Electromagnetic wave in vacuum - Do we have always ${\bf E}\cdot {\bf B} = 0$ and $E = cB$?

Question

In vacuum, but not necessarily into free space, e.g. inside a waveguide, does a EM wave always fulfill the relations $${\bf E}\cdot {\bf B} = 0, \quad E = cB,$$ with $\bf E$ and $\bf B$ the electric and magnetic fields resp, and $c $ the speed of light in vacuum?

Answer 1

Although $E\cdot B,\,E^2-c^2B^2$ are invariant (they're respectively proportional to $\epsilon^{\mu\nu\rho\sigma}F_{\mu\nu}F_{\rho\sigma},\,F_{\mu\nu}F^{\mu\nu}$), neither is $0$ for all solutions of Maxwell's equations. To take a simple but somewhat unrealistic example, any spacetime-constant values of the vectors $E,\,B$ are compatible with$$\nabla\cdot E=0,\,\nabla\times E=-\dot{B},\,\nabla\cdot B=0,\,\nabla\times B=c^{-2}\dot{E}.$$But these invariant quantities are at least in that example spacetime-constant, which isn't true in general either.

More generally, if $E=E_1,\,B=B_1$ and $E=E_2,\,B=B_2$ are realistic and fit your conjecture, $E=E_1+E_2,\,B_1+B_2$ are realistic and in general don't.