High frequency limit of Helmholtz equations: refraction by sharp interfaces
The author composes the high frequency limit of the Helmholtz equation with source term, in the case of a refraction index that is discontinuous along a sharp interface between two unbounded media. The asymptotic propagation of energy is studied using Wigner measures. Her result is twofold. First, in the general case, assuming some geometrical hypotheses on the index and assuming that the interface does not capture energy asymptotically, she proves that the limiting Wigner measure satisfies a stationary transport equation with source term. As a consequence, the Wigner measure is characterized as the integral, along the rays of geometrical optics and up to infinite time, of the energy source. This result encodes the refraction phenomenon. Secondly, the author studies the particular case when the index is constant in each medium, for which the analysis goes further: she proves that the interface does not capture energy asymptotically in this case. The content of the chapters is as follows: In chapter 2 the so-called Wigner transform is introduced. In chapter 3 uniform bounds on a certain sequence \((\mu^{\varepsilon})\) are established. In chapter 4, the transport equations are established. In chapter 5 the refraction results are proved in the case of two homogeneous media. Chapter 6 is devoted to the proof of the main result in the case of a general index of refraction. The proof uses the so-called Snell-Descartes semigroup. Chapter 7 is devoted to the proof of the radiation conditions at infinity in the homogeneous case.
scientific article; zbMATH DE number 5134238