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