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Pré-Publication, Document De Travail Année : 2023

Reflection and transmission problems for high-frequency waves at a randomly perturbed interface: generalized Snell's laws

Résumé

In this paper we consider the problem of reflexion and transmission of a high-frequency pulse at a rapidly oscillating rough interface with general mixing properties. Using an asymptotic analysis based on a separation of scale technique, corresponding to the paraxial (parabolic) scaling, the specular and speckle (diffusive) components of the reflected and transmitted signals are precisely characterized. A critically scaled interface is considered, in the sense that the amplitudes of the interface fluctuations and the central wavelength are of the same order. If the correlation length of the interface fluctuations and the beam width are of the same order, random specular components are observed for the reflected and transmitted wave, but no speckle component. That is, the reflected and transmitted fields are to leading order confined essentially to the specular cones and these fields exhibit relatively small random intensity fluctuations. The situation with a correlation length smaller than the beam width leads to a specular homogenization situation so that the rough interface can be approximated by an effective flat interface providing deterministic specular reflected and transmitted paraxial wave cones. However, in this situation, there are now relatively broad cones (containing the specular cones) where the wavefields form speckle patterns whose total energy is leading order and we give the two-points correlation functions for these speckle patterns. We also present a central limit theorem type result for the speckle patterns and show that they can be modeled as Gaussian random fields. These descriptions allow the derivation of generalized Snell's laws of refraction and transmission depending on an effective scattering operator at the interface.
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Dates et versions

hal-04317227 , version 1 (01-12-2023)

Identifiants

  • HAL Id : hal-04317227 , version 1

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Christophe Gomez, Knut Sølna. Reflection and transmission problems for high-frequency waves at a randomly perturbed interface: generalized Snell's laws. 2023. ⟨hal-04317227⟩
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