Dirichlet-to-Neumann map for a Helmholtz half-space (${\partial }_z\mathcal{E}g{|}_0=i\mathcal{B}g$)

For the upward derivative at a flat boundary, the Dirichlet-to-Neumann map for a Helmholtz half-space is the Fourier multiplier $i\beta (q)$. The outward normal of the upper half-space points downward and has the opposite sign. Its evanescent multiplier grows like $|q|$, explaining why fine roughness matters even when the height is small.