Sommerfeld's half-plane diffraction problem

Problem

Let $k > 0$, let $\Gamma = \set{(x, 0) : x \ge 0}$ be a half-line in $\R^2$, and let $u_i(x, y) = e^{i k (x \cos\alpha + y \sin\alpha)}$ be an incident plane wave with $0 < \alpha < \pi$. Find $u = u_i + u_s$ on $\R^2 \setminus \Gamma$ such that:

1. $u$ solves the Helmholtz equation $\Delta u + k^2 u = 0$ in $\R^2 \setminus \Gamma$;
2. $u = 0$ on both sides of $\Gamma$ (the Dirichlet case), or $\partial u / \partial y = 0$ on both sides of $\Gamma$ (the Neumann case);
3. the scattered field $u_s$ is outgoing: away from the directions of the reflected wave and of the shadow boundary, it satisfies the Sommerfeld radiation condition;
4. $u$ has finite energy near the edge: $u$ and $\nabla u$ are square integrable on every bounded set.