Speed of sound

Pressure-pulse or compression-type wave (longitudinal wave) confined to a plane. This is the only type of sound wave that travels in fluids (gases and liquids). A pressure-type wave may also travel in solids, along with other types of waves (transverse waves, see below)

Transverse wave affecting atoms initially confined to a plane. This additional type of sound wave (additional type of elastic wave) travels only in solids, for it requires a sideways shearing motion which is supported by the presence of elasticity in the solid. The sideways shearing motion may take place in any direction which is at right-angle to the direction of wave-travel (only one shear direction is shown here, at right angles to the plane). Furthermore, the right-angle shear direction may change over time and distance, resulting in different types of polarization of 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, and is the same process in gases and liquids, with an analogous compression-type wave in solids. Only compression waves are supported in gases and liquids. An additional type of wave, the transverse wave, also called a shear wave, occurs only in solids because only solids support elastic deformations. It is due to elastic deformation of the medium perpendicular to the direction of wave travel; the direction of shear-deformation is called the "polarization" of this type of wave. In general, transverse waves occur as a pair 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 a fluid is determined by the medium's compressibility and density. In solids, the compression waves are analogous to those in fluids, depending on compressibility and density, but with the additional factor of shear modulus which affects compression waves due to off-axis elastic energies which are able to influence effective tension and relaxation in a compression. The speed of shear waves, which can occur only in solids, is determined simply by the solid material's shear modulus and density.

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

$${\displaystyle K=\gamma \cdot p.}$$

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

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

where

• γ is the adiabatic index also known as the isentropic expansion factor. It is the ratio of the specific heat of a gas at constant pressure to that of a gas at constant volume (${\displaystyle C_{p}/C_{v}}$) 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;
• ρ 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

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

where

• c_ideal is the speed of sound in an ideal gas;
• R is the molar gas constant;
• k is the Boltzmann constant;
• γ (gamma) is the adiabatic index. At room temperature, where thermal energy is fully partitioned into rotation (rotations are fully excited) but quantum effects prevent excitation of vibrational modes, the value is 7/5 = 1.400 for diatomic gases (such as oxygen and nitrogen), according to kinetic theory. Gamma is actually experimentally measured over a range from 1.3991 to 1.403 at 0 °C, for air. Gamma is exactly 5/3 = 1.667 for monatomic gases (such as argon) and it is 4/3 = 1.333 for triatomic molecule gases that, like H
  ₂O, are not co-linear (a co-linear triatomic gas such as CO
  ₂ is equivalent to a diatomic gas for our purposes here);
• T is the absolute temperature;
• M is the molar mass of the gas. The mean molar mass for dry air is about 0.02897 kg/mol (28.97 g/mol);
• n is the number of moles;
• m is the mass of a single molecule.

This equation applies only when the sound wave is a small perturbation on the ambient condition, and the certain other noted conditions are fulfilled, as noted below. Calculated values for c_air have been found to vary slightly from experimentally determined values.[10]

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 γ but was otherwise correct.

Numerical substitution of the above values gives the ideal gas approximation of sound velocity for gases, which is accurate at relatively low gas pressures and densities (for air, this includes standard Earth sea-level conditions). Also, for diatomic gases the use of γ = 1.4000 requires that the gas exists in a temperature range high enough that rotational heat capacity is fully excited (i.e., molecular rotation is fully used as a heat energy "partition" or reservoir); but at the same time the temperature must be low enough that molecular vibrational modes contribute no heat capacity (i.e., insignificant heat goes into vibration, as all vibrational quantum modes above the minimum-energy-mode have energies that are too high to be populated by a significant number of molecules at this temperature). For air, these conditions are fulfilled at room temperature, and also temperatures considerably below room temperature (see tables below). See the section on gases in specific heat capacity for a more complete discussion of this phenomenon.

For air, we introduce the shorthand

$${\displaystyle R_{*}=R/M_{\mathrm {air} }.}$$

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

In addition, we switch to the Celsius temperature ${\displaystyle \theta }$ = T − 273.15 K, which is useful to calculate air speed in the region near 0 °C (273 K). Then, for dry air,

$${\displaystyle {\begin{aligned}c_{\mathrm {air} }&={\sqrt {\gamma \cdot R_{*}\cdot T}}={\sqrt {\gamma \cdot R_{*}\cdot (\theta +273.15\,\mathrm {K} )}},\\c_{\mathrm {air} }&={\sqrt {\gamma \cdot R_{*}\cdot 273.15\,\mathrm {K} }}\cdot {\sqrt {1+{\frac {\theta }{273.15\,\mathrm {K} }}}}.\end{aligned}}}$$

Substituting numerical values

$${\displaystyle R=8.314\,462\,618\,153\,24~\mathrm {J/(mol{\cdot }K)} }$$

$${\displaystyle M_{\mathrm {air} }=0.028\,964\,5~\mathrm {kg/mol} }$$

and using the ideal diatomic gas value of γ = 1.4000, we have

$${\displaystyle c_{\mathrm {air} }\approx 331.3\,\mathrm {m/s} \times {\sqrt {1+{\frac {\theta }{273.15\,\mathrm {K} }}}}.}$$

Finally, Taylor expansion of the remaining square root in ${\displaystyle \theta }$ yields

$${\displaystyle {\begin{aligned}c_{\mathrm {air} }&\approx 331.3\,\mathrm {m/s} \times \left(1+{\frac {\theta }{2\times 273.15\,\mathrm {K} }}\right),\\&\approx 331.3\,\mathrm {m/s} +\theta \times 0.606\,\mathrm {(m/s)/^{\circ }C} .\end{aligned}}}$$

A graph comparing results of the two equations is to the right, using the slightly more accurate value of 331.5 m/s (1,088 ft/s) for the speed of sound at 0 °C.[11]^: 120-121

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.[9] Wind shear of 4 m/(s · km) can produce refraction equal to a typical temperature lapse rate of 7.5 °C/km.[12] Higher values of wind gradient will refract sound downward toward the surface in the downwind direction,[13] eliminating the acoustic shadow on the downwind side. This will increase the audibility of sounds downwind. This downwind refraction effect occurs because there is a wind gradient; the fact that sound is carried along by the wind is not important.[14]

For sound propagation, the exponential variation of wind speed with height can be defined as follows:[15]

$${\displaystyle {\begin{aligned}U(h)&=U(0)h^{\zeta },\\{\frac {\mathrm {d} U}{\mathrm {d} H}}(h)&=\zeta {\frac {U(h)}{h}},\end{aligned}}}$$

where

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

In the 1862 American Civil War 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,[16] because they could not hear the sounds of battle only 10 km (six miles) downwind.[17]

In the standard atmosphere:

• T₀ is 273.15 K (= 0 °C = 32 °F), giving a theoretical value of 331.3 m/s (= 1086.9 ft/s = 1193 km/h = 741.1 mph = 644.0 kn). Values ranging from 331.3 to 331.6 m/s may be found in reference literature, however;
• T₂₀ is 293.15 K (= 20 °C = 68 °F), giving a value of 343.2 m/s (= 1126.0 ft/s = 1236 km/h = 767.8 mph = 667.2 kn);
• T₂₅ is 298.15 K (= 25 °C = 77 °F), giving a value of 346.1 m/s (= 1135.6 ft/s = 1246 km/h = 774.3 mph = 672.8 kn).

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

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

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.[18]

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.[10] The standard equations for the speed of sound apply with reasonable accuracy only to situations in which the wavelength of the sound wave is considerably longer than the mean free path of molecules in a gas.

The molecular composition of the gas contributes both as the mass (M) of the molecules, and their heat capacities, and so both have an influence on speed of sound. In general, at the same molecular mass, monatomic gases have slightly higher speed of sound (over 9% higher) because they have a higher γ (5/3 = 1.66...) than diatomics do (7/5 = 1.4). Thus, at the same molecular mass, the speed of sound of a monatomic gas goes up by a factor of

$${\displaystyle {c_{\mathrm {gas,monatomic} } \over c_{\mathrm {gas,diatomic} }}={\sqrt {{5/3} \over {7/5}}}={\sqrt {25 \over 21}}=1.091\ldots }$$

This gives the 9% difference, and would be a typical ratio for speeds of sound at room temperature 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 travels at about 70% of the mean molecular speed in gases; the figure is 75% in monatomic gases and 68% in diatomic gases).

Note that in this example we have assumed that temperature is low enough that heat capacities are not influenced by molecular vibration (see heat capacity). However, vibrational modes simply cause gammas which decrease toward 1, since vibration modes in a polyatomic gas give the gas additional ways to store heat which do not affect temperature, and thus do not affect molecular velocity and sound velocity. Thus, the effect of higher temperatures and vibrational heat capacity acts to increase the difference between the speed of sound in monatomic vs. polyatomic molecules, with the speed remaining greater in monatomics.

By far, the most important factor influencing the speed of sound in air is temperature. The speed is proportional to the square root of the absolute temperature, giving an increase of about 0.6 m/s per degree Celsius. For this reason, the pitch of a musical wind instrument increases as its temperature increases.

The speed of sound is raised by humidity. The difference between 0% and 100% humidity is about 1.5 m/s at standard pressure and temperature, but the size of the humidity effect increases dramatically with temperature.

The dependence on frequency and pressure are normally insignificant in practical applications. In dry air, the speed of sound increases by about 0.1 m/s as the frequency rises from 10 Hz to 100 Hz. For audible frequencies above 100 Hz it is relatively constant. Standard values of the speed of sound are quoted in the limit of low frequencies, where the wavelength is large compared to the mean free path.[19]

As shown above, the approximate value 1000/3 = 333.33... m/s is exact a little below 5 °C and is a good approximation for all "usual" outside temperatures (in temperate climates, at least), hence the usual rule of thumb to determine how far lightning has struck: count the seconds from the start of the lightning flash to the start of the corresponding roll of thunder and divide by 3: the result is the distance in kilometers to the nearest point of the lightning bolt.

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.

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 = fλ.

The effect of impurities can be significant when making high-precision measurements. Chemical desiccants can be used to dry the air, but will, in turn, contaminate the sample. The air can be dried cryogenically, but this has the effect of removing the carbon dioxide as well; therefore many high-precision measurements are performed with air free of carbon dioxide rather than with natural air. A 2002 review[20] found that a 1963 measurement by Smith and Harlow using a cylindrical resonator gave "the most probable value of the standard speed of sound to date." The experiment was done with air from which the carbon dioxide had been removed, but the result was then corrected for this effect so as to be applicable to real air. The experiments were done at 30 °C but corrected for temperature in order to report them at 0 °C. The result was 331.45 ± 0.01 m/s for dry air at STP, for frequencies from 93 Hz to 1,500 Hz.

In a solid, there is a non-zero stiffness both for volumetric deformations and shear deformations. Hence, it is possible to generate sound waves with different velocities dependent on the deformation mode. Sound waves generating volumetric deformations (compression) and shear deformations (shearing) are called pressure waves (longitudinal waves) and shear waves (transverse waves), respectively. In earthquakes, the corresponding seismic waves are called P-waves (primary waves) and S-waves (secondary waves), respectively. The sound velocities of these two types of waves propagating in a homogeneous 3-dimensional solid are respectively given by[11]

$${\displaystyle c_{\mathrm {solid,p} }={\sqrt {\frac {K+{\frac {4}{3}}G}{\rho }}}={\sqrt {\frac {E(1-\nu )}{\rho (1+\nu )(1-2\nu )}}},}$$

$${\displaystyle c_{\mathrm {solid,s} }={\sqrt {\frac {G}{\rho }}},}$$

where

• K is the bulk modulus of the elastic materials;
• G is the shear modulus of the elastic materials;
• E is the Young's modulus;
• ρ is the density;
• ν is Poisson's ratio.

The last quantity is not an independent one, as E = 3K(1 − 2ν). Note that the speed of pressure waves depends both on the pressure and shear resistance properties of the material, while the speed of shear waves depends on the shear properties only.

Typically, pressure waves travel faster in materials than do shear waves, and in earthquakes this is the reason that the onset of an earthquake is often preceded by a quick upward-downward shock, before arrival of waves that produce a side-to-side motion. For example, for a typical steel alloy, K = 170 GPa, G = 80 GPa and ρ = 7,700 kg/m³, yielding a compressional speed c_solid,p of 6,000 m/s.[11] This is in reasonable agreement with c_solid,p measured experimentally at 5,930 m/s for a (possibly different) type of steel.[21] The shear speed c_solid,s is estimated at 3,200 m/s using the same numbers.

Speed of sound in semiconductor solids can be very sensitive to the amount of electronic dopant in them.[22]

The speed of sound for pressure waves in stiff materials such as metals is sometimes given for "long rods" of the material in question, in which the speed is easier to measure. In rods where their diameter is shorter than a wavelength, the speed of pure pressure waves may be simplified and is given by:[11]^: 70

$${\displaystyle c_{\mathrm {solid} }={\sqrt {\frac {E}{\rho }}},}$$

where E is Young's modulus. This is similar to the expression for shear waves, save that Young's modulus replaces the shear modulus. This speed of sound for pressure waves in long rods will always be slightly less than the same speed in homogeneous 3-dimensional solids, and the ratio of the speeds in the two different types of objects depends on Poisson's ratio for the material.

Speed of sound in water vs temperature.

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

$${\displaystyle c_{\mathrm {fluid} }={\sqrt {\frac {K}{\rho }}},}$$

where K is the bulk modulus of the fluid.

In fresh water, sound travels at about 1481 m/s at 20 °C (see the External Links section below for online calculators).[23] Applications of underwater sound can be found in sonar, acoustic communication and acoustical oceanography.

Speed of sound as a function of depth at a position north of Hawaii in the Pacific Ocean derived from the 2005 World Ocean Atlas. The SOFAR channel spans the minimum in the speed of sound at about 750-m depth.

In salt water that is free of air bubbles or suspended sediment, sound travels at about 1500 m/s (1500.235 m/s at 1000 kilopascals, 10 °C and 3% salinity by one method).[24] 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 the speed of sound from these variables.[25][26] Other factors affecting the speed of sound are minor. Since in most ocean regions temperature decreases with depth, the profile of the speed of sound with depth decreases to a minimum at a depth of several hundred metres. Below the minimum, sound speed increases again, as the effect of increasing pressure overcomes the effect of decreasing temperature (right).[27] For more information see Dushaw et al.[28]

An empirical equation for the speed of sound in sea water is provided by Mackenzie:[29]

$${\displaystyle c(T,S,z)=a_{1}+a_{2}T+a_{3}T^{2}+a_{4}T^{3}+a_{5}(S-35)+a_{6}z+a_{7}z^{2}+a_{8}T(S-35)+a_{9}Tz^{3},}$$

where

• T is the temperature in degrees Celsius;
• S is the salinity in parts per thousand;
• z is the depth in metres.

The constants a₁, a₂, ..., a₉ are

$${\displaystyle {\begin{aligned}a_{1}&=1,448.96,&a_{2}&=4.591,&a_{3}&=-5.304\times 10^{-2},\\a_{4}&=2.374\times 10^{-4},&a_{5}&=1.340,&a_{6}&=1.630\times 10^{-2},\\a_{7}&=1.675\times 10^{-7},&a_{8}&=-1.025\times 10^{-2},&a_{9}&=-7.139\times 10^{-13},\end{aligned}}}$$

with check value 1550.744 m/s for T = 25 °C, S = 35 parts per thousand, z = 1,000 m. This equation has a standard error of 0.070 m/s for salinity between 25 and 40 ppt. See for an online calculator.

(Note: The Sound Speed vs. Depth graph does not correlate directly to the MacKenzie formula. This is due to the fact that the temperature and salinity varies at different depths. When T and S are held constant, the formula itself is always increasing with depth.)

Other equations for the speed of sound in sea water are accurate over a wide range of conditions, but are far more complicated, e.g., that by V. A. Del Grosso[30] and the Chen-Millero-Li Equation.[28][31]

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)

$${\displaystyle c_{s}=\left({\frac {\gamma ZkT_{\mathrm {e} }}{m_{\mathrm {i} }}}\right)^{1/2}=\left({\frac {\gamma ZT_{e}}{\mu }}\right)^{1/2}\times 90.85~\mathrm {m/s} ,}$$

where

• m_i is the ion mass;
• μ is the ratio of ion mass to proton mass μ = m_i/m_p;
• T_e is the electron temperature;
• Z is the charge state;
• k is Boltzmann constant;
• γ is the adiabatic index.

In contrast to a gas, the pressure and the density are provided by separate species: the pressure by the electrons and the density by the ions. The two are coupled through a fluctuating electric field.