### Ray Theory: Characteristics and Asymptotics

Andrej Bóna

Department of Exploration Geophysics, Curtain University

Perth, Australia

A.Bona@curtain.edu.au

Michael Slawinski

Department of Earth Science, Memorial University

Newfoundland, Canada

mslawins@mun.ca

In these lecture notes we strive to explain the understanding of the
underpinnings of ray theory. These notes are intended for senior
undergraduate and graduate students interested in the modern treatment
of ray theory expressed in mathematical language. We assume that the
reader is familiar with linear algebra, differential and integral
calculus, vector calculus as well as tensor analysis. To investigate
seismic wave propagation, we often use the concepts of rays and
wavefronts. These concepts result from studying the elastodynamic
equations using the method of characteristics or using the
high-frequency approximation. Characteristics of the elastodynamic
equations are given by the eikonal function whose level sets are
wavefronts. Characteristic equations of the eikonal equation are the
Hamilton ray equations whose solutions are rays. Hence, rays are
bicharacteristics of the elastodynamic equations. Characteristics are
entities that are associated with differential equations in a way that
is invariant under a change of coordinates. This property illustrates
the fact that characteristics possess information about the physical
essence of a given phenomenon.

This book is intended for graduate
students as well as scientists interested in theoretical
seismology. We assume that the reader is familiar with undergraduate
mathematics and physics. The chapters of this book are intended to be
studied in sequence. In that manner, the entire book can be used as a
manual for a one-semester graduate course. Each chapter begins with a
section called Preliminary remarks, where we provide the motivation
for the specific concepts discussed therein, outline the structure of
the chapter and provide links to other chapters. Each chapter ends
with a section called Closing remarks, which emphasizes the importance
of the discussed concepts and show their relevance to other
chapters. Each chapter is followed by Exercises and their solutions.

Entire text as a 5 MB PDF file

Back to Samizdat Press
home page.