= Scalar polarization reduction at a conducting corrugation
{title2=$E_y|_\Gamma=0,\quad\partial_nH_y|_\Gamma=0$}
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 $\mathbf E=i\nabla\times(H_y\widehat{\mathbf y})/(\omega\epsilon)$, so the same condition gives $(\partial_z-h'\partial_x)H_y=0$. This explains why the TE problem agrees with scalar Dirichlet acoustics while the TM problem has different rough-surface boundary forcing.
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