On the orthogonality of eigenwaves in a semi-open elastic waveguide

We consider a two-layer elastic waveguide structure such that its one layer is unbounded in the lateral direction and at the boundary of the other one the standard boundary conditions are stated. We study several kinds of eigenwaves of this structure. In general, they have a complex-valued longitudinal propagation constant (the spectral parameter). On the base of the Green formula we introduce the scalar product of two waves and prove that the set of eigenwaves of the semiopen elastic waveguide is orthogonal. We construct families of waves which belong to discrete and continuous parts of the spectrum.