Acoustics/Sound Speed

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Acoustics Sound Speed.jpg

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). In conventional use and in scientific literature sound velocity v is the same as sound speed c. Sound velocity c or velocity of sound should not be confused with sound particle velocity v, which is the velocity of the individual particles.

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:


c_{\mathrm{air}} = (331{.}5 + (0{.}6 \cdot \theta)) \ \mathrm{m/s}\,

where \theta\, (theta) is the temperature in degrees Celsius.

Details[edit]

A more accurate expression for the speed of sound is


c = \sqrt {\kappa \cdot R\cdot T}

where

  • R (287.05 J/(kg·K) for air) is the gas constant for air: the universal gas constant R, which units of J/(mol·K), 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. Any qualification of the speed of sound being "at sea level" is also irrelevant. Speed of sound varies with altitude (height) only because of the changing temperature!

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. For audio sound range air is a non-dispersive medium. We should also note that air contains CO2 which is a dispersive medium and it introduces dispersion to air at ultrasound frequencies (28kHz).
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


c = \sqrt{\frac{C}{\rho}}

where

C is a coefficient of stiffness
\rho 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


c = \sqrt {\frac{K}{\rho}}

where

K is the adiabatic bulk modulus

For a gas, K is approximately given by


K=\kappa \cdot p

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:


c = \sqrt {{\kappa \cdot p}\over\rho}

which using the ideal gas law is identical to:


c = \sqrt {\kappa \cdot R\cdot T}

(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 κ but was otherwise correct.)

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:


c = \sqrt{\frac{E}{\rho}}

where

E is Young's modulus
\rho (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 by 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:


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

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 c is given by


c^2=\frac{\partial p}{\partial\rho}

where differentiation is taken with respect to adiabatic change.

If relativistic effects are important, the speed of sound S is given by:


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

(Note that  e= \rho (c^2+e^C) \, is the relativisic internal energy density).

This formula differs from the classical case in that \rho has been replaced by e/c^2 \,.

Speed of sound in air[edit]

Impact of temperature
θ in °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).

Sound in solids[edit]

In solids, the velocity of sound depends on density of the material, not its temperature. Solid materials, such as steel, conduct sound much faster than air.

Experimental methods[edit]

In air a range of different methods exist for the measurement of sound.

Single-shot timing methods[edit]

The simplest concept is the measurement made using two microphones and a fast recording device such as a digital storage scope. This method uses the following idea.

If a sound source and two microphones are arranged in a straight line, with the sound source at one end, then the following can be measured:

  1. The distance between the microphones (x)
  2. The time delay between the signal reaching the different microphones (t)

Then v = x/t

An older method is to create a sound at one end of a field with an object that can be seen to move when it creates the sound. When the observer sees the sound-creating device act they start a stopwatch and when the observer hears the sound they stop their stopwatch. Again using v = x/t you can calculate the speed of sound. A separation of at least 200 m between the two experimental parties is required for good results with this method.

Other methods[edit]

In these methods the time measurement has been replaced by a measurement of the inverse of time (frequency).

Kundt's tube is an example of an experiment which can be used to measure the speed of sound in a small volume, it has the advantage of being able to measure the speed of sound in any gas. This method uses a powder to make the nodes and antinodes visible to the human eye. This is an example of a compact experimental setup.

A tuning fork can be held near the mouth of a long pipe which is dipping into a barrel of water, in this system it is the case that the pipe can be brought to resonance if the length of the air column in the pipe is equal to ( {1+2n}/λ ) where n is an integer. As the antinodal point for the pipe at the open end is slightly outside the mouth of the pipe it is best to find two or more points of resonance and then measure half a wavelength between these.

Here it is the case that v = fλ

External links[edit]

Fundamentals of Psychoacoustics · Filter Design and Implementation