= Specular first-order mean reflection
{title2=$\langle\psi_{\rm sc}\rangle=(-1+2iq_i\langle f\rangle)e^{ip_ix+iq_iz}$}
For incident amplitude one, $\psi_i=e^{ip_ix-iq_iz}$ with $q_i>0$, Taylor expansion of a small-height <Dirichlet boundary condition> gives $\psi_{\rm sc}^{[1]}(x,0)=2iq_if(x)e^{ip_ix}$. If the mean height is constant, its mean transform is supported only at tangential wavenumber $p_i$, giving the displayed specular mean through first order. Small height alone is insufficient: deterministic mean height $a\cos Kx$ produces mean sidebands $p_i\pm K$. Stationary roughness has constant mean and therefore satisfies the first-order requirement.
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