentrations of ozone . This layer absorbs 97–99% of the Sun's high frequency ultraviolet light, which is potentially damaging to the life forms on Earth...

, producing a positive sound speed gradient.

## Practical formula for dry air

The approximate speed of sound in dry (0% humidity) air, in meters per second (m·s−1), at temperatures near 0 °C, can be calculated from:$c_\left\{\mathrm\left\{air\right\}\right\} = \left(331\left\{.\right\}3 + 0\left\{.\right\}606 \cdot \vartheta\right) \ \mathrm\left\{m \cdot s^\left\{-1\right\}\right\}\,$ where $\vartheta$ is the temperature in degrees Celsius
Celsius
Celsius is a scale and unit of measurement for temperature. It is named after the Swedish astronomer Anders Celsius , who developed a similar temperature scale two years before his death...

(°C). This equation is derived from the first two terms of the Taylor expansion of the following more accurate equation: $c_\left\{\mathrm\left\{air\right\}\right\} = 331.3\,\mathrm\left\{m \cdot s^\left\{-1\right\}\right\} \sqrt\left\{1+\frac\left\{\vartheta\right\}\left\{273.15\,^\left\{\circ\right\}\mathrm\left\{C\right\}\right\}\right\}$ Dividing the first part, and multiplying the second part, on the right hand side, by $\sqrt\left\{273.15\right\}$ gives the exactly equivalent form: $c_\left\{\mathrm\left\{air\right\}\right\} = 20.0457\,\mathrm\left\{m \cdot s^\left\{-1\right\}\right\} \sqrt\left\{\left\{\vartheta\right\}+ \left\{273.15\;\right\}\right\}$ The value of 331.3 m/s, which represents the 0 °C speed, is based on theoretical (and some measured) values of the heat capacity ratio
Heat capacity ratio
The heat capacity ratio or adiabatic index or ratio of specific heats, is the ratio of the heat capacity at constant pressure to heat capacity at constant volume . It is sometimes also known as the isentropic expansion factor and is denoted by \gamma or \kappa . The latter symbol kappa is...

, $\gamma$, as well as on the fact that at 1 atm
Atmosphere (unit)
The standard atmosphere is an international reference pressure defined as 101325 Pa and formerly used as unit of pressure. For practical purposes it has been replaced by the bar which is 105 Pa...

real air is very well described by the ideal gas approximation. Commonly found values for the speed of sound at 0 °C may vary from 331.2 to 331.6 due to the assumptions made when it is calculated. If ideal gas $\gamma$ is assumed to be 7/5 = 1.4 exactly, the 0 °C speed is calculated (see section below) to be 331.3 m/s, 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
Stratosphere
The stratosphere is the second major layer of Earth's atmosphere, just above the troposphere, and below the mesosphere. It is stratified in temperature, with warmer layers higher up and cooler layers farther down. This is in contrast to the troposphere near the Earth's surface, which is cooler...

. 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
Mean free path
In physics, the mean free path is the average distance covered by a moving particle between successive impacts which modify its direction or energy or other particle properties.-Derivation:...

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 331.5 m/s for the speed of sound °C.

### Speed in ideal gases and in air

For a gas, K (the bulk modulus
Bulk modulus
The bulk modulus of a substance measures the substance's resistance to uniform compression. It is defined as the pressure increase needed to decrease the volume by a factor of 1/e...

in equations above, equivalent to C, the coefficient of stiffness in solids) is approximately given by$K = \gamma \cdot p\,$ thus$c = \sqrt\left\{\gamma \cdot \left\{p \over \rho\right\}\right\}\,$ Where:$\gamma$ is the adiabatic index also known as the isentropic expansion factor. It is the ratio of specific heats of a gas at a constant-pressure to a gas at a constant-volume($C_p/C_v$), 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.NEWLINE
NEWLINE
p is the pressure
Pressure
Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...

.$\rho$ is the density
Density
The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...

NEWLINE Using the ideal gas
Ideal gas
An ideal gas is a theoretical gas composed of a set of randomly-moving, non-interacting point particles. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is amenable to analysis under statistical mechanics.At normal conditions such as...

law to replace $p$ with nRT/V, and replacing ρ with nM/V, the equation for an ideal gas becomes:$c_\left\{\mathrm\left\{ideal\right\}\right\} = \sqrt\left\{\gamma \cdot \left\{p \over \rho\right\}\right\} = \sqrt\left\{\gamma \cdot R \cdot T \over M\right\}= \sqrt\left\{\gamma \cdot k \cdot T \over m\right\}\,$ whereNEWLINE
NEWLINE
• $c_\left\{\mathrm\left\{ideal\right\}\right\}$ is the speed of sound in an ideal gas.
• NEWLINE
• $R$ (approximately 8.3145 J·mol−1·K−1) is the molar gas constant.
• NEWLINE
• $k$ is the Boltzmann constant
• NEWLINE
• $\gamma$ (gamma) is the adiabatic index (sometimes assumed 7/5 = 1.400 for diatomic molecules from kinetic theory, assuming from quantum theory a temperature range at which thermal energy is fully partitioned into rotation (rotations are fully excited), but none into vibrational modes. Gamma is actually experimentally measured over a range from 1.3991 to 1.403 at 0 degrees Celsius, for air. Gamma is assumed from kinetic theory to be exactly 5/3 = 1.6667 for monoatomic molecules such as noble gas
Noble gas
The noble gases are a group of chemical elements with very similar properties: under standard conditions, they are all odorless, colorless, monatomic gases, with very low chemical reactivity...

es).
• NEWLINE
• $T$ is the absolute temperature in kelvin
Kelvin
The kelvin is a unit of measurement for temperature. It is one of the seven base units in the International System of Units and is assigned the unit symbol K. The Kelvin scale is an absolute, thermodynamic temperature scale using as its null point absolute zero, the temperature at which all...

.
• NEWLINE
• $M$ is the molar mass in kilogram
Kilogram
The kilogram or kilogramme , also known as the kilo, is the base unit of mass in the International System of Units and is defined as being equal to the mass of the International Prototype Kilogram , which is almost exactly equal to the mass of one liter of water...

s per mole
Mole (unit)
The mole is a unit of measurement used in chemistry to express amounts of a chemical substance, defined as an amount of a substance that contains as many elementary entities as there are atoms in 12 grams of pure carbon-12 , the isotope of carbon with atomic weight 12. This corresponds to a value...

. The mean molar mass for dry air is about 0.0289645 kg/mol.
• NEWLINE
• $m$ is the mass of a single molecule in kilograms.
NEWLINE 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 $c_\left\{\mathrm\left\{air\right\}\right\}$ have been found to vary slightly from experimentally determined values. Newton
Isaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...

famously considered the speed of sound before most of the development of thermodynamics
Thermodynamics
Thermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...

and so incorrectly used isothermal calculations instead of adiabatic. His result was missing the factor of $\gamma$ 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 $\ \gamma\, = 1.4000$ requires that the gas exist 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., insigificant 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 for a more complete discussion of this phenomenon. For air, we use a simplified symbol $\ R_* = R/M_\left\{\mathrm\left\{air\right\}\right\}$. Additionally, if temperatures in degrees Celsius
Celsius
Celsius is a scale and unit of measurement for temperature. It is named after the Swedish astronomer Anders Celsius , who developed a similar temperature scale two years before his death...

(°C) are to be used to calculate air speed in the region near 273 kelvin, then Celsius temperature $\vartheta = T - 273.15$ may be used. Then:$c_\left\{\mathrm\left\{ideal\right\}\right\} = \sqrt\left\{\gamma \cdot R_* \cdot T\right\} = \sqrt\left\{\gamma \cdot R_* \cdot \left(\vartheta + 273.15\;^\left\{\circ\right\}\mathrm\left\{C\right\}\right)\right\}\,$$c_\left\{\mathrm\left\{ideal\right\}\right\} = \sqrt\left\{\gamma \cdot R_* \cdot 273.15\right\} \cdot \sqrt\left\{1+\frac\left\{\vartheta\right\}\left\{273.15\;^\left\{\circ\right\}\mathrm\left\{C\right\}\right\}\right\}\,$ For dry air, where $\vartheta\,$ (theta) is the temperature in degrees Celsius
Celsius
Celsius is a scale and unit of measurement for temperature. It is named after the Swedish astronomer Anders Celsius , who developed a similar temperature scale two years before his death...

(°C). Making the following numerical substitutions:$\ R = 8.314510 \cdot \mathrm\left\{J \cdot mol^\left\{-1\right\}\right\} \cdot K^\left\{-1\right\}\,$ is the molar gas constant
Gas constant
The gas constant is a physical constant which is featured in many fundamental equations in the physical sciences, such as the ideal gas law and the Nernst equation. It is equivalent to the Boltzmann constant, but expressed in units of energy The gas constant (also known as the molar, universal,...

in J/mole/Kelvin;$\ M_\left\{\mathrm\left\{air\right\}\right\} = 0.0289645 \cdot \mathrm\left\{kg \cdot mol^\left\{-1\right\}\right\}\,$ is the mean molar mass of air, in kg; and using the ideal diatomic gas value of $\ \gamma\, = 1.4000\,$ Then:$c_\left\{\mathrm\left\{air\right\}\right\} = 331.3 \ \mathrm\left\{m \cdot s^\left\{-1\right\}\right\} \sqrt\left\{1+\frac\left\{\vartheta^\left\{\circ\right\}\mathrm\left\{C\right\}\right\}\left\{273.15\;^\left\{\circ\right\}\mathrm\left\{C\right\}\right\}\right\}\,$ Using the first two terms of the Taylor expansion:$c_\left\{\mathrm\left\{air\right\}\right\} = 331.3 \ \mathrm\left\{m \cdot s^\left\{-1\right\}\right\} \left(1 + \frac\left\{\vartheta^\left\{\circ\right\}\mathrm\left\{C\right\}\right\}\left\{2 \cdot 273.15\;^\left\{\circ\right\}\mathrm\left\{C\right\}\right\}\right)\,$ $c_\left\{\mathrm\left\{air\right\}\right\} = \left( 331\left\{.\right\}3 + 0\left\{.\right\}606\;^\left\{\circ\right\}\mathrm\left\{C\right\}^\left\{-1\right\} \cdot \vartheta\right)\ \mathrm\left\{ m \cdot s^\left\{-1\right\}\right\}\,$ The derivation includes the first two equations given in the Practical formula for dry air section above.

### Effects due to wind shear

The speed of sound varies with temperature. Since temperature and sound velocity normally decrease with increasing altitude, sound is refracted
Refraction
Refraction is the change in direction of a wave due to a change in its speed. It is essentially a surface phenomenon . The phenomenon is mainly in governance to the law of conservation of energy. The proper explanation would be that due to change of medium, the phase velocity of the wave is changed...

upward, away from listeners on the ground, creating an acoustic shadow
An acoustic shadow is an area through which sound waves fail to propagate, due to topographical obstructions or disruption of the waves via phenomena such as wind currents. A gobo refers to a movable acoustic isolation panel and that makes an acoustic shadow. As one website refers to it, "an...

at some distance from the source. Wind shear of 4 m·s−1·km−1 can produce refraction equal to a typical temperature lapse rate
Lapse rate
The lapse rate is defined as the rate of decrease with height for an atmospheric variable. The variable involved is temperature unless specified otherwise. The terminology arises from the word lapse in the sense of a decrease or decline; thus, the lapse rate is the rate of decrease with height and...

of 7.5 °C/km. Higher values of wind gradient will refract sound downward toward the surface in the downwind direction, eliminating the acoustic shadow on the downwind side. This will increase the audibility of sounds downwind. This downwind refraction effect occurs because there is a wind gradient; the 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\left(h\right) = U\left(0\right) h ^ \zeta\,$$\ \frac \left\{dU\right\} \left\{dH\right\} = \zeta \frac \left\{U\left(h\right)\right\} \left\{h\right\}\,$ where: $\ U\left(h\right)$ = speed of the wind at height $\ h$, and $\ U\left(0\right)$ is a constant$\ \zeta$ = exponential coefficient based on ground surface roughness, typically between 0.08 and 0.52$\ \frac \left\{dU\right\} \left\{dH\right\}$ = expected wind gradient at height $h$ In the 1862 American Civil War
American Civil War
The American Civil War was a civil war fought in the United States of America. In response to the election of Abraham Lincoln as President of the United States, 11 southern slave states declared their secession from the United States and formed the Confederate States of America ; the other 25...

Battle of Iuka
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 Maj. Gen. William S. Rosecrans stopped the advance of the army of Confederate Maj. Gen. Sterling Price.Maj. Gen. Ulysses S...

, 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 10 km (six miles) downwind.

### Tables

In the standard atmosphere:NEWLINE
NEWLINE
• T0 is 273.15 K (= 0 °C = 32 °F), giving a theoretical value of 331.3 m·s−1 (= 1086.9 ft/s = 1193 km·h−1 = 741.1 mph = 644.0 knots). Values ranging from 331.3-331.6 may be found in reference literature, however.
• NEWLINE
• T20 is 293.15 K (= 20 °C = 68 °F), giving a value of 343.2 m·s−1 (= 1126.0 ft/s = 1236 km·h−1 = 767.8 mph = 667.2 knots).
• NEWLINE
• T25 is 298.15 K (= 25 °C = 77 °F), giving a value of 346.1 m·s−1 (= 1135.6 ft/s = 1246 km·h−1 = 774.3 mph = 672.8 knots).
NEWLINE In fact, assuming an ideal gas
Ideal gas
An ideal gas is a theoretical gas composed of a set of randomly-moving, non-interacting point particles. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is amenable to analysis under statistical mechanics.At normal conditions such as...

, 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. {{Temperature_effect}} Given normal atmospheric conditions, the temperature, and thus speed of sound, varies with altitude: NEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINE
 Altitude Temperature m·s−1 km·h−1 mph knots Sea level 15 °C (59 °F) 340 1225 761 661 11 000 m−20 000 m(Cruising altitude of commercial jets,and first supersonic flightBell X-1The Bell X-1, originally designated XS-1, was a joint NACA-U.S. Army/US Air Force supersonic research project built by Bell Aircraft. Conceived in 1944 and designed and built over 1945, it eventually reached nearly 1,000 mph in 1948...) −57 °C (−70 °F) 295 1062 660 573 29 000 m (Flight of X-43ABoeing X-43The X-43 is an unmanned experimental hypersonic aircraft with multiple planned scale variations meant to test various aspects of hypersonic flight. It was part of NASA's Hyper-X program. It has set several airspeed records for jet-propelled aircraft....) −48 °C (−53 °F) 301 1083 673 585
NEWLINENEWLINE

## Effect of frequency and gas composition

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 the mean free path
Mean free path
In physics, the mean free path is the average distance covered by a moving particle between successive impacts which modify its direction or energy or other particle properties.-Derivation:...

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 soundwave 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 sound speeds (over 9% higher) because they have a higher $\gamma$ (5/3 = 1.66...) than diatomics do (7/5 = 1.4). Thus, at the same molecular mass, the sound speed of a monatomic gas goes up by a factor of $\left\{ c_\left\{\mathrm\left\{gas: monatomic\right\}\right\} \over c_\left\{\mathrm\left\{gas: diatomic\right\}\right\} \right\} = \sqrt \left\{\left\{other uses\right\}\right\}$
The speed of sound is the distance travelled during a unit of time by a sound wave propagating through an elastic medium. In dry air at 20 °C (68 °F), the speed of sound is 343.2 metres per second (1,126 ft/s). This is 1236 kilometres per hour (768 mph), or about one kilometer in three seconds or approximately one mile in five seconds. In fluid dynamics
Fluid dynamics
In physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...

, the speed of sound in a fluid medium (gas or liquid) is used as a relative measure of speed itself. The speed of an object (in distance per time) divided by the speed of sound in the fluid is called the Mach number
Mach number
Mach number is the speed of an object moving through air, or any other fluid substance, divided by the speed of sound as it is in that substance for its particular physical conditions, including those of temperature and pressure...

. Objects moving at speeds greater than {{gaps|Mach|1}} are traveling at supersonic
Supersonic
Supersonic speed is a rate of travel of an object that exceeds the speed of sound . For objects traveling in dry air of a temperature of 20 °C this speed is approximately 343 m/s, 1,125 ft/s, 768 mph or 1,235 km/h. Speeds greater than five times the speed of sound are often...

speeds. The speed of sound in an ideal gas
Ideal gas
An ideal gas is a theoretical gas composed of a set of randomly-moving, non-interacting point particles. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is amenable to analysis under statistical mechanics.At normal conditions such as...

is independent of frequency, but it weakly depends on frequency for all real physical situations. It is a function of the square root of temperature, but is nearly independent of pressure
Pressure
Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...

or density
Density
The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...

for a given gas. For different gases, the speed of sound is inversely dependent on square root of the mean molecular weight of the gas, and affected to a lesser extent by the number of ways in which the molecules of the gas can store heat
Heat
In physics and thermodynamics, heat is energy transferred from one body, region, or thermodynamic system to another due to thermal contact or thermal radiation when the systems are at different temperatures. It is often described as one of the fundamental processes of energy transfer between...

from compression, since sound in gases is a type of compression. Although, in the case of gases only, the speed of sound may be expressed in terms of a ratio of both density and pressure, these quantities are not fully independent of each other, and canceling their common contributions from physical conditions, leads to a velocity expression using the independent variables of temperature, composition, and heat capacity noted above. In common everyday speech, speed of sound refers to the speed of sound waves in air
Earth's atmosphere
The atmosphere of Earth is a layer of gases surrounding the planet Earth that is retained by Earth's gravity. The atmosphere protects life on Earth by absorbing ultraviolet solar radiation, warming the surface through heat retention , and reducing temperature extremes between day and night...

. However, the speed of sound varies from substance to substance. Sound travels faster in liquids and non-porous solids than it does in air. It travels about 4.3 times faster in water
Water
Water is a chemical substance with the chemical formula H2O. A water molecule contains one oxygen and two hydrogen atoms connected by covalent bonds. Water is a liquid at ambient conditions, but it often co-exists on Earth with its solid state, ice, and gaseous state . Water also exists in a...

(1,484 m/s), and nearly 15 times as fast in iron (5,120 m/s), than in air at 20 degrees Celsius. In solids, sound waves propagate as two different types. A longitudinal wave
Longitudinal wave
Longitudinal waves, as known as "l-waves", are waves that have the same direction of vibration as their direction of travel, which means that the movement of the medium is in the same direction as or the opposite direction to the motion of the wave. Mechanical longitudinal waves have been also...

is associated with compression and decompression in the direction of travel, which is the same process as all sound waves in gases and liquids. A transverse wave
Transverse wave
A transverse wave is a moving wave that consists of oscillations occurring perpendicular to the direction of energy transfer...

, often called shear wave, is due to elastic deformation of the medium perpendicular to the direction of wave travel; the direction of shear-deformation is called the "polarization" of this type of wave. In general, transverse waves occur as a pair of orthogonal polarizations. 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
Earthquake
An earthquake is the result of a sudden release of energy in the Earth's crust that creates seismic waves. The seismicity, seismism or seismic activity of an area refers to the frequency, type and size of earthquakes experienced over a period of time...

, where sharp compression waves arrive first, and rocking transverse waves seconds later. The speed of an elastic wave in any medium is determined by the medium's compressibility and density
Density
The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...

. The speed of shear waves, which can occur only in solids, is determined by the solid material's stiffness
Stiffness
Stiffness is the resistance of an elastic body to deformation by an applied force along a given degree of freedom when a set of loading points and boundary conditions are prescribed on the elastic body.-Calculations:...

, compressibility and density. {{Sound measurements}}

## Basic concept

The transmission of sound can be illustrated by using a toy model
Toy model
In physics, a toy model is a simplified set of objects and equations relating them that can nevertheless be used to understand a mechanism that is also useful in the full, non-simplified theory....

consisting of an array of balls interconnected by springs. For real material the balls represent molecules and the springs represent the bonds between them. Sound passes through the model by compressing and expanding the springs, transmitting energy to neighboring balls, which transmit energy to their springs, and so on. The speed of sound through the model depends on the stiffness of the springs (stiffer springs transmit energy more quickly). Effects like dispersion and reflection can also be understood using this model. In a real material, the stiffness of the springs is called the elastic modulus
Elastic modulus
An elastic modulus, or modulus of elasticity, is the mathematical description of an object or substance's tendency to be deformed elastically when a force is applied to it...

, and the mass corresponds to the density
Density
The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...

. All other things being equal, sound will travel more slowly in spongy materials, and faster in stiffer ones. For instance, sound will travel much faster in steel than soft iron, due to the greater stiffness of steel at about the same density. Similarly, sound travels about 1.41 times faster in light hydrogen (protium) gas than in heavy hydrogen (deuterium
Deuterium
Deuterium, also called heavy hydrogen, is one of two stable isotopes of hydrogen. It has a natural abundance in Earth's oceans of about one atom in of hydrogen . Deuterium accounts for approximately 0.0156% of all naturally occurring hydrogen in Earth's oceans, while the most common isotope ...

) 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 increasing density. This is usually illustrated by presenting data for three materials, such as air, water and steel, which also have vastly different compressibilities which more than make 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.

## Basic formula

In general, the speed of sound c is given by the Newton-Laplace equation:$c = \sqrt\left\{\frac\left\{P\right\}\left\{\rho\right\}\right\}\,$ whereNEWLINE
NEWLINE
P is a coefficient of stiffness, the bulk modulus
Bulk modulus
The bulk modulus of a substance measures the substance's resistance to uniform compression. It is defined as the pressure increase needed to decrease the volume by a factor of 1/e...

(or the modulus of bulk elasticity for gas mediums),$\rho$ is the density
Density
The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...

NEWLINE 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 the density. For general equations of state, if classical mechanics is used, the speed of sound $c$ is given by$c^2=\frac\left\{\partial p\right\}\left\{\partial\rho\right\}$ where differentiation is taken with respect to adiabatic change.NEWLINE
NEWLINE
where $p$ is the pressure and $\rho$ is the density
NEWLINE If relativistic
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...

effects are important, the speed of sound may be calculated from the relativistic Euler equations
Relativistic Euler equations
In fluid mechanics and astrophysics, the relativistic Euler equations are a generalization of the Euler equations that account for the effects of special relativity....

. In a non-dispersive medium sound speed is independent of sound frequency, so the speeds of energy transport and sound propagation are the same. For audible sounds, a mixture of oxygen and nitrogen constitutes a non-dispersive medium. But air does contain a small amount of CO2 which is a dispersive medium, and it introduces dispersion to air at ultrasonic
Ultrasound
Ultrasound is cyclic sound pressure with a frequency greater than the upper limit of human hearing. Ultrasound is thus not separated from "normal" sound based on differences in physical properties, only the fact that humans cannot hear it. Although this limit varies from person to person, it is...

frequencies (> 28 kHz). In a dispersive medium sound speed is a function of sound frequency, through the dispersion relation
Dispersion relation
In physics and electrical engineering, dispersion most often refers to frequency-dependent effects in wave propagation. Note, however, that there are several other uses of the word "dispersion" in the physical sciences....

. The spatial and temporal distribution of a propagating disturbance will continually change. Each frequency component propagates at its own phase velocity
Phase velocity
The phase velocity of a wave is the rate at which the phase of the wave propagates in space. This is the speed at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave will appear to travel at the phase velocity...

, while the energy of the disturbance propagates at the group velocity
Group velocity
The group velocity of a wave is the velocity with which the overall shape of the wave's amplitudes — known as the modulation or envelope of the wave — propagates through space....

. The same phenomenon occurs with light waves; see optical dispersion for a description.

## Dependence on the properties of the medium

The speed of sound is variable and depends on the properties of the substance through of which the wave is travelling. In solids, the speed of longitudinal waves depend on the stiffness to tensile stress, and the density of the medium. In fluids, the medium's compressibility and density are the important factors. In gases, compressibility and density are related, making other compositional effects and properties important, such as temperature and molecular composition. In low molecular weight gases, such as helium
Helium
Helium is the chemical element with atomic number 2 and an atomic weight of 4.002602, which is represented by the symbol He. It is a colorless, odorless, tasteless, non-toxic, inert, monatomic gas that heads the noble gas group in the periodic table...

, sound propagates faster compared to heavier gases, such as xenon
Xenon
Xenon is a chemical element with the symbol Xe and atomic number 54. The element name is pronounced or . A colorless, heavy, odorless noble gas, xenon occurs in the Earth's atmosphere in trace amounts...

(for monatomic gases the speed of sound is about 75% of the mean speed that molecules move in the gas). For a given ideal gas
Ideal gas
An ideal gas is a theoretical gas composed of a set of randomly-moving, non-interacting point particles. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is amenable to analysis under statistical mechanics.At normal conditions such as...

the sound speed depends only on its temperature
Temperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...

. At a constant temperature, the ideal gas pressure
Pressure
Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...

has no effect on the speed of sound, because pressure and density
Density
The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...

(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 (see derivations below). Thus, for a single given gas (where molecular weight does not change) and over a small temperature range (where heat capacity is relatively constant), the speed of sound becomes dependent on only the temperature of the gas. In non-ideal gases, such as a van der Waals gas
Van der Waals equation
The van der Waals equation is an equation of state for a fluid composed of particles that have a non-zero volume and a pairwise attractive inter-particle force It was derived by Johannes Diderik van der Waals in 1873, who received the Nobel prize in 1910 for "his work on the equation of state for...

, 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 sound speed (causing it to increase by about 0.1%-0.6%), because oxygen
Oxygen
Oxygen is the element with atomic number 8 and represented by the symbol O. Its name derives from the Greek roots ὀξύς and -γενής , because at the time of naming, it was mistakenly thought that all acids required oxygen in their composition...

and nitrogen
Nitrogen
Nitrogen is a chemical element that has the symbol N, atomic number of 7 and atomic mass 14.00674 u. Elemental nitrogen is a colorless, odorless, tasteless, and mostly inert diatomic gas at standard conditions, constituting 78.08% by volume of Earth's atmosphere...

molecules of the air are replaced by lighter molecules of water
Water
Water is a chemical substance with the chemical formula H2O. A water molecule contains one oxygen and two hydrogen atoms connected by covalent bonds. Water is a liquid at ambient conditions, but it often co-exists on Earth with its solid state, ice, and gaseous state . Water also exists in a...

. This is a simple mixing effect.

## Implications for atmospheric acoustics

In the Earth's atmosphere
Earth's atmosphere
The atmosphere of Earth is a layer of gases surrounding the planet Earth that is retained by Earth's gravity. The atmosphere protects life on Earth by absorbing ultraviolet solar radiation, warming the surface through heat retention , and reducing temperature extremes between day and night...

, the most important factor affecting the speed of sound is the temperature
Temperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...

(see Details below). Since temperature and thus the speed of sound normally decrease with increasing altitude, sound is refracted
Refraction
Refraction is the change in direction of a wave due to a change in its speed. It is essentially a surface phenomenon . The phenomenon is mainly in governance to the law of conservation of energy. The proper explanation would be that due to change of medium, the phase velocity of the wave is changed...

upward, away from listeners on the ground, creating an acoustic shadow
An acoustic shadow is an area through which sound waves fail to propagate, due to topographical obstructions or disruption of the waves via phenomena such as wind currents. A gobo refers to a movable acoustic isolation panel and that makes an acoustic shadow. As one website refers to it, "an...

at some distance from the source. The decrease of the sound speed with height is referred to as a negative sound speed gradient
In acoustics, the sound speed gradient is the rate of change of the speed of sound with distance, for example with depth in the ocean,or height in the Earth's atmosphere. A sound speed gradient leads to refraction of sound wavefronts in the direction of lower sound speed, causing the sound rays to...

. However, in the stratosphere
Stratosphere
The stratosphere is the second major layer of Earth's atmosphere, just above the troposphere, and below the mesosphere. It is stratified in temperature, with warmer layers higher up and cooler layers farther down. This is in contrast to the troposphere near the Earth's surface, which is cooler...

, the speed of sound increases with height due to heating within the ozone layer
Ozone layer
The ozone layer is a layer in Earth's atmosphere which contains relatively high concentrations of ozone . This layer absorbs 97–99% of the Sun's high frequency ultraviolet light, which is potentially damaging to the life forms on Earth...

, producing a positive sound speed gradient.

## Practical formula for dry air

The approximate speed of sound in dry (0% humidity) air, in meters per second (m·s−1), at temperatures near 0 °C, can be calculated from:$c_\left\{\mathrm\left\{air\right\}\right\} = \left(331\left\{.\right\}3 + 0\left\{.\right\}606 \cdot \vartheta\right) \ \mathrm\left\{m \cdot s^\left\{-1\right\}\right\}\,$ where $\vartheta$ is the temperature in degrees Celsius
Celsius
Celsius is a scale and unit of measurement for temperature. It is named after the Swedish astronomer Anders Celsius , who developed a similar temperature scale two years before his death...

(°C). This equation is derived from the first two terms of the Taylor expansion of the following more accurate equation: $c_\left\{\mathrm\left\{air\right\}\right\} = 331.3\,\mathrm\left\{m \cdot s^\left\{-1\right\}\right\} \sqrt\left\{1+\frac\left\{\vartheta\right\}\left\{273.15\,^\left\{\circ\right\}\mathrm\left\{C\right\}\right\}\right\}$ Dividing the first part, and multiplying the second part, on the right hand side, by $\sqrt\left\{273.15\right\}$ gives the exactly equivalent form: $c_\left\{\mathrm\left\{air\right\}\right\} = 20.0457\,\mathrm\left\{m \cdot s^\left\{-1\right\}\right\} \sqrt\left\{\left\{\vartheta\right\}+ \left\{273.15\;\right\}\right\}$ The value of 331.3 m/s, which represents the 0 °C speed, is based on theoretical (and some measured) values of the heat capacity ratio
Heat capacity ratio
The heat capacity ratio or adiabatic index or ratio of specific heats, is the ratio of the heat capacity at constant pressure to heat capacity at constant volume . It is sometimes also known as the isentropic expansion factor and is denoted by \gamma or \kappa . The latter symbol kappa is...

, $\gamma$, as well as on the fact that at 1 atm
Atmosphere (unit)
The standard atmosphere is an international reference pressure defined as 101325 Pa and formerly used as unit of pressure. For practical purposes it has been replaced by the bar which is 105 Pa...

real air is very well described by the ideal gas approximation. Commonly found values for the speed of sound at 0 °C may vary from 331.2 to 331.6 due to the assumptions made when it is calculated. If ideal gas $\gamma$ is assumed to be 7/5 = 1.4 exactly, the 0 °C speed is calculated (see section below) to be 331.3 m/s, 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
Stratosphere
The stratosphere is the second major layer of Earth's atmosphere, just above the troposphere, and below the mesosphere. It is stratified in temperature, with warmer layers higher up and cooler layers farther down. This is in contrast to the troposphere near the Earth's surface, which is cooler...

. 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
Mean free path
In physics, the mean free path is the average distance covered by a moving particle between successive impacts which modify its direction or energy or other particle properties.-Derivation:...

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 331.5 m/s for the speed of sound °C.

### Speed in ideal gases and in air

For a gas, K (the bulk modulus
Bulk modulus
The bulk modulus of a substance measures the substance's resistance to uniform compression. It is defined as the pressure increase needed to decrease the volume by a factor of 1/e...

in equations above, equivalent to C, the coefficient of stiffness in solids) is approximately given by$K = \gamma \cdot p\,$ thus$c = \sqrt\left\{\gamma \cdot \left\{p \over \rho\right\}\right\}\,$ Where:$\gamma$ is the adiabatic index also known as the isentropic expansion factor. It is the ratio of specific heats of a gas at a constant-pressure to a gas at a constant-volume($C_p/C_v$), 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.NEWLINE
NEWLINE
p is the pressure
Pressure
Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...

.$\rho$ is the density
Density
The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...

NEWLINE Using the ideal gas
Ideal gas
An ideal gas is a theoretical gas composed of a set of randomly-moving, non-interacting point particles. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is amenable to analysis under statistical mechanics.At normal conditions such as...

law to replace $p$ with nRT/V, and replacing ρ with nM/V, the equation for an ideal gas becomes:$c_\left\{\mathrm\left\{ideal\right\}\right\} = \sqrt\left\{\gamma \cdot \left\{p \over \rho\right\}\right\} = \sqrt\left\{\gamma \cdot R \cdot T \over M\right\}= \sqrt\left\{\gamma \cdot k \cdot T \over m\right\}\,$ whereNEWLINE
NEWLINE
• $c_\left\{\mathrm\left\{ideal\right\}\right\}$ is the speed of sound in an ideal gas.
• NEWLINE
• $R$ (approximately 8.3145 J·mol−1·K−1) is the molar gas constant.
• NEWLINE
• $k$ is the Boltzmann constant
• NEWLINE
• $\gamma$ (gamma) is the adiabatic index (sometimes assumed 7/5 = 1.400 for diatomic molecules from kinetic theory, assuming from quantum theory a temperature range at which thermal energy is fully partitioned into rotation (rotations are fully excited), but none into vibrational modes. Gamma is actually experimentally measured over a range from 1.3991 to 1.403 at 0 degrees Celsius, for air. Gamma is assumed from kinetic theory to be exactly 5/3 = 1.6667 for monoatomic molecules such as noble gas
Noble gas
The noble gases are a group of chemical elements with very similar properties: under standard conditions, they are all odorless, colorless, monatomic gases, with very low chemical reactivity...

es).
• NEWLINE
• $T$ is the absolute temperature in kelvin
Kelvin
The kelvin is a unit of measurement for temperature. It is one of the seven base units in the International System of Units and is assigned the unit symbol K. The Kelvin scale is an absolute, thermodynamic temperature scale using as its null point absolute zero, the temperature at which all...

.
• NEWLINE
• $M$ is the molar mass in kilogram
Kilogram
The kilogram or kilogramme , also known as the kilo, is the base unit of mass in the International System of Units and is defined as being equal to the mass of the International Prototype Kilogram , which is almost exactly equal to the mass of one liter of water...

s per mole
Mole (unit)
The mole is a unit of measurement used in chemistry to express amounts of a chemical substance, defined as an amount of a substance that contains as many elementary entities as there are atoms in 12 grams of pure carbon-12 , the isotope of carbon with atomic weight 12. This corresponds to a value...

. The mean molar mass for dry air is about 0.0289645 kg/mol.
• NEWLINE
• $m$ is the mass of a single molecule in kilograms.
NEWLINE 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 $c_\left\{\mathrm\left\{air\right\}\right\}$ have been found to vary slightly from experimentally determined values. Newton
Isaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...

famously considered the speed of sound before most of the development of thermodynamics
Thermodynamics
Thermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...

and so incorrectly used isothermal calculations instead of adiabatic. His result was missing the factor of $\gamma$ 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 $\ \gamma\, = 1.4000$ requires that the gas exist 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., insigificant 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 for a more complete discussion of this phenomenon. For air, we use a simplified symbol $\ R_* = R/M_\left\{\mathrm\left\{air\right\}\right\}$. Additionally, if temperatures in degrees Celsius
Celsius
Celsius is a scale and unit of measurement for temperature. It is named after the Swedish astronomer Anders Celsius , who developed a similar temperature scale two years before his death...

(°C) are to be used to calculate air speed in the region near 273 kelvin, then Celsius temperature $\vartheta = T - 273.15$ may be used. Then:$c_\left\{\mathrm\left\{ideal\right\}\right\} = \sqrt\left\{\gamma \cdot R_* \cdot T\right\} = \sqrt\left\{\gamma \cdot R_* \cdot \left(\vartheta + 273.15\;^\left\{\circ\right\}\mathrm\left\{C\right\}\right)\right\}\,$$c_\left\{\mathrm\left\{ideal\right\}\right\} = \sqrt\left\{\gamma \cdot R_* \cdot 273.15\right\} \cdot \sqrt\left\{1+\frac\left\{\vartheta\right\}\left\{273.15\;^\left\{\circ\right\}\mathrm\left\{C\right\}\right\}\right\}\,$ For dry air, where $\vartheta\,$ (theta) is the temperature in degrees Celsius
Celsius
Celsius is a scale and unit of measurement for temperature. It is named after the Swedish astronomer Anders Celsius , who developed a similar temperature scale two years before his death...

(°C). Making the following numerical substitutions:$\ R = 8.314510 \cdot \mathrm\left\{J \cdot mol^\left\{-1\right\}\right\} \cdot K^\left\{-1\right\}\,$ is the molar gas constant
Gas constant
The gas constant is a physical constant which is featured in many fundamental equations in the physical sciences, such as the ideal gas law and the Nernst equation. It is equivalent to the Boltzmann constant, but expressed in units of energy The gas constant (also known as the molar, universal,...

in J/mole/Kelvin;$\ M_\left\{\mathrm\left\{air\right\}\right\} = 0.0289645 \cdot \mathrm\left\{kg \cdot mol^\left\{-1\right\}\right\}\,$ is the mean molar mass of air, in kg; and using the ideal diatomic gas value of $\ \gamma\, = 1.4000\,$ Then:$c_\left\{\mathrm\left\{air\right\}\right\} = 331.3 \ \mathrm\left\{m \cdot s^\left\{-1\right\}\right\} \sqrt\left\{1+\frac\left\{\vartheta^\left\{\circ\right\}\mathrm\left\{C\right\}\right\}\left\{273.15\;^\left\{\circ\right\}\mathrm\left\{C\right\}\right\}\right\}\,$ Using the first two terms of the Taylor expansion:$c_\left\{\mathrm\left\{air\right\}\right\} = 331.3 \ \mathrm\left\{m \cdot s^\left\{-1\right\}\right\} \left(1 + \frac\left\{\vartheta^\left\{\circ\right\}\mathrm\left\{C\right\}\right\}\left\{2 \cdot 273.15\;^\left\{\circ\right\}\mathrm\left\{C\right\}\right\}\right)\,$ $c_\left\{\mathrm\left\{air\right\}\right\} = \left( 331\left\{.\right\}3 + 0\left\{.\right\}606\;^\left\{\circ\right\}\mathrm\left\{C\right\}^\left\{-1\right\} \cdot \vartheta\right)\ \mathrm\left\{ m \cdot s^\left\{-1\right\}\right\}\,$ The derivation includes the first two equations given in the Practical formula for dry air section above.

### Effects due to wind shear

The speed of sound varies with temperature. Since temperature and sound velocity normally decrease with increasing altitude, sound is refracted
Refraction
Refraction is the change in direction of a wave due to a change in its speed. It is essentially a surface phenomenon . The phenomenon is mainly in governance to the law of conservation of energy. The proper explanation would be that due to change of medium, the phase velocity of the wave is changed...

upward, away from listeners on the ground, creating an acoustic shadow
An acoustic shadow is an area through which sound waves fail to propagate, due to topographical obstructions or disruption of the waves via phenomena such as wind currents. A gobo refers to a movable acoustic isolation panel and that makes an acoustic shadow. As one website refers to it, "an...

at some distance from the source. Wind shear of 4 m·s−1·km−1 can produce refraction equal to a typical temperature lapse rate
Lapse rate
The lapse rate is defined as the rate of decrease with height for an atmospheric variable. The variable involved is temperature unless specified otherwise. The terminology arises from the word lapse in the sense of a decrease or decline; thus, the lapse rate is the rate of decrease with height and...

of 7.5 °C/km. Higher values of wind gradient will refract sound downward toward the surface in the downwind direction, eliminating the acoustic shadow on the downwind side. This will increase the audibility of sounds downwind. This downwind refraction effect occurs because there is a wind gradient; the 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\left(h\right) = U\left(0\right) h ^ \zeta\,$$\ \frac \left\{dU\right\} \left\{dH\right\} = \zeta \frac \left\{U\left(h\right)\right\} \left\{h\right\}\,$ where: $\ U\left(h\right)$ = speed of the wind at height $\ h$, and $\ U\left(0\right)$ is a constant$\ \zeta$ = exponential coefficient based on ground surface roughness, typically between 0.08 and 0.52$\ \frac \left\{dU\right\} \left\{dH\right\}$ = expected wind gradient at height $h$ In the 1862 American Civil War
American Civil War
The American Civil War was a civil war fought in the United States of America. In response to the election of Abraham Lincoln as President of the United States, 11 southern slave states declared their secession from the United States and formed the Confederate States of America ; the other 25...

Battle of Iuka
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 Maj. Gen. William S. Rosecrans stopped the advance of the army of Confederate Maj. Gen. Sterling Price.Maj. Gen. Ulysses S...

, 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 10 km (six miles) downwind.

### Tables

In the standard atmosphere:NEWLINE
NEWLINE
• T0 is 273.15 K (= 0 °C = 32 °F), giving a theoretical value of 331.3 m·s−1 (= 1086.9 ft/s = 1193 km·h−1 = 741.1 mph = 644.0 knots). Values ranging from 331.3-331.6 may be found in reference literature, however.
• NEWLINE
• T20 is 293.15 K (= 20 °C = 68 °F), giving a value of 343.2 m·s−1 (= 1126.0 ft/s = 1236 km·h−1 = 767.8 mph = 667.2 knots).
• NEWLINE
• T25 is 298.15 K (= 25 °C = 77 °F), giving a value of 346.1 m·s−1 (= 1135.6 ft/s = 1246 km·h−1 = 774.3 mph = 672.8 knots).
NEWLINE In fact, assuming an ideal gas
Ideal gas
An ideal gas is a theoretical gas composed of a set of randomly-moving, non-interacting point particles. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is amenable to analysis under statistical mechanics.At normal conditions such as...

, 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. {{Temperature_effect}} Given normal atmospheric conditions, the temperature, and thus speed of sound, varies with altitude: NEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENEWLINE
 Altitude Temperature m·s−1 km·h−1 mph knots Sea level 15 °C (59 °F) 340 1225 761 661 11 000 m−20 000 m(Cruising altitude of commercial jets,and first supersonic flightBell X-1The Bell X-1, originally designated XS-1, was a joint NACA-U.S. Army/US Air Force supersonic research project built by Bell Aircraft. Conceived in 1944 and designed and built over 1945, it eventually reached nearly 1,000 mph in 1948...) −57 °C (−70 °F) 295 1062 660 573 29 000 m (Flight of X-43ABoeing X-43The X-43 is an unmanned experimental hypersonic aircraft with multiple planned scale variations meant to test various aspects of hypersonic flight. It was part of NASA's Hyper-X program. It has set several airspeed records for jet-propelled aircraft....) −48 °C (−53 °F) 301 1083 673 585
NEWLINENEWLINE

## Effect of frequency and gas composition

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 the mean free path
Mean free path
In physics, the mean free path is the average distance covered by a moving particle between successive impacts which modify its direction or energy or other particle properties.-Derivation:...

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 soundwave 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 sound speeds (over 9% higher) because they have a higher $\gamma$ (5/3 = 1.66...) than diatomics do (7/5 = 1.4). Thus, at the same molecular mass, the sound speed of a monatomic gas goes up by a factor of $\left\{ c_\left\{\mathrm\left\{gas: monatomic\right\}\right\} \over c_\left\{\mathrm\left\{gas: diatomic\right\}\right\} \right\} = \sqrt\left\{\left\{\left\{\left\{5 / 3\right\} \over \left\{7 / 5\right\}\right\}\right\}\right\} = \sqrt\left\{25 \over 21\right\}$ = 1.091... This gives the 9 % difference, and would be a typical ratio for sound speeds at room temperature in helium
Helium
Helium is the chemical element with atomic number 2 and an atomic weight of 4.002602, which is represented by the symbol He. It is a colorless, odorless, tasteless, non-toxic, inert, monatomic gas that heads the noble gas group in the periodic table...

vs. deuterium
Deuterium
Deuterium, also called heavy hydrogen, is one of two stable isotopes of hydrogen. It has a natural abundance in Earth's oceans of about one atom in of hydrogen . Deuterium accounts for approximately 0.0156% of all naturally occurring hydrogen in Earth's oceans, while the most common isotope ...

, 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 generally travels at about 70% of the mean molecular speed in 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
Heat capacity , or thermal capacity, is the measurable physical quantity that characterizes the amount of heat required to change a substance's temperature by a given amount...

). However, vibrational modes simply cause gammas which decrease toward 1, since vibration modes in a polyatomic gas gives 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 sound speed in monatomic vs. polyatomic molecules, with the speed remaining greater in monatomics.

## Mach number

{{main|Mach number}}
Mach number, a useful quantity in aerodynamics, is the ratio of air speed
Speed
In kinematics, the speed of an object is the magnitude of its velocity ; it is thus a scalar quantity. The average speed of an object in an interval of time is the distance traveled by the object divided by the duration of the interval; the instantaneous speed is the limit of the average speed as...

to the local speed of sound. At altitude, for reasons explained, Mach number is a function of temperature. Aircraft flight instruments
Flight instruments
Flight instruments are the instruments in the cockpit of an aircraft that provide the pilot with information about the flight situation of that aircraft, such as height, speed and altitude...

, 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
Pitot tube
A pitot tube is a pressure measurement instrument used to measure fluid flow velocity. The pitot tube was invented by the French engineer Henri Pitot Ulo in the early 18th century and was modified to its modern form in the mid-19th century by French scientist Henry Darcy...

is dependent on altitude as well as speed. Assuming air to be an ideal gas
Ideal gas
An ideal gas is a theoretical gas composed of a set of randomly-moving, non-interacting point particles. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is amenable to analysis under statistical mechanics.At normal conditions such as...

, the formula to compute Mach number in a subsonic compressible flow is derived from Bernoulli's equation for M<1:$\left\{M\right\}=\sqrt\left\{5\left\left[\left\left(\frac\left\{q_c\right\}\left\{P\right\}+1\right\right)^\frac\left\{2\right\}\left\{7\right\}-1\right\right]\right\}\,$ where$M$ is Mach number$q_c$ is dynamic pressure and$P$ is static pressure. The formula to compute Mach number in a supersonic compressible flow is derived from the Rayleigh
Rayleigh number
In fluid mechanics, the Rayleigh number for a fluid is a dimensionless number associated with buoyancy driven flow...

Supersonic Pitot equation: $\left\{M\right\}=0.88128485\sqrt\left\{\left\left(\frac\left\{q_c\right\}\left\{P\right\}+1\right\right)\left\left(1-\frac\left\{1\right\}\left\{7M^2\right\}\right\right)^\left\{2.5\right\}\right\}$ where$M$ is Mach number$q_c$ is dynamic pressure measured behind a normal shock$P$ is static pressure. As can be seen, M appears on both sides of the equation. The easiest method to solve the supersonic M calculation is to enter both the subsonic and supersonic equations into a computer spreadsheet such as Microsoft Excel
Microsoft Excel
Microsoft Excel is a proprietary commercial spreadsheet application written and distributed by Microsoft for Microsoft Windows and Mac OS X. It features calculation, graphing tools, pivot tables, and a macro programming language called Visual Basic for Applications...

, OpenOffice.org Calc, or some equivalent program. First determine if M is indeed greater than 1.0 by calculating M from the subsonic equation. If M is greater than 1.0 at that point, then use the value of M from the subsonic equation as the initial condition in the supersonic equation. Then perform a simple iteration of the supersonic equation, each time using the last computed value of M, until M converges to a value—usually in just a few iterations.

## 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 by William Derham
William Derham
William Derham was an English clergyman and natural philosopher. He produced the earliest, reasonably accurate estimate of the speed of sound.-Life:...

, and acknowledged by Isaac Newton
Isaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...

. Derham had a telescope at the top of the tower of the Church of St Laurence
Church of St Laurence, Upminster
The church of St Laurence, Upminster is the Church of England parish church in Upminster, England. It is a Grade I listed building. It is the historic minster or church from which Upminster derives its name, meaning 'upper church', probably signifying 'church on higher ground'...

in Upminster
Upminster
Upminster is a suburban town in northeast London, England, and part of the London Borough of Havering. Located east-northeast of Charing Cross, it is one of the locally important district centres identified in the London Plan, and comprises a number of shopping streets and a large residential...

, 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 noise using a half-second pendulum. The distance from where the gun was fired was found by triangulation, and simple division (time / distance) provided velocity. Lastly, by making many observations, using a range of different distances, the inaccuracy of the half-second pendulum could be averaged out, giving his final estimate of the speed of sound. Modern stopwatches enable this method to be used today over distances as short as 200–400 meters, and not needing something as loud as a shotgun.

### Single-shot timing methods

The simplest concept is the measurement made using two microphone
Microphone
A microphone is an acoustic-to-electric transducer or sensor that converts sound into an electrical signal. In 1877, Emile Berliner invented the first microphone used as a telephone voice transmitter...

s and a fast recording device such as a digital
Digital
A digital system is a data technology that uses discrete values. By contrast, non-digital systems use a continuous range of values to represent information...

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

### Other methods

In these methods the time
Time
Time is a part of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify rates of change such as the motions of objects....

measurement has been replaced by a measurement of the inverse of time (frequency
Frequency
Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...

). Kundt's tube
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...

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
Node (physics)
A node is a point along a standing wave where the wave has minimal amplitude. For instance, in a vibrating guitar string, the ends of the string are nodes. By changing the position of the end node through frets, the guitarist changes the effective length of the vibrating string and thereby the...

and antinodes visible to the human eye. This is an example of a compact experimental setup. A tuning fork
Tuning fork
A tuning fork is an acoustic resonator in the form of a two-pronged fork with the prongs formed from a U-shaped bar of elastic metal . It resonates at a specific constant pitch when set vibrating by striking it against a surface or with an object, and emits a pure musical tone after waiting a...

can be held near the mouth of a long pipe
Pipe (material)
A pipe is a tubular section or hollow cylinder, usually but not necessarily of circular cross-section, used mainly to convey substances which can flow — liquids and gases , slurries, powders, masses of small solids...

which is dipping into a barrel of water
Water
Water is a chemical substance with the chemical formula H2O. A water molecule contains one oxygen and two hydrogen atoms connected by covalent bonds. Water is a liquid at ambient conditions, but it often co-exists on Earth with its solid state, ice, and gaseous state . Water also exists in a...

. In this system it is the case that the pipe can be brought to resonance if the length of the air column in the pipe is equal to ({1+2n}λ/4) where n is an integer. As the antinodal point for the pipe at the open end is slightly outside the mouth of the pipe it is best to find two or more points of resonance and then measure half a wavelength between these. Here it is the case that v = fλ

#### Three dimensional solids

In a solid, there is a non-zero stiffness both for volumetric and shear deformations. Hence, it is possible to generate sound waves with different velocities dependent on the deformation mode. Sound waves generating volumetric deformations (compressions) and shear deformations are called longitudinal waves and shear waves, respectively. In earthquake
Earthquake
An earthquake is the result of a sudden release of energy in the Earth's crust that creates seismic waves. The seismicity, seismism or seismic activity of an area refers to the frequency, type and size of earthquakes experienced over a period of time...

s, the corresponding seismic waves are called P-wave
P-wave
P-waves are a type of elastic wave, also called seismic waves, that can travel through gases , solids and liquids, including the Earth. P-waves are produced by earthquakes and recorded by seismographs...

s and S-wave
S-wave
A type of seismic wave, the S-wave, secondary wave, or shear wave is one of the two main types of elastic body waves, so named because they move through the body of an object, unlike surface waves....

s, respectively. The sound velocities of these two type waves propagating in a homogeneous 3-dimensional solid are respectively given by: $c_\left\{\mathrm\left\{l\right\}\right\} = \sqrt \left\{\frac\left\{K+\frac\left\{4\right\}\left\{3\right\}G\right\}\left\{\rho\right\}\right\} = \sqrt \left\{\frac\left\{Y \left(1-\nu\right)\right\}\left\{\rho \left(1+\nu\right)\left(1 - 2 \nu\right)\right\}\right\}$ $c_\left\{\mathrm\left\{s\right\}\right\} = \sqrt \left\{\frac\left\{G\right\}\left\{\rho\right\}\right\}$ where K and G are the bulk modulus
Bulk modulus
The bulk modulus of a substance measures the substance's resistance to uniform compression. It is defined as the pressure increase needed to decrease the volume by a factor of 1/e...

and shear modulus of the elastic materials, respectively, Y is the Young's modulus
Young's modulus
Young's modulus is a measure of the stiffness of an elastic material and is a quantity used to characterize materials. It is defined as the ratio of the uniaxial stress over the uniaxial strain in the range of stress in which Hooke's Law holds. In solid mechanics, the slope of the stress-strain...

, and $\nu$ is Poisson's ratio
Poisson's ratio
Poisson's ratio , named after Siméon Poisson, is the ratio, when a sample object is stretched, of the contraction or transverse strain , to the extension or axial strain ....

. The last quantity is not an independent one, as $Y = 3K\left(1-2\nu\right)$. Note that the speed of longitudinal/compression waves depends both on the compression and shear resistance properties of the material, while the speed of shear waves depends on the shear properties only. Typically, compression or P-waves travel faster in materials than do shear waves, and in earthquakes this is the reason that onset of an earthquake is often preceded by a quick upward-downward shock, before arrival of waves that produce a side-to-side motion. For example, for a typical steel alloy, K = 170 GPa, G = 80 GPa and $\rho$ = 7700 kg/m3, yielding a longitudinal velocity cl of 6000 m/s. This is in reasonable agreement with cl=5930 m/s measured experimentally for a (possibly different) type of steel. The shear velocity
Shear velocity
Shear velocity, also called friction velocity, is a form by which a shear stress may be re-written in units of velocity. It is useful as a method in fluid mechanics to compare true velocities, such as the velocity of a flow in a stream, to a velocity that relates shear between layers of flow.Shear...

cs is estimated at 3200 m/s using the same numbers.

#### Long rods

The speed of sound for shear waves in stiff materials such as metals is sometimes given for "long, thin rods" of the material in question, in which the speed is easier to measure. In rods, the speed of shear waves is given by: $c_\left\{\mathrm\left\{s\right\}\right\} = \sqrt \left\{\frac\left\{Y\right\}\left\{\rho\right\}\right\}$ This is similar to the expression for compression waves, save that Young's modulus
Young's modulus
Young's modulus is a measure of the stiffness of an elastic material and is a quantity used to characterize materials. It is defined as the ratio of the uniaxial stress over the uniaxial strain in the range of stress in which Hooke's Law holds. In solid mechanics, the slope of the stress-strain...

replaces the bulk modulus
Bulk modulus
The bulk modulus of a substance measures the substance's resistance to uniform compression. It is defined as the pressure increase needed to decrease the volume by a factor of 1/e...

. This speed is of about the same value as for shear wave speed in 3-D materials, but the ratio of the speeds in the two different types of objects depends on Poisson's ratio
Poisson's ratio
Poisson's ratio , named after Siméon Poisson, is the ratio, when a sample object is stretched, of the contraction or transverse strain , to the extension or axial strain ....

for the material.

### Speed of sound in liquids

In a fluid the only non-zero stiffness
Stiffness
Stiffness is the resistance of an elastic body to deformation by an applied force along a given degree of freedom when a set of loading points and boundary conditions are prescribed on the elastic body.-Calculations:...

is to volumetric deformation (a fluid does not sustain shear forces). Hence the speed of sound in a fluid is given by$c_\left\{\mathrm\left\{fluid\right\}\right\} = \sqrt \left\{\frac\left\{K\right\}\left\{\rho\right\}\right\}$ whereNEWLINE
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K is the bulk modulus
Bulk modulus
The bulk modulus of a substance measures the substance's resistance to uniform compression. It is defined as the pressure increase needed to decrease the volume by a factor of 1/e...

of the fluid
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#### Water

The speed of sound in water is of interest to anyone using underwater sound
Underwater acoustics
Underwater acoustics is the study of the propagation of sound in water and the interaction of the mechanical waves that constitute sound with the water and its boundaries. The water may be in the ocean, a lake or a tank. Typical frequencies associated with underwater acoustics are between 10 Hz and...

as a tool, whether in a laboratory, a lake or the ocean. Examples are sonar
Sonar
Sonar is a technique that uses sound propagation to navigate, communicate with or detect other vessels...

, acoustic communication and acoustical oceanography
Acoustical oceanography
Acoustical oceanography is the use of underwater sound to study the sea, its boundaries and its contents.-History:The earliest and most widespread use of sound and sonar technology to study the properties of the sea is the use of an echo sounder to measure water depth...

. See Discovery of Sound in the Sea for other examples of the uses of sound in the ocean (by both man and other animals). In fresh water, sound travels at about 1497 m/s at 25 °C. See Technical Guides - Speed of Sound in Pure Water for an online calculator.

#### Seawater

In salt water that is free of air bubbles or suspended sediment, sound travels at about 1560 m/s. The speed of sound in seawater depends on pressure (hence depth), temperature (a change of 1 °C ~ 4 m/s), and salinity
Salinity
Salinity is the saltiness or dissolved salt content of a body of water. It is a general term used to describe the levels of different salts such as sodium chloride, magnesium and calcium sulfates, and bicarbonates...

(a change of 1‰ ~ 1 m/s), and empirical equations have been derived to accurately calculate sound speed from these variables. Other factors affecting sound speed are minor. Since temperature decreases with depth while pressure and generally salinity increase, the profile of sound speed with depth generally shows a characteristic curve which decreases to a minimum at a depth of several hundred meters, then increases again with increasing depth (right). For more information see Dushaw et al. A simple empirical equation for the speed of sound in sea water with reasonable accuracy for the world's oceans is due to Mackenzie:NEWLINE
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c(T, S, z) = a1 + a2T + a3T2 + a4T3 + a5(S - 35) + a6z + a7z2 + a8T(S - 35) + a9Tz3
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a1 = 1448.96, 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
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### Speed in plasma

The speed of sound in a plasma
Plasma (physics)
In physics and chemistry, plasma is a state of matter similar to gas in which a certain portion of the particles are ionized. Heating a gas may ionize its molecules or atoms , thus turning it into a plasma, which contains charged particles: positive ions and negative electrons or ions...

for the common case that the electrons are hotter than the ions (but not too much hotter) is given by the formula (see here)$c_s = \left(\gamma ZkT_e/m_i\right)^\left\{1/2\right\} = 9.79\times10^3\,\left(\gamma ZT_e/\mu\right)^\left\{1/2\right\}\,\mbox\left\{m/s\right\}\,$ where $m_i$ is the ion
Ion
An ion is an atom or molecule in which the total number of electrons is not equal to the total number of protons, giving it a net positive or negative electrical charge. The name was given by physicist Michael Faraday for the substances that allow a current to pass between electrodes in a...

mass, $\mu$ is the ratio of ion mass to proton
Proton
The proton is a subatomic particle with the symbol or and a positive electric charge of 1 elementary charge. One or more protons are present in the nucleus of each atom, along with neutrons. The number of protons in each atom is its atomic number....

mass $\mu = m_i/m_p$; $T_e$ is the electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...

temperature; Z is the charge state; k is Boltzmann's constant; K is wavelength; and $\gamma$ is the adiabatic index. In contrast to a gas, the pressure and the density are provided by separate species, the pressure by the electrons and the density by the ions. The two are coupled through a fluctuating electric field.

{{main|sound speed gradient}} When sound spreads out evenly in all directions in three dimensions, the intensity drops in proportion to the inverse square of the distance. However, in the ocean there is a layer called the 'deep sound channel' or SOFAR channel
Sofar channel
The SOFAR channel , or deep sound channel , is a horizontal layer of water in the ocean at which depth the speed of sound is minimal. The SOFAR channel acts as a waveguide for sound, and low frequency sound waves within the channel may travel thousands of miles before dissipating...

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 index
Refractive index
In optics the refractive index or index of refraction of a substance or medium is a measure of the speed of light in that medium. It is expressed as a ratio of the speed of light in vacuum relative to that in the considered medium....

, sound waves will refract
Refraction
Refraction is the change in direction of a wave due to a change in its speed. It is essentially a surface phenomenon . The phenomenon is mainly in governance to the law of conservation of energy. The proper explanation would be that due to change of medium, the phase velocity of the wave is changed...

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 in a sheet of glass or optical fiber
Optical fiber
An optical fiber is a flexible, transparent fiber made of a pure glass not much wider than a human hair. It functions as a waveguide, or "light pipe", to transmit light between the two ends of the fiber. The field of applied science and engineering concerned with the design and application of...

. 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
Project Mogul
Project Mogul was a top secret project by the US Army Air Forces involving microphones flown on high altitude balloons, whose primary purpose was long-distance detection of sound waves generated by Soviet atomic bomb tests. The project was carried out from 1947 until early 1949...

successfully used this effect to detect a nuclear explosion
Nuclear explosion
A nuclear explosion occurs as a result of the rapid release of energy from an intentionally high-speed nuclear reaction. The driving reaction may be nuclear fission, nuclear fusion or a multistage cascading combination of the two, though to date all fusion based weapons have used a fission device...

at a considerable distance.

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• Elastic wave
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• Second sound
Second sound
Second sound is a quantum mechanical phenomenon in which heat transfer occurs by wave-like motion, rather than by the more usual mechanism of diffusion. Heat takes the place of pressure in normal sound waves. This leads to a very high thermal conductivity...

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• Sound barrier
Sound barrier
The sound barrier, in aerodynamics, is the point at which an aircraft moves from transonic to supersonic speed. The term, which occasionally has other meanings, came into use during World War II, when a number of aircraft started to encounter the effects of compressibility, a collection of several...

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• Underwater acoustics
Underwater acoustics
Underwater acoustics is the study of the propagation of sound in water and the interaction of the mechanical waves that constitute sound with the water and its boundaries. The water may be in the ocean, a lake or a tank. Typical frequencies associated with underwater acoustics are between 10 Hz and...

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• Vibrations
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