Waves and Rays in Elastic Continua

Michael Slawinski

Department of Earth Science, Memorial University

Newfoundland, Canada

mslawins@mac.com

This book, which is the second edition of the book published by Elsevier in 2003, emphasizes the interdependence of mathematical formulation and physical meaning in the description of seismic phenomena. Herein, we use aspects of continuum mechanics, wave theory and ray theory to explain phenomena resulting from the propagation of seismic waves. The book is divided into three main parts: Elastic continua, Waves and rays and Variational formulation of rays. There is also a fourth part, which consists of Appendices.

In Part 1, we use continuum mechanics to describe the material through which seismic waves propagate, and to formulate a system of equations to study the behaviour of such a material. In Part 2, we use these equations to identify the types of body waves propagating in elastic continua as well as to express their velocities and displacements in terms of the properties of these continua. To solve the equations of motion in anisotropic inhomogeneous continua, we use the high-frequency approximation and, hence, establish the concept of a ray. In Part 3, we show that, in elastic continua, a ray is tantamount to a trajectory along which a seismic signal propagates in accordance with the variational principle of stationary traveltime. Consequently, many seismic problems in elastic continua can be conveniently formulated and solved using the calculus of variations. In Part IV, we describe two mathematical concepts that are used in the book; namely, homogeneity of a function and Legendre’s transformation. This book is intended for senior undergraduate and graduate students as well as scientists interested in quantitative seismology. We assume that the reader is familiar with linear algebra, differential and integral calculus, vector calculus, tensor analysis, as well as ordinary and partial differential equations.

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 two-semester course. If the variational formulation of ray theory is not to be included in such a course, the entire Part 3 can be omitted. Each part begins with an Introduction, which situates the topics discussed therein in the overall context of the book as well as in a broader scientific context. Each chapter begins with Preliminary remarks, which state the motivation for the specific concepts discussed therein, outline the structure of the chapter and provide links to other chapters in the book. Each chapter ends with Closing remarks, which specify the limitations of the concepts discussed and direct the reader to related chapters. Each chapter is followed by Exercises and their solutions.