Connections between seismology, waveguide physics and quantum mechanics

It is well-known that surface waves represent usually the main part of seismograms because of the 2D-character of their propagation. Their interaction with lateral heterogeneities is important not only in seismology but also in non-destructive testing (NDT), solid state physics and other disciplines. If we consider an elastic half-space the differential operator from the Navier equation has an infinite domain of definition with the consequence of a mixed spectrum: the discrete part represented by surface waves and the continuous part represented by body waves. We can identify the surface waves with the bound states of an atom and the body waves with the unbound ones. Conceptions of quantum mecanics like Dirac´s perturbation theory and others can be profitably used for the theory of disturbed wave-guides. We present a bra-ket formalism for the Rayleigh and Love waves and observe that contrary to the Love waves, which follow from a standard Sturm-Liouville problem, the eigenvalue problem for Rayleigh waves is a generalized and complicated one. The discrete eigenfunction system of the surface waves is incomplete and it is not yet proven wether the surface waves and body aves together build up a complete system. It is not easy to determine expansion coefficients within any expansion containing surface waves because of the mixed character of the governing operator´s eigenspectrum. An approximation technique is proposed to determine the coefficients by minimizing a special functional F₂ and the results are compared with an impedance jump problem, the only configuration with an exact solution for reflection and transmission coefficients for surface waves.