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, waves can exist in plasmas or confined spaces, called plasma waves, which can be longitudinal, transverse, or a mixture of both. Plasma waves can also occur in force-free magnetic fields.
In the early development of electromagnetism, there were some like Alexandru Proca (1897-1955)known for developing relativistic quantum field equations bearing his name (Proca's equations) for the massive, vector spin-1 mesons. In recent decades some extended electromagnetic theorists such as Jean-Pierre Vigier, and Bo Lehnert of the Swedish Royal Society have used the Proca equation in an attempt to demonstrate photon mass as a longitudinal electromagnetic component of Maxwell's equations suggesting that longitudinal electromagnetic waves could exist in a Dirac polarized 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 Maxwell's equations do lead to the appearance of longitudinal waves under some circumstances, for example, in 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 modes of a resonant cavity are the particular standing wave patterns formed by waves confined in a cavity. The longitudinal modes correspond to those 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 a longitudinal electromagnetic (light) wave in free space, and this wave can propagate without divergence for a few wavelengths.
Read more about this topic: Longitudinal Wave