Speed of sound

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Sound is a vibration that travels through an elastic medium as a wave. The speed of sound describes how much distance such a wave travels in a given amount of time. In dry air, at a temperature of 21 °C (70 °F) the speed of sound is 344 m/s (1238 km/h, or 769 mph, or 1128 ft/s).

Although the term is commonly used to refer specifically to air, the speed of sound can be measured in virtually any material. The speed of sound in liquids and solids is much higher than that in air.

In the Earth's atmosphere, the speed varies with atmospheric conditions; the most important factor is the temperature. Air pressure has almost no effect on sound speed. It has no effect at all in an ideal gas approximation, 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, creating a negative sound speed gradient. In the stratosphere, the speed of sound increases with height due to heating within the ozone layer, producing a positive sound speed gradient.

Humidity has a small, but measurable effect on sound speed. Sound travels slightly (0.1%-0.6%) faster in humid air. The approximate speed of sound in 0% humidity (dry) air, in metres per second (m·s^-1), at temperatures near 0 °C, can be calculated from:

where is the temperature in degrees Celsius (°C).

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

The value of 331.3 m/s, which represents the 0 °C speed, is probably the most defensible based on theoretical (and some measured) values of the specific heat ratio, γ. Commonly found values for the speed of sound at 0 °C 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 7/5 = 1.4 exactly, the 0 °C speed is calculated (see section below) to be 331.3 m/s, the coefficient used above.

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. It gives good predictions in relatively dry, cold, low pressure conditions, such as the Earth's stratosphere. A derivation of these equations will be given in a later section.

[edit] 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 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). Effects like dispersion and reflection can also be understood using this model.

In a real material, the stiffness of the springs is called the elastic modulus, and the mass 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 iron 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 iron than hydrogen, because the internal bonds in a solid like iron 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.

[edit] Details

In general, the speed of sound c is given by

where

C is a coefficient of stiffness

ρ 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

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).^{[citation needed]}

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 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.

[edit] 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:

where

E is Young's modulus

ρ (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:

[edit] Speed 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

where

K is the bulk modulus of the fluid

[edit] water

The speed of sound in water is of interest to anyone using underwater sound as a tool, whether in a laboratory, a lake or the ocean. Examples are sonar, acoustic communication and acoustical oceanography. See Discovery of Sound in the Sea for other examples of the uses of sound in the ocean (by both man and other animals). In fresh water, sound travels at about 1497 m/s at 25 °C. See Technical Guides - Speed of Sound in Pure Water for an online calculator.

[edit] seawater

In salt water that is free of air bubbles or suspended sediment, sound travels at about 1500 m/s. 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 sound speed from these variables. Other factors affecting sound speed are minor. For more information see Dushaw et al. (1993).

A simple empirical equation for the speed of sound in sea water with reasonable accuracy for the world's oceans is due to Mackenzie (1981)

c(T, S, z) = a₁ + a₂T + a₃T² + a₄T³ + a₅(S - 35) + a₆z + a₇z² + a₈T(S - 35) + a₉Tz³

where T, S, and z are temperature in degrees Celsius, salinity in parts per thousand and depth in metres, respectively. The constants a₁, a₂, ..., a₉ are:

a₁ = 1448.96, a₂ = 4.591, a₃ = -5.304×10^-2, a₄ = 2.374×10^-4, a₅ = 1.340, a₆ = 1.630×10^-2, a₇ = 1.675×10^-7, a₈ = -1.025×10^-2, a₉ = -7.139×10^-13

with check value 1550.744 m/s for T=25 °C, S=35‰, z=1000 m. This equation is accurate to O(0.2 m/s).^{[citation needed]} See Technical Guides - Speed of Sound in Sea-Water for an online calculator.

Other equations for sound speed in sea water have slightly greater accuracy, but are far more complicated, e.g., that by V. A. Del Grosso (1974) and the Chen-Millero-Li Equation (1994).^{[1] [2]}

[edit] Speed in ideal gases and in air

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

thus

Where:

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

ρ is the density

Using the ideal gas law to replace p with NRT/V, and replacing ρ with NM/V, the equation for an ideal gas becomes:

where

• c_ideal is the speed of sound in an ideal gas.
• R (approximately 8.3145 J·mol^-1·K^-1) is the molar gas constant.[1]
• k is the Boltzmann constant
• γ (gamma) is the adiabatic index (sometimes assumed 7/5 = 1.400 for diatomic molecules from kinetic theory, assuming from quantum theory a temperature range at which thermal energy is fully partitioned into rotation (rotations are fully excited), but none into vibrational modes. Gamma is actually experimentally measured over a range from 1.3991 to 1.403 at 0 degrees Celsius, for air. Gamma is assumed from kinetic theory to be exactly 5/3 = 1.6667 for monoatomic molecules such as noble gases).
• T is the absolute temperature in kelvins.
• M is the molar mass in kilograms per mole. The mean molar mass for dry air is about .0289645 kg/mole.
• m is the mass of a single molecule in kilograms.

This equation applies only when the sound wave is a small perturbation on the ambient condition, and the certain other noted conditions are fulfilled, as noted below. Calculated values for c_air have been found to vary slightly from experimentally determined values.:^[3]

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 (for air, this includes standard Earth sea-level conditions). Also, for diatomic gases the use of requires that the gas exist in a temperature range high enough that rotational heat capacity is fully excited (i.e., molecular rotation is fully used as a heat energy "partition" or reservoir); but at the same time the temperature must be low enough that molecular vibrational modes contribute no heat capacity (i.e., insigificant heat goes into vibration, as all vibrational quantum modes above the minimum-energy-mode, have energies too high to be populated by a significant number of molecules at this temperature). For air, these conditions are fulfilled at room temperature, and also temperatures considerably below room temperature (see tables below). See the section on gases in heat capacity for a more complete discussion of this phenomenon.

If temperatures in degrees Celsius(°C) are to be used to calculate air speed in the region near 273 kelvins, then Celsius temperature may be used.

For dry air, where (theta) is the temperature in degrees Celsius(°C).

Making the following numerical substitutions: , where is the molar gas constant, , and using the ideal diatomic gas value of

Then:

Using the first two terms of the Taylor expansion:

The derivation includes the two approximate equations which were given in the introduction. For Celsius temperatures which are negative, the second term of the equation right hand side, is negative.

[edit] Tables

In the standard atmosphere:

T₀ is 273.15 K (= 0 °C = 32 °F), giving a theoretical value of 331.3 m·s^-1 (= 1086.9 ft/s = 1193 km·h^-1 = 741.1 mph = 644.0 knots). Values ranging from 331.3-331.6 may be found in reference literature, however.
T₂₀ is 293.15 K (= 20 °C = 68 °F), giving a value of 343.2 m·s^-1 (= 1126.0 ft/s = 1236 km·h^-1 = 767.8 mph = 667.2 knots).
T₂₅ is 298.15 K (= 25 °C = 77 °F), giving a value of 346.1 m·s^-1 (= 1135.6 ft/s = 1246 km·h^-1 = 774.3 mph = 672.8 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
in °C	c in m·s-1	ρ in kg·m-3	Z in N·s·m-3
−10	325.2	1.342	436.1
−5	328.3	1.317	432.0
0	331.3	1.292	428.4
+5	334.3	1.269	424.3
+10	337.3	1.247	420.6
+15	340.3	1.225	416.8
+20	343.2	1.204	413.2
+25	346.1	1.184	409.8
+30	349.0	1.165	406.3

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 characteristic acoustic impedance in N·s·m^-3 (Z=ρ·c)

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

[edit] Effect of frequency and gas composition

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.^[4]

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 (except for frequencies approaching zero) progressively loses its range of applicability at high altitudes.:^[3]

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 γ (5/3 = 1.67) 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

= 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 sound wave 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.

[edit] Mach number

Main article: 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.

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 impact pressure sensed by a Pitot tube is dependent on altitude as well as speed.

Assuming air to be an ideal gas, the formula to compute Mach number in a subsonic compressible flow is derived from Bernoulli's equation for M<1:^[5]

where

M is Mach number

q_c is impact pressure and

P is static pressure.

The formula to compute Mach number in a supersonic compressible flow is derived from the Rayleigh Supersonic Pitot equation:

where

M is Mach number

q_c is impact pressure measured behind a normal shock

P is static pressure.

As can be seen, M appears on both sides of the equation. The easiest method to solve the supersonic M calculation is to enter both the subsonic and supersonic equations into a computer spread sheet such as Microsoft Excel, OpenOffice.org Calc, or some equivalent program. First determine if M is indeed greater than 1.0 by calculating M from the subsonic equation. If M is greater than 1.0 at that point, then use the value of M from the subsonic equation as the initial condition in the supersonic equation. Then perform a simple iteration of the supersonic equation, each time using the last computed value of M, until M converges to a value--usually in just a few iterations.^[5]

[edit] Experimental methods

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

[edit] 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.

[edit] 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/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

[edit] Gradients

When sound spreads out evenly in all directions, 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, allowing them to travel much further. 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 in a sheet of glass or optical fiber.

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