Editing Speed of sound (section)

===Speed of sound in ideal gases and air===
<!-- This section is linked from [[Supersonic]] -->
For an ideal gas, ''K'' (the [[bulk modulus]] in equations above, equivalent to ''C'', the coefficient of stiffness in solids) is given by
<math display="block">K = \gamma \cdot p .</math>
Thus, from the Newton–Laplace equation above, the speed of sound in an ideal gas is given by
<math display="block">c = \sqrt{\gamma \cdot {p \over \rho}},</math>
where
* ''γ'' is the [[adiabatic index]] also known as the ''isentropic expansion factor''. It is the ratio of the specific heat of a gas at constant pressure to that of a gas at constant volume (<math>C_p/C_v</math>) and arises because a classical sound wave induces an adiabatic compression, in which the heat of the compression does not have enough time to escape the pressure pulse, and thus contributes to the pressure induced by the compression;
* ''p'' is the [[pressure]];
* ''ρ'' is the [[density]].

Using the [[ideal gas]] law to replace ''p'' with ''nRT''/''V'', and replacing ''ρ'' with ''nM''/''V'', the equation for an ideal gas becomes
<math display="block">c_{\mathrm{ideal}} = \sqrt{\gamma \cdot {p \over \rho}} = \sqrt{\gamma \cdot R \cdot T \over M} = \sqrt{\gamma \cdot k \cdot T \over m},</math>
where
* ''c''<sub>ideal</sub> is the speed of sound in an [[ideal gas]];
* ''R'' is the specific [[molar gas constant]];
* ''k'' is the [[Boltzmann constant]];
* ''γ'' (gamma) is the [[adiabatic index]]. At room temperature, where thermal energy is fully partitioned into rotation (rotations are fully excited) but quantum effects prevent excitation of vibrational modes, the value is {{math|1=7/5 = 1.400}} for diatomic gases (such as [[oxygen]] and [[nitrogen]]), according to kinetic theory. Gamma is actually experimentally measured over a range from 1.3991 to 1.403 at {{val|0|u=degC}}, for air. Gamma is exactly {{math|1=5/3 = 1.667}} for monatomic gases (such as [[argon]]) and it is {{math|1=4/3 = 1.333}} for triatomic molecule gases that, like {{chem|link=water|H|2|O}}, are not co-linear (a co-linear triatomic gas such as {{chem|C|O|2}} is equivalent to a diatomic gas for our purposes here);
* ''T'' is the absolute temperature;
* ''M'' is the molar mass of the gas. The mean molar mass for dry air is about {{cvt|0.02897|kg/mol|g/mol}};
* ''n'' is the number of moles;
* ''m'' is the mass of a single molecule.

This equation applies only when the sound wave is a small perturbation on the ambient condition, and the certain other noted conditions are fulfilled, as noted below. Calculated values for ''c''<sub>air</sub> have been found to vary slightly from experimentally determined values.<ref name=USSA1976>U.S. Standard Atmosphere, 1976, U.S. Government Printing Office, Washington, D.C., 1976.</ref>

[[Isaac Newton|Newton]] famously considered the speed of sound before most of the development of [[thermodynamics]] and so incorrectly used [[isothermal]] calculations instead of [[adiabatic]]. His result was missing the factor of ''γ'' but was otherwise correct.

Numerical substitution of the above values gives the ideal gas approximation of sound velocity for gases, which is accurate at relatively low gas pressures and densities (for air, this includes standard Earth sea-level conditions). Also, for diatomic gases the use of {{math|1=''γ'' = 1.4000}} requires that the gas exists in a temperature range high enough that rotational heat capacity is fully excited (i.e., molecular rotation is fully used as a heat energy "partition" or reservoir); but at the same time the temperature must be low enough that molecular vibrational modes contribute no heat capacity (i.e., insignificant heat goes into vibration, as all vibrational quantum modes above the minimum-energy-mode have energies that are too high to be populated by a significant number of molecules at this temperature). For air, these conditions are fulfilled at room temperature, and also temperatures considerably below room temperature (see tables below). See the section on gases in [[specific heat capacity]] for a more complete discussion of this phenomenon.

For air, we introduce the shorthand
<math display="block">R_* = R/M_{\mathrm{air}}.</math>

[[File:Speed of sound in dry air.svg|thumb|Approximation of the speed of sound in dry air based on the [[heat capacity ratio]] (in green) against the truncated [[Taylor expansion]] (in red)]]

In addition, we switch to the [[Celsius]] temperature {{math|1=''θ'' = ''T'' − {{val|273.15|u=K}}}}, which is useful to calculate air speed in the region near {{val|0|u=degC}} ({{val|273|u=K}}). Then, for dry air,
<math display=block>\begin{align}
c_{\mathrm{air}} &= \sqrt{\gamma \cdot R_* \cdot T} = \sqrt{\gamma \cdot R_* \cdot (\theta + 273.15\,\mathrm{K})},\\
c_{\mathrm{air}} &= \sqrt{\gamma \cdot R_* \cdot 273.15\,\mathrm{K}} \cdot \sqrt{1 + \frac{\theta}{273.15\,\mathrm{K}}} .
\end{align}</math>

Substituting numerical values
<math display=block>R = 8.314\,462\,618\,153\,24~\mathrm{J/(mol{\cdot}K)}</math>
<math display=block>M_{\mathrm{air}} = 0.028\,964\,5~\mathrm{kg/mol}</math>
and using the ideal diatomic gas value of {{math|1=''γ'' = 1.4000}}, we have
<math display=block>c_{\mathrm{air}} \approx 331.3\,\mathrm{m/s} \times \sqrt{1 + \frac{\theta}{273.15\,\mathrm{K}}} .</math>

Finally, Taylor expansion of the remaining square root in <math>\theta</math> yields
<math display=block>\begin{align}
c_{\mathrm{air}} & \approx 331.3\,\mathrm{m/s} \times \left(1 + \frac{\theta}{2 \times 273.15\,\mathrm{K}}\right),\\
                 & \approx 331.3\,\mathrm{m/s} + \theta \times 0.606 \,\mathrm{(m/s)/^\circ C} .
\end{align}</math>

A graph comparing results of the two equations is to the right, using the slightly more accurate value of {{cvt|331.5|m/s}} for the speed of sound at {{val|0|u=degC}}.<ref name="Kinsler2000">{{cite book| last1=Kinsler|first1=L.E.| title=Fundamentals of Acoustics | edition = 4th | url=https://archive.org/details/fundamentalsacou00kins_265| url-access=limited| last2=Frey|first2=A.R.| last3=Coppens|first3=A.B.| last4=Sanders|first4=J.V. | publisher=John Wiley & Sons|year=2000|isbn=0-471-84789-5|location=New York}}</ref>{{rp|pages=[https://archive.org/details/fundamentalsacou00kins_265/page/n135 120]-121}}