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Transverse and longitudinal waves are classified according to the direction of propagation of wave motion and direction of oscillations of particles of the medium. But in the case of non-mechanical waves there is no medium, so there are no particles of a medium to oscillate. In this case how does one classify longitudinal and transverse waves?

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Usually you consider a test particle and describe the wave based on how that test particle would move within the wave. For instance EM waves you imagine a charged particle within the field and the direction that charged particle moves in would determine the classification, Transverse or longitudinal for instance.

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Even for non-mechanical waves, there is still something that varies in time and space and fulfills a wave equation. If that something has a direction, we can classify transverse and orthogonal waves.

The obvious example is the electromagnetic field: In an electromagnetic wave, the $\vec{E}$ and $\vec{B}$ fields are vector fields, i.e.\ they have a direction, and they are orthogonal to the direction of propagation.

A direct consequence of the transversality is that these waves can be polarised: Since there are two directions transversal (in three spatial dimensions) to a give direction, there can be two independent transverse waves. A longitudinal wave, on the other hand cannot be polarised, since there is only one direction.

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