Speed of sound
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, the speed of sound in air is about, or in or one mile in. It depends strongly on temperature as well as the medium through which a sound wave is propagating.
At, the speed of sound in dry air is about.
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 in air, it travels in fresh water at at a temperature of and at in iron. In an exceptionally stiff material such as diamond, sound travels at 12,000 m/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 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 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 is called the object's Mach number. Objects moving at speeds greater than the speed of sound are said to be traveling at supersonic speeds.
Earth
In Earth's atmosphere, the speed of sound varies greatly from about at high altitudes to about at high temperatures.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. This is too low by about 15%. The discrepancy is due primarily to neglecting the effect of rapidly fluctuating temperature in a sound wave. 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 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. 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.. 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, which propagates faster than normal sound waves.
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. 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 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:
where
- is a coefficient of stiffness, the isentropic bulk modulus ;
- is the density.
Thus, the speed of sound increases with the stiffness 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 through a pipe aligned with the axis and with a cross-sectional area of. In time interval it moves length. In steady state, the mass flow rate must be the same at the two ends of the tube, therefore the mass flux is constant and. Per Newton's second law, the pressure-gradient force provides the acceleration:
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.
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.