There is alot of material out there on explicit resonant frequencies and modes for Dirichlet and Neumann boundary value problems (BVPs), but I can't seem to find anything that covers the case of transmission BVPS. Say we have the Helmholtz equation in some domain $D$, $$ \Delta u(x) + k^2 u(x) = 0, \quad \quad x \in D, $$ If we apply Dirichlet or Neumann boundary conditions we can find the eigenfunctions and eigenvalues, and the eigenvalues. The eigenvalues represent resonant frequencies and the corresponding eigenfunctions are modes for the domain, which are standing waves that feature and integer number of cycles.

But what about the case when we a domain $D$ in freespace, and the boundary conditions are transmission conditions, that is

\begin{align} & \Delta u(x) + k_1^2 u(x) = 0, \quad \quad x \in \mathbb{R}^d\setminus D, \\ & \Delta u(x) + k_2^2 u(x) = 0, \quad \quad x \in D, \\ & u|_+ = u|_-, \quad \quad \frac{1}{\rho_1}\frac{\partial u}{\partial n}|_+ = \frac{1}{\rho_2}\frac{\partial u}{\partial n}|_-, \\ & u^s = u - u^{in}, \end{align} where $u^s$ is the scattered wave and $u^{in}$ is the incident wave. In this case the function $u$ is not fixed at the boundary so do we still have eigenfunctions which correspond to standing waves featuring an integer number of cycles?

If so, how can we find the eigenfrequencies and eigenfunctions in the simple one-dimensional case where $D$ is the interval $[0,1]$ and the incident wave is $u^{in} = u_0e^{ik_1 x}$? Can an analytic solution be found or can this only be done numerically?


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