Speed of sound

The speed of sound $c_{{\text{S}}}$ is the speed at which sound waves propagate in a medium. Its SI unit is meters per second (m/s).

It is not to be confused with the sound velocity , i.e. the instantaneous velocity with which the individual particles of the medium move in order to build up and break down the deformation associated with the sound wave.

The speed of sound is generally dependent on the medium (in particular elasticity and density) and its temperature, in fluids additionally on the pressure and in solids significantly on the wave type (longitudinal wave, shear wave, Rayleigh wave, Lamb wave etc.) and on the frequency. In anisotropic media, it is also direction-dependent. In gases or gas mixtures, e.g. in normal air, only the temperature dependence plays a significant role.

The speed of sound in dry air at 20 °C is 343.2 m/s (1236 km/h).

For the relation between sound velocity and frequency of a monochromatic sound wave of wavelength λ $\lambda$ holds as for all such waves:

Speed of sound in liquids and gases

In liquids and gases, only pressure or density waves can propagate, in which the individual particles move back and forth in the direction of the wave propagation (longitudinal wave). The speed of sound is a function of the density ρ $\rho$ and the (adiabatic) compression modulus and is calculated like this:

$c_{\,{\text{Flüssigkeit, Gas}}}={\sqrt {\frac {K}{\rho }}}$

Speed of sound in solids

Sound waves in solids can propagate as longitudinal waves (where the direction of vibration of the particles is parallel to the direction of propagation) or as transverse waves (direction of vibration perpendicular to the direction of propagation).

For longitudinal waves, in the general case, the speed of sound in solids depends on the density ρ $\rho$ , the Poisson's ratio ν $\nu$ and the elastic modulus of the solid. The following applies

$c_{\text{Festkörper, longitudinal}}={\sqrt {\frac {E\,(1-\nu )}{\rho \,(1-\nu -2\nu ^{2})}}}$

$c_{\text{Festkörper, transversal}}={\sqrt {\frac {E}{2\,\rho \,(1+\nu )}}}={\sqrt {\frac {G}{\rho }}}$

With the shear modulus $G={\frac {E}{2\,(1+\nu )}}$ .

For a surface wave on an extended solid (Rayleigh wave) holds:

$c_{\text{Festkörper, Oberfläche}}\approx 0{,}922\cdot c_{\text{Festkörper, transversal}}$

The expression $M={\frac {E\,(1-\nu )}{(1-\nu -2\nu ^{2})}}$ is also called the longitudinal modulus, so that for the longitudinal wave we also have

$c_{\text{Festkörper, longitudinal}}={\sqrt {\frac {M}{\rho }}}$

can be written.

In the special case of a long rod whose diameter is much smaller than the wavelength of the sound wave, the influence of transverse contraction can be neglected (i.e. ν $\nu =0$ ), and we obtain:

$c_{{\text{langer Stab, longitudinal}}}={\sqrt {{\frac {E}{\rho }}}}$

$c_{\text{langer Stab, transversal}}={\sqrt {\frac {E}{2\rho }}}$

The theoretical limit for sound velocity in solids is

$c_{S}^{\mathrm {max} }=c\cdot \alpha \cdot {\sqrt {\frac {m_{\mathrm {e} }}{2m_{\mathrm {p} }}}}=38\;\mathrm {km/s} \,,$

where _me and mp are the masses of electron and neutron, c is the speed of light and α is the fine structure constant.

Speed of sound

Speed of sound in liquids and gases

Speed of sound in solids

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