A wave along the length of a stretched Slinky toy, where the distance between coils increases and decreases, is a good visualization.

[4] For longitudinal harmonic sound waves, the frequency and wavelength can be described by the formula where: The quantity

[5] This is caused by the scattering of the wave at interfaces, the loss of energy due to the friction between molecules, or geometric divergence.

[5] The study of attenuation of elastic waves in materials has increased in recent years, particularly within the study of polycrystalline materials where researchers aim to "nondestructively evaluate the degree of damage of engineering components" and to "develop improved procedures for characterizing microstructures" according to a research team led by R. Bruce Thompson in a Wave Motion publication.

Due to the difference in crystal structure and properties of these grains, when a wave propagating through a poly-crystal crosses a grain boundary, a scattering event occurs causing scattering based attenuation of the wave.

[8] A current prediction for modeling attenuation of waves in polycrystalline materials with elongated grains is the second-order approximation (SOA) model which accounts the second order of inhomogeneity allowing for the consideration multiple scattering in the crystal system.

[9] The equations for sound in a fluid given above also apply to acoustic waves in an elastic solid.

[11] In May 2022, NASA reported the sonification (converting astronomical data associated with pressure waves into sound) of the black hole at the center of the Perseus galaxy cluster.

[17] Maxwell's equations, as we now understand them, retain that conclusion: in free-space or other uniform isotropic dielectrics, electro-magnetic waves are strictly transverse.

In recent decades some other 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[19] as a longitudinal electromagnetic component of Maxwell's equations, suggesting that longitudinal electromagnetic waves could exist in a Dirac polarized vacuum.

However photon rest mass is strongly doubted by almost all physicists and is incompatible with the Standard Model of physics.