= Solution
Write the unit-amplitude incident <plane wave> as $\psi_i=e^{ip_ix-iq_iz}$, with $q_i=\sqrt{k^2-p_i^2}>0$. The flat Dirichlet reflected field is $\psi_s^{[0]}=-e^{ip_ix+iq_iz}$, so the flat total field vanishes at $z=0$ and has <derivative> $\partial_z\psi^{[0]}(x,0)=-2iq_ie^{ip_ix}$.
For a controlled height expansion take $f=\varepsilon h$ with fixed regular bounded shape $h$. <Taylor expansion> at the moving boundary yields
$$
0=\psi^{[0]}(x,0)+f(x)\partial_z\psi^{[0]}(x,0)+\psi_s^{[1]}(x,0)+O(\varepsilon^2).
$$
Thus the first-order scattered trace is $\psi_s^{[1]}(x,0)=2iq_if(x)e^{ip_ix}$, and its transform follows by the <Translation property of the Fourier transform>:
$$
\boxed{\widehat\psi_s^{[1]}(\nu)=2iq_i\widehat f(\nu-p_i),\qquad
\widehat\psi_{\rm sc}(\nu)=-2\pi\delta(\nu-p_i)+2iq_i\widehat f(\nu-p_i)+O(\varepsilon^2)}.
$$
The first formula is the roughness correction; the second includes the zeroth-order reflection. The small-height assumption controls the <Taylor expansion> for a fixed profile; arbitrarily fine surface scales can require additional <derivative> control, not merely a small height at fixed $k$.
If the mean roughness height is a constant $\bar f$, as for stationary roughness, then $\langle\widehat f(\nu-p_i)\rangle=2\pi\bar f\delta(\nu-p_i)$. Consequently the <specular first-order mean reflection> is
$$
\boxed{\langle\psi_{\rm sc}(x,z)\rangle
=(-1+2iq_i\bar f)e^{ip_ix+iq_iz}+O(\varepsilon^2)}.
$$
Its tangential <wavenumber> is unchanged and its normal <wavenumber> is reversed relative to incidence, which is precisely <specular reflection>. For zero mean height the first-order mean correction is zero, but the full mean reflected field is not zero: it still contains the flat reflection.
Small height alone does not imply the statistical conclusion. For example, deterministic mean height $\bar f(x)=a\cos Kx$ with $|ka|\ll1$ produces first-order mean sidebands at $p_i\pm K$ and is not purely specular. Constant mean is sufficient at first order; <stationarity> supplies it and, with suitable translation-invariant statistics, constrains higher-order coherent reflection as well.
Back to article page
