WIAS Preprint No. 1565, (2010)

Scattering of plane elastic waves by three-dimensional diffraction gratings


  • Elschner, Johannes
  • Hu, Guanghui

2010 Mathematics Subject Classification

  • 74J20 74B05 35J55 35B27


  • Elastic waves, diffraction gratings, Navier equation, variational formulation




The reflection and transmission of a time-harmonic plane wave in an isotropic elastic medium by a three-dimensional diffraction grating is investigated. If the diffractive structure involves an impenetrable surface, we study the first, second, third and fourth kind boundary value problems for the Navier equation in an unbounded domain by the variational approach. Based on the Rayleigh expansions, a radiation condition for quasi-periodic solutions is proposed. Existence of solutions in Sobolev spaces is established if the grating profile is a two dimensional Lipschitz surface, while uniqueness is proved only for small frequencies or for all frequencies excluding a discrete set. Similar solvability results are obtained for multilayered transmission gratings in the case of an incident pressure wave. Moreover, by a periodic Rellich identity, uniqueness of the solution to the first kind (Dirichlet) boundary value problem is established for all frequencies under the assumption that the impenetrable surface is given by the graph of a Lipschitz function.

Appeared in

  • Math. Models Methods Appl. Sci., 22 (2012) pp. 1150019-1--1150019-34.

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