= Second-order rough-surface scattered field
{title2=$\psi_s^{[2]}(x,0)=2k h\mathcal B h$}
For normal plane-wave incidence, the second-order <Dirichlet boundary condition> gives $\psi_s^{[2]}=-h\partial_z\psi_s^{[1]}$ at the mean plane, since $\partial_z^2\psi^{[0]}$ vanishes there. The <Dirichlet-to-Neumann map for a Helmholtz half-space> then yields $2k h\mathcal B h$. This nonlocal expression is not generally $2k^2h^2$; that local limit needs roughness varying slowly compared with the wavelength.
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