Longitudinal waves are waves that have the same direction of vibration as their direction of travel, which means that the movement of the medium is in the same direction as or the opposite direction to the motion of the wave. Mechanical longitudinal waves have been also referred to as compressional waves or compression waves.
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Nonelectromagnetic
Longitudinal waves include sound waves (alternation in pressure, particle displacement, or particle velocity propagated in an elastic material) and seismic Pwaves (created by earthquakes and explosions).
Sound waves
In the case of longitudinal harmonic sound waves, the frequency and wavelength can be described with the formula
where:
 y is the displacement of the point on the traveling sound wave;
 x is the distance the point has traveled from the wave's source;
 t is the time elapsed;
 y_{0} is the amplitude of the oscillations,
 c is the speed of the wave; and
 ω is the angular frequency of the wave.
The quantity x/c is the time that the wave takes to travel the distance x.
The ordinary frequency (f) of the wave is given by
For sound waves, the amplitude of the wave is the difference between the pressure of the undisturbed air and the maximum pressure caused by the wave.
Sound's propagation speed depends on the type, temperature and pressure of the medium through which it propagates.
Pressure waves
In an elastic medium with rigidity, a harmonic pressure wave oscillation has the form,
where:
 y_{0} is the amplitude of displacement,
 k is the wavenumber,
 x is distance along the axis of propagation,
 ω is angular frequency,
 t is time, and
 φ is phase difference.
The force acting to return the medium to its original position is provided by the medium's bulk modulus.^{[1]}
Electromagnetic
Maxwell's equations lead to the prediction of electromagnetic waves in a vacuum, which are transverse (in that the electric fields and magnetic fields vary perpendicularly to the direction of propagation).^{[2]} However, waves can exist in plasma or confined spaces. These are called plasma waves and can be longitudinal, transverse, or a mixture of both.^{[2]}^{[3]} Plasma waves can also occur in forcefree magnetic fields.
In the early development of electromagnetism there was some suggesting that longitudinal electromagnetic waves existed in a vacuum. After Heaviside's attempts to generalize Maxwell's equations, Heaviside came to the conclusion that electromagnetic waves were not to be found as longitudinal waves in "free space" or homogeneous media.^{[4]} But it should be stated that Maxwell's equations do lead to the appearance of longitudinal waves under some circumstances in either plasma waves or guided waves. Basically distinct from the "freespace" waves, such as those studied by Hertz in his UHF experiments, are Zenneck waves.^{[5]} The longitudinal mode of a resonant cavity is a particular standing wave pattern formed by waves confined in a cavity. The longitudinal modes correspond to the wavelengths of the wave which are reinforced by constructive interference after many reflections from the cavity's reflecting surfaces. Recently, Haifeng Wang et al. proposed a method that can generate longitudinal electromagnetic (light) wave in free space, and this wave can propagate without divergence for a few wavelengths.^{[6]}
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