Diffraction of elastic waves by edges
Annales de l'Institut Fourier, Volume 68 (2018) no. 4, p. 1447-1517
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.
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.
Received : 2016-11-14
Revised : 2017-07-26
Accepted : 2017-09-14
Published online : 2018-11-23
DOI : https://doi.org/10.5802/aif.3192
Classification:  58J47,  35A21,  35L51
Keywords: Propagation of singularities, elastic wave equation, diffraction
@article{AIF_2018__68_4_1447_0,
     author = {Katsnelson, Vitaly},
     title = {Diffraction of elastic waves by edges},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {68},
     number = {4},
     year = {2018},
     pages = {1447-1517},
     doi = {10.5802/aif.3192},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2018__68_4_1447_0}
}
Diffraction of elastic waves by edges. Annales de l'Institut Fourier, Volume 68 (2018) no. 4, pp. 1447-1517. doi : 10.5802/aif.3192. https://aif.centre-mersenne.org/item/AIF_2018__68_4_1447_0/

[1] Dencker, Nils On the propagation of polarization sets for systems of real principal type, J. Funct. Anal., Tome 46 (1982) no. 3, pp. 351-372 | Article | MR 661876 | Zbl 0487.58028

[2] Dencker, Nils On the propagation of polarization in conical refraction, Duke Math. J., Tome 57 (1988) no. 1, pp. 85-134 | Article | MR 952227 | Zbl 0669.35116

[3] Duistermaat, Johannes Jisse; Hörmander, Lars Fourier integral operators. II, Acta Math., Tome 128 (1972) no. 3-4, pp. 183-269 | MR 0388464 | Zbl 0232.47055

[4] Haber, Nick; Vasy, András Propagation of singularities around a Lagrangian submanifold of radial points, Bull. Soc. Math. Fr., Tome 143 (2015) no. 4, pp. 679-726 | Zbl 1336.35015

[5] Hörmander, Lars The analysis of linear partial differential operators. IV: Fourier integral operators, Springer, Classics in Mathematics (2009), viii+352 pages (Reprint of the 1994 edition) | Article | MR 2512677 | Zbl 1178.35003

[6] Katsnelson, Vitaly Diffraction of Elastic Waves By Edges (2015) (Ph. D. Thesis)

[7] Mazzeo, Rafe Elliptic theory of differential edge operators. I, Commun. Partial Differ. Equations, Tome 16 (1991) no. 10, pp. 1615-1664 | Article | MR 1133743 | Zbl 0745.58045

[8] Melrose, Richard; Vasy, András; Wunsch, Jared Propagation of singularities for the wave equation on edge manifolds, Duke Math. J., Tome 144 (2008) no. 1, pp. 109-193 | Article | MR 2429323 | Zbl 1147.58029

[9] Melrose, Richard; Vasy, András; Wunsch, Jared Diffraction of singularities for the wave equation on manifolds with corners, Société Mathématique de France, Astérisque, Tome 351 (2013), vi+135 pages | MR 3100155 | Zbl 1277.35004

[10] Melrose, Richard; Wunsch, Jared Propagation of singularities for the wave equation on conic manifolds, Invent. Math., Tome 156 (2004) no. 2, pp. 235-299 | Article | MR 2052609 | Zbl 1088.58011

[11] Nazaikinskii, Vladimir E.; Anton Yu. Savin; Schulze, Bert-Wolfgang; Boris Yu. Sternin Elliptic theory on singular manifolds, Chapman & Hall/CRC, Differential and Integral Equations and Their Applications, Tome 7 (2006), xiv+356 pages | MR 2167050 | Zbl 1084.58007

[12] Taylor, Michael E. Reflection of singularities of solutions to systems of differential equations, Commun. Pure Appl. Math., Tome 28 (1975) no. 4, pp. 457-478 | MR 0509098 | Zbl 0332.35058

[13] Taylor, Michael E. Rayleigh waves in linear elasticity as a propagation of singularities phenomenon, Partial differential equations and geometry (Proc. Conf., Park City, Utah, 1977), Dekker (Lecture Notes in Pure and Appl. Math.) Tome 48 (1979), pp. 273-291 | MR 535598 | Zbl 0432.73021

[14] Taylor, Michael E. Partial differential equations II. Qualitative studies of linear equations, Springer, Applied Mathematical Sciences, Tome 116 (2011), xxii+614 pages | Article | MR 2743652 | Zbl 1206.35003

[15] Vasy, András Propagation of singularities for the wave equation on manifolds with corners, Ann. Math., Tome 168 (2008) no. 3, pp. 749-812 (erratum in ibid. 177 (2013), no. 2, p. 783-785) | Article | MR 2456883 | Zbl 1171.58007

[16] Vasy, András Diffraction at corners for the wave equation on differential forms, Commun. Partial Differ. Equations, Tome 35 (2010) no. 7, pp. 1236-1275 | Article | MR 2753634 | Zbl 1208.58026

[17] Vasy, András The wave equation on asymptotically anti de Sitter spaces, Anal. PDE, Tome 5 (2012) no. 1, pp. 81-144 | Article | MR 2957552

[18] Yamamoto, Kazuhiro Reflective elastic waves at the boundary as a propagation of singularities phenomenon, J. Math. Soc. Japan, Tome 42 (1990) no. 1, pp. 1-11 | Article | MR 1027537 | Zbl 0739.73013