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- W1966123669 abstract "In this paper, we compute 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. Our 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, we prove 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. Second, we study the particular case when the index is constant in each media, for which the analysis goes further: we prove that the interface does not capture energy asymptotically in this case. Dans cet article, nous calculons la limite haute fréquence de l'équation de Helmholtz avec terme source, dans le cas d'un indice de réfraction discontinu le long d'une interface franche séparant deux milieux non bornés. La propagation asymptotique de l'énergie est étudiée en terme de mesures de Wigner. Nous obtenons deux résultats. D'abord, dans le cas général, en faisant des hypothèses géométriques sur l'indice et en supposant qu'il n'y ait pas d'énergie captée par l'interface, nous montrons que la mesure de Wigner vérifie une équation de transport stationnaire avec terme source. Ainsi, cette mesure est caractérisée comme l'intégrale, le long des rayons de l'optique géometrique et jusqu'en temps infini, de la source d'énergie. Ce résultat code le phénomène de réfraction. Ensuite, nous étudions le cas particulier où l'indice est constant dans chaque milieu, pour lequel nous montrons que l'interface ne capte pas d'énergie asymptotiquement." @default.
- W1966123669 created "2016-06-24" @default.
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- W1966123669 date "2007-02-01" @default.
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- W1966123669 title "High frequency limit of Helmholtz equations: refraction by sharp interfaces" @default.
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- W1966123669 doi "https://doi.org/10.1016/j.matpur.2006.11.002" @default.
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