The Earth-flattening problem: Genesis, exact solving methods, and expansion into series
We present exact solutions of the Earth-flattening problem, properties of these solutions, and their expansions into power series in the reciprocal of the Earth's radius. Although the exact Earth-flattening transformation for SH vibrations has been known for more than a quarter of a century, all attempts to find its analogue for $P$-$SV$ vibrations have not been successful. Various approximation methods were used instead. We considered inhomogeneous media and succeeded in deriving the exact flattening transformation based on the following result. Equations of linear elastodynamics for plane, cylindrically, and spherically layered media are transformed to a certain matrix Sturm-Liouville form. In all these cases the part of the differential operator that does not depend on wavenumber is represented as the composition of two first-order differential operators. By virtue of this representation, the respective spherical operator reduces to a plane one through a matrix transformation. This change offers the possibility of computing Rayleigh-like vibrations of a cylindrically or spherically symmetric body by methods developed for plane layered media.