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

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.