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The Speed of Sound

August 19, 2010 by  


tufail

The speed of sound is the rate of travel of a sound wave through an elastic medium. In dry air at 20 °C (68 °F), the speed of sound is 343 metres per second (1,125 ft/s). This equates to 1,236 kilometres per hour (768 mph), or about one kilometer in three seconds and about one mile in five seconds. This figure increases with temperature (equations are given below), but is nearly independent of pressure or density for a given gas. For different gases, the speed of sound is dependent on the mean molecular weight of the gas, and to a lesser extent upon the ways in which the molecules of the gas can store heat energy from compression (since sound in gases is a type of compression). The speed of sound in air is referred to as Mach 1 by aerospace engineers.

Although “the speed of sound” is commonly used to refer specifically to the speed of sound waves in air, the speed of sound can be measured in virtually any substance. Sound travels faster in liquids and non-porous solids (5,120 m/s in iron) than it does in air, traveling about 4.3 times faster in water (1,484 m/s) than in air at 20 degrees Celsius.

Additionally, in solids, there occurs the possibility of two different types of sound waves: one type (called “longitudinal waves” when in solids) is associated with compression (the same as all sound waves in fluids) and the other is associated with shear stresses, which cannot occur in fluids. These two types of waves have different speeds, and (for example in an earthquake) may thus be initiated at the same time but arrive at distant points at appreciably different times. The speed of compression-type waves in all media is determined by the medium’s compressibility and density, and the speed of shear waves in solids is determined by the material’s stiffness, compressibility and density.

The transmission of sound can be illustrated by using a toy model consisting of an array of balls interconnected by springs. For 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 spongy materials, and faster in stiffer ones. For instance, sound will travel much faster in steel than soft iron, due to the greater stiffness of steel at about the same density. Similarly, sound travels about = about 1.41 times faster in light hydrogen (protium) gas than in heavy hydrogen (deuterium) 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 increasing density. This is usually illustrated by presenting data for three materials, such as air, water and steel, which also have vastly different compressibilities which more than make 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 in the two media.

The speed of sound is variable and depends on the properties of the substance through of which the wave is travelling. In solids, the speed of longitudinal waves depend on the stiffness to tensile stress, and the density of the medium. In fluids, the medium’s compressibility and density are the important factors.

In gases, compressibility and density are related, making other compositional effects and properties important, such as temperature and molecular composition. In low molecular weight gases, such as helium, sound propagates faster compared to heavier gases, such as xenon (for monatomic gases the speed of sound is about 68% of the mean speed that molecules move in the gas). For a given ideal gas the sound speed depends only on its temperature. At a constant temperature, the ideal gas pressure has no effect on the speed of sound, because 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 (see derivations below). Thus, for a single given gas (where molecular weight does not change) and over a small temperature range (where heat capacity is relatively constant), the speed of sound becomes dependent on only the temperature of the gas.

In non-ideal gases, such as a van der Waals gas, 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 sound speed (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.

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