Rayleigh waves in perturbed transversely isotropic half-space

Abstract

This thesis work is focused on the study of Rayleigh wave propagation in perturbed transversely isotropic half-space. These perturbations can be in the form of cracks, obstacles, or non-homogeneous part of the medium. The existence and uniqueness of Rayleigh surface wave in transversely isotropic half-space and the propagation of this wave in perturbed isotropic half-space have already been studied. The key result used in this work is the limiting absorption principle. Proof of the limiting absorption principle is derived for both the self-adjoint operators governing wave propagation in unperturbed and perturbed transversely isotropic half-space. Using this principle, the gen eralized eigenfunction for Rayleigh wave propagating in perturbed transversely isotropic half-space is also derived. This thesis is divided into four chapters along with the bibliography at the end. In chapter-1, introduction to some basic concepts of seismic waves and continuum mechanics is given. Also, the definitions of Lp space and Sobolev space are given. We then define the basic concepts of spectral theory which are used throughout this work. In chapter-2, the primary result, the limiting absorption principle which claims that specific limitations exist for the resolvent operator in a suitable uniform operator topology is demonstrated for operator M.T hisoperato r controls the wave propagation in transversely isotropic half-space. Concepts of weighted Lebesgue space, Sobolev space trace operators are employed in the proof. Prior to providing the proof for the main result, we establish few inequalities and construct the trace operators for the given eigenvalue of the operator M. In chapter-3, the pertubed half-space is defined. The operator M1,which is responsible for wave propagation in perturbed transversely isotropic half space is then shown to be self-adjoint. We presume that our domain satisfies the Korn inequality. Additionally, we demonstrate the main result of the limiting absorption principle for the operator M1. Finally, we obtain the equation of the generalized Rayleigh wave eigenfunction in the perturbed transversely isotropic half-space using the limiting absorption principle. Chapter-4 contains some conclusions and future scopes of the thesis work.