= Solution
Let the stationary height have <covariance function> $C_h(\xi)=\langle h(x)h(x+\xi)\rangle$, with $C_h(0)=\sigma^2$. Define the <power spectrum of surface height> by
$$
S_h(q)=\int C_h(\xi)e^{-iq\xi}d\xi,\qquad C_h(\xi)=\frac1{2\pi}\int S_h(q)e^{iq\xi}dq.
$$
For real stationary heights this spectrum is even and nonnegative. The <Fourier multiplier> identity in part (c) gives
$$
\langle h(x)\mathcal B h(x)\rangle=\frac1{2\pi}\int\beta(q)S_h(q)dq.
$$
Thus <coherent reflection from a stationary rough surface> at normal incidence is
$$
\boxed{\langle\psi_s(x,0)\rangle_{\text{through second order}}=-1+\frac{k}{\pi}\int_{\mathbb R}\beta(q)S_h(q)dq.}
$$
Require 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:
$$
\langle\psi_s(x,0)\rangle=-1+\frac{k}{\pi}\int_{|q|<k}\sqrt{k^2-q^2}\,S_h(q)dq+\frac{ik}{\pi}\int_{|q|>k}\sqrt{q^2-k^2}\,S_h(q)dq
$$
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 $\beta\simeq k$ and
$$
\langle\psi_s(x,0)\rangle\simeq-1+2k^2\sigma^2.
$$
This 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.
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