This paper gives a brief overview of some configurations in which high-frequency wave propagation modelled by Helmholtz equation gives rise to solutions that vary rapidly across thin layers. The configurations are grouped according to their mathematical structure and tractability and one of them concerns a famous open problem of mathematical physics.

In this article we are interested in the rigorous construction of WKB expansions for hyperbolic boundary value problems in the strip $\mathbb{R}^{d-1}\times [0,1]$. In this geometry, a new inversibility condition has to be imposed to construct the WKB expansion. This new condition is due to selfinteraction phenomenon which naturally appear when several boundary conditions are imposed. More precisely, by selfinteraction we mean that some rays can regenerated themselves after some rebounds against the sides of the strip. This phenomenon is not new and has already been studied in Benoit (Geometric optics expansions for hyperbolic corner problems, I: self-interaction phenomenon, Anal. PDE9(6) (2016), 1359–1418), Sarason and Smoller (Geometrical optics and the corner problem, Arch. Rat. Mech. Anal.56 (1974/75), 34–69) in the corner geometry. In this framework the existence of such selfinteracting rays is linked to specific geometries of the characteristic variety and may seem to be somewhat anecdotal. However for the strip geometry such rays become generic. The new inversibility condition, used to construct the WKB expansion, is a microlocalized version of the one characterizing the uniform in time strong well-posedness (Benoit, Lower exponential strong well-posedness of hyperbolic boundary value problems in a strip (preprint)). It is interesting to point here that such a situation already occurs in the half space geometry (Kreiss, Initial boundary value problems for hyperbolic systems, Comm. Pure Appl. Math.23 (1970), 277–298).

The paper describes some qualitative properties of minimizers on a manifold $\mathcal M$ endowed with a discontinuous metric. The discontinuity occurs on a hypersurface $\Sigma$ disconnecting $\mathcal M$. Denote by $\Omega_1$ and $\Omega_2$ the open subsets of $\mathcal M$ such that $\mathcal M\setminus\Sigma=\Omega_1\cup\Omega_2$. Assume that $\overline\Omega_1$ and $\overline\Omega_2$ are endowed with metrics $\left\langle\cdot,\cdot\right\rangle_{\left(1\right)}$ and $\left\langle\cdot,\cdot\right\rangle_{\left(2\right)}$, respectively, such that $\overline\Omega_i (i=1, 2)$ is convex or concave. The existence of a minimizer of the length functional on curves joining two given points of $\mathcal M$ is proved. The qualitative properties obtained allows the refraction law in a very general situation to be described.

We propose the following problem. Given an array of vertical mirrors that simultaneously transmit and reflect, and a single incoming ray of light, describe the configuration of all light rays that are generated. We solve the problem here for a certain infinite configuration of mirrors; the solution involves the winning positions (a(n), b(n)) of Wythoff's game.