Scalar polarization reduction at a conducting corrugation ($E_y{|}_{\Gamma }=0,{\partial }_n{H}_y{|}_{\Gamma }=0$)

If a metal surface is $z=h(x)$ and the incident field is independent of $y$, Maxwell's equations separate into TE polarization and TM polarization. The metal condition that the tangential electric field vanish sets $E_y=0$ in the TE sector. In the TM sector $\mathbf{E}=i\nabla \times (H_y\hat{\mathbf{y}})/(\omega ϵ)$, so the same condition gives $({\partial }_z-h^{\prime }{\partial }_x)H_y=0$. This explains why the TE problem agrees with scalar Dirichlet acoustics while the TM problem has different rough-surface boundary forcing.