Decomposition into factors and Sturm-Liouville's form of equations for P-SV vibrations of layered media
We suggest a theory of linear elastodynamics for the cases of isotropic continuously layered media of three kinds: plane, spherically, and cylindrically layered. The known transformations of the general solution lead to its decomposition into axially symmetrical solutions. On separation of variables, we obtain a second-order two-dimensional matrix operator for $P$-$SV$ vibrations and a scalar operator for $SH$ vibrations. The two-dimensional operator reduces to a matrix Sturm-Liouville form which is a composition of explicitly represented first-order operators in the case where the spectral parameter vanishes. This is the result of the theory. By using this result, we find canonical forms of the operator, which follows from various matrix transformations. We show that, given a fixed frequency and a plane, spherically, or cylindrically layered medium, the local interpretation problem for a given operator is solvable. This result is necessary for solving the inverse problem of seismology; besides, it provides for an exact solution of the Earth-flattening problem for Rayleigh waves.