Editing Speed of sound (section)

==Non-gaseous media==

===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.

===Speed of sound in liquids===
[[File:Speed of sound in water.svg|thumb|right|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
<math display="block">c_{\mathrm{fluid}} = \sqrt{\frac{K}{\rho}},</math>
where {{math|''K''}} is the [[bulk modulus]] of the fluid.

====Water====
In fresh water, sound travels at about {{val|1481|u=m/s}} at {{val|20|u=degC}} (see the External Links section below for online calculators).<ref>{{cite web
| url = http://www.engineeringtoolbox.com/sound-speed-water-d_598.html
| title = Speed of Sound in Water at Temperatures between 32–212 oF (0–100 oC) — imperial and SI units
| website = The Engineering Toolbox
}}</ref> Applications of [[underwater acoustics|underwater sound]] can be found in [[sonar]], [[Underwater acoustics#Underwater communication|acoustic communication]] and [[acoustical oceanography]].

====Seawater====
{{See also|Sound speed profile}}
[[File:Underwater speed of sound.svg|thumb|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 {{val|1500|u=m/s}} ({{val|1500.235|u=m/s}} at {{val|1000|ul=kilopascals}}, {{val|10|u=degC}} and 3% [[salinity]] by one method).<ref>{{cite journal|last1=Wong|first1=George S. K.|last2=Zhu|first2=Shi-ming|title=Speed of sound in seawater as a function of salinity, temperature, and pressure|journal=The Journal of the Acoustical Society of America | date=1995 | volume=97 | issue=3|page=1732|doi=10.1121/1.413048|bibcode=1995ASAJ...97.1732W}}</ref> The speed of sound in seawater depends on pressure (hence depth), temperature (a change of {{val|1|u=degC}} ~ {{val|4|u=m/s}}), and [[salinity]] (a change of 1[[Per mil|‰]] ~ {{val|1|u=m/s}}), and empirical equations have been derived to accurately calculate the speed of sound from these variables.<ref>[http://webarchive.loc.gov/all/20030402194852/http://handle.dtic.mil/100.2/ADB199453 APL-UW TR 9407 High-Frequency Ocean Environmental Acoustic Models Handbook], pp. I1-I2.</ref><ref>{{cite web |last1=Robinson|first1=Stephen |title=Technical Guides – Speed of Sound in Sea-Water|url=http://resource.npl.co.uk/acoustics/techguides/soundseawater/content.html |website=National Physical Laboratory|access-date=7 December 2016|date=22 September 2005 | archive-date=29 April 2017|archive-url=https://web.archive.org/web/20170429192345/http://resource.npl.co.uk/acoustics/techguides/soundseawater/content.html|url-status=dead}}</ref> 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).<ref>{{cite web
| url = http://www.dosits.org/science/soundmovement/speedofsound/
| title = How Fast Does Sound Travel?
| work = Discovery of Sound in the Sea
| publisher = University of Rhode Island
| access-date = 30 November 2010
| archive-date = 20 May 2017
| archive-url = https://web.archive.org/web/20170520205400/http://www.dosits.org/science/soundmovement/speedofsound/
| url-status = dead
}}</ref> For more information see Dushaw et al.<ref name=Dushaw93/>

An empirical equation for the speed of sound in sea water is provided by Mackenzie:<ref>{{cite journal
| last = Kenneth V.
| first = Mackenzie
| title = Discussion of sea-water sound-speed determinations
| year = 1981
| journal = Journal of the Acoustical Society of America
| volume = 70
| issue = 3
| pages = 801–806
| doi = 10.1121/1.386919
|bibcode = 1981ASAJ...70..801M}}</ref>
<math display="block">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 T z^3,</math>
where
* ''T'' is the temperature in degrees Celsius;
* ''S'' is the salinity in parts per thousand;
* ''z'' is the depth in metres.

The constants ''a''<sub>1</sub>, ''a''<sub>2</sub>, ..., ''a''<sub>9</sub> are
<math display="block">\begin{align}
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{align}</math>
with check value {{val|1550.744|u=m/s}} for {{math|1=''T'' = {{val|25|u=degC}}}}, {{nobreak|1=''S'' = 35 parts per thousand}}, {{nobreak|1=''z'' = 1,000 m}}. This equation has a standard error of {{val|0.070|u=m/s}} for salinity between 25 and 40 [[Parts per thousand|ppt]]. See [http://resource.npl.co.uk/acoustics/techguides/soundseawater/] for an online calculator.

(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<ref>{{cite journal
| last = Del Grosso
| first = V. A.
| title = New equation for speed of sound in natural waters (with comparisons to other equations)
| year = 1974
| journal = Journal of the Acoustical Society of America
| volume = 56
| issue = 4
| pages = 1084–1091
| doi = 10.1121/1.1903388 | bibcode = 1974ASAJ...56.1084D| doi-access = free
}}</ref> and the Chen-Millero-Li Equation.<ref name=Dushaw93>{{cite journal
| last1 = Dushaw
| first1 = Brian D.
| last2 = Worcester
| first2 = P. F.
| last3 = Cornuelle
| first3 = B. D.
| last4 = Howe
| first4 = B. M.
| title = On Equations for the Speed of Sound in Seawater
| year = 1993
| journal = Journal of the Acoustical Society of America
| volume = 93
| issue = 1
| pages = 255–275
| doi = 10.1121/1.405660|bibcode = 1993ASAJ...93..255D}}</ref><ref>{{cite journal
| last1 = Meinen
| first1 = Christopher S.
| last2 = Watts
| first2 = D. Randolph
| title = Further Evidence that the Sound-Speed Algorithm of Del Grosso Is More Accurate Than that of Chen and Millero
| year = 1997
| journal = Journal of the Acoustical Society of America
| volume = 102
| issue = 4
| pages = 2058–2062
| doi = 10.1121/1.419655|bibcode = 1997ASAJ..102.2058M
| s2cid = 38144335
| url = https://digitalcommons.uri.edu/cgi/viewcontent.cgi?article=1249&context=gsofacpubs
}}</ref>

===Speed of sound in plasma===
The speed of sound in a [[Plasma (physics)|plasma]] for the common case that the electrons are hotter than the ions (but not too much hotter) is given by the formula (see [[Plasma parameters#Velocities|here]])
<math display="block">c_s = \left(\frac{\gamma Z k T_\mathrm{e}}{m_\mathrm{i}}\right)^{1/2} = \left(\frac{\gamma Z T_e}{\mu} \right)^{1/2} \times 90.85~\mathrm{m/s},</math>
where
* ''m''<sub>i</sub> is the [[ion]] mass;
* ''μ'' is the ratio of ion mass to [[proton]] mass {{nobreak|1=''μ'' = ''m''<sub>i</sub>/''m''<sub>p</sub>}};
* ''T''<sub>e</sub> 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.