speed of sound

History

The Pythagorean Archytas taught that higher pitched sound travels faster, an opinion accepted by some subsequent philosophers, such as those of the Academy and the Peripatos, including possibly Aristotle.

Sir Isaac Newton's 1687 Principia includes a computation of the speed of sound in air as 979 feet per second (298 m/s). This is too low by about 15%. The discrepancy is due primarily to neglecting the (then unknown) effect of rapidly fluctuating temperature in a sound wave (in modern terms, sound wave compression and expansion of air is an adiabatic process, not an isothermal process). Newton then invented various fudge factors, such as the "crassitude of the solid particles of the air", until the number agreed with the experimental measurement. Lagrange and Euler both attempted and failed to explain the discrepancy. This discrepancy was finally correctly explained by Pierre-Simon Laplace. In Traité de mécanique céleste, he used the result from the Clément-Desormes experiment of 1819, which measured the heat capacity ratio of air to be 1.35. This produced a near agreement between theory and experiment for the speed of sound. The modern value of 1.40 was found some years later, leading to complete agreement.

During the 17th century there were several attempts to measure the speed of sound accurately. Marin Mersenne in 1630 found two values. When measuring the time (of a seconds pendulum) between seeing the flash of a gun and hearing its sound over a known distance, he found a value of 1,380 Parisian feet/second (448 m/s). When he measured the time between firing a gun and hearing its echo from a reflecting surface of a known distance, however, he found 970 Paris feet per second. This led to some to theorize that echoed sound is slower than unechoed sound. Most subsequent experimenters used only his first method.

Pierre Gassendi in 1635 found 1,473 Parisian feet/second, and Robert Boyle 1,125 Parisian feet/second. In 1650, G. A. Borelli and V. Viviani of the Accademia del Cimento found 350 m/s. In 1709, the Reverend William Derham, Rector of Upminster, published a more accurate measure of the speed of sound, at 1,072 Parisian feet per second. (The Parisian foot was 325 mm. This is longer than the standard "international foot" in common use today, which was officially defined in 1959 as 304.8 mm, making the speed of sound at 20 °C (68 °F) 1,055 Parisian feet per second). See for a table of more speeds of sound measured in the 1636 to 1791 period.

Compression and shear waves

In a gas or liquid, sound consists of compression waves. In solids, waves propagate as two different types. A longitudinal wave is associated with compression and decompression in the direction of travel, and is the same process in gases and liquids, with an analogous compression-type wave in solids. Only compression waves are propagated through fluids (gases and liquids). An additional type of wave, the transverse wave, also called a shear wave, occurs only in solids because only solids support elastic deformations. It is due to elastic deformation of the medium perpendicular to the direction of wave travel; the direction of shear deformation is called the "polarization" of this type of wave. In general, transverse waves occur as a pair of orthogonal polarizations.

These different waves (compression waves and the different polarizations of shear waves) may have different speeds at the same frequency. Therefore, they arrive at an observer at different times, an extreme example being an earthquake, where sharp compression waves arrive first and rocking transverse waves seconds later.

The speed of a compression wave in a fluid is determined by the medium's compressibility and density. In solids, the compression waves are analogous to those in fluids, depending on compressibility and density, but with the additional factor of shear modulus, which affects compression waves due to off-axis elastic energies that are able to influence effective tension and relaxation in a compression. The speed of shear waves, which can occur only in solids, is determined simply by the solid material's shear modulus and density.

Equations

The speed of sound in mathematical notation is conventionally represented by c, from the Latin celeritas meaning "swiftness".

For fluids in general, the speed of sound c is given by the Newton–Laplace equation:

c

=

K

s

ρ

,

{\displaystyle c={\sqrt {\frac {K_{s}}{\rho }}},}

where

K

s

{\displaystyle K_{s}}

is a coefficient of stiffness, the isentropic bulk modulus (or the modulus of bulk elasticity for gases);

ρ

{\displaystyle \rho }

is the density.

K

s

=

ρ

(

∂

P

∂

ρ

)

s

{\displaystyle K_{s}=\rho \left({\frac {\partial P}{\partial \rho }}\right)_{s}}

Dependence on the properties of the medium

The speed of sound is variable and depends on the properties of the substance through which the wave is travelling. In solids, the speed of transverse (or shear) waves depends on the shear deformation under shear stress (called the shear modulus), and the density of the medium. Longitudinal (or compression) waves in solids depend on the same two factors with the addition of a dependence on compressibility.

In fluids, only the medium's compressibility and density are the important factors, since fluids do not transmit shear stresses. In heterogeneous fluids, such as a liquid filled with gas bubbles, the density of the liquid and the compressibility of the gas affect the speed of sound in an additive manner, as demonstrated in the hot chocolate effect.

In gases, adiabatic compressibility is directly related to pressure through the heat capacity ratio (adiabatic index), while pressure and density are inversely related to the temperature and molecular weight, thus making only the completely independent properties of temperature and molecular structure important (heat capacity ratio may be determined by temperature and molecular structure, but simple molecular weight is not sufficient to determine it).

Sound propagates faster in low molecular weight gases such as helium than it does in heavier gases such as xenon. For monatomic gases, the speed of sound is about 75% of the mean speed that the atoms move in that gas.

For a given ideal gas the molecular composition is fixed, and thus the speed of sound depends only on its temperature. At a constant temperature, the gas pressure has no effect on the speed of sound, since the density will increase, and since pressure and density (also proportional to pressure) have equal but opposite effects on the speed of sound, and the two contributions cancel out exactly. In a similar way, compression waves in solids depend both on compressibility and density—just as in liquids—but in gases the density contributes to the compressibility in such a way that some part of each attribute factors out, leaving only a dependence on temperature, molecular weight, and heat capacity ratio, which can be independently derived from temperature and molecular composition (see derivations below). Thus, for a single given gas (assuming the molecular weight does not change) and over a small temperature range (for which the heat capacity is relatively constant), the speed of sound becomes dependent on only the temperature of the gas.

Altitude variation and implications for atmospheric acoustics

In the Earth's atmosphere, the chief factor affecting the speed of sound is the temperature. For a given ideal gas with constant heat capacity and composition, the speed of sound is dependent solely upon temperature; see § Details below. In such an ideal case, the effects of decreased density and decreased pressure of altitude cancel each other out, save for the residual effect of temperature.

Since temperature (and thus the speed of sound) decreases with increasing altitude up to 11 km, sound is refracted upward, away from listeners on the ground, creating an acoustic shadow at some distance from the source. The decrease of the speed of sound with height is referred to as a negative sound speed gradient.

There are variations in this trend above 11 km. In particular, in the stratosphere above about 20 km, the speed of sound increases with height, due to an increase in temperature from heating within the ozone layer. This produces a positive speed-of-sound gradient in this region. Still another region of positive gradient occurs at very high altitudes, in the thermosphere above 90 km.

Details

Speed of sound in ideal gases and air

For an ideal gas, K (the bulk modulus in equations above, equivalent to C, the coefficient of stiffness in solids) is given by

K

=

γ

⋅

p

.

{\displaystyle K=\gamma \cdot p.}

Thus, from the Newton–Laplace equation above, the speed of sound in an ideal gas is given by

c

=

γ

⋅

p

ρ

,

{\displaystyle c={\sqrt {\gamma \cdot {p \over \rho }}},}

where

γ is the adiabatic index also known as the isentropic expansion factor. It is the ratio of the specific heat of a gas at constant pressure to that of a gas at constant volume (

C

p

/

C

v

{\displaystyle 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;

Effects due to wind shear

The speed of sound varies with temperature. Since temperature and sound velocity normally decrease with increasing altitude, sound is refracted upward, away from listeners on the ground, creating an acoustic shadow at some distance from the source. Wind shear of 4 m/(s · km) can produce refraction equal to a typical temperature lapse rate of 7.5 °C/km. Higher values of wind gradient will refract sound downward toward the surface in the downwind direction, eliminating the acoustic shadow on the downwind side. This will increase the audibility of sounds downwind. This downwind refraction effect occurs because there is a wind gradient; the fact that sound is carried along by the wind is not important.

For sound propagation, the exponential variation of wind speed with height can be defined as follows:

U

(

h

)

=

U

(

0

)

h

ζ

,

d

U

d

h

(

h

)

=

ζ

U

(

h

)

Tables

In the standard atmosphere:

T0 is 273.15 K (= 0 °C = 32 °F), giving a theoretical value of 331.3 m/s (= 1086.9 ft/s = 1193 km/h = 741.1 mph = 644.0 kn). Values ranging from 331.3 to 331.6 m/s may be found in reference literature, however;

T20 is 293.15 K (= 20 °C = 68 °F), giving a value of 343.2 m/s (= 1126.0 ft/s = 1236 km/h = 767.8 mph = 667.2 kn);

T25 is 298.15 K (= 25 °C = 77 °F), giving a value of 346.1 m/s (= 1135.6 ft/s = 1246 km/h = 774.3 mph = 672.8 kn).

In fact, assuming an ideal gas, the speed of sound c depends on temperature and composition 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.

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

Effect of frequency and gas composition

General physical considerations

The medium in which a sound wave is travelling does not always respond adiabatically, and as a result, the speed of sound can vary with frequency.

The limitations of the concept of speed of sound due to extreme attenuation are also of concern. The attenuation that exists at sea level for high frequencies applies to successively lower frequencies as atmospheric pressure decreases, or as the mean free path increases. For this reason, the concept of speed of sound (except for frequencies approaching zero) progressively loses its range of applicability at high altitudes. The standard equations for the speed of sound apply with reasonable accuracy only to situations in which the wavelength of the sound wave is considerably longer than the mean free path of molecules in a gas.

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 speed of sound (over 9% higher) because they have a higher γ (5/3 = 1.66...) than diatomics do (7/5 = 1.4). Thus, at the same molecular mass, the speed of sound of a monatomic gas goes up by a factor of

c

g

a

s

,

m

o

n

a

t

o

m

i

c

c

g

a

Practical application to air

By far, the most important factor influencing the speed of sound in air is temperature. The speed is proportional to the square root of the absolute temperature, giving an increase of about 0.6 m/s per degree Celsius. For this reason, the pitch of a musical wind instrument increases as its temperature increases.

The speed of sound is raised by humidity. The difference between 0% and 100% humidity is about 1.5 m/s at standard pressure and temperature, but the size of the humidity effect increases dramatically with temperature.

The dependence on frequency and pressure are normally insignificant in practical applications. In dry air, the speed of sound increases by about 0.1 m/s as the frequency rises from 10 Hz to 100 Hz. For audible frequencies above 100 Hz it is relatively constant. Standard values of the speed of sound are quoted in the limit of low frequencies, where the wavelength is large compared to the mean free path.

As shown above, the approximate value 1000/3 = 333.33... m/s is exact a little below 5 °C and is a good approximation for all "usual" outside temperatures (in temperate climates, at least), hence the usual rule of thumb to determine how far lightning has struck: count the seconds from the start of the lightning flash to the start of the corresponding roll of thunder and divide by 3: the result is the distance in kilometers to the nearest point of the lightning bolt. Or divide the number of seconds by 5 for an approximate distance in miles.

Mach number

Mach number, a useful quantity in aerodynamics, is the ratio of air speed to the local speed of sound. At altitude, for reasons explained, Mach number is a function of temperature.

Aircraft flight instruments, however, operate using pressure differential to compute Mach number, not temperature. The assumption is that a particular pressure represents a particular altitude and, therefore, a standard temperature. Aircraft flight instruments need to operate this way because the stagnation pressure sensed by a Pitot tube is dependent on altitude as well as speed.

Experimental methods

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

The earliest reasonably accurate estimate of the speed of sound in air was made by William Derham and acknowledged by Isaac Newton. Derham had a telescope at the top of the tower of the Church of St Laurence in Upminster, England. On a calm day, a synchronized pocket watch would be given to an assistant who would fire a shotgun at a pre-determined time from a conspicuous point some miles away, across the countryside. This could be confirmed by telescope. He then measured the interval between seeing gunsmoke and arrival of the sound using a half-second pendulum. The distance from where the gun was fired was found by triangulation, and simple division (distance/time) provided velocity. Lastly, by making many observations, using a range of different distances, the inaccuracy of the half-second pendulum could be averaged out, giving his final estimate of the speed of sound. Modern stopwatches enable this method to be used today over distances as short as 200–400 metres, and not needing something as loud as a shotgun.

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:

The distance between the microphones (x), called microphone basis.

The time of arrival between the signals (delay) reaching the different microphones (t).

Then v = x/t.

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 that 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 that is dipped 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/4 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.

High-precision measurements in air

The effect of impurities can be significant when making high-precision measurements. Chemical desiccants can be used to dry the air, but will, in turn, contaminate the sample. The air can be dried cryogenically, but this has the effect of removing the carbon dioxide as well; therefore many high-precision measurements are performed with air free of carbon dioxide rather than with natural air. A 2002 review found that a 1963 measurement by Smith and Harlow using a cylindrical resonator gave "the most probable value of the standard speed of sound to date". The experiment was done with air from which the carbon dioxide had been removed, but the result was then corrected for this effect so as to be applicable to real air. The experiments were done at 30 °C but corrected for temperature in order to report them at 0 °C. The result was 331.45 ± 0.01 m/s for dry air at STP, for frequencies from 93 Hz to 1,500 Hz.

Non-gaseous media

Speed of sound in solids

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

on the deformation mode. Sound waves generating volumetric deformations (compression) and shear deformations (shearing) are called pressure waves (longitudinal waves) and shear waves (transverse waves), respectively. In earthquakes, the corresponding seismic waves are called P-waves (primary waves) and S-waves (secondary waves), respectively. The sound velocities of these two types of waves propagating in a homogeneous 3-dimensional solid are respectively given by

c

solid,p

=

K

+

4

3

G

ρ

=

E

(

1

−

ν

)

ρ

(

1

+

ν

)

(

1

−

2

ν

Speed of sound in liquids

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

f

l

u

i

d

=

K

ρ

,

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

where K is the bulk modulus of the fluid.

In fresh water, sound travels at about 1481 m/s at 20 °C (see the External Links section below for online calculators). Applications of underwater sound can be found in sonar, acoustic communication and acoustical oceanography.

In salt water that is free of air bubbles and suspended sediment, sound travels at about 1500 m/s (1500.235 m/s at 1000 kilopascals, 10 °C and 3% salinity by one method). The speed of sound in seawater depends on pressure (hence depth), temperature (a change of 1 °C ~ 4 m/s), and salinity (a change of 1‰ ~ 1 m/s), and empirical equations have been derived to accurately calculate the speed of sound from these variables. Other factors affecting the speed of sound are minor. Since in most ocean regions temperature decreases with depth, the profile of the speed of sound with depth decreases to a minimum at a depth of several hundred metres. Below the minimum, sound speed increases again, as the effect of increasing pressure overcomes the effect of decreasing temperature (right). For more information see Dushaw et al.

An empirical equation for the speed of sound in sea water is provided by Mackenzie:

Speed of sound in plasma

The speed of sound in a plasma for the common case that the electrons are hotter than the ions (but not too much hotter) is given by the formula (see here)

c

s

=

(

γ

Z

k

T

e

m

i

)

1

/

2

=

(

γ

Z

T

e

μ

)

1

/

2

×

90.85

m

/

s

,

{\displaystyle c_{s}=\left({\frac {\gamma ZkT_{\mathrm {e} }}{m_{\mathrm {i} }}}\right)^{1/2}=\left({\frac {\gamma ZT_{e}}{\mu }}\right)^{1/2}\times 90.85~\mathrm {m/s} ,}

Mars

The speed of sound on Mars varies as a function of frequency. Higher frequencies travel faster than lower frequencies. Higher frequency sound from lasers travels at 250 m/s (820 ft/s), while low frequency sound travels at 240 m/s (790 ft/s).

Gradients

When sound spreads out evenly in all directions in three dimensions, the intensity drops in proportion to the inverse square of the distance. In the ocean, however, there is a layer called the 'deep sound channel' or SOFAR channel that can confine sound waves at a particular depth.

In the SOFAR channel, the speed of sound is lower than that in the layers above and below. Just as light waves will refract towards a region of higher refractive index, sound waves will refract towards a region where their speed is reduced. The result is that sound gets confined in the layer, much the way light can be confined to a sheet of glass or optical fiber. Thus, the sound is confined in essentially two dimensions. In two dimensions the intensity drops in proportion to only the inverse of the distance. This allows waves to travel much further before being undetectably faint.

A similar effect occurs in the atmosphere. Project Mogul successfully used this effect to detect a nuclear explosion at a considerable distance.