# Poynting Vector Always Perpendicular to Planes of same phase?

I'm calculating this in a linear isotropic medium. For plane waves, the poynting vector is perpendicular to planes with the same phase: Let $$\vec{E}(\vec{x}, t) = \vec{E}_0 e^{i(\vec{k}\vec{x} - \omega t)}$$ Then $\vec{S}$, the poyntingvector, will be parallel to $\vec{k}$. What about the case of a more general, monochromatic wave: $$\vec{E}(\vec{x}, t) = \vec{E}_0(\vec{x}) e^{i(k \chi(\vec{x}) - \omega t)}$$ Will $\vec{S}$ still be paralell to $\nabla \chi$ in this case?

• I was hoping for some kind of calculation. I tried to calculate the poynting vector for a given decomposition $\vec{E} = \vec{E}_0(\vec{x}) e^{i(k\chi(\vec{x})-\omega t}$, but I fail to show the paralellity to $\nabla \chi$ – Quantumwhisp Nov 28 '16 at 17:36