= Dirichlet-to-Neumann map for a Helmholtz half-space
{c}
{title2=$\partial_z\mathcal E g\big|_0=i\mathcal B g$}
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 $\xi$, gives
$$
\widehat u(\xi,z)=e^{i\beta(\xi)z}\widehat g(\xi),\qquad
\beta(\xi)=\begin{cases}\sqrt{k^2-|\xi|^2},&|\xi|\leq k,\\i\sqrt{|\xi|^2-k^2},&|\xi|>k.\end{cases}
$$
Thus 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.
Back to article page
