Existence and uniqueness for the reflection-transmission process originated in a viscoelastic solid layer are investigated. Wave propagation is framed within the Fourier-transform domain and the oblique incidence is modelled by a factor involving a transverse wave vector. The backward-forward propagation in the axial direction is ascertained through the sign of an energy flux. Next, a connection is established between the energy flux and an Hermitian matrix whose eigenvalues are half positive and half negative. The proof is given that if the matrix has two diagonal blocks, one of which is positive definite and the other is negative definite, the solution to the reflection-transmission problem exists and is unique. The condition on the blocks is found to hold, e.g., for obliquely propagating homogeneous waves in anisotropic elasticity or normally propagating waves in isotropic viscoelasticity.
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Let x be a q-variate (weakly) stationary process over a locally compact Abelian group G, and J a family of subsets of G invariant under translation. We show that the set of all regular non-negative Hermitian matrix-valued measures M not exceeding the (non-stochastic) spectral measure of x and such that the Hilbert space L2(M) is J-regular contains a unique maximal element. Moreover, this maximal element coincides with the spectral measure of the J-regular part of the Wold decomposition of x.
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