| A longitudinal wave is one where the vibration at any point is in the same direction as the wave itself is moving. The waveform (pattern of crests and troughs) moves forward transmitting energy, while the particles of which the medium is composed do not move long distances, but oscillate to and fro. This creates regions of compression and rarefaction in the medium. Sound waves are a familiar example of longitudinal waves. Contrast with transverse waves, where the direction of vibration is perpendicular to that of travel.
Longitudinal waves can be diffracted or reflected but can not be polarized.
The velocity of a longitudinal wave in a solid is given by v = sqrt(E / ρ), where E is its Young's modulus of elasticity and ρ is its density. In a liquid the same formula is applicable but E stands for its bulk modulus.
The same formula applies in a gas too, with E being the gas's bulk modulus, but with the complication that E has different values depending on whether the propagation of the wave is isothermal or adiabatic. Since the temperature of a gas varies with its pressure (by the Ideal Gas Law PV = nRT) the oscillating regions of compression and rarefaction cause oscillations of temperature, that is heat must be redistributed. If this is done fast compared to the period of the wave, the propagation is isothermal and the bulk modulus E is equal to the pressure P. This was calculated by Newton and he found it was wrong: it gives a value 1.4 times too low.
The solution was provided by Laplace in 1816, who suggested sound was propagated adiabatically. The adiabatic bulk modulus is γP, where γ = CP / CV, the ratio of molar heat capacities, and P is the pressure. The γ of air is about 1.40 and this is termed Laplace's correction. As the density of air is ρ = 1.29 kg m-3 at standard temperature and pressure, this gives a velocity v = sqrt(γP / ρ) = about 330 m s-1.
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