Diffraction of elastic waves by edges

Katsnelson, Vitaly

Annales de l'Institut Fourier, Tome 68 (2018) no. 4, pp. 1447-1517.

Nous examinons la diffraction des singularités associées aux solutions de l’équation élastique linéaire dans des domaines contenant des singularités aux bords. Ces domaines sont définis comme le produit d’un domaine régulier et d’un cône sur une fibre compacte. Concernant la solution fondamentale, le pôle initial génère une onde de pression (p-wave) ainsi qu’une onde de cisaillement secondaire plus lente (s-wave). Si le pôle initial est situé près du bord, nous montrons que lorsque l’onde de pression frappe le bord, les ondes de pression et de cisaillement diffractées (c.a.d en quelques mots, leurs singularités ainsi que les singularités des ondes de pression incidentes ne sont pas les limites des rayons associés à la vélocité des ondes de pression manquant le bord de peu) sont plus faible au sens de Sobolev que les ondes de pression incidentes. Nous montrons de plus un résultat analogue pour une onde de cisaillement frappant le bord, et nous donnons des résultats pour des situations plus générales.

We investigate the diffraction of singularities of solutions to the linear elastic equation on manifolds with edge singularities. Such manifolds are modeled on the product of a smooth manifold and a cone over a compact fiber. For the fundamental solution, the initial pole generates a pressure wave (p-wave), and a secondary, slower shear wave (s-wave). If the initial pole is appropriately situated near the edge, we show that when a p-wave strikes the edge, the diffracted p-waves and s-waves (i.e. loosely speaking, their singularities together with the singularities of the incoming $p$-wave are not limits of rays associated to the pressure wave speed which just miss the edge) are weaker in a Sobolev sense than the incident p-wave. We also show an analogous result for an s-wave that hits the edge, and provide results for more general situations.

Classification : 58J47, 35A21, 35L51
Keywords: Propagation of singularities, elastic wave equation, diffraction
Mot clés : propagation de singularités, l’équation élastique, diffraction

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