Generated by the Physics Derivation Graph. Eq. \ref{eq:4757562} is an initial equation. $$\vec{ \nabla} \times \vec{E} = -\mu_0\frac{\partial \vec{H}}{\partial t} \label{eq:4757562}$$ Eq. \ref{eq:1199299} is an initial equation. $$\vec{ \nabla} \times \vec{H} = \epsilon_0 \frac{\partial }{\partial t}\vec{E} \label{eq:1199299}$$ Eq. \ref{eq:4857731} is an initial equation. $$\vec{ \nabla} \cdot \vec{E} = \rho/\epsilon_0 \label{eq:4857731}$$ Partially differentiate Eq. \ref{eq:1199299} with respect to $$t$$; yields Eq. \ref{eq:4642245}. $$\vec{ \nabla} \times \frac{\partial \vec{H}}{\partial t} = \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2} \label{eq:4642245}$$ Apply curl to both sides of Eq. \ref{eq:4757562}; yields Eq. \ref{eq:2392932}. $$\vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = -\mu_0\vec{ \nabla} \times \frac{\partial \vec{H}}{\partial t} \label{eq:2392932}$$ Substitute LHS of Eq. \ref{eq:2392932} into Eq. \ref{eq:4642245}; yields Eq. \ref{eq:2962831}. $$\vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = -\mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2} \label{eq:2962831}$$ Eq. \ref{eq:3984852} is an assumption. $$\rho = 0 \label{eq:3984852}$$ Substitute LHS of Eq. \ref{eq:4857731} into Eq. \ref{eq:3984852}; yields Eq. \ref{eq:2837471}. $$\vec{ \nabla} \cdot \vec{E} = 0 \label{eq:2837471}$$ Eq. \ref{eq:1939485} is an identity. $$\vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E}) \label{eq:1939485}$$ Substitute LHS of Eq. \ref{eq:1939485} into Eq. \ref{eq:2837471}; yields Eq. \ref{eq:3738373}. $$\vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = - \nabla^2 \vec{E} \label{eq:3738373}$$ Substitute LHS of Eq. \ref{eq:3738373} into Eq. \ref{eq:2962831}; yields Eq. \ref{eq:4758592}. $$\nabla^2 \vec{E} = \mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2} \label{eq:4758592}$$ Eq. \ref{eq:4758592} is one of the final equations.