Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 70 / 3 / c / Solution

Translation invariance in $y$, together with a $y$-independent incident field, permits a $y$-independent solution. In homogeneous space Maxwell's equations then separate into two scalar sectors.

For a TE wave, set $\mathbf{E}=\hat{\mathbf{y}}{E}_y(x,z)$. Its component satisfies $({\partial }_x^2+{\partial }_z^2+k^2)E_y=0$. The $y$ direction is tangent to the conducting surface, so vanishing tangential electric field imposes $E_y(x,h(x))=0$. Therefore TE scattering is exactly the two-dimensional Dirichlet scalar problem, with $E_y$ replacing the acoustic scalar. Its first-order boundary expansion and angular-spectrum formula are those above.

For a TM wave, set $\mathbf{H}=\hat{\mathbf{y}}{H}_y(x,z)$. Maxwell's equation $\nabla \times \mathbf{H}=-i\omega ϵ\mathbf{E}$ givesA surface tangent is $(1,0,h^{\prime })$, so vanishing tangential electric field givesThus TM also reduces to two dimensions, but to a Neumann problem, not the Dirichlet problem in part (a). This is scalar polarization reduction at a conducting corrugation.

For example the flat TM reflection has positive sign, $H^{[0]}=A_i{e}^{i{p}_{i}x}(e^{-i{q}_{i}z}+e^{i{q}_{i}z})$. Its linearized rough forcing isIn the transform domain this becomes $2A_i(k^2-p_i\xi )\hat{h}(\xi -p_i)$, and propagation divides by $i\beta (\xi )$ rather than prescribing the electric Dirichlet trace. The explicit slope term and different reflection sign show why simply reusing the TE boundary formula for TM would be incorrect.