Can anyone give me an idea of what happens in the following situation involving Robin boundary conditions with a complex coefficient.
Lets say we have an incident electromagnetic plane wave $u(x) = e^{i d \cdot x}$ in two dimensions, where $d$ denotes the incident direction, governed by the Helmholtz equation
$$\nabla u + k^2 u = 0,$$ in the upper half space $\mathbb{R}_+^2 = \{x: x_2 > 0\}$ with Robin boundary conditions on the boundary $\partial \mathbb{R}_+^2 = \{x: x_2= 0\}$. The Robin boundary conditions are
$$u + \beta \frac{\partial u}{\partial \nu} = 0,$$
where $\nu$ denotes the normal derivative in the positive $x_2$ direction. Now $\beta$ can be complex and this leads me to believe this boundary condition can be considered lossy and can increase absorption?
I am interested in what kind of happens to the waves when we let $|\beta|$ be very small and then increase it so that $|\beta|$ becomes very large. Can anyone explain how this will affect the scattering/reflection of the incident wave?
If $u(x,y)=Ae^{-i(k_1x_1+k_2x_2)}+Be^{-i(k_1x_1-k_2y_2 )}$, then on the boundary we have
$$A+B+\beta(A(-ik_2)-B(-ik_2))=0$$
whence we have
$$\begin{align} B&=-\frac{1-ik_2\beta}{1+ik_2\beta}A\\\\ &=-\left(\frac{\left(1+k_2\text{Im}(\beta)\right)-i\left(k_2\text{Re}(\beta)\right)}{\left(1-k_2\text{Im}(\beta)\right)+i\left(k_2\text{Re}(\beta)\right)}\right)\,A \end{align}$$
When $\beta =0$, $B=-A$ and as $\beta \to \infty$, $B\to A$.
Note that if $k_2$ is a real number with $k_2\le 0$, then it is easy to see that $\text{Im}(\beta)\ge 0$ in order to ensure that $|B|\le|A|$.
Note that when $k_2=-i/\beta$, $B=0$ and the incident wave is fully absorbed. Since we assume that $k_2\in \mathbb{R}$, this can only happen if $\text{Re}(\beta)=0$.