Wave transmission in dispersive media
Résumé
The aim of this paper is to study the reflection-transmission of geometrical optic rays described by semi-linear symmetric hyperbolic systems such as the Maxwell–Lorentz equations with the anharmonic model of polarisation. The framework is both that of Donnat and Williams since we consider dispersive media and profiles with hyperbolic (imaginary) phases and elliptic phases (complex with non-null real part). We first give hypothesis close to the Maxwell equation. Then we introduce a decomposition for both profile into boundary (tangential) and normal part and we solve the so-called "microscopic" equation of the small scales for each boundary frequency. Then we show that the non-linearities generate harmonics which interact at the boundary and generate new resonant profiles with harmonic tangential frequency. Lastly we make a WKB expansion at any order and give a precise description of the correctors.