# Two different forms of the electromagnetic wave equation

Wikipedia says that the electromagnetic wave equation written in terms of the electric field $$\mathbf{E}$$ is

{\displaystyle {\begin{aligned}\left(v_{ph}^{2}\nabla ^{2}-{\frac {\partial ^{2}}{\partial t^{2}}}\right)\mathbf {E} = \mathbf{0} \end{aligned}}}

However, I have seen the electromagnetic wave equation written as

$$\nabla^2 E + k^2 E = 0$$

in the context of rays.

I would appreciate it if someone could please take the time to explain why these two equations are different, and what is the connection between them?

The second equation is the Fourier transform of the first one with the frequency symbol taken as $$k$$.
You get your second equation from your first one if you choose $$E$$ in such a way that all dependence on $$t$$ is presented by a factor $$\exp(-i\omega t)$$ or $$\sin(\omega t +\varphi)$$.