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Gap opening in the essential spectrum of the elasticity theory problem in a periodic half-layer. (English. Russian original) Zbl 1202.35326
St. Petersbg. Math. J. 21, No. 2, 281-307 (2010); translation from Algebra Anal. 21, No. 2, 166-204 (2009).
The spectrum of an elastic half layer with rigidly clamped faces and periodic traction free top is studied. It is shown that the essential spectrum has a band structure. An example is presented in which a gap in the essential spectrum occurs.
The half layer is decomposed into integer translates of a periodicity cell. Via the Gelfand transformation (discrete Fourier transform) the elastic spectral problem becomes a family of spectral problems in variational form for the periodicity cell. The cell is unbounded because its depth is. Using Fourier transformation with respect to the depth variable, the spectral problems for the periodicity cell are reduced to parameter dependent spectral problems for the cross section (projection) of the cell. The eigenfunctions obtained decay exponentially with depth, which is typical of Rayleigh waves.
MSC:
35Q74 PDEs in connection with mechanics of deformable solids
35P05 General topics in linear spectral theory for PDEs
35A22 Transform methods (e.g., integral transforms) applied to PDEs
47A10 Spectrum, resolvent
47F05 General theory of partial differential operators (should also be assigned at least one other classification number in Section 47-XX)
74B05 Classical linear elasticity
74J15 Surface waves in solid mechanics
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