Editing Speed of sound (section)

==Equations==
The speed of sound in mathematical notation is conventionally represented by ''c'', from the Latin ''celeritas'' meaning "swiftness".

For fluids in general, the speed of sound ''c'' is given by the Newton–Laplace equation:
<math display="block">c = \sqrt{\frac{K_s}{\rho}},</math>
where
* <math>K_s</math> is a coefficient of stiffness, the isentropic [[bulk modulus]] (or the modulus of bulk elasticity for gases);
* <math>\rho</math> is the [[density]].

<math>K_s = \rho \left(\frac{\partial P}{\partial\rho}\right)_s</math>, where <math>P</math> is the pressure and the [[derivative]] is taken isentropically, that is, at constant [[entropy]] ''s''. This is because a sound wave travels so fast that its propagation can be approximated as an [[Adiabatic process#Various applications of the adiabatic assumption|adiabatic process]], meaning that there isn't enough time, during a pressure cycle of the sound, for significant heat conduction and radiation to occur.

Thus, the speed of sound increases with the stiffness (the resistance of an elastic body to deformation by an applied force) of the material and decreases with an increase in density. For ideal gases, the bulk modulus ''K'' is simply the gas pressure multiplied by the dimensionless [[adiabatic index]], which is about 1.4 for air under normal conditions of pressure and temperature.

For general [[equations of state]], if [[classical mechanics]] is used, the speed of sound ''c'' can be derived<ref>{{Cite web|url=https://courses.lumenlearning.com/suny-osuniversityphysics/chapter/17-2-speed-of-sound/|title=17.2 Speed of Sound {{!}} University Physics Volume 1|website=courses.lumenlearning.com|access-date=2020-01-24}}</ref> as follows:

Consider the sound wave propagating at speed <math>v</math> through a pipe aligned with the <math>x</math> axis and with a cross-sectional area of <math>A</math>. In time interval <math>dt</math> it moves length <math>dx = v \, dt</math>. In [[steady state]], the [[mass flow rate]] <math>\dot m = \rho v A </math> must be the same at the two ends of the tube, therefore the [[Mass flux#Equations for fluids|mass flux]] <math>j=\rho v </math> is constant and <math>v \, d\rho = -\rho \, dv</math>. Per [[Newton's second law]], the [[pressure-gradient force]] provides the acceleration:
<math display="block">\begin{align}
\frac{dv}{dt}
&=-\frac{1}{\rho}\frac{dP}{dx} \\[1ex]
\rightarrow
dP&=(-\rho \,dv)\frac{dx}{dt}=(v \, d\rho)v \\[1ex]
\rightarrow
v^2& \equiv c^2=\frac{dP}{d\rho}
\end{align}
</math>

And therefore:

<math display="block">c = \sqrt{\left(\frac{\partial P}{\partial\rho}\right)_s} = \sqrt{\frac{K_s}{\rho}},</math>

If [[special relativity|relativistic]] effects are important, the speed of sound is calculated from the [[relativistic Euler equations]].

In a [[Acoustic dispersion|non-dispersive medium]], the speed of sound is independent of [[sound frequency]], so the speeds of energy transport and sound propagation are the same for all frequencies. Air, a mixture of oxygen and nitrogen, constitutes a non-dispersive medium. However, air does contain a small amount of CO<sub>2</sub> which ''is'' a dispersive medium, and causes dispersion to air at [[ultrasound|ultrasonic]] frequencies (greater than {{val|28|ul=kHz}}).<ref>Dean, E. A. (August 1979). [https://web.archive.org/web/20120531095150/http://handle.dtic.mil/100.2/ADA076060 Atmospheric Effects on the Speed of Sound], Technical report of Defense Technical Information Center</ref>

In a [[Acoustic dispersion|dispersive medium]], the speed of sound is a function of sound frequency, through the [[dispersion relation]]. Each frequency component propagates at its own speed, called the [[phase velocity]], while the energy of the disturbance propagates at the [[group velocity]]. The same phenomenon occurs with light waves; see [[Dispersion (optics)#Group velocity dispersion|optical dispersion]] for a description.