Electromagnetic wave equation with time varying phase velocity

Question

I am attempting to solve Maxwell's equations for the electric field of an electromagnetic wave, with time varying phase velocity, propagating within a medium within unity permeability but time varying permittivity. So far I have come up with the following derivations \begin{align} \nabla \times(\nabla\times E(t)) &= -\mu_0\frac{\partial}{\partial t}\left(\nabla \times H\right)\\ \nabla^2 E &= \mu_0 \frac{\partial^2\varepsilon E}{\partial t^2}\\ &= \frac{\partial^2}{\partial t^2}\left(v_{ph}^2E\right) \end{align}

Are there any well-known solutions to this form of the electromagnetic wave equation? Also if there are any texts that review this form of the electromagnetic wave equation I would be ever so thankful if someone could list them.

Answer 1

Some suggestions: (1) Keep the time-dependence of the permittivity and evaluate the time derivative of the product $\epsilon(t) E(t)$. (2) Expand the time dependence of $\epsilon(t)$ as a sum of single frequency functions. The resulting differential equation is still linear and should not be too difficult to solve.