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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?

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The second equation is the Fourier transform of the first one with the frequency symbol taken as $k$.

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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 $\sin(\omega t +\varphi)$.

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