Plane Electromagnetic Waves in Vacuum

We will now derive the wave equations for electromagnetic fields in vacuum, i.e., in the absence of charges and currents within an infinite volume of space. 

Eq and Bq are constant complex amplitudes, and e- and 2 shall be constant unit vectors. As a consequence the electric field vector always points in the direction of Cy and such an electromagnetic wave is said to be linearly polarized. By superpositions of two such linearly polarized waves with different phases and amplitudes so-called elliptically polarized waves may be constructed. However, we do not need to further discuss this possibility here. 

The wave vector k defines the direction of propagation of the waves and is related to the angular frequency u by 

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