WIAS Preprint No. 1466, (2009)

Variational approach to scattering of plane elastic waves by diffraction gratings


  • Elschner, Johannes
  • Hu, Guanghui

2010 Mathematics Subject Classification

  • 74J20 74B05 35J55 35Q72


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




The scattering of a time-harmonic plane elastic wave by a two-dimensional periodic structure is studied. The grating profile is given by a Lipschitz curve on which the displacement vanishes. Using a variational formulation in a bounded periodic cell involving a nonlocal boundary operator, existence of solutions in quasi-periodic Sobolev spaces is investigated by establishing the Fredholmness of the operator generated by the corresponding sesquilinear form. Moreover, by a Rellich identity, uniqueness is proved under the assumption that the grating profile is given by a Lipschitz graph. The direct scattering problem for transmission gratings is also investigated. In this case, uniqueness is proved except for a discrete set of frequencies.

Appeared in

  • Math. Methods Appl. Sci., 33 (2010) pp. 1924--1941.

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