Earth-flattening transformation for P-SV waves

Abstract

Abstract We consider a layered nongravitating spherical Earth where in each layer P- and S-wave velocities are proportional to r and Lame's elastic parameters λ and μ are proportional to rp, where r is the radial distance and p is an arbitrary constant. In this article, the solutions of equations of motion in each layer are obtained in terms of exponential functions, and the expression for the stress-displacement matrix becomes similar to that for a flat-layered Earth. Thus, widely used formulations of a flat-layered Earth can be applied to study P-SV waves in a spherical-layered Earth through the transformation proposed here. When p = −2, explicit expressions are obtained for eigenvalues appearing in the solutions, and this makes the transformation further simplified. The transformation is applied, as an example, to find (1) the roots of the secular function and (2) eigenfunctions of Rayleigh waves in a spherical Earth through available formulations for a flat Earth.