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

y(x,t) = y_0 \sin\Bigg( \omega \left(t-\frac{x}{c} \right) \Bigg)

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
• 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, in Hertz, of the wave can be found using

f = \frac{\omega}{2 \pi}.

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,

y(x,t)\, = y_0 \cos(k x - \omega t +\phi)

where:

• y₀ 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.

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).David J. Griffiths, Introduction to Electrodynamics, ISBN 0-13-805326-X 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.