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electric field wave equation: from time dependent to time independent

Generated by the Physics Derivation Graph. Eq. \ref{eq:4758592} is an initial equation. \begin{equation} \nabla^2 \vec{E} = \mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2} \label{eq:4758592} \end{equation} Eq. \ref{eq:9393848} is an initial equation. \begin{equation} \vec{E} = E( \vec{r},t) \label{eq:9393848} \end{equation} Judicious choice as a guessed solution to Eq. \ref{eq:4758592} is \begin{equation} E( \vec{r},t) = E( \vec{r})\exp(i \omega t) \label{eq:3994928} \end{equation} Substitute LHS of Eq. \ref{eq:9393848} into Eq. \ref{eq:4758592}; yields Eq. \ref{eq:3839493}. \begin{equation} \nabla^2 E( \vec{r},t) = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E( \vec{r},t) \label{eq:3839493} \end{equation} Substitute LHS of Eq. \ref{eq:3994928} into Eq. \ref{eq:3839493}; yields Eq. \ref{eq:1029393}. \begin{equation} \nabla^2 E( \vec{r})\exp(i \omega t) = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E( \vec{r})\exp(i \omega t) \label{eq:1029393} \end{equation} Differentiate Eq. \ref{eq:1029393} with respect to \(t\); yields Eq. \ref{eq:2939392}. \begin{equation} \nabla^2 E( \vec{r})\exp(i \omega t) = i \omega \mu_0 \epsilon_0 \frac{\partial}{\partial t} E( \vec{r})\exp(i \omega t) \label{eq:2939392} \end{equation} Differentiate Eq. \ref{eq:2939392} with respect to \(t\); yields Eq. \ref{eq:4958573}. \begin{equation} \nabla^2 E( \vec{r})\exp(i \omega t) = - \omega^2 \mu_0 \epsilon_0 E( \vec{r})\exp(i \omega t) \label{eq:4958573} \end{equation} Eq. \ref{eq:4949582} is an initial equation. \begin{equation} \epsilon_0 \mu_0 = \frac{1}{c^2} \label{eq:4949582} \end{equation} Substitute LHS of Eq. \ref{eq:4958573} into Eq. \ref{eq:4949582}; yields Eq. \ref{eq:9495903}. \begin{equation} \nabla^2 E( \vec{r})\exp(i \omega t) = - \frac{\omega^2}{c^2} E( \vec{r})\exp(i \omega t) \label{eq:9495903} \end{equation} Simplify Eq. \ref{eq:9495903}; yields Eq. \ref{eq:3949492}. \begin{equation} \nabla^2 E( \vec{r}) = - \frac{\omega^2}{c^2} E( \vec{r}) \label{eq:3949492} \end{equation} Eq. \ref{eq:3949492} is one of the final equations.