Speed of sound
The speed of sound is the distance travelled per unit time by a sound wave as it propagates through an elastic medium. At, the speed of sound in air is about, or a kilometre in or a mile in. It depends strongly on temperature as well as the medium through which a sound wave is propagating.
The speed of sound in an ideal gas depends only on its temperature and composition. The speed has a weak dependence on frequency and pressure in ordinary air, deviating slightly from ideal behavior.
In common everyday speech, speed of sound refers to the speed of sound waves in air. However, the speed of sound varies from substance to substance: typically sound travels most slowly in gases; it travels faster in liquids; and faster still in solids. For example,, sound travels at in air; it travels at in water ; and at in iron. In an exceptionally stiff material such as diamond, sound travels at about 35 times as fast as in airwhich is around the maximum speed that sound will travel under normal conditions.
Sound waves in solids are composed of compression waves, and a different type of sound wave called a shear wave, which occurs only in solids. Shear waves in solids usually travel at different speeds, as exhibited in seismology. The speed of compression waves in solids is determined by the medium's compressibility, shear modulus and density. The speed of shear waves is determined only by the solid material's shear modulus and density.
In fluid dynamics, the speed of sound in a fluid medium is used as a relative measure for the speed of an object moving through the medium. The ratio of the speed of an object to the speed of sound in the fluid is called the object's Mach number. Objects moving at speeds greater than are said to be traveling at supersonic speeds.
History
's 1687 Principia includes a computation of the speed of sound in air as. This is too low by about 15%. The discrepancy is due primarily to neglecting the effect of rapidly-fluctuating temperature in a sound wave. This error was later rectified by Laplace.During the 17th century there were several attempts to measure the speed of sound accurately, including attempts by Marin Mersenne in 1630, Pierre Gassendi in 1635 and Robert Boyle. 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..
Derham used a telescope from the tower of the church of St Laurence, Upminster to observe the flash of a distant shotgun being fired, and then measured the time until he heard the gunshot with a half-second pendulum. Measurements were made of gunshots from a number of local landmarks, including North Ockendon church. The distance was known by triangulation, and thus the speed that the sound had travelled was calculated.
Basic concepts
The transmission of sound can be illustrated by using a model consisting of an array of spherical objects interconnected by springs.In real material terms, the spheres represent the material's molecules and the springs represent the bonds between them. Sound passes through the system by compressing and expanding the springs, transmitting the acoustic energy to neighboring spheres. This helps transmit the energy in-turn to the neighboring sphere's springs, and so on.
The speed of sound through the model depends on the stiffness/rigidity of the springs, and the mass of the spheres. As long as the spacing of the spheres remains constant, stiffer springs/bonds transmit energy quicker, while larger spheres transmit the energy slower.
In a real material, the stiffness of the springs is known as the "elastic modulus", and the mass corresponds to the material density. Given that all other things being equal, sound will travel slower in spongy materials, and faster in stiffer ones. Effects like dispersion and reflection can also be understood using this model.
For instance, sound will travel 1.59 times faster in nickel than in bronze, due to the greater stiffness of nickel at about the same density. Similarly, sound travels about 1.41 times faster in light hydrogen gas than in heavy hydrogen gas, since deuterium has similar properties but twice the density. At the same time, "compression-type" sound will travel faster in solids than in liquids, and faster in liquids than in gases, because the solids are more difficult to compress than liquids, while liquids in turn are more difficult to compress than gases.
Some textbooks mistakenly state that the speed of sound increases with density. This notion is illustrated by presenting data for three materials, such as air, water and steel, they each have vastly different compressibility, which more than makes up for the density differences. An illustrative example of the two effects is that sound travels only 4.3 times faster in water than air, despite enormous differences in compressibility of the two media. The reason is that the larger density of water, which works to slow sound in water relative to air, nearly makes up for the compressibility differences in the two media.
A practical example can be observed in Edinburgh when the "One o' Clock Gun" is fired at the eastern end of Edinburgh Castle. Standing at the base of the western end of the Castle Rock, the sound of the Gun can be heard through the rock, slightly before it arrives by the air route, partly delayed by the slightly longer route. It is particularly effective if a multi-gun salute such as for "The Queen's Birthday" is being fired.
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 supported in 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 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 which 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 "velocity".For fluids in general, the speed of sound c is given by the Newton–Laplace equation:
where
- K_{s} is a coefficient of stiffness, the isentropic bulk modulus ;
- is the density.
For general equations of state, if classical mechanics is used, the speed of sound c can be derived as follows:
Consider the sound wave propagating through a pipe with cross-sectional area. In time interval it moves through a tube of length. In steady state, the mass flow rate must be the same at the two ends of the tube, therefore the mass flux. Per Newton's second law, the pressure-gradient force provides the acceleration:
And therefore:
where
- p is the pressure;
- is the density and the derivative is taken isentropically, that is, at constant entropy s. This is because sound wave travels so fast that its propagation can be approximated as adiabatic process.
In a non-dispersive medium, the speed of sound is independent of sound frequency, so the speeds of energy transport and sound propagation are the same for all frequencies. Air, a mixture of oxygen and nitrogen, constitutes a non-dispersive medium. However, air does contain a small amount of CO_{2} which is a dispersive medium, and causes dispersion to air at ultrasonic frequencies.
In a dispersive medium, the speed of sound is a function of sound frequency, through the dispersion relation. Each frequency component propagates at its own speed, called the phase velocity, while the energy of the disturbance propagates at the group velocity. The same phenomenon occurs with light waves; see optical dispersion for a description. No
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 waves depends on the shear deformation under shear stress, and the density of the medium. Longitudinal 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, 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.
In low molecular weight gases such as helium, sound propagates faster as compared to 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 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. Thus, for a single given gas and over a small temperature range, the speed of sound becomes dependent on only the temperature of the gas.
In non-ideal gas behavior regimen, for which the van der Waals gas equation would be used, the proportionality is not exact, and there is a slight dependence of sound velocity on the gas pressure.
Humidity has a small but measurable effect on the speed of sound, because oxygen and nitrogen molecules of the air are replaced by lighter molecules of water. This is a simple mixing effect.
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 decreases with increasing altitude up to, 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.
However, there are variations in this trend above. In particular, in the stratosphere above about, 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 aptly-named thermosphere above.
Practical formula for dry air
The approximate speed of sound in dry air, in metres per second, at temperatures near, can be calculated fromwhere is the temperature in degrees Celsius.
This equation is derived from the first two terms of the Taylor expansion of the following more accurate equation:
Dividing the first part, and multiplying the second part, on the right hand side, by gives the exactly equivalent form
which can also be written as
where T denotes the thermodynamic temperature.
The value of, which represents the speed at , is based on theoretical values of the heat capacity ratio, γ, as well as on the fact that at 1 atm real air is very well described by the ideal gas approximation. Commonly found values for the speed of sound at may vary from 331.2 to 331.6 due to the assumptions made when it is calculated. If ideal gas γ is assumed to be exactly, the speed is calculated to be, the coefficient used above.
This equation is correct to a much wider temperature range, but still depends on the approximation of heat capacity ratio being independent of temperature, and for this reason will fail, particularly at higher temperatures. It gives good predictions in relatively dry, cold, low-pressure conditions, such as the Earth's stratosphere. The equation fails at extremely low pressures and short wavelengths, due to dependence on the assumption that the wavelength of the sound in the gas is much longer than the average mean free path between gas molecule collisions. A derivation of these equations will be given in the following section.
A graph comparing results of the two equations is at right, using the slightly different value of for the speed of sound at.
Details
Speed of sound in ideal gases and air
For an ideal gas, K is given bythus, from the Newton–Laplace equation above, the speed of sound in an ideal gas is given by
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 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;
- ρ is the density.
where
- c_{ideal} is the speed of sound in an ideal gas;
- R is the molar gas constant ;
- k is the Boltzmann constant;
- γ is the adiabatic index. At room temperature, where thermal energy is fully partitioned into rotation but quantum effects prevent excitation of vibrational modes, the value is for diatomic molecules, according to kinetic theory. Gamma is actually experimentally measured over a range from 1.3991 to 1.403 at, for air. Gamma is exactly for monatomic gases such as noble gases and it is approximately 1.3 for triatomic molecule gases;
- T is the absolute temperature;
- M is the molar mass of the gas. The mean molar mass for dry air is about ;
- n is the number of moles;
- m is the mass of a single molecule.
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 γ but was otherwise correct.
Numerical substitution of the above values gives the ideal gas approximation of sound velocity for gases, which is accurate at relatively low gas pressures and densities. Also, for diatomic gases the use of requires that the gas exists in a temperature range high enough that rotational heat capacity is fully excited ; but at the same time the temperature must be low enough that molecular vibrational modes contribute no heat capacity. For air, these conditions are fulfilled at room temperature, and also temperatures considerably below room temperature. See the section on gases in specific heat capacity for a more complete discussion of this phenomenon.
For air, we introduce the shorthand
In addition, we switch to the Celsius temperature, which is useful to calculate air speed in the region near
0 °C. Then, for dry air,
where is the temperature in degrees Celsius.
Substituting numerical values
for the molar gas constant in J/mole/Kelvin, and
for the mean molar mass of air, in kg; and using the ideal diatomic gas value of, we have
Finally, Taylor expansion of the remaining square root in yields
The above derivation includes the first two equations given in the "Practical formula for dry air" section above.
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/ can produce refraction equal to a typical temperature lapse rate of. 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 sound is not being carried along by the wind.For sound propagation, the exponential variation of wind speed with height can be defined as follows:
where
- U is the speed of the wind at height h;
- ζ is the exponential coefficient based on ground surface roughness, typically between 0.08 and 0.52;
- dU/dH is the expected wind gradient at height h.
Tables
In the standard atmosphere:- T_{0} is , giving a theoretical value of . Values ranging from 331.3 to 331.6 m/s may be found in reference literature, however;
- T_{20} is , giving a value of ;
- T_{25} is , giving a value of .
Given normal atmospheric conditions, the temperature, and thus speed of sound, varies with altitude:
Altitude | Temperature | m/s | km/h | mph | kn |
Sea level | 340 | 1,225 | 761 | 661 | |
− | 295 | 1,062 | 660 | 573 | |
29,000 m | 301 | 1,083 | 673 | 585 |
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 which 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 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 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 because they have a higher γ than diatomics do. Thus, at the same molecular mass, the speed of sound of a monatomic gas goes up by a factor of
This gives the 9% difference, and would be a typical ratio for speeds of sound 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 travel faster in a sound wave and transmit sound faster..
Note that in this example we have assumed that temperature is low enough that heat capacities are not influenced by molecular vibration. However, vibrational modes simply cause gammas which decrease toward 1, since vibration modes in a polyatomic gas give 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 the speed of sound in monatomic vs. polyatomic molecules, with the speed remaining greater in monatomics.
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 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 but decreased by carbon dioxide. The difference between 0% and 100% humidity is about at standard pressure and temperature, but the size of the humidity effect increases dramatically with temperature. The carbon dioxide content of air is not fixed, due to both carbon pollution and human breath.
The dependence on frequency and pressure are normally insignificant in practical applications. In dry air, the speed of sound increases by about as the frequency rises from to. For audible frequencies above 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, 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.
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 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 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, called microphone basis.
- The time of arrival between the signals reaching the different microphones.
Other methods
In these methods, the time measurement has been replaced by a measurement of the inverse of time.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 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 but corrected for temperature in order to report them at. The result was for dry air at STP, for frequencies from to.Non-gaseous media
Speed of sound in solids
Three-dimensional 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 dependenton the deformation mode. Sound waves generating volumetric deformations and shear deformations are called pressure waves and shear waves, respectively. In earthquakes, the corresponding seismic waves are called P-waves and S-waves, respectively. The sound velocities of these two types of waves propagating in a homogeneous 3-dimensional solid are respectively given by
where
- K is the bulk modulus of the elastic materials;
- G is the shear modulus of the elastic materials;
- E is the Young's modulus;
- ρ is the density;
- ν is Poisson's ratio.
Typically, pressure waves travel faster in materials than do shear waves, and in earthquakes this is the reason that the onset of an earthquake is often preceded by a quick upward-downward shock, before arrival of waves that produce a side-to-side motion. For example, for a typical steel alloy,, and, yielding a compressional speed c_{solid,p} of. This is in reasonable agreement with c_{solid,p} measured experimentally at for a type of steel. The shear speed c_{solid,s} is estimated at using the same numbers.
One-dimensional solids
The speed of sound for pressure waves in stiff materials such as metals is sometimes given for "long rods" of the material in question, in which the speed is easier to measure. In rods where their diameter is shorter than a wavelength, the speed of pure pressure waves may be simplified and is given by:where E is Young's modulus. This is similar to the expression for shear waves, save that Young's modulus replaces the shear modulus. This speed of sound for pressure waves in long rods will always be slightly less than the same speed in homogeneous 3-dimensional solids, and the ratio of the speeds in the two different types of objects depends on Poisson's ratio for the material.
Speed of sound in liquids
In a fluid, the only non-zero stiffness is to volumetric deformation.Hence the speed of sound in a fluid is given by
where K is the bulk modulus of the fluid.
Water
In fresh water, sound travels at about at . Applications of underwater sound can be found in sonar, acoustic communication and acoustical oceanography.Seawater
In salt water that is free of air bubbles or suspended sediment, sound travels at about . The speed of sound in seawater depends on pressure, temperature, and salinity, 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. For more information see Dushaw et al.An empirical equation for the speed of sound in sea water is provided by Mackenzie:
where
- T is the temperature in degrees Celsius;
- S is the salinity in parts per thousand;
- z is the depth in metres.
with check value for,,. This equation has a standard error of for salinity between 25 and 40 ppt. See for an online calculator.
Other equations for the speed of sound in sea water are accurate over a wide range of conditions, but are far more complicated, e.g., that by V. A. Del Grosso and the Chen-Millero-Li Equation.
Speed of sound in plasma
The speed of sound in a plasma for the common case that the electrons are hotter than the ions is given by the formulawhere
- m_{i} is the ion mass;
- μ is the ratio of ion mass to proton mass ;
- T_{e} is the electron temperature;
- Z is the charge state;
- k is Boltzmann constant;
- γ is the adiabatic index.
Gradients
When sound spreads out evenly in all directions in three dimensions, the intensity drops in proportion to the inverse square of the distance. However, in the ocean, there is a layer called the 'deep sound channel' or SOFAR channel which 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 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.