Editing Speed of sound (section)

===Speed of sound in solids===

====Three-dimensional solids====
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 [[earthquake]]s, the corresponding seismic waves are called [[P-wave]]s (primary waves) and [[S-wave]]s (secondary waves), respectively. The sound velocities of these two types of waves propagating in a homogeneous 3-dimensional solid are respectively given by<ref name="Kinsler2000"/>
<math display="block">c_{\mathrm{solid,p}} = \sqrt{\frac{K + \frac{4}{3}G}{\rho}} = \sqrt{\frac{E(1 - \nu)}{\rho (1 + \nu)(1 - 2 \nu)}},</math>
<math display="block">c_{\mathrm{solid,s}} = \sqrt{\frac{G}{\rho}},</math>
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 {{nobreak|1=E = 3K(1 − 2ν)}}. 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, {{nobreak|1=''K'' = 170 GPa}}, {{nobreak|1=''G'' = 80 GPa}} and {{math|1=''p'' = {{val|7700|u=kg/m3}}}}, yielding a compressional speed ''c''<sub>solid,p</sub> of {{nobreak|6,000 m/s}}.<ref name="Kinsler2000"/> This is in reasonable agreement with ''c''<sub>solid,p</sub> measured experimentally at {{nobreak|5,930 m/s}} for a (possibly different) type of steel.<ref>J. Krautkrämer and H. Krautkrämer (1990), ''Ultrasonic testing of materials'', 4th fully revised edition, Springer-Verlag, Berlin, Germany, p. 497</ref> The shear speed ''c''<sub>solid,s</sub> is estimated at {{nobreak|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.<ref>{{cite journal| doi=10.1016/j.joule.2021.03.009|title=Charge-carrier-mediated lattice softening contributes to high zT in thermoelectric semiconductors|journal=Joule|volume=5|issue=5|page=1168-1182 | year=2021| last1=Slade|first1=Tyler |last2=Anand|first2=Shashwat|last3=Wood|first3=Max|last4=Male|first4=James|last5=Imasato|first5=Kazuki | last6=Cheikh|first6=Dean | last7=Al Malki|first7=Muath | last8=Agne|first8=Matthias | last9=Griffith|first9=Kent | last10=Bux|first10=Sabah | last11=Wolverton|first11=Chris|last12=Kanatzidis|first12=Mercouri | last13=Snyder|first13=Jeff | s2cid=233598665 |doi-access=free|bibcode=2021Joule...5.1168S }}</ref>

====One-dimensional solids====
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:<ref name="Kinsler2000"/>{{rp|pages=[https://archive.org/details/fundamentalsacou00kins_265/page/n85 70]}}
<math display="block">c_{\mathrm{solid}} = \sqrt{\frac{E}{\rho}},</math>
where {{math|''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.