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Speed of sound

From Academic Kids

"Speed of Sound" is also a single by Coldplay.

The speed of sound c (from Latin celeritas, "velocity") 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 per second) can be calculated from:

<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 (In this case, the gas constant R, which normally has units of J/(molK), is divided by the molar mass of air, as is common practice in aerodynamics)
κ (kappa) is the adiabatic index (1.402 for air), sometimes noted γ
T is the absolute temperature in kelvins.

In the standard atmosphere:

T0 is 273.15 K (= 0 C = 32 F), giving a value of 331.5 m/s (= 1087.6 ft/s = 1193 km/h = 741.5 mph = 643.9 knots).
T20 is 293.15 K (= 20 C = 68 F), giving a value of 343.4 m/s (= 1126.6 ft/s = 1236 km/h = 768.2 mph = 667.1 knots).
T25 is 298.15 K (= 25 C = 77 F), giving a value of 346.3 m/s (= 1136.2 ft/s = 1246 km/h = 774.7 mph = 672.7 knots).

In fact, assuming an ideal gas, the speed of sound c depends on temperature only, not on the pressure. 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.

Altitude Temperature m/s km/h mph knots
Sea level 15 C (59 F) 340 1225 761 661
11,000 m–20,000 m
(Cruising altitude of commercial jets,
and first supersonic flight)
-57 C (-70 F) 295 1062 660 573
29,000 m (Flight of X-43A) -48 C (-53 F) 301 1083 673 585

In a Non-Dispersive Medium – Sound speed is independent of frequency, therefore the speed of energy transport and sound propagation are the same. Air is a non-dispersive medium.
In a Dispersive Medium – Sound speed is a function of frequency. The spatial and temporal distribution of a propagating disturbance will continually change. Each frequency component propagates at each its own phase speed, while the energy of the disturbance propagates at the group velocity. Water is an example of a dispersive medium.

In general, the speed of sound c is given by

<math>

c = \sqrt{\frac{C}{\rho}} <math> where

C is a coefficient of stiffness
<math>\rho<math> is the density

Thus the speed of sound increases with the stiffness of the material, and decreases with the density.

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>

c = \sqrt {\frac{K}{\rho}} <math> where

K is the adiabatic bulk modulus

For a gas, K is approximately given by

<math>

K=\kappa \cdot p <math>

where

κ is the adiabatic index, sometimes called γ.
p is the pressure.

Thus, for a gas the speed of sound can be calculated using:

<math>

c = \sqrt {{\kappa \cdot p}\over\rho} <math> which using the ideal gas law is identical to:

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

(Newton famously used isothermal calculations and omitted the κ from the numerator.)

In a solid, there is a non-zero stiffness both for volumetric and shear deformations. Hence, in a solid it is possible to generate sound waves with different velocities dependent on the deformation mode.

In a solid rod (with thickness much smaller than the wavelength) the speed of sound is given by:

<math>

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

where

E is Young's modulus
<math>\rho<math> (rho) is density

Thus in steel the speed of sound is approximately 5100 m/s.

In a solid with lateral dimensions much larger than the wavelength, the sound velocity is higher. It is found be replacing Young's modulus with the plane wave modulus, which can be expressed in terms of the Young's modulus and Poisson's ratio as:

<math>

M = E \frac{1-\nu}{1-\nu-2\nu^2} <math>

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
<math>\vartheta<math> in C c in m/s ρ in kg/m Z in N·s/m
 -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).

Template:Sound measurements

External links

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