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Eigenfunction expansions for elastic wave propagation problems in stratified media \(R^ 3\). (English) Zbl 0841.35074
Stationary problems are considered for elastic wave propagation in a stratified two-sided medium in \(\mathbb{R}^3\). Eigenfunction expansion is given in terms of a family of generalized eigenfunctions corresponding to incident, reflected, refracted and Stonely waves. Expansions are derived using methods due to S. Makabayashi.
Wave equations with continuity conditions at the interface \(z= 0\) are interpreted as an abstract wave equation involving a non-negative selfadjoint operator defined in a Hilbert space. A proposed decomposition of the operator permits the author to get an explicit representation of the Green functions. The existence of Stonely waves is due to the zero of the Lopatinsky determinant. A careful determination of residues relative to the integral representation of projectors in appropriate regions in the complex plane leads to the eigenfunctions determination. Later the completeness is shown.

35P10 Completeness of eigenfunctions and eigenfunction expansions in context of PDEs
74B99 Elastic materials
74H99 Dynamical problems in solid mechanics
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