Mathematical analysis of the propagation of elastic guided waves in heterogeneous media (Q750749)

This article is concerned with the propagation of elastic waves in isotropic heterogeneous media, invariant under translation in one direction. The authors give a theoretical analysis of the existence of guided waves and of their properties. By definition, a guided wave (or guided mode) is a solution of the elastodynamic equations in the form \[ U_ j(x,t)=\tilde u_ j(x_ 1,x_ 2)\exp i(\omega t-\beta x_ 3),\quad j=1,2,3, \] where \(\omega >0\) is the pulsation of the mode, \(\beta >0\) is the wave number, and \((\tilde u_ j(x_ 1,x_ 2)\), \(j=1,2,3)\) is a complex valued vector field which must satisfy \[ 0<\sum^{3}_{j=1}\int_{R^ 2}| \tilde u_ j(x_ 1,x_ 2)|^ 2 dx_ 1 dx_ 2<+\infty. \] Such solutions can appear in a nonhomogeneous medium, if and only if \(\omega\) and \(\beta\) satisfy a dispersion relation. The problem is reduced to research the eigenvalues \(\omega^ 2\) and the eigenfunctions \(\tilde u\) of a selfadjoint operator \({\mathcal A}(\beta)\) in the Hilbert space \(L^ 2(R^ 2,\rho dx_ 1 dx_ 2)\). All the results stem from the spectral analysis of \({\mathcal A}(\beta)\) and the main tool of the analysis is the spectral theory of selfadjoint operators and more specifically the Max-Min principle.