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Speed of sound

The speed of sound varies depending on the medium through which the sound waves pass. It is usually quoted in describing properties of substances (e.g. see the article on sodium).

More commonly the term refers to the speed of sound in air. The speed varies depending on atmospheric conditions; the most important factor is the temperature. The humidity has very little effect on the speed of sound, while the static sound pressure (air pressure) has none. Sound travels slower with an increased altitude (elevation if you are on solid earth), primarily as a result of temperature and humidity changes. An approximate speed (in metres/second) can be calculated from: (The proposal to take the letter v for speed of sound instead of c for speed of light is not generally accepted.)

<math>

c_{\mathrm{air}} = (331{.}5 + 0{.}6 \cdot \vartheta) \ \mathrm{m/s} <math>

where <math>\vartheta<math> (theta) is the temperature in degrees Celsius.

A more accurate expression is

<math>

c = \sqrt {\kappa \cdot R\cdot T} <math>

where R (287.05 J/kgK for air) is the universal gas constant R divided by the molar mass of air, κ (kappa) is the adiabatic index (1.402 for air), sometimes called γ, and T is the absolute temperature in kelvin. In the standard atmosphere:
T₀ is 273.15 K (= 0°C = 32°F), giving a value of 331.5 m/s (= 1193.4 km/h = 741.541 mph = 643.95 knots).
T₂₀ is 293.15 K (= 20°C = 68°F), giving a value of 343.421 m/s (= 1139.319 km/h = 768.209 mph = 667.109 knots).
T₂₅ is 298.15 K (= 25°C = 77°F), giving a value of 346.338 m/s (= 1246.818 km/h = 774.732 mph = 672.774 knots).

In fact, assuming a perfect gas the speed of sound depends on temperature only, not on the pressure. Air is almost a perfect 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).

Altitude	Temperature	m/s	km/h	mph	knots
Sea level	15°C (59°F)	340	1225	761	661
11000m-20000m (Cruising altitude of commercial jets, and first supersonic flight)	-57°C (-70°F)	295	1062	660	573
29000m (Flight of X-43A)	-48°C (-53°F)	301	1083	673	585

In fluids, using the theory of compressible flow, the speed of sound can be calculated using:

<math>

c = \sqrt{{\kappa \cdot p}\over\rho} <math>

This is correct for adiabatic flow; Newton famously used isothermal calculations and omitted the κ from the numerator.

In solids the speed of sound is given by:

<math>

c = \sqrt{\frac{E}{\rho}} <math>

where E is Young's modulus and ρ (rho) is density. Thus in steel the speed of sound is approximately 5100 m/s.
For air, see density of air.

The speed of sound in water is of interest to those mapping the ocean floor. In saltwater, sound travels at about 1500 m/s and in freshwater 1435 m/s. These speeds vary due to pressure, depth, temperature, salinity and other factors.

For general equations of state, if classical mechanics is used, the speed of sound <math>c<math> is given by

<math>

c^2=\frac{\partial p}{\partial\rho}<math> where differentiation is taken with respect to adiabatic change. If relativistic effects are important, the speed of sound <math>S<math> is given by

<math>

S^2=c^2 \left. \frac{\partial p}{\partial e} \right|_{\rm adiabatic}.<math>

(note that <math>e=\rho (c^2+e^C)<math> is the relativisic internal energy density; see relativistic Euler equations). This formula differs from the classical case in that <math>\rho<math> has been replaced by <math>e/c^2<math>.

Table - Speed of sound in air c, density of air ρ and acoustic impedance Z vs. temperature °C

Impact of temperature
°C	c in m/s	ρ in kg/m³	Z in N·s/m³
- 10	325.4	1.341	436.5
- 5	328.5	1.316	432.4
0	331.5	1.293	428.3
+ 5	334.5	1.269	424.5
+ 10	337.5	1.247	420.7
+ 15	340.5	1.225	417.0
+ 20	343.4	1.204	413.5
+ 25	346.3	1.184	410.0
+ 30	349.2	1.164	406.6

Mach number is the ratio of the object's speed to the speed of sound in air (medium).