Some Algebraic Aspects of the Lamb Problem.
COLORADO SCHOOL OF MINES GOLDEN CENTER FOR WAVE PHENOMENA
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This paper discusses the classical Lamb problem for the elastic wave equation. The motivation, for the authors, is to be able to conveniently construct Greens functions matrices for later use in formulating and solving various problems. For example, we will want to be able to solve for perturbations from constant reference densities ad Lamb parameters. Hence, Greens function for the homogeneous isotropic equation is discussed. Due to the scattering taking place in inverse problems it is usually impossible to retain the P-SV and SH decoupling hence, we do not persue this decoupling in the formation of the Greens functions herein. While nothing conceptually new is presented here, the approach is a bit different and, we believe, is helpful in isolating some important issues. The approach is algebraic in nature and makes heavy use of several simple facts from linear algebra for example, the spectral decomposition of special matrices. This approach facilitates some helpful decoupling, particularly in solving for reflection coefficients.
- Theoretical Mathematics