The speed of sound is the speed to which the sound waves move. It varies according to the propagation medium, and is in the following way defined:

$c\; =\; \backslash \; omega/k\; \backslash ,$ (in m/S)

with

• $\backslash \; omega$, the Frequency of the wave (in rad /s)
• K the standard of sound Vector of wave (in rad/m)

That corresponds to the definition of its Speed of phase. If the medium is dispersive, it is different from the Speed of group, which is the propagation velocity of sound energy. This difference can play a part when one measures the speed of sound (see low). One should not either confuse this speed with that of the Molécule S constituting material, nor that of the fluid particles in the case of a Fluide.

The independent factor exploiting the value speed of sound is the Densité propagation medium: in a Gas, its speed is lower than in a Liquide. For example, the sound is propagated roughly to 340 m/s (1224 km/h) in the air with 15°C to 1.435 m/s (5166 km/h) in soft Eau and approximately 1.500 m/s (5400 km/h) in the Sea water.

This property is in particular used to determine the quality of a Béton, because a faster propagation means than the concrete contains few bubbles of air (the speed of sound in the concrete is much higher than in the air).

History

The first experiments aiming at measuring the speed of sound are the work of Marin Mersenne and Pierre Gassendi during the Renaissance. However an excessive value will be given: 1  473 feet a second.

During the 17th century of other experiments are carried out by Edmond Halley and Robert Boyle like by Giovanni Cassini and Christian Huygens, but the results are contradictory. The French Academy of Science then decides to organize new experiments in 1738. Using blows of gun drawn the night (to see the flames leaving the mouth of the weapon) between the Observatory from Paris, Montmartre, Fontenay-Aux-Roses and Montlhéry, one estimates the speed of sound at 333  m/s in a temperature of the air with 0  °C. Once more, the results are contradictory with the repetition of the experiment in Germany.

In 1822, François Arago and Riche of Prony carries out new more rigorous experiments, on order of the Bureau of longitudes. This time it decides to use crossed shootings, between Villejuif and Montlhéry. The blows of guns will be drawn at the same time, in this manner, the experimenters hope to limit the disturbances due to the rate of Hygrométrie, speed of the wind, of pressure and temperature, which they think of being the cause of the failure of the preceding experiment. Moreover, stop watches much more precise are used. The experiments take place in the nights of the 21 and June 22nd 1822. The results give the value of 340,88  m/s at a temperature of 15,9  °C. After correction, speed with 0  °C is of 330,9  m/s.

The speed of sound is also given in other environments, as in 1808 in the solids by Jean-Baptiste Biot and in 1828 in the water of the Lac Léman by Jean-Daniel Colladon and Charles Sturm.

Speed of sound in a solid body

In a solid, the speed of the mechanical waves is dependant on the Density ρ and the modulus of elasticity. In the case of the compression waves, it is the Modulus Young E who enters in account, and speed is calculated as follows:

$c\_\; \{\backslash \; mathrm\; \{solid\}\}\; =\; \backslash \; sqrt\; \{\backslash \; frac\; \{E\}\; \{\backslash \; rho\}\}$.

Let us note that the waves of shearing are not propagated in the fluids.

Speed of sound in an unspecified fluid

Without wave of shearing, the speed of sound is propagated only by compression. If the sound is not too strong ($\backslash \; sound\; Delta\; P\_\; \{\}\; \backslash \; L\; ambient\; P\_\; \{\}$), the compression and the relaxation of the fluid can be taken isentropic and the speed of sound is:

$$

c_ {\ mathrm {fluid}} = \ sqrt {\ frac {\ partial p} {\ partial \ rho}} _S . The square root of the derivative partial of the pressure by the density with constant entropy.

Speed of sound in a liquid

The speed of sound in a liquid is a function of the density ρ and adiabatic coefficient of compressibility χ and is calculated as follows:

$$

c_ {\ mathrm {liquid}} = \ frac {1} {\ sqrt {\ rho \; \ chi}} .

Speed of sound in a perfect gas

The speed of sound in a Perfect gas is function of the isentropic coefficient γ (gamma), of the density ρ as well as pressure p of gas, and is calculated as follows:

$$

c_ {\ mathrm {gas}} = \ sqrt {\ frac {\ gamma \ cdot p} {\ rho}} with

$\backslash \; gamma\; =\; \backslash \; frac\; \{c\_p\}\; \{c\_v\}$

c_p and c_v being the mass heat capacities isobar and isochoric.

The speed of sound can be also calculated using the equation of state, of the coefficient Adiabatique γ (gamma), of the specific constant of the gas R_s and the Température T (K in Kelvin).

$$

c_ {\ mathrm {gas}} = \ sqrt {\ gamma \ cdot R_s \ cdot T}

With for the air:

• γ = 1,4
• R_s = 287 J/kg/K

The coefficient Adiabatique γ depends little on the Température T , the constant R is a size independent of the Température.

This speed is correlated at the mean velocity < v > of the molecules. Indeed, the equation of perfect gases connects p to the temperature T and volume V , and there is

statement ^γ = constant

What makes it possible to thus express C according to T only, and of < v > In the case of a monoatomic perfect gas (γ = 5/3), one a:

$c\_\; \{\backslash \; mathrm\; \{gases\}\}\; =\; \backslash \; sqrt\; \{\backslash \; frac\; \{5p\}\; \{3\; \backslash \; rho\}\}\; =\; \backslash \; sqrt\; \{\backslash \; frac\; \{5kT\}\; \{3m\}\}\; =\; \backslash \; sqrt\; \{\backslash \; frac\; \{5\; \backslash \; pi\}\; \{24\}\}\; \backslash \; langle\; v\; \backslash \; rangle$

C _gaz ≈ 0,81·< v' >

m being mass of a molecule.

This relation indicates that in the field of perfect gases (i.e. moderate pressures), the speed of sound is proportional to the speed of the molecules, i.e. to the square root of the absolute temperature.

In the case of the air (compound in majority of diatomic perfect gases), the speed of sound can be approximate by the following linearization:

C _air = (331,5 + 0,6·θ) m/s

where θ (theta) is the temperature in degrees Celsius:

θ = T -273,15

T being in K. This approximate formula makes it possible to obtain -20°C with +40°C an error lower than 0,2%.

Influence other factors

The moisture of the air influences little.

Fluids diphasic

In the case of a diphasic fluid (bubble of air in water for example), the speed of sound is strongly modified. The calculation speed of sound is then rather complex and depends in particular on the relations which link the two fluids (for example, in the case of a liquid with vapor bubbles, it will faura to take into account the phase shifts). Nevertheless, a general result little to be drawn. The speed of sound of this mixture is quite lower than smallest of both. For example, for a mixture water/vapor, the speed of sound is around 30 m/s for a presence rate of 0,5.

Count of the properties of the air according to the temperature

The following table presents the evolution of some properties of the air under a pressure of a atmosphere according to the temperature.

Experimental methods

There exist several ways of measuring the speed of sound:

; By measurement of a travel time While sending since a transmitter of the sound impulses and by detecting them using a Microphone, one can measure time that puts the impulse to traverse the distance separating them. That thus corresponds to measure the speed of sound energy, i.e. the Speed of group.

; By measurement frequency and wavelength By successively measuring the frequency and the Wavelength of the sound, one obtains his speed by multiplying these two sizes. That corresponds to the Speed of phase. There exist several methods allowing these measurements:

• For example, a Tube of Kundt consists of a tube stopped with the one of the ends, and joined with a Haut-parleur with the other. The sound resulting from this loudspeaker is considered by the side of the tube, and it settles a Standing wave inside. By moving a microphone in the tube, one can detect of them the bellies (maximum) and the nodes (minima), which makes it possible to measure the wavelength, then speed of sound.
• One can also carry out standing waves in the liquids, but it is then impossible to use a microphone to detect them. However, these waves act on the light in the same way that a optical Réseau. It is thus possible, thanks to an optical assembly, to measure the speed of sound.

The principal difference between these two methods is the result obtained: on the one hand the speed of phase, and on the other hand the speed of group. The difference between these two sizes is however visible only when the Dispersion of the medium is important, which is seldom the case.

Examples speeds of sound for various materials

The following table gives some examples for some materials to a temperature of 20°C and under an atmosphere.

It should be noticed that there is no speed of sound in the vacuum, since there is no particle which can be used as support with the sound waves.

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