Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 72 / 2

Work above the surface, with $e^{-i\omega t}$ time dependence. Define the scattered field by $\psi ={\psi }_i+{\psi }_s$, so it includes the reflection from a flat surface. For small-height Dirichlet scattering, write $h=\epsilon \eta$ and expandThe zeroth-order total field ${\psi }^{[0]}={\psi }_i+{\psi }_s^{[0]}$ satisfies the Dirichlet boundary condition ${\psi }^{[0]}(x,0)=0$. Taylor expansion at the perturbed boundary givesThus the first-order rough-surface scattered field has mean-plane boundary data $g_1(x)=-h(x){\partial }_z{\psi }^{[0]}(x,0)$.

Use the outgoing angular spectrum to solve this boundary-value problem. For $\hat{g}(q)=\int g(x)e^{-iqx}dx$, letThe branch ensures upward propagation or upward evanescent decay. Each component solves the Helmholtz equation, and $\mathcal{E}g$ has trace $g$ at $z=0$. ConsequentlyThis gives the scattered field through first order by adding the two contributions. If one reserves “rough scattered field” for the non-specular correction, it is ${\psi }_s^{[1]}$ alone; the convention here keeps the flat reflection as well.

The expansion is in height for a fixed sufficiently regular profile. The condition $|kh|\ll 1$ controls the incident wave's height expansion, but very short spatial scales can create large evanescent normal derivatives. The surface regularity and relevant spectral moments must also control the subsequent boundary expansions; small amplitude alone is not a uniform guarantee for arbitrarily fine roughness.

For the incident acoustic plane wave, set $q_0=k\sin \theta$ and ${\beta }_0=k\cos \theta >0$. The flat reflected field and total field areTherefore ${\partial }_z{\psi }^{[0]}(x,0)=-2i{\beta }_0{e}^{i{q}_{0}x}$ and the first-order rough-surface scattered field isIt is linear in the height. Since $⟨h(x)⟩=0$, its mean vanishes, with the expectation interpreted through finite windows or stationary spectral distributions when needed. HenceThe coherent first-order reflection is the flat-surface reflection. If the field symbol is instead used only for the rough correction, its first-order mean is zero. Stationarity ensures the coherent reflection retains the incident horizontal wavenumber, but zero mean height already explains the vanishing linear correction. The nonzero root mean square height does not enter this mean at first order; it does enter the fluctuating reflected field and its intensity.

For the second-order rough-surface scattered field, the Dirichlet boundary condition expanded at the mean plane isAt normal incidence, ${\psi }^{[0]}=e^{-ikz}-e^{ikz}$, so its second normal derivative vanishes at zero. Define the Dirichlet-to-Neumann map for a Helmholtz half-space through the Fourier multiplier $\mathcal{B}$:The first-order trace is $2ikh$, hence ${\partial }_z{\psi }_s^{[1]}(x,0)=-2k\mathcal{B}h(x)$. It follows thatwhere $h=O(\epsilon )$ with fixed regular profile. In integral notation the quadratic contribution isThe second-order field above the mean plane is $\mathcal{E}[2kh\mathcal{B}h]$ added to $-e^{ikz}+\mathcal{E}[2ikh]$. No local replacement of $\beta (q)$ by $k$ has been made; such a replacement would be an additional long-spatial-scale approximation.

The physical surface trace and reference-plane trace are distinct. At the actual rough boundary $z=h(x)$, the condition itself says ${\psi }_s(x,h(x))=-e^{-ikh(x)}$, soThe first boxed expression is the reference-plane trace needed in part (d), at $z=0$, using the perturbative continuation where that plane lies below the actual boundary; the second answers the literal “at the surface” wording if it means the physical boundary. Taylor-expanding the first expression and its normal derivatives from $z=0$ to $z=h$ reproduces the second, so there is no contradiction between them.

Let the stationary height have covariance function $C_h(\xi )=⟨h(x)h(x+\xi )⟩$, with $C_h(0)={\sigma }^2$. Define the power spectrum of surface height byFor real stationary heights this spectrum is even and nonnegative. The Fourier multiplier identity in part (c) givesThus coherent reflection from a stationary rough surface at normal incidence isRequire the corresponding weighted spectral moment to exist. If the stationary process has a spectral measure rather than a density, the same formula uses that measure with the matching normalization.

Splitting the propagating and evanescent parts makes the effect clear:through second order. The positive real correction reduces the magnitude of the initially negative unit coherent reflection at this order, as some reflection becomes diffuse. The evanescent part produces a coherent wave phase correction. In contrast, the first-order mean was exactly the flat reflected wave.

The height-correlation dependence of coherent reflection cannot generally be determined from $\sigma$ alone: the quadratic term weights the whole spectrum by $\beta (q)$, while ${\sigma }^2=(2\pi {)}^{-1}\int S_h(q)dq$ is unweighted. If the roughness varies only on scales much longer than the wavelength, so its spectrum is concentrated at $|q|\ll k$, then $\beta \simeq k$ andThis is a useful limiting formula, not the general second-order answer under only small-height assumptions. Also the mean field sampled at the moving physical boundary is $-1+k^2{\sigma }^2/2$ through second order, from part (c), and is a different observable.