Speed of sound in ideal gases and air For an ideal gas,
K (the
bulk modulus in equations above, equivalent to
C, the coefficient of stiffness in solids) is given by K = \gamma \cdot p . Thus, from the Newton–Laplace equation above, the speed of sound in an ideal gas is given by c = \sqrt{\gamma \cdot {p \over \rho}}, 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 (C_p/C_v) 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 \begin{align} c_{\mathrm{ideal}} &= \sqrt{\frac{\gamma p}{\rho}} = \sqrt{\frac{\gamma RT}{M}} = \sqrt{\frac{\gamma kT}{m}}\\ &= \sqrt{\frac{1.380649 \cdot 10^{-23} \cdot \gamma T}{m}} \approx \sqrt{\frac{8.314\gamma T}{M}}\\ & \begin{cases} &= \sqrt{\frac{1.9329086 \cdot 10^{-23} \cdot T}{m}} \approx \sqrt{\frac{11.640 T}{M}}, &&\gamma = \frac{7}{5}\\ &\approx \sqrt{\frac{2.301 \cdot 10^{-23} \cdot T}{m}} \approx \sqrt{\frac{13.857 T}{M}}, &&\gamma = \frac{5}{3}\\ &\approx \sqrt{\frac{1.840 \cdot 10^{-23} \cdot T}{m}} \approx \sqrt{\frac{11.086 T}{M}}, &&\gamma = \frac{4}{3}\\ \end{cases}\\ c_{\mathrm{dry\ air}} &\approx 20.04687087513010149970678963\sqrt{T} \end{align} where •
cideal is the speed of sound in an
ideal gas; •
p is the
pressure; •
ρ is the
density; •
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 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 , for air. Gamma is exactly for monatomic gases (such as
argon) and it is for triatomic molecule gases that, like , are not co-linear (a co-linear triatomic gas such as is equivalent to a diatomic gas for our purposes here); •
γ is, itself, temperature-dependent, with a greater value at lower temperatures and a lower value at higher temperatures. For dry air, for example, it is about 1.404 at 258.15 K, 1.400 at 293.15 K, and 1.398 at 473.15 K. •
R is the
molar gas constant, ; •
k is the
Boltzmann constant, ; •
T is the absolute temperature; •
M is the molar mass of the gas. The mean molar mass for dry air is about •
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 certain other conditions are fulfilled as noted below. Calculated values for
cair have been found to vary slightly from experimentally determined values.
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 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 R_* = R/M_{\mathrm{air}}. (in green) against the truncated
Taylor expansion (in red) In addition, we switch to the
Celsius temperature , which is useful to calculate air speed in the region near . Then, for dry air, \begin{align} c_{\text{air}} &= \sqrt{\gamma \cdot R_* \cdot T} = \sqrt{\gamma \cdot R_* \cdot (\theta + 273.15)}\\ &= \sqrt{\gamma \cdot R_* \cdot 273.15\,\mathrm{^\circ K}} \cdot \sqrt{1 + \frac{\theta}{273.15}}\\ R &= 8.314\,462\,618\,153\,24~\frac{\text{J}}{\mathrm{mol \cdot {^\circ K}}}\\ M_{\text{air}} &= 0.028\,964\,7~\frac{\text{kg}}{\text{mol}}\\ R_* &\approx 287.055\,022\,773\,853\,725\,396\,776\,076\\ \gamma &= 1.4000~\text{(Ideal diatomic gas)} \\ c_{\text{air}} &\approx 331.32\frac{\text{m}}{\text{s}} \times \sqrt{1 + \frac{\theta}{273.15}}\\ \end{align} Finally, the
binomial approximation (assuming is very close to 0) of the remaining square root yields \begin{align} c_{\mathrm{air}} & \approx 331.32\frac{\text{m}}{\text{s}} \times \left(1 + \frac{\theta}{546.3}\right)\\ & \approx 331.32\frac{\text{m}}{\text{s}} + 0.606\theta\frac{\mathrm{m}}{\text{s}} \end{align} At zero degrees Celsius, the
binomial approximation introduces no inaccuracy, but does - perfectly dry
air at that temperature actually has a
heat capacity ratio of about 1.403. Corrected: \begin{align} \gamma &= 1.403~(\text{dry air at }\theta=0)\\ c_{\mathrm{air}} & \approx 331.67\frac{\text{m}}{\text{s}} \times \left(1 + \frac{\theta}{546.3}\right)\\ & \approx 331.67\frac{\text{m}}{\text{s}} + 0.607\theta\frac{\mathrm{m}}{\text{s}} \end{align} A graph comparing results of the two equations is to the right, using the slightly more accurate value of for the speed of sound at .
Effects due to wind shear The speed of sound varies with temperature. Since temperature and sound velocity normally decrease with increasing altitude, sound is
refracted upward, away from listeners on the ground, creating an
acoustic shadow at some distance from the source. Higher values of
wind gradient will refract sound downward toward the surface in the downwind direction, eliminating the acoustic shadow on the downwind side. This will increase the audibility of sounds downwind. This downwind refraction effect occurs because there is a wind gradient; the fact that sound is carried along by the wind is not important. For sound propagation, the exponential variation of wind speed with height can be defined as follows: \begin{align} U(h) &= U(0) h^\zeta, \\ \frac{\mathrm{d}U}{\mathrm{d}h}(h) &= \zeta \frac{U(h)}{h}, \end{align} where •
U(
h) is the speed of the wind at height
h; •
ζ is the exponential coefficient based on ground surface roughness, typically between 0.08 and 0.52; •
dU/
dh(
h) is the rate of change of the wind speed with respect to the height h, which is proportional to the expected wind gradient at fixed height
h. In the 1862
American Civil War Battle of Iuka, an acoustic shadow, believed to have been enhanced by a northeast wind, kept two divisions of Union soldiers out of the battle, because they could not hear the sounds of battle only (six miles) downwind.
Tables In the
standard atmosphere: •
T0 is (= = ), giving a theoretical value of (= = = = ). Values ranging from 331.3 to may be found in reference literature, however; •
T20 is (= = ), giving a value of (= = = = ); •
T25 is (= = ), giving a value of (= = = = ). In fact, assuming an
ideal gas, the speed of sound
c depends on temperature and composition only,
not on the pressure or
density (since these change in lockstep for a given temperature and cancel out). Air is almost an ideal gas. The temperature of the air varies with altitude, giving the following variations in the speed of sound using the standard atmosphere—
actual conditions may vary. Given normal atmospheric conditions, the temperature, and thus speed of sound, varies with altitude: ==Effect of frequency and gas composition==