Longitudinal wave
Longitudinal waves are waves that have vibrations along or parallel to their direction of travel; that is, waves in which the motion of the medium is in the same direction as the motion of the wave. Mechanical longitudinal waves have been also referred to as compressional waves or compression waves.

# Non-electromagnetic
Examples of non-electromagnetic longitudinal waves include sound waves (alternation in pressure, particle displacement, or particle velocity propagated in an elastic material) and seismic P-waves (created by earthquakes and explosions).

## Sound waves
In the case of longitudinal harmonic sound waves, the frequency and wavelength can be described with the equation
where:
- y(x,t) is the displacement of particles from the stable position, in the direction of propagation of the wave;
- x is the displacement from the source of the wave to the point under consideration;
- t is the time elapsed;
- y_0 is the amplitude of the oscillations,
- c is the speed of the wave; and
-  is the angular frequency of the wave.
The quantity x/c is the time that the wave takes to travel the distance x.
The  (nonangular) frequency of the wave can be found using the formula
where f is the frequency of the wave, usually measured in Hz.
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:
- y0 is the amplitude of displacement,
- k is the wave number,
- x is distance along the axis of propagation,
-  is angular frequency,
- t is time, and
-  is phase difference.
The force acting to return the medium to its original position is provided by the medium's bulk modulus.

# 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). However, in a plasma or a confined space, there can exist waves which are either longitudinal or transverse, or a mixture of both. In plasma waves, there exists some examples and these plasma waves can occur in the situation of force-free magnetic fields.
In the early development of electromagnetism, there was some controversy, in that Helmholtz' theory led to the prediction of longitudinal waves. Oliver Heaviside examined this problem as there was no evidence 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. But it should be stated that longitudinal waves can exist along the interface between differing media (such as the various layers of the Earth's atmosphere and the surface of the Earth or as in the Schumann resonance).
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 "free-space" waves, such as those studied by Hertz in his UHF experiments, are Zenneck waves. 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.

# Media