# Time dependent permittivity/permeability

Hello,

I am following up on the questions I have already asked you here : How to create time varying material property ?

I need more clarification if possible.

I am working on a project where we study the propagation of an electromagnetic wave in a medium where the dielectric permittivity and the magnetic permeability vary according to time and space.

The problem consists of a rectangle where there is a sinusoidal electric field excitation on the left edge, PMC conditions on the top and bottom edges and an absorbing condition on the right edge (cf. image). The wave crosses 3 media with the following properties :

- ε0, μ0
- ε = ε(x,t) = ε0*(εr0+δε
*cos(ωt-kx)), μ = μ(x,t) = μ0*(μr0+δμ*cos(ωt-kx)) - ε0, μ0

I wish to adapt the variational formulation for this problem but I have some difficulties. Here is my weak formulation :

My questions are therefore:

- Is it enough to take into account the electric excitation on the left edge as a constraint? Or should we add a term in the variational formulation?
- I don't see the possibility of writing the Laplacian operator, is there one? If not, do I have to go through a Green formula?
- How to write the temporal derivatives? I understand that "DtDof" only applies to the unknown in "Dof" and we cannot use "Dt".
- I have defined permittivity and permeability in "Function" as follows:
`epsilon [Domain 1] = eps0;`

`epsilon [Domain 2] = eps0 * (epsr0 + d_eps*Cos[w*$Time-k*X[]];`

`epsilon [Domain 3] = eps0;`

Is this the right way to go? I used X[] instead of $X because $X didn't seem to work. - Should we work in BF_Node or in BF_PerpendicularEdge?

Best regards,

KOENIG Jules