PEKERIS SUBSTITUTION AND SOME SPECTRAL PROPERTIES OF THE RAYLEIGH BOUNDARY VALUE PROBLEMV. M. Markushevich
Inverse problems in seismology are usually considered in a framework of ray theory describing propagation of a pulse. Pulse sources are found by deconvolution of broad band signals. However, there is another method to investigate a medium, by using a fixed frequency, or a monochromatic source. It excites a stationary vibration field. Inversion of the field is based on representation of the elastic vibration equations in the Sturm-Liouville form. This paper presents two different substitutions into the system for the Rayleigh-type vibrations reducing it to two mutually transposed Sturm-Liouville's boundary problems in matrix form. The property is used to derive orthonormality conditions for normal modes with complex, as well as real eigenvalues. Some relations are established between complex eigenvalues and energy of the corresponding modes. The formula of Weyl's matrix function residues is obtained from the same property of the Rayleigh system. Representation of the Rayleigh wave equations in the Sturm-Liouville form is important because it allows one to introduce the concept of a potential. Then we can use the theory of scattering by potentials to solve direct and inverse problems for Rayleigh waves. Unfortunately, the theory for nonsymmetric matrix potentials is far from being complete.