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

Consider the Helmholtz equation $(\Delta +k^2)u=0$ in $z>0$, with $k>0$, boundary value $u(x,0)=g(x)$, and upward outgoing or decaying propagation. Taking the tangential Fourier transform, with frequency $\xi$, givesThus the upward normal derivative is the Fourier multiplier $i\beta (\xi )$. The outward unit normal of the upper half-space points downward, so its Dirichlet-to-Neumann map has multiplier $-i\beta (\xi )$ instead. The normal orientation and the square-root branch determine both signs.

For $|\xi |<k$ the modes propagate; for $|\xi |>k$ they are evanescent waves decaying with height. At high frequency $\beta (\xi )\sim i|\xi |$, so the map grows in magnitude like one tangential derivative. Small-amplitude boundary roughness can therefore have a large derivative effect when it varies on a short spatial scale.