= Neumann representation for a radiating Helmholtz half-space
{c}
{title2=$u(x)=-\frac1{2\pi}\int_{\mathbb R^2}\partial_3u(y',0)\frac{e^{ik|x-(y',0)|}}{|x-(y',0)|}\,d^2y'$}
Let $(\Delta+k^2)u=0$ for $x_3>0$, with real $k>0$, $u=O(r^{-1})$ and $r(\partial_ru-iku)\to0$ uniformly in direction. With sufficient boundary regularity, <Green's second identity> on an expanding half-ball gives the title formula. The plane integral is interpreted as the limit over expanding disks when it is not absolutely convergent. Its sign uses the upward derivative $\partial_3u$; the outward <normal derivative> is $-\partial_3u$.
For $G=e^{ik|x-y|}/(4\pi|x-y|)$, $G=O(r^{-1})$ and $(\partial_r-ik)G=O(r^{-2})$ at large $|y|=r$. The hemispherical integrand can be written $u(\partial_r-ik)G-G(\partial_r-ik)u$, whose integral tends to zero by the <Sommerfeld radiation condition>. Add the boundary identities for a source point $x$ and its reflected point $x^*=(x_1,x_2,-x_3)$. On the plane $G(x,y)=G(x^*,y)$ and their normal derivatives are opposite. The terms involving boundary values of $u$ cancel, leaving twice $G\partial_3u$ and hence the title formula. In particular, zero <Neumann boundary data> imply $u=0$ in this radiating class.
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