The speed of sound is the distance travelled per unit of time by a sound wave as it propagates through an elastic medium. More simply, the speed of sound is how fast vibrations travel. At 20 °C (68 °F), the speed of sound in air is about 343 m/s (1,125 ft/s; 1,235 km/h; 767 mph; 667 kn), or 1 km in 2.92 s or one mile in 4.69 s. It depends strongly on temperature as well as the medium through which a sound wave is propagating. At 0 °C (32 °F), the speed of sound in dry air (sea level 14.7 psi) is about 331 m/s (1,086 ft/s; 1,192 km/h; 740 mph; 643 kn). 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 dry air, deviating slightly from ideal behavior. In colloquial speech, speed of sound refers to the speed of sound waves in air. The speed of sound varies from substance to substance, however: typically, sound travels most slowly in gases, faster in liquids, and fastest in solids. For example, while sound travels at 343 m/s in air, it travels in fresh water at 1481 m/s at a temperature of 20 °C (68 °F) (slightly more than 4.3 times as fast) and at 5120 m/s in iron (almost 15 times as fast). In an exceptionally stiff material such as diamond, sound travels at 12,000 m/s (39,000 ft/s), – about 35 times its speed in air and about the fastest it can travel under normal conditions. In theory, the speed of sound is actually the speed of vibrations. Sound waves in solids are composed of compression waves (just as in gases and liquids) 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 than compression waves, 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 (gas or liquid) 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 same medium) is called the object's Mach number. Objects moving at speeds greater than the speed of sound (Mach1) are said to be traveling at supersonic speeds.
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. 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. He measured this many times under many circumstances, to find the dependence of the speed on wind, barometric pressure, temperature, and humidity. For example, he found that if wind is blowing towards the observer, the speed of sound is faster, and vice versa. He thought temperature did not affect it, because the speed was the same in summer and winter. He was also mistaken in finding that rain and fog reduced the speed, a conclusion that was accepted until Tyndall disproved it. Early measurements found that the speeds of sound did not agree, and it was suspected that the speed of wind and temperature may change the speed of sound. In 1740, G. L. Bianconi showed that the speed of sound in air increases with temperature. The Academy of Sciences of Paris in 1738 used cannon as the source sound, and found that when there is no wind, the speed of sound at 0 °C was 332 m/s, which is within 1% of the modern accepted value. Chladni measured the speed of sound in solids by comparing the pitch of sound in a tube of air and a solid bar, and found that the speed of sound in tin is about 7.5 times greater than in air, while in copper it was about 12 times greater. Biot in 1808 measured the speed of sound in an iron pipe about 1000 m long, and found it was 10.5 times that of air, though he thought it was only an order of magnitude estimate, since his time-measurement had an accuracy of 0.5 seconds, longer than the time actually necessary for sound to propagate through the pipe. The first measurement of speed of sound in water was done by Jean-Daniel Colladon and Charles Sturm at Lake Geneva in 1826. They were on two boats separated by 10 km. Colladon repeatedly pressed a lever that would, simultaneously, both ignite a bit of gunpowder above water and ring a bell in water. Sturm would listen for the bell with an underwater tube and measure the time until the sound is heard. They found a value of 1437.8 m/s in water at 8 C. This differs from the modern value by 1 m/s. They presented the result in a monograph. Samuel Earnshaw reported in 1860 that he was at an experiment in 1822, where the sound of cannon fire came before the officer standing next to it shouting "fire". He hypothesized that this meant a loud enough sound would create discontinuity in the air (a shock wave in modern language), which propagates faster than normal sound waves. To further support his theory, he showed that ideal fluid cannot propagate waves uniformily, this became known as the Earnshaw paradox. In the 1980's, over the course of 18 months, George Wong, a senior research officer of the National Research Council of Canada, investigated the subject because his experiments indicated that the speed was actually slower than the textbooks said. The textbook speed was 331.45 meters per second; Wong's results were 331.29 meters per second. The textbook speed was from a series of papers indirectly and directly based on a 1942 paper by H. C. Hardy, which made an error in "correcting" the results of experiments done on dried air which had had the carbon dioxide removed. To restore the observed value to that of "standard air", Dr. Hardy made an error in his "correction" due to not compensating for the very slight loss of the oxygen as well as carbon dioxide in his experiment.
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}} , where P {\displaystyle P} is the pressure and the derivative is taken isentropically, that is, at constant entropy s. This is because a sound wave travels so fast that its propagation can be approximated as an adiabatic process, meaning that there isn't enough time, during a pressure cycle of the sound, for significant heat conduction and radiation to occur. Thus, the speed of sound increases with the stiffness (the resistance of an elastic body to deformation by an applied force) of the material and decreases with an increase in density. For ideal gases, the bulk modulus K is simply the gas pressure multiplied by the dimensionless adiabatic index, which is about 1.4 for air under normal conditions of pressure and temperature. 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 at speed v {\displaystyle v} through a pipe aligned with the x {\displaystyle x} axis and with a cross-sectional area of A {\displaystyle A} . In time interval d t {\displaystyle dt} it moves length d x = v d t {\displaystyle dx=v\,dt} . In steady state, the mass flow rate m ˙ = ρ v A {\displaystyle {\dot {m}}=\rho vA} must be the same at the two ends of the tube, therefore the mass flux j = ρ v {\displaystyle j=\rho v} is constant and v d ρ = − ρ d v {\displaystyle v\,d\rho =-\rho \,dv} . Per Newton's second law, the pressure-gradient force provides the acceleration:
d v d t = − 1 ρ d P d x → d P = ( − ρ d v ) d x d t = ( v d ρ ) v → v 2 ≡ c 2 = d P d ρ → c = ( ∂ P ∂ ρ ) s = K s ρ {\displaystyle {\begin{aligned}{\frac {dv}{dt}}&=-{\frac {1}{\rho }}{\frac {dP}{dx}}\\[1ex]\rightarrow dP&=(-\rho \,dv){\frac {dx}{dt}}=(v\,d\rho )v\\[1ex]\rightarrow v^{2}&\equiv c^{2}={\frac {dP}{d\rho }}\\[1ex]\rightarrow c&={\sqrt {\left({\frac {\partial P}{\partial \rho }}\right)_{s}}}={\sqrt {\frac {K_{s}}{\rho }}}\\\end{aligned}}}
If relativistic effects are important, the speed of sound is calculated from the relativistic Euler equations. 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. Nonetheless, air does contain a small amount of CO2, which is a dispersive medium, and causes dispersion to air at ultrasonic frequencies (greater than 28 kHz). 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.
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. 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 (causing it to increase by about 0.1%–0.6%), 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 (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; 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
c i d e a l = γ p ρ = γ R T M = γ k T m = 1.380649 ⋅ 10 − 23 ⋅ γ T m ≈ 8.314 γ T M { = 1.9329086 ⋅ 10 − 23 ⋅ T m ≈ 11.640 T M , γ = 7 5 ≈ 2.301 ⋅ 10 − 23 ⋅ T m ≈ 13.857 T M , γ = 5 3 ≈ 1.840 ⋅ 10 − 23 ⋅ T m ≈ 11.086 T M , γ = 4 3 c d r y a i r ≈ 20.04687087513010149970678963 T {\displaystyle {\begin{aligned}c_{\mathrm {ideal} }&={\sqrt {\frac {\gamma p}{\rho }}}={\sqrt {\frac {\gamma RT}{M}}}={\sqrt {\frac {\gamma kT}{m}}}\\&={\sqrt {\frac {1.380649\cdot 10^{-23}\cdot \gamma T}{m}}}\approx {\sqrt {\frac {8.314\gamma T}{M}}}\\&{\begin{cases}&={\sqrt {\frac {1.9329086\cdot 10^{-23}\cdot T}{m}}}\approx {\sqrt {\frac {11.640T}{M}}},&&\gamma ={\frac {7}{5}}\\&\approx {\sqrt {\frac {2.301\cdot 10^{-23}\cdot T}{m}}}\approx {\sqrt {\frac {13.857T}{M}}},&&\gamma ={\frac {5}{3}}\\&\approx {\sqrt {\frac {1.840\cdot 10^{-23}\cdot T}{m}}}\approx {\sqrt {\frac {11.086T}{M}}},&&\gamma ={\frac {4}{3}}\\\end{cases}}\\c_{\mathrm {dry\ air} }&\approx 20.04687087513010149970678963{\sqrt {T}}\end{aligned}}}
where
cideal is the speed (in m/s) of sound in an ideal gas; p is the pressure; ρ is the density; γ (gamma) is the adiabatic index. At room temperature, where thermal energy is fully partitioned into rotation (rotations are fully excited) but quantum effects prevent excitation of vibrational modes, the value is 7/5 = 1.400 for diatomic gases (such as oxygen and nitrogen), according to kinetic theory. Gamma is actually experimentally measured over a range from 1.3991 to 1.403 at 0 °C, for air. Gamma is exactly 5/3 = 1.667 for monatomic gases (such as argon) and it is 4/3 = 1.333 for triatomic molecule gases that, like H2O, are not co-linear (a co-linear triatomic gas such as CO2 is equivalent to a diatomic gas for our purposes here); γ is, itself, temperature-dependent, with a greater value at lower temperatures and a lower value at higher temperatures. For dry air, for example, it is about 1.404 at 258.15 K, 1.400 at 293.15 K, and 1.398 at 473.15 K. R is the molar gas constant, 8.31446261815324 J⋅mol−1⋅K−1; k is the Boltzmann constant, 1.380649×10−23 J⋅K−1; T is the absolute temperature; M is the molar mass of the gas. The mean molar mass for dry air is about 28.9647 g/mol (0.0289647 kg/mol) n is the number of moles; m is the mass of a single molecule. This equation applies only when the sound wave is a small perturbation on the ambient condition, and certain other conditions are fulfilled as noted below. Calculated values for cair have been found to vary slightly from experimentally determined values. 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 γ = 1.4000 requires that the gas exists 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., insignificant heat goes into vibration, as all vibrational quantum modes above the minimum-energy-mode have energies that are 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 specific heat capacity for a more complete discussion of this phenomenon. For air, we introduce the shorthand
R ∗ = R / M a i r . {\displaystyle R_{*}=R/M_{\mathrm {air} }.}
In addition, we switch to the Celsius temperature θ = T − 273.15 K, which is useful to calculate air speed in the region near 0 °C (273.15 K). Then, for dry air,
c air = γ ⋅ R ∗ ⋅ T = γ ⋅ R ∗ ⋅ ( θ + 273.15 ) = γ ⋅ R ∗ ⋅ 273.15 ∘ K ⋅ 1 + θ 273.15 R = 8.314 462 618 153 24 J m o l ⋅ ∘ K M air = 0.028 964 7 kg mol R ∗ ≈ 287.055 022 773 853 725 396 776 076 γ = 1.4000 (Ideal diatomic gas) c air ≈ 331.32 m s × 1 + θ 273.15