# Acoustics/Sound Speed

![](Acoustics_Sound_Speed.jpg "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. Sodium\'s
Speed of Sound is listed under *Other
Properties*). In conventional use and 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. 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

A more accurate expression for the speed of sound is

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

-   *R* is the gas constant (287.05 J/(kg·K) for air). It is derived by
    dividing the universal gas constant $R$ (J/(mol·K)) by the molar
    mass of air (kg/mol), as is common practice in aerodynamics.
-   *κ* (kappa) is the adiabatic index (1.402 for air), sometimes noted
    as *γ* (gamma).
-   *T* is the absolute temperature in kelvins.

In the standard atmosphere:

*T*~0~ 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).\
*T*~20~ 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).\
*T*~25~ 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!

+----------+----------+---------+----------+---------+----------+
| **Al     | **Tempe  | **m/s** | **km/h** | **mph** | *        |
| titude** | rature** |         |          |         | *knots** |
+----------+----------+---------+----------+---------+----------+
| Sea      | 15 °C    | 340     | 1225     | 761     | 661      |
| level    | (59 °F)  |         |          |         |          |
| (?)      |          |         |          |         |          |
+----------+----------+---------+----------+---------+----------+
| 11,000   | -57 °C   | 295     | 1062     | 660     | 573      |
| m        | (-70 °F) |         |          |         |          |
| --20,000 |          |         |          |         |          |
| m\       |          |         |          |         |          |
| (        |          |         |          |         |          |
| Cruising |          |         |          |         |          |
| altitude |          |         |          |         |          |
| of       |          |         |          |         |          |
| co       |          |         |          |         |          |
| mmercial |          |         |          |         |          |
| jets,\   |          |         |          |         |          |
| and      |          |         |          |         |          |
| first    |          |         |          |         |          |
| su       |          |         |          |         |          |
| personic |          |         |          |         |          |
| flight)  |          |         |          |         |          |
+----------+----------+---------+----------+---------+----------+
| 29,000 m | -48 °C   | 301     | 1083     | 673     | 585      |
| (Flight  | (-53 °F) |         |          |         |          |
| of       |          |         |          |         |          |
| X-43A)   |          |         |          |         |          |
+----------+----------+---------+----------+---------+----------+
|          |          |         |          |         |          |
+----------+----------+---------+----------+---------+----------+

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 (\~28 kHz).\
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 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 are 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 relativistic 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

  **Impact of temperature**
  ---------------------------
  θ in °C
  −10
  −5
  0
  +5
  +10
  +15
  +20
  +25
  +30

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

## Sound in solids

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

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

### Single-shot timing methods

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

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

-   Calculation: Speed of sound in air and the
    temperature
-   The speed of sound, the temperature, and \... **not** the air
    pressure
-   Properties Of The U.S. Standard Atmosphere
    1976

