Speed of sound

The speed of sound is a term used to describe the speed of sound waves passing through an elastic medium. The speed varies with the medium employed (for example, sound waves move faster through water than through air), as well as with the properties of the medium, especially temperature. It is sometimes used in describing the nature of substances (see the article on sodium). In conventional use and in scientific literature sound velocity, v, and sound speed, c, are used synonymously and should not be confused with sound particle velocity (also symbolized as v), which is the velocity of the individual particles.

The term is commonly used to refer specifically to the speed of sound in air. The speed varies depending on atmospheric conditions; the most important factor is the temperature. Humidity has little effect on the speed of sound, nor does air pressure per se. (Pressure has no effect at all in an ideal gas approximation. This is because pressure and density both contribute to sound velocity equally, and in an ideal gas the two effects cancel out, leaving only the effect of temperature.) Sound usually travels more slowly with greater altitude, due to reduced temperature. An approximate speed of sound in 0% humidity (dry) air, in meters per second, at temperatures near 0 °C, can be calculated from:

c_{\mathrm{air}} = (331{.}5 + (0{.}6 \cdot \vartheta)) \ \mathrm{ms^{-1}}\,

where $\vartheta\,$ (theta) is the temperature in degrees Celsius(°C), not Kelvins.

This equation is derived from the first two terms of the Taylor expansion of the equation:

c_{\mathrm{air}} = 331.5 \sqrt{1+\frac{\vartheta}{273.15}}

This equation is correct to a wider temperature range, but still depends on the approximation of heat capacity being independent of temperature, and will fail particularly at higher temperatures. A derivation of these equations will be given in a later section.

Basic concept

The transmission of sound can be explained using a toy model consisting of an array of balls interconnected by springs. (For a real material the balls represent atoms or molecules and the springs represent the bonds between them.) Sound passes through the model by compressing and expanding the springs, transmitting energy to neighboring balls, which transmit energy to their springs, and so on. The speed of sound through the model depends on the stiffness of the springs (stiffer springs transmit energy more quickly) and the mass and spacing of the balls (more massive balls move more slowly, as do a larger number of balls in the same space). Effects like dispersion and reflection can also be understood using this model.

In a real material, the stiffness of the springs is a referred to as a modulus, and the mass and spacing of the balls corresponds to the density. All other things being equal, sound will travel more slowly in denser materials, and faster in stiffer 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, because the internal bonds in a solid like aluminium are much stronger than the gaseous bonds between hydrogen molecules. In general, solids will have a higher speed of sound than liquids, and liquids will have a higher speed of sound than gases.

Some textbooks mistakenly state that the speed of sound increases with increasing density. This is usually illustrated by presenting data for three materials, such as air, water and steel. With only these three examples it indeed appears that speed is correlated to density, yet including only a few more examples would show this assumption to be incorrect.

Details

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. 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 may be calculated from the relativistic Euler equations.

In a non-dispersive medium sound speed is independent of sound frequency, so the speeds of energy transport and sound propagation are the same. For audible sounds air is a non-dispersive medium. But air does contain a small amount of CO₂ which is a dispersive medium, and it introduces dispersion to air at ultrasonic frequencies (> 28 kHz).

In a dispersive medium sound speed is a function of sound 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. A suspension of small particles in a fluid is an example of a dispersive medium.

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:

c_{\mathrm{solids}} = \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^-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:

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

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

c_{\mathrm{fluid}} = \sqrt {\frac{K}{\rho}}

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.

Speed in ideal gases and in air

For a gas, K is approximately given by

K = \kappa \cdot p

where

κ is the adiabatic index also known as the isentropic expansion factor and sometimes called γ (Greek letter gamma). It is the ratio of constant-pressure to constant-volume heat capacities of the gas (

C_p/C_v

), and arises because a classical sound wave induces an adiabatic compression, in which the heat of the compression does not have enough time to escape the pressure pulse, and thus contributes to the pressure induced by the compression.

p is the pressure.

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

c_{\mathrm{gas}} = \sqrt{\kappa \cdot {p \over \rho}} = \sqrt{\kappa \cdot R \cdot T}

where

R (287.05 J·kg^-1·K^-1 for air) is the gas constant for air: the universal gas constant $R$ , with 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 331.6 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 or density (since these change in lockstep for a given temperature and cancel out). 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.

Effect of temperature
$\vartheta$ 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

$\vartheta$ 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 (Z=ρ·c)

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

Effect of frequency and gas composition

With increasing frequency the sound wave compression approaches a perfect adiabatic because there is less and less time for heat to escape in the compression process. For this reason, sound waves in air, particularly ultrasound, approach the theoretical relation given above very closely, as frequency rises.

The molecular composition of the gas contributes both as the mass (M) of the molecules, and their heat capacities, and so both have an influence on speed of sound. In general, at the same molecular mass, monatomic gases have slightly higher sound speeds (over 9% higher) due to the fact that they have a higher gamma (5/3 = 1.6) than diatomics do (7/5 = 1.4). Thus, at the same molecular mass, the sound speed of a monatomic gas goes up by a factor of

$c_{\mathrm{gas}} = \sqrt}$ = 1.09

This gives the 9% difference, and would be a typical ratio for sound speeds at room temperature in helium vs. deuterium, each with a molecular weight of 4. Sound travels faster in helium than deuterium because adiabatic compression heats helium more, since the helium molecules can store heat energy from compression only in translation, but not rotation. Thus helium molecules (monatomic molecules) travel faster in a soundwave and transmit sound faster. (Sound generally travels at about 70 % of the mean molecular velocity in gases).

Note that in this example we have assumed that temperature is low enough that heat capacities are not influenced by molecular vibration (see heat capacity). However, vibrational modes simply cause gammas which decrease toward 1, since vibration modes in a polyatomic gas gives the gas additional ways to store heat which do not affect temperature, and thus do not affect molecular velocity and sound velocity. Thus, the effect of higher temperatures and vibrational heat capacity acts to increase the difference between sound speed in monatomic vs. polyatomic molecules, with the speed remaining greater in monatomics.

Mach number

Mach number, a useful quantity in aerodynamics, is the ratio of an object's speed to the speed of sound in the medium through which it is passing (again, usually air). At altitude, for reasons explained, Mach number is a function of temperature.

Experimental methods

A range of different methods exist for the measurement of sound in air.

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), called microphone basis. 2. The time of arrival between the signals (delay) 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 = f · λ

External links

Chemical properties | Fluid dynamics | Acoustics | Units of velocity

Gourt Home::Gourt :: Speed of sound

Basic concept

Details

Speed in solids

Speed in a fluid

Speed in ideal gases and in air

Effect of frequency and gas composition

Mach number

Experimental methods

Single-shot timing methods

Other methods

External links