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# Sound speed

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### Sound speed

For other uses, see Speed of sound (disambiguation).
Sound measurements
Sound pressure p, SPL
Particle velocity v, SVL
Particle displacement ξ
Sound intensity I, SIL
Sound power Pac
Sound power level SWL
Sound energy
Sound exposure E
Sound exposure level SEL
Sound energy density E
Sound energy flux q
Acoustic impedance Z
Speed of sound
Audio frequency AF

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 1,236 kilometres per hour (768 mph), or about a kilometre in three seconds or a mile in five seconds.

In fluid dynamics, 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 divided by the speed of sound in the fluid is called the Mach number. Objects moving at speeds greater than Mach1 are traveling at supersonic speeds.

The speed of sound in an ideal gas is independent of frequency, but does vary slightly with frequency in a real gas. It is proportional to the square root of the absolute temperature, but is independent of pressure or density for a given ideal gas. Sound speed in air varies slightly with pressure only because air is not quite an ideal gas. In addition, for different gases, the speed of sound is inversely proportional to the mean molecular weight of the gas, and is affected to a lesser extent by the number of ways in which the molecules of the gas can store heat from compression, since sound in gases is a compression wave. Although (in the case of gases only) the speed of sound is expressed in terms of a ratio of both density and pressure, these quantities cancel in ideal gases at any given temperature, composition, and heat capacity. This leads to a velocity formula for ideal gases which includes only the latter independent variables.

In common everyday speech, speed of sound refers to the speed of sound waves in air. However, the speed of sound varies from substance to substance. Sound travels faster in liquids and non-porous solids than it does in air. It travels about 4.3 times as fast in water (1,484 m/s), and nearly 15 times as fast in iron (5,120 m/s), than in air at 20 degrees Celsius. Sound waves in solids are composed of compression waves (just as in gases and liquids), but there is also a different type of sound wave called a shear wave, which occurs only in solids. These different types of waves in solids usually travel at different speeds, as exhibited in seismology. The speed of a compression sound wave in solids is determined by the medium's compressibility, shear modulus and density. The speed of shear waves is determined only by the solid material's shear modulus and density.

## Basic concept

The transmission of sound can be illustrated by using a toy model 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, and the mass corresponds to the density. All other things being equal (ceteris paribus), sound will travel more slowly in spongy materials, and faster in stiffer ones. For instance, sound will travel 1.59 times faster in nickel than in bronze, due to the greater stiffness of nickel at about the same density. Similarly, sound travels about 1.41 times faster in light hydrogen (protium) gas than in heavy hydrogen (deuterium) gas, since deuterium has similar properties but twice the density. At the same time, "compression-type" sound will travel faster in solids than in liquids, and faster in liquids than in gases, because the solids are more difficult to compress than liquids, while liquids in turn are more difficult to compress than gases.

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.

### Compression and shear waves

In a gas or liquid, sound consists of compression waves. In solids, waves propagate as two different types. A longitudinal wave is associated with compression and decompression in the direction of travel, which is the same process as all sound waves in gases and liquids. A transverse wave, called a shear wave in solids, 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, where sharp compression waves arrive first, and rocking transverse waves seconds later.

The speed of a compression wave in fluid is determined by the medium's compressibility and density. In solids, the compression waves are analogous to those in fluids, depending on compressibility, density, and the additional factor of shear modulus. The speed of shear waves, which can occur only in solids, is determined simply by the solid material's shear modulus and density.

## Basic

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



c = \sqrt{\frac{K}{\rho}}\,

where

K is a coefficient of stiffness, the bulk modulus (or the modulus of bulk elasticity for gases),
$\rho$ is the density

Thus the speed of sound increases with the stiffness (the resistance of an elastic body to deformation by an applied force) of the material, and decreases with the density. For ideal gases the bulk modulus P is simply the gas pressure multiplied by the adiabatic index.

For general equations of state, if classical mechanics is used, the speed of sound $c$ is given by



c^2=\left(\frac{\partial p}{\partial\rho}\right)_s where $p$ is the pressure and $\rho$ is the density and the derivative is taken adiabatically, that is, at constant entropy per particle (s).

If relativistic effects are important, the speed of sound is calculated from the relativistic Euler equations.

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 all sound frequencies. For audible sounds, the 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 frequencies (> 28 kHz).[1]

In a dispersive medium sound speed is a function of sound frequency, through the dispersion relation. The spatial and temporal distribution of a propagating disturbance will continually change. Each frequency component propagates at its own phase velocity, while the energy of the disturbance propagates at the group velocity. The same phenomenon occurs with light waves; see optical dispersion for a description.

## Dependence on the properties of the medium

The speed of sound is variable and depends on the properties of the substance through which the wave is travelling. In solids, the speed of transverse (or shear) waves depend on the shear deformation under shear stress (called the shear modulus), and the density of the medium. Longitudinal (or compression) waves in solids depend on the same two factors with the addition of a dependence on compressibility.

In fluids, only the medium's compressibility and density are the important factors, since fluids do not tolerate shear stresses. In heterogeneous fluids, such as a liquid filled with gas bubbles, the density of the liquid and the compressibility of the gas affect the speed of sound in an additive manner, as demonstrated in the hot chocolate effect.

In gases, adiabatic compressibility is directly related to pressure through the heat capacity ratio (adiabatic index), and pressure and density are inversely related at a given temperature and composition, thus making only the latter independent properties (temperature, molecular composition, and heat capacity ratio) important. In low molecular weight gases, such as helium, sound propagates faster compared to heavier gases, such as xenon (for monatomic gases the speed of sound is about 75% of the mean speed that molecules move in the gas). For a given ideal gas the sound speed depends only on its temperature. At a constant temperature, the ideal gas pressure has no effect on the speed of sound, because pressure and density (also proportional to pressure) have equal but opposite effects on the speed of sound, and the two contributions cancel out exactly. In a similar way, compression waves in solids depend both on compressibility and density—just as in liquids—but in gases the density contributes to the compressibility in such a way that some part of each attribute factors out, leaving only a dependence on temperature, molecular weight, and heat capacity ratio (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, 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 and nitrogen molecules of the air are replaced by lighter molecules of water. This is a simple mixing effect.

## Altitude variation and implications for atmospheric acoustics

In the Earth's atmosphere, the chief factor affecting the speed of sound is the temperature. For a given ideal gas with constant heat capacity and composition, sound speed is dependent solely upon temperature; see Details below. In such an ideal case, the effects of decreased density and decreased pressure of altitude cancel each other out, save for the residual effect of temperature.

Since temperature (and thus the speed of sound) decreases with increasing altitude up to 11 km, sound is refracted upward, away from listeners on the ground, creating an acoustic shadow at some distance from the source.[2] The decrease of the sound speed with height is referred to as a negative sound speed gradient.

However, there are variations in this trend above 11 km. In particular, in the stratosphere above about 20 km, the speed of sound increases with height, due to an increase in temperature from heating within the ozone layer. This produces a positive sound speed gradient in this region. Still another region of positive gradient occurs at very high altitudes, in the aptly-named thermosphere above 90 km.

## 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_{\mathrm{air}} = (331{.}3 + 0{.}606 \cdot \vartheta) \ \mathrm{m \cdot s^{-1}}\,

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

This equation is derived from the first two terms of the Taylor expansion of the following more accurate equation:

$c_\left\{\mathrm\left\{air\right\}\right\} = 331.3\,\mathrm\left\{m \cdot s^\left\{-1\right\}\right\} \sqrt\left\{1+\frac\left\{\vartheta\right\}\left\{273.15\right\}\right\}$

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:

$c_\left\{\mathrm\left\{air\right\}\right\} = 20.0457\,\mathrm\left\{m \cdot s^\left\{-1\right\}\right\} \sqrt$

The value of 331.3 m/s, which represents the speed at 0 °C (or 273.15 K), is based on theoretical (and some measured) values of the heat capacity ratio, $\gamma$, as well as on the fact that at 1 atm real air is very well described by the ideal gas approximation. Commonly found values for the speed of sound at 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. 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 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 at 0°C.

## Details

### Speed in ideal gases and in air

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



K = \gamma \cdot p\,

thus



c = \sqrt{\gamma \cdot {p \over \rho}}\,

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.
p is the pressure.
$\rho$ is the density

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



c_{\mathrm{ideal}} = \sqrt{\gamma \cdot {p \over \rho}} = \sqrt{\gamma \cdot R \cdot T \over M}= \sqrt{\gamma \cdot k \cdot T \over m}\,

where

• $c_\left\{\mathrm\left\{ideal\right\}\right\}$ is the speed of sound in an ideal gas.
• $R$ (approximately 8.3145 J·mol−1·K−1) is the molar gas constant.[3]
• $k$ is the Boltzmann constant
• $\gamma$ (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 monatomic molecules such as noble gases).
• $T$ is the absolute temperature in kelvin.
• $M$ is the molar mass in kilograms per mole. The mean molar mass for dry air is about 0.0289645 kg/mol.
• $m$ is the mass of a single molecule in kilograms.

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.[4]

Newton famously considered the speed of sound before most of the development of thermodynamics and so incorrectly used isothermal calculations instead of adiabatic. His result was missing the factor of $\gamma$ 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., insignificant heat goes into vibration, as all vibrational quantum modes above the minimum-energy-mode, have energies too high to be populated by a significant number of molecules at this temperature). For air, these conditions are fulfilled at room temperature, and also temperatures considerably below room temperature (see tables below). See the section on gases in specific heat capacity 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(°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_{\mathrm{ideal}} = \sqrt{\gamma \cdot R_* \cdot T} = \sqrt{\gamma \cdot R_* \cdot (\vartheta + 273.15)}\,



c_{\mathrm{ideal}} = \sqrt{\gamma \cdot R_* \cdot 273.15} \cdot \sqrt{1+\frac{\vartheta}{273.15}}\,

For dry air, where $\vartheta\,$ (theta) is the temperature in degrees Celsius(°C).

Making the following numerical substitutions:



\ R = 8.314510 \cdot \mathrm{J \cdot mol^{-1}} \cdot K^{-1}\,

is the molar gas constant in J/mole/Kelvin;



\ M_{\mathrm{air}} = 0.0289645 \cdot \mathrm{kg \cdot mol^{-1}}\,

is the mean molar mass of air, in kg; and using the ideal diatomic gas value of $\ \gamma\, = 1.4000\,$

Then:



c_{\mathrm{air}} = 331.3 \ \mathrm{\frac{m}{s}} \sqrt{1+\frac{\vartheta^{\circ}\mathrm{C}}{273.15\;^{\circ}\mathrm{C}}}\,

Using the first two terms of the Taylor expansion:



c_{\mathrm{air}} = 331.3 \ \mathrm{\frac{m}{s}} (1 + \frac{\vartheta^{\circ}\mathrm{C}}{2 \cdot 273.15\;^{\circ}\mathrm{C}})\,



c_{\mathrm{air}} = ( 331{.}3 + 0{.}606\;^{\circ}\mathrm{C}^{-1} \cdot \vartheta)\ \mathrm{ \frac{m}{s}}\,

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 upward, away from listeners on the ground, creating an acoustic shadow at some distance from the source.[2] Wind shear of 4 m·s−1·km−1 can produce refraction equal to a typical temperature lapse rate of 7.5 °C/km.[5] Higher values of wind gradient will refract sound downward toward the surface in the downwind direction,[6] 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.[7]

For sound propagation, the exponential variation of wind speed with height can be defined as follows:[8]



\ U(h) = U(0) h ^ \zeta\,



\ \frac {dU} {dH} = \zeta \frac {U(h)} {h}\,

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 Battle of Iuka, an acoustic shadow, believed to have been enhanced by a northeast wind, kept two divisions of Union soldiers out of the battle,[9] because they could not hear the sounds of battle only 10 km (six miles) downwind.[10]

### Tables

In the standard atmosphere:

• 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.
• 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).
• 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).

In fact, assuming an ideal gas, 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.

Effect of temperature on properties of air
Temperature T in °C Speed of sound c in m·s−1 Density of air ρ in kg·m−3 Acoustic impedance Z in N·s·m−3
+35 351.88 1.1455 403.2
+30 349.02 1.1644 406.5
+25 346.13 1.1839 409.4
+20 343.21 1.2041 413.3
+15 340.27 1.2250 416.9
+10 337.31 1.2466 420.5
+5 334.32 1.2690 424.3
0 331.30 1.2922 428.0
−5 328.25 1.3163 432.1
−10 325.18 1.3413 436.1
−15 322.07 1.3673 440.3
−20 318.94 1.3943 444.6
−25 315.77 1.4224 449.1

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

 Altitude Temperature m·s−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 flight) −57 °C (−70 °F) 295 1062 660 573 29,000 m (Flight of X-43A) −48 °C (−53 °F) 301 1083 673 585

## 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.[11]

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 increases. For this reason, the concept of speed of sound (except for frequencies approaching zero) progressively loses its range of applicability at high altitudes.[4] 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

$\left\{ c_\left\{\mathrm\left\{gas: monatomic\right\}\right\} \over c_\left\{\mathrm\left\{gas: diatomic\right\}\right\} \right\} = \sqrt\right\}\right\} = \sqrt\left\{25 \over 21\right\}$ = 1.091...

This gives the 9% difference, and would be a typical ratio for sound speeds at room temperature in helium vs. deuterium, each with a molecular weight of 4. Sound travels faster in helium than deuterium because adiabatic compression heats helium more, since the helium molecules can store heat energy from compression only in translation, but not rotation. Thus helium molecules (monatomic molecules) travel faster in a sound wave and transmit sound faster. (Sound 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). 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 article: Mach number

Mach number, a useful quantity in aerodynamics, is the ratio of air speed to the local speed of sound. At altitude, for reasons explained, Mach number is a function of temperature. Aircraft flight instruments, however, operate using pressure differential to compute Mach number, not temperature. The assumption is that a particular pressure represents a particular altitude and, therefore, a standard temperature. Aircraft flight instruments need to operate this way because the stagnation pressure sensed by a Pitot tube is dependent on altitude as well as speed.

## Experimental methods

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

The earliest reasonably accurate estimate of the speed of sound in air was made by William Derham, and acknowledged by Isaac Newton. Derham had a telescope at the top of the tower of the Church of St Laurence in Upminster, England. On a calm day, a synchronized pocket watch would be given to an assistant who would fire a shotgun at a pre-determined time from a conspicuous point some miles away, across the countryside. This could be confirmed by telescope. He then measured the interval between seeing gunsmoke and arrival of the 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 microphones and a fast recording device such as a digital 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 measurement has been replaced by a measurement of the inverse of time (frequency).

Kundt's tube 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 and antinodes visible to the human eye. This is an example of a compact experimental setup.

A tuning fork can be held near the mouth of a long pipe which is dipping into a barrel of water. In this system it is the case that the pipe can be brought to resonance if the length of the air column in the pipe is equal to ({1+2n}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 =

## Non-gaseous media

### Speed of sound in solids

#### 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 earthquakes, the corresponding seismic waves are called P-waves and S-waves, respectively. The sound velocities of these two type waves propagating in a homogeneous 3-dimensional solid are respectively given by:[13]

$c_\left\{\mathrm\left\{l\right\}\right\} = \sqrt \left\{\frac\left\{K+\frac\left\{4\right\}\left\{3\right\}G\right\}\left\{\rho\right\}\right\} = \sqrt \left\{\frac\left\{Y \left(1-\nu\right)\right\}\left\{\rho \left(1+\nu\right)\left(1 - 2 \nu\right)\right\}\right\}$

$c_\left\{\mathrm\left\{s\right\}\right\} = \sqrt \left\{\frac\left\{G\right\}\left\{\rho\right\}\right\}$

where K and G are the bulk modulus and shear modulus of the elastic materials, respectively, Y is the Young's modulus, and $\nu$ is Poisson's ratio. 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.[13] This is in reasonable agreement with cl=5930 m/s measured experimentally for a (possibly different) type of steel.[14]

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

#### Long rods

The speed of sound for longitudinal 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 where their diameter is shorter than a wavelength, the speed of pure longitudinal waves may be simplified and is given by:

$c_\left\{\mathrm\left\{l\right\}\right\} = \sqrt \left\{\frac\left\{Y\right\}\left\{\rho\right\}\right\}$

This is similar to the expression for shear waves, save that Young's modulus replaces the shear modulus. This speed of sound for longitudinal waves in long, thin rods will always be slightly less than the 3-D, longitudinal wave speed in an isotropic materials, and the ratio of the speeds in the two different types of objects depends on Poisson's ratio for the material.

### Speed of sound in liquids

In a fluid the only non-zero stiffness is to volumetric deformation (a fluid does not sustain shear forces).

Hence the speed of sound in a fluid is given by



c_{\mathrm{fluid}} = \sqrt {\frac{K}{\rho}}

where K is the bulk modulus of the fluid. This value typically decreases with temperature for non-polar fluids: the speed of sound in ultra-wave frequency range is inverse proportional to the cube of the volume of a fixed amount of the fluid.[15]

#### Water

The speed of sound in water is of interest to anyone using 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 (a change of 1 ~ 1 m/s), and empirical equations have been derived to accurately calculate sound speed from these variables.[16] 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).[17] For more information see Dushaw et al.[18]

A simple empirical equation for the speed of sound in sea water with reasonable accuracy for the world's oceans is due to Mackenzie:[19]

c(T, S, z) = a1 + a2T + a3T2 + a4T3 + a5(S - 35) + a6z + a7z2 + a8T(S - 35) + a9Tz3

where T, S, and z are temperature in degrees Celsius, salinity in parts per thousand and depth in meters, respectively. The constants a1, a2, ..., a9 are:

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

with check value 1550.744 m/s for T=25 °C, S=35 parts per thousand, z=1000 m. This equation has a standard error of 0.070 m/s for salinity between 25 and 40 Technical Guides - Speed of Sound in Sea-Water for an online calculator.

Other equations for sound speed in sea water are accurate over a wide range of conditions, but are far more complicated, e.g., that by V. A. Del Grosso[20] and the Chen-Millero-Li Equation.[18][21]

### Speed in plasma

The speed of sound in a plasma 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 = (\gamma ZkT_e/m_i)^{1/2} = 9.79\times10^3\,(\gamma ZT_e/\mu)^{1/2}\,\mbox{m/s}\,

where $m_i$ is the ion mass, $\mu$ is the ratio of ion mass to proton mass $\mu = m_i/m_p$; $T_e$ is the electron 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.

When sound spreads out evenly in all directions in three dimensions, the intensity drops in proportion to the inverse square of the distance. However, in the ocean there is a layer called the 'deep sound channel' or SOFAR channel which can confine sound waves at a particular depth.

In the SOFAR channel, the speed of sound is lower than that in the layers above and below. Just as light waves will refract towards a region of higher index, sound waves will refract towards a region where their speed is reduced. The result is that sound gets confined in the layer, much the way light can be confined in a sheet of glass or optical fiber. Thus, the sound is confined in essentially two dimensions. In two dimensions the intensity drops in proportion to only the inverse of the distance. This allows waves to travel much further before being undetectably faint.

A similar effect occurs in the atmosphere. Project Mogul successfully used this effect to detect a nuclear explosion at a considerable distance.