= Small-height Neumann scattering
{title2=$\partial_z\psi-h'\partial_x\psi=0\quad(z=h(x))$}
For a rigid rough boundary and incident pressure $Ae^{i\alpha x-i\beta z}$, use time dependence $e^{-i\omega t}$ and $\alpha^2+\beta^2=k^2$. The flat reflected wave is $Ae^{i\alpha x+i\beta z}$. Expanding the <Neumann boundary condition> at $z=0$ gives $\partial_z\psi_1=2Ae^{i\alpha x}(\beta^2h+i\alpha h')$. For height transform $\widehat h(q)=\int h(x)e^{-iqx}dx$, the outgoing correction is $(2A/(2\pi i))\int (k^2-\alpha q)\widehat h(q-\alpha)e^{iqx+i\beta(q)z}/\beta(q)\,dq$, 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.
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