= Electromagnetic wave equation
{title2=$\nabla^2\mathbf E-c^{-2}\partial_t^2\mathbf E=0$}
{wiki}
In source-free vacuum the <electric field> and <magnetic field> obey
$$
\nabla^2\mathbf E-c^{-2}\partial_t^2\mathbf E=0,\qquad
\nabla^2\mathbf B-c^{-2}\partial_t^2\mathbf B=0.
$$
A brief derivation takes the <curl> of <Faraday's law> and substitutes the <Ampère-Maxwell equation>. The <vector calculus> identity $\nabla\times(\nabla\times\mathbf E)=\nabla(\nabla\cdot\mathbf E)-\nabla^2\mathbf E$, together with source-free <Gauss's law>, leaves $\nabla^2\mathbf E=\mu_0\epsilon_0\partial_t^2\mathbf E$. The same argument gives the <magnetic field> equation. The operator $\nabla^2$ is the <Laplacian>, applied componentwise.
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