Small-height Neumann scattering (${\partial }_z\psi -h^{\prime }{\partial }_x\psi =0(z=h(x))$)

For a rigid rough boundary and incident pressure $A{e}^{i\alpha x-i\beta z}$, use time dependence $e^{-i\omega t}$ and ${\alpha }^2+{\beta }^2=k^2$. The flat reflected wave is $A{e}^{i\alpha x+i\beta z}$. Expanding the Neumann boundary condition at $z=0$ gives ${\partial }_z{\psi }_1=2A{e}^{i\alpha x}({\beta }^{2}h+i\alpha h^{\prime })$. For height transform $\hat{h}(q)=\int h(x)e^{-iqx}dx$, the outgoing correction is $(2A/(2\pi i))\int (k^2-\alpha q)\hat{h}(q-\alpha )e^{iqx+i\beta (q)z}/\beta (q)dq$, with $\beta (q)=\sqrt{k^2-q^2}$ on the outgoing sheet. Controlled height expansions also require adequate slope and spectral regularity; small incident phase displacement alone does not bound every normal-derivative correction.

For a zero-mean stationary rough height, the mean first-order correction vanishes. At second order, the boundary derivative is $-h{\partial }_z^2{\psi }_1+h^{\prime }{\partial }_x{\psi }_1$. 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, $r_N\simeq 1-2{\beta }^2{\sigma }^2$. A local random-height phase average needs this additional approximation and is not a general consequence of stationarity.