= Oblique coherent reflection from a stationary rigid rough surface
{title2=$r_N=1-\frac1{\pi\beta}\int\frac{(k^2-\alpha q)^2}{\beta(q)}S_h(q-\alpha)\,dq+\cdots$}
For a zero-mean stationary rough height, the mean first-order correction vanishes. At second order, the boundary derivative is $-h\partial_z^2\psi_1+h'\partial_x\psi_1$. Averaging this expression using the <power spectrum of surface height> gives the displayed coherent reflection coefficient. Thus the <root mean square> height alone does not fix the coherent mean: the height correlation spectrum matters. For roughness varying slowly enough that $q\simeq\alpha$ in the integral, $r_N\simeq1-2\beta^2\sigma^2$. A local random-height phase average needs this additional approximation and is not a general consequence of stationarity.
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