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

Take $e^{-i\omega t}$ time dependence and measure angles from the upward normal. Write $p_i=k\sin {\theta }_i$, $q_i=k\cos {\theta }_i>0$ and choose incident amplitude $A_i$. Separate the flat Dirichlet reflection from the rough correction:Taylor expansion of the boundary condition about $z=0$ givesSince ${\partial }_z{\psi }^{[0]}(x,0)=-2i{q}_i{A}_i{e}^{i{p}_{i}x}$, the induced trace isUse $\hat{h}(Q)=\int h(x)e^{-iQx}dx$ and $\beta (\xi )=\sqrt{k^2-{\xi }^2}$ with the outgoing branch: $\beta >0$ for $|\xi |<k$ and $\mathrm{Im}\beta >0$ outside. The outgoing angular spectrum solves the Helmholtz equation with that trace, giving the first-order rough-surface scattered fieldIf “scattered field” includes the specular reflection, add $-A_i{e}^{i{p}_{i}x+i{q}_{i}z}$ to this rough correction. Fourier transforms of an infinite random surface are generalized objects; finite-window transforms give an ordinary integral and are also needed for its far-field intensity. The expansion assumes a fixed regular surface shape scaled to small height. Highly oscillatory profiles can require additional control of slopes and spectral moments; small height alone is not a uniform error estimate for every profile.

Illuminate a surface window of length $D$, and define ${\hat{h}}_D(Q)={\int }_{-D/2}^{D/2}h(x)e^{-iQx}dx$. In the far field put $(x,z)=r(\sin {\theta }_s,\cos {\theta }_s)$, with $|{\theta }_s|<\pi /2$ away from grazing. The spectral phase $\xi \sin {\theta }_s+\beta (\xi )\cos {\theta }_s$ has stationary point ${\xi }_s=k\sin {\theta }_s$, value $k$, and second derivative $-1/(k{\cos }^2{\theta }_s)$. Stationary phase therefore givesLet $C_h(s)=⟨h(x+s)h(x)⟩$ and $S_h(Q)=\int C_h(s)e^{-iQs}ds$. Statistical stationarity gives exactlyIf $C_h$ is integrable, division by $D$ and dominated convergence yield $⟨|{\hat{h}}_D(Q){|}^2⟩/D\to S_h(Q)$. Thus the leading rough-surface far-field intensity per illuminated length isThis uses the field-intensity convention $I=|\psi {|}^2$; if $\psi$ is acoustic pressure, multiply by $1/(2\rho c)$ for physical far-field energy flux. The variance normalization is ${\sigma }^2=C_h(0)=(2\pi {)}^{-1}\int S_h(Q)dQ$. Consequently the diffuse intensity is of order ${\sigma }^2$ and samples the surface spectrum at the tangential momentum transfer. Its value is not determined by the r.m.s. height alone.

The finite window must first be in its far field, for example $r\gg k{D}^2$, before interpreting the large-window intensity density. An infinitely long stationary surface has infinite illuminated power and no finite total far-field intensity at a fixed point. For nonintegrable correlations the analogous result is a spectral measure, possibly with atoms. Zero mean height makes the linear rough mean field zero, but does not make its mean intensity zero; the flat specular reflection remains separate.

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.