Mode series expansions at vertical boundaries in elastic waveguides

Abstract

This paper investigates the mathematical foundations of modal expansion series of the Love and Rayleigh type at vertical boundaries in elastic waveguides. A functional analytic treatment in a product Hilbert space of displacement–stress vector functions is given. For Love modes, an expansion theorem is achieved. The Love mode series are shown to be Fourier-like expansion series, and this provides a clear insight into their convergence behaviour. For the Rayleigh modes, a formulation of the eigenvalue problem is introduced and a completeness proof is obtained. It is shown how the orthogonality relation, which is known from seismology, can be represented in the product Hilbert space. This also leads to the normability of Rayleigh modes due to the orthogonality relation.