Speed of sound

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This page is about the physical constant that describes the speed of sound waves in a medium. For the Coldplay single off their album, X&Y, see Speed of Sound (single).

Template:Sound measurements 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 air pressure has an effect (see equations later in the page). 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 meters per second) can be calculated from:

<math>

c_{\mathrm{air}} = (331{.}5 + (0{.}6 \cdot \theta)) \ \mathrm{ms^{-1}}\, </math>

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

Contents

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Basic concept

One can understand the transmission of sound using a simple toy model of materials, one that consists of a number of atoms or molecules represented by balls, and the bonds between them as springs. Sound compresses and expands the springs, transmitting that energy to the balls around it. Effects like dispersion and reflection can be understood simply under this model.

The speed of sound in this model is effected primarily by two factors, the number of balls that need to be moved, and strength of the springs. If there are more balls to move the sound will travel more slowly. Stronger springs, on the other hand, will speed the transmission up.

In a real material, the former measure is referred to as density, and the later a modulus. All other things being equal, sound will travel slower in denser materials, and faster in "springier" ones. For instance, sound will travel faster in aluminium than uranium, and faster in hydrogen than nitrogen, due to the lower density of the first material of each set. At the same time, sound will travel faster in aluminium than hydrogen, as the internal bonds in aluminium are much stronger. Generally solids will have a higher speed of sound than liquids or gases.

Note that many textbooks claim that the speed of sound increases with increasing density. They typically present only three data points to refer to, steel, water and air. Given only these three examples it indeed appears that speed is related to density, yet including only a few more demonstrates this to be incorrect. This is an extremely misleading claim, yet is commonly taught in grade school texts.

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Details

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. 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>.

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 each its own phase speed, while the energy of the disturbance propagates at the group velocity. Water is an example of a dispersive medium.

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Speed in solids

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_{\mathrm{solids}} = \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-1.

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:

<math>

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

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Speed in a fluid

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_{\mathrm{fluid}} = \sqrt {\frac{K}{\rho}} </math>

where

K is the adiabatic bulk modulus

The speed of sound in water is of interest to those mapping the ocean floor. In saltwater, sound travels at about 1500 m·s-1 and in freshwater 1435 m·s-1. These speeds vary due to pressure, depth, temperature, salinity and other factors.

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Speed in ideal gases and in air

For a gas, K is approximately given by

<math>

K = \kappa \cdot p </math>

where

κ is the adiabatic index also known as the isentropic expansion factor and sometimes called γ.
p is the pressure.

Using the ideal gas law the speed of sound is identical to:

<math>

c_{\mathrm{gas}} = \sqrt{\kappa \cdot {p \over \rho}} = \sqrt{\kappa \cdot \frac{R \cdot T}{M}} </math>

where

  • R (287.05 J·kg-1·K-1 for air) is the gas constant for air: the universal gas constant <math>R</math>, which units of J·mol-1·K-1, 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.

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 the standard atmosphere:

T0 is 273.15 K (= 0 °C = 32 °F), giving a value of 331.5 m·s-1 (= 1087.6 ft/s = 1193 km·h-1 = 741.5 mph = 643.9 knots).
T20 is 293.15 K (= 20 °C = 68 °F), giving a value of 343.4 m·s-1 (= 1126.6 ft/s = 1236 km·h-1 = 768.2 mph = 667.1 knots).
T25 is 298.15 K (= 25 °C = 77 °F), giving a value of 346.3 m·s-1 (= 1136.2 ft/s = 1246 km·h-1 = 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.

Impact of temperature
θ in °C c in m·s-1 ρ in kg·m-3 Z in N·s·m-3
−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
θ is the temperature in °C
c is the speed of sound in m·s-1
ρ is the density in kg·m-3
Z is the acoustic impedance in N·s·m-3

Given normal atmospheric conditions, the temperature, and thus speed of sound, varies with altitude:

Altitude Temperature m·s-1 km·h-1 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

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

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Experimental methods

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

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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 obsever 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.

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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

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External links

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