# Books

Henning Hoeber et al.

ISBN: 978-1-56080-142-9

1. edition: July 2007

Soft cover, 545 pages

## Classics of Elastic Wave Theory

Society of Exploration GeophysicsISBN: 978-1-56080-142-9

1. edition: July 2007

Soft cover, 545 pages

### Description

"Classics of Elastic Wave Theory" (SEG Geophysics Reprint Series No. 24) brings together for the first time 16 key papers on elastic wave theory, starting with Hooke's 1678 "Potentiva Restitutiva, or Spring," and grouped into five historical periods through the mid-20th century. These papers, which often are hard to find, all are presented here in English.This volume explores fundamentals of theory by Hooke, Navier, Cauchy, Green, Kelvin, Stokes, and Betti; key early papers on anisotropy by Christoffel and Rudzki; fundamentals of reflection and transmission amplitudes by Knott and Zoeppritz; surface waves by Rayleigh, Love, and Stoneley; the first synthetic seismogram by Lamb; and the defining paper on poroelesticity by Gassmann. In addition, a reprint of A. E. H. Love's "Historical Introduction" from his classic "Treatise on the Mathematical Theory of Elasticity" provides an excellent overview of elastic wave theory to the end of the 19th century.

Each paper is accompanied by an author biography. An editors' introduction provides a modern connection to the many fields of science to which the papers are fundamental today. The book includes extensive references and is an invaluable source for anyone working in elastic wave theory or for those with a general interest in the development and evolution of wave propagation.

### Preface

Applied elastic wave theory has influenced the modern industrial world profoundly, with applications ranging from microscopic to planetary. The theory is central to the hydrocarbon exploration and production industry and plays a very important role in medicine, aeronautics, civil engineering, materials science, planetary exploration, and defense.In this volume, we offer a broad perspective of some of the key works in the development of elastic wave theory. The reprinted papers, spanning the 17th through mid-20th centuries, are from authors whose works are so fundamental that their contributions are an intrinsic part of the technical language – Hooke's law, the Navier equation, the Christoffel equation, Betti's relation, Rayleigh waves, and Stoneley waves.

This volume is a tribute to A. E. H. Love's "Historical Introduction" from his classic Treatise on the Mathematical Theory of Elasticity. The initial concept of this reprint volume was to combine Love's introduction with a collection of key historical papers referenced in his work. The articles reprinted in this volume are drawn almost exclusively from his extensive bibliography, the main exceptions being those which postdate his book.

Access to the papers reprinted in this volume is limited. Many remain unavailable online or can be difficult to find. The frequent lack of explicit references in modern publications to the works of the pioneers is also a factor that motivated this volume. A statement of Hooke's law, for example, is accompanied only rarely by any reference to Hooke's definitive 1678 article, "Potentia Restitutiva, or Spring," on which the law is based. We hope the reprinting of these papers also will uncover some lesser-known but useful insights or relations that inadvertently have been filtered out over time.

The classic papers present a language barrier. Many of the early fundamentals of the mathematical theory were developed in Paris, and further development and consolidation of the theory were the result of contributions from throughout Europe. This volume includes papers translated from French, German, and Italian.

The papers reprinted in this volume represent five periods, loosely defined, in the development of the theory. The beginnings of the theory are traced to Robert Hooke, c.1648. The next period, almost two centuries later, includes the basic mathematical formulation of the theory in the 1820s. The papers by Claude-Louis-Marie-Henri Navier and Augustin-Louis Cauchy are among the key contributions from that early period. The works of George Green, George Gabriel Stokes, Lord Kelvin, Enrico Betti, and Elwin Bruno Christoffel are representative of the rapid pace of further development during the next half century. In that period, the cornerstones of the theory were consolidated. The fact that the basic theory has held fast throughout the next century, which was characterized by spectacular advances in mathematical physics, confirms the superb contributions of that third period.

The fourth period, from the 1880s through the 1920s, saw the early practical implementation of the theory, characterizing various types of wave propagation through the works of Lord Rayleigh, Maurycy Pius Rudzki, Cargill Gilston Knott, Horace Lamb, Augustus Edward Hough Love, Karl Zoeppritz, and Robert Stoneley. The applied focus of much of their work was earthquake seismology. Here we can see some of the early efforts in the interpretation of seismic data.

The fifth period of development, from the 1930s through the 1960s, provided the framework for reservoir-scale applications. That period includes Fritz Gassmann's classic paper on poroelasticity. Gassmann uses the concept of a homogeneous equivalent medium for a heterogeneous medium. The study of how porosity, rock matrix, and fluid content of a reservoir affect elastic wave propagation contrasts with the solid-earth geophysics focus of many of the earlier developments. Another important development was the periodic thin-layering representation for seismic anisotropy. That was based also on the principle of a homogenous equivalent medium.

The early works on elastic wave theory contain, of course, many elements that have not stood the test of time. For example, Robert Hooke's article addressing what is now known as Hooke's law also contains hypotheses on the constitution of matter which are hardly in accordance with modern theories. The molecular theory of material points employed by Claude-Louis-Marie-Henri Navier to derive the elastic wave equation has proved inadequate, as has his use of a single elastic constant. However, the equation itself, now commonly referred to as the Navier equation, constitutes one of the fundamental relations of the mathematical theory, fully substantiated by the laws of thermodynamics.

Although the mathematical notation of the pioneering articles can be criticized easily when it is measured against the much more elegant, compact, and versatile modern tensor notation, the foundations of the modern notation nevertheless can be seen clearly in these early articles. In reading the articles, we should be reminded that during time, our own recent advances in the elastic wave theory and its applications will be seen to have a similar balance of strengths and weaknesses. Certainly one of the great strengths communicated by the reprinted articles is the analytical technique, employed in an impressive variety of forms.

In compiling this volume, we have strived to collate a selection of those works that, taken together, provide the cornerstones of the present-day theory of elastic wave propagation. Our hope is that the sampling is sufficient to span the entire theory. Unfortunately, we could not include all the great contributions to the theory in a single volume. Among the authors omitted who certainly have a claim for inclusion are Siméon Denis Poisson, Thomas Young, Gabriel Lamé, Gustav Robert Kirchhoff, Christiaan Huygens, and Augustin Jean Fresnel. Love's "Historical Introduction" gives a more complete list of those who have contributed to the theory during the centuries. Many of them also are cited in the reprinted papers. If the current volume proves to be of sufficient interest to the elastic-wave community, we could include some of those authors in subsequent reprint volumes.

This volume is organized into four chapters. Chapter 1 contains a brief review of the reprinted articles, highlighting the principal contributions. The objective is to provide anchor points to well-known equations or wave phenomena. To help bridge the notation gap, some of the key relations from the reprinted papers are identified and given in modern notation. In a few cases, we also attempt to relate the equations to some current applications.

Chapter 2 is the "Historical Introduction" by A. E. H. Love, which provides an excellent overview of the development of the theory up to the end of the 19th century.

Chapter 3 comprises the reprinted technical papers, presented in chronological order. The material has been typeset using a uniform text font, but every effort has been made to retain the original mathematical symbols and paragraph layouts.

Chapter 4 contains biographical sketches for the authors whose works are represented in Chapter 3. Most of the biographies are reprinted from the Dictionary of Scientific Biography, the exceptions being those of Gassmann, Rudzki, Stoneley, and Zoeppritz.

We conclude this volume with a brief epilogue in which we try to extrapolate the pattern of development of the elastic theory and its applications, to speculate on future development relative to the exploration and production industry.