Seismic Waves And Rays In Elastic Media

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Seismic Waves and Rays in Elastic Media

Author : M.A. Slawinski
Publisher : Elsevier
ISBN 13 : 0080540899
Total Pages : 425 pages
Book Rating : 4.0/5 (85 download)

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Book Synopsis Seismic Waves and Rays in Elastic Media by : M.A. Slawinski

Download or read book Seismic Waves and Rays in Elastic Media written by M.A. Slawinski and published by Elsevier. This book was released on 2003-08-04 with total page 425 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book seeks to explore seismic phenomena in elastic media and emphasizes the interdependence of mathematical formulation and physical meaning. The purpose of this title - which is intended for senior undergraduate and graduate students as well as scientists interested in quantitative seismology - is to use aspects of continuum mechanics, wave theory and ray theory to describe phenomena resulting from the propagation of waves. The book is divided into three parts: Elastic continua, Waves and rays, and Variational formulation of rays. In Part I, continuum mechanics are used to describe the material through which seismic waves propagate, and to formulate a system of equations to study the behaviour of such material. In Part II, these equations are used 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, the high-frequency approximation is used and establishes the concept of a ray. In Part III, it is shown 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 travel time.

Seismic Waves and Rays in Elastic Media

Author : M.A. Slawinski
Publisher : Elsevier
ISBN 13 : 9780080439303
Total Pages : 424 pages
Book Rating : 4.4/5 (393 download)

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Book Synopsis Seismic Waves and Rays in Elastic Media by : M.A. Slawinski

Download or read book Seismic Waves and Rays in Elastic Media written by M.A. Slawinski and published by Elsevier. This book was released on 2003-08-04 with total page 424 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book seeks to explore seismic phenomena in elastic media and emphasizes the interdependence of mathematical formulation and physical meaning. The purpose of this title - which is intended for senior undergraduate and graduate students as well as scientists interested in quantitative seismology - is to use aspects of continuum mechanics, wave theory and ray theory to describe phenomena resulting from the propagation of waves. The book is divided into three parts: Elastic continua, Waves and rays, and Variational formulation of rays. In Part I, continuum mechanics are used to describe the material through which seismic waves propagate, and to formulate a system of equations to study the behaviour of such material. In Part II, these equations are used 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, the high-frequency approximation is used and establishes the concept of a ray. In Part III, it is shown 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 travel time.

Waves And Rays In Elastic Continua

Author : Michael A Slawinski
Publisher : World Scientific Publishing Company
ISBN 13 : 9813107677
Total Pages : 614 pages
Book Rating : 4.8/5 (131 download)

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Book Synopsis Waves And Rays In Elastic Continua by : Michael A Slawinski

Download or read book Waves And Rays In Elastic Continua written by Michael A Slawinski and published by World Scientific Publishing Company. This book was released on 2010-09-09 with total page 614 pages. Available in PDF, EPUB and Kindle. Book excerpt: The present book — which is the second, and significantly extended, edition of the textbook originally published by Elsevier Science — 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 sections: Elastic Continua, Waves and Rays and Variational Formulation of Rays. There is also a fourth part, which consists of appendices.In Elastic Continua, 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 Waves and Rays, 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 invoke the concept of a ray. In Variational Formulation of Rays, 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 the Appendices, we describe two mathematical concepts that are used in the book; namely, homogeneity of a function and Legendre's transformation. This section also contains a list of symbols.

Seismic Waves and Rays in Elastic Media

Author : M.A. Slawinski
Publisher : Elsevier Science
ISBN 13 : 9780080439303
Total Pages : 424 pages
Book Rating : 4.4/5 (393 download)

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Book Synopsis Seismic Waves and Rays in Elastic Media by : M.A. Slawinski

Download or read book Seismic Waves and Rays in Elastic Media written by M.A. Slawinski and published by Elsevier Science. This book was released on 2003-08-18 with total page 424 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book seeks to explore seismic phenomena in elastic media and emphasizes the interdependence of mathematical formulation and physical meaning. The purpose of this title - which is intended for senior undergraduate and graduate students as well as scientists interested in quantitative seismology - is to use aspects of continuum mechanics, wave theory and ray theory to describe phenomena resulting from the propagation of waves. The book is divided into three parts: Elastic continua, Waves and rays, and Variational formulation of rays. In Part I, continuum mechanics are used to describe the material through which seismic waves propagate, and to formulate a system of equations to study the behaviour of such material. In Part II, these equations are used 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, the high-frequency approximation is used and establishes the concept of a ray. In Part III, it is shown 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 travel time.

Elastic Waves in the Earth

Author : Walter L. Pilant
Publisher : Elsevier
ISBN 13 : 0444601945
Total Pages : 506 pages
Book Rating : 4.4/5 (446 download)

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Book Synopsis Elastic Waves in the Earth by : Walter L. Pilant

Download or read book Elastic Waves in the Earth written by Walter L. Pilant and published by Elsevier. This book was released on 2012-12-02 with total page 506 pages. Available in PDF, EPUB and Kindle. Book excerpt: Elastic Waves in the Earth provides information on the relationship between seismology and geophysics and their general aspects. The book offers elastodynamic equations and derivative equations that can be used in the propagation of elastic waves. It also covers major topics in detail, such as the fundamentals of elastodynamics; the Lamb's problem, which includes the Cagniard-de Hoop theory; rays and modes in a radially inhomogeneous earth and in multilayered media, which includes the Thomson-Haskell theory; the elastic wave dissipation; the seismic source and noise; and the seismographs. The book consists of 33 chapters. The first 16 chapters include basic material related to the propagation of elastic waves. Topics covered by these chapters include scalars, vectors, and tensors in cartesian coordinates, stress and strain analysis, equations of elasticity and motion, plane waves, Rayleigh waves, plane-wave theory, and fluid-fluid and solid-solid interfaces. The second half of the book covers various ray and mode theories, elastic wave dissipation, and the observations and theories of seismic source and seismic noise. It concludes by discussing earthquake seismology and different seismographs, like the pendulum seismometer and the strain seismometer.

Waves And Rays In Seismology: Answers To Unasked Questions (Second Edition)

Author : Michael A Slawinski
Publisher : World Scientific
ISBN 13 : 9813239891
Total Pages : 577 pages
Book Rating : 4.8/5 (132 download)

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Book Synopsis Waves And Rays In Seismology: Answers To Unasked Questions (Second Edition) by : Michael A Slawinski

Download or read book Waves And Rays In Seismology: Answers To Unasked Questions (Second Edition) written by Michael A Slawinski and published by World Scientific. This book was released on 2018-05-04 with total page 577 pages. Available in PDF, EPUB and Kindle. Book excerpt: 'In summary, Professor Slawinski has written an engaging volume covering an unfamiliar topic in a highly accessible fashion. Non-specialists will gain a significant appreciation of the unique complexities associated with seismology.'Contemporary PhysicsThe author dedicates this book to readers who are concerned with finding out the status of concepts, statements and hypotheses, and with clarifying and rearranging them in a logical order. It is thus not intended to teach tools and techniques of the trade, but to discuss the foundations on which seismology — and in a larger sense, the theory of wave propagation in solids — is built. A key question is: why and to what degree can a theory developed for an elastic continuum be used to investigate the propagation of waves in the Earth, which is neither a continuum nor fully elastic. But the scrutiny of the foundations goes much deeper: material symmetry, effective tensors, equivalent media; the influence (or, rather, the lack thereof) of gravitational and thermal effects and the rotation of the Earth, are discussed ab initio. The variational principles of Fermat and Hamilton and their consequences for the propagation of elastic waves, causality, Noether's theorem and its consequences on conservation of energy and conservation of linear momentum are but a few topics that are investigated in the process to establish seismology as a science and to investigate its relation to subjects like realism and empiricism in natural sciences, to the nature of explanations and predictions, and to experimental verification and refutation.In the second edition, new sections, figures, examples, exercises and remarks are added. Most importantly, however, four new appendices of about one-hundred pages are included, which can serve as a self-contained continuum-mechanics course on finite elasticity. Also, they broaden the scope of elasticity theory commonly considered in seismology.

Propagation of Transient Elastic Waves in Stratified Anisotropic Media

Author : J.H.M.T. van der Hijden
Publisher : Elsevier
ISBN 13 : 1483295907
Total Pages : 299 pages
Book Rating : 4.4/5 (832 download)

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Book Synopsis Propagation of Transient Elastic Waves in Stratified Anisotropic Media by : J.H.M.T. van der Hijden

Download or read book Propagation of Transient Elastic Waves in Stratified Anisotropic Media written by J.H.M.T. van der Hijden and published by Elsevier. This book was released on 2016-05-24 with total page 299 pages. Available in PDF, EPUB and Kindle. Book excerpt: Seismic waves are one of the standard diagnostic tools used to determine the mechanical parameters (volume density of mass, compressibility, elastic stiffness) in the interior of the earth and the geometry of subsurface structures. There is increasing evidence that in the interpretation of seismic data - especially shear-wave data - the influence of anisotropy must be taken into account.This volume presents a method to compute the seismic waves that are generated by an impulsive source in a stratified anisotropic medium. Although written with the seismic applications in mind, the method that is developed is not limited to solid-earth geophysics. In fact, the methods discussed in this monograph are applicable wherever waves propagate in stratified, anisotropic media. The standard approach to this problem is to employ Fourier transformations with respect to time and with respect to the horizontal spatial coordinates. To obtain numerical results, the relevant inverse transformations then have to be evaluated numerically. In this monograph the problem is, in contrast to the standard approach, solved by applying the Cagniard-de Hoop method and by representing the wave field as a sum of generalized rays. With this method, the computational results can be obtained relatively easily with any degree of accuracy, and with considerably less computation time. For completeness, analysis of acoustic waves in stratified isotropic media is included. Furthermore, for large horizontal or vertical source-receiver separations very efficient approximations are derived. Several examples and applications are given.

Transient Waves in Visco-Elastic Media

Author : Norman Ricker
Publisher : Elsevier
ISBN 13 : 0444601694
Total Pages : 289 pages
Book Rating : 4.4/5 (446 download)

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Book Synopsis Transient Waves in Visco-Elastic Media by : Norman Ricker

Download or read book Transient Waves in Visco-Elastic Media written by Norman Ricker and published by Elsevier. This book was released on 2012-12-02 with total page 289 pages. Available in PDF, EPUB and Kindle. Book excerpt: Developments in Solid Earth Geophysics 10: Transient Waves in Visco-Elastic Media deals with the propagation of transient elastic disturbances in visco-elastic media. More specifically, it explores the visco-elastic behavior of a medium, whether gaseous, liquid, or solid, for very-small-amplitude disturbances. This volume provides a historical overview of the theory of the propagation of elastic waves in solid bodies, along with seismic prospecting and the nature of seismograms. It also discusses the seismic experiments, the behavior of waves propagated in accordance with the Stokes wave equation, and wavelet functions and their polynomials. The book explains the laws of propagation of seismic wavelets and seismic ray paths, as well as the equations of wavelet propagation, the velocity-type seismic wavelet, and the spectrum of the wavelet. It discusses the motion of a mechanical seismograph disturbed by extraneous forces or motions. It also provides information on the differential equation describing the motion of a galvanometer, laboratory studies of wavelet contraction, and characteristics of a wavelet-contractor amplifier. Furthermore, the book explains the experimental studies of the primary seismic disturbance and internal friction. This monograph is a valuable source of information for physicists, students who want to pursue a career in geophysics or selenophysics, and those who actively working in these fields.

Seismic Wave Propagation in Non-Homogeneous Elastic Media by Boundary Elements

Author : George D. Manolis
Publisher : Springer
ISBN 13 : 3319452061
Total Pages : 301 pages
Book Rating : 4.3/5 (194 download)

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Book Synopsis Seismic Wave Propagation in Non-Homogeneous Elastic Media by Boundary Elements by : George D. Manolis

Download or read book Seismic Wave Propagation in Non-Homogeneous Elastic Media by Boundary Elements written by George D. Manolis and published by Springer. This book was released on 2016-09-23 with total page 301 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book focuses on the mathematical potential and computational efficiency of the Boundary Element Method (BEM) for modeling seismic wave propagation in either continuous or discrete inhomogeneous elastic/viscoelastic, isotropic/anisotropic media containing multiple cavities, cracks, inclusions and surface topography. BEM models may take into account the entire seismic wave path from the seismic source through the geological deposits all the way up to the local site under consideration. The general presentation of the theoretical basis of elastodynamics for inhomogeneous and heterogeneous continua in the first part is followed by the analytical derivation of fundamental solutions and Green's functions for the governing field equations by the usage of Fourier and Radon transforms. The numerical implementation of the BEM is for antiplane in the second part as well as for plane strain boundary value problems in the third part. Verification studies and parametric analysis appear throughout the book, as do both recent references and seminal ones from the past. Since the background of the authors is in solid mechanics and mathematical physics, the presented BEM formulations are valid for many areas such as civil engineering, geophysics, material science and all others concerning elastic wave propagation through inhomogeneous and heterogeneous media. The material presented in this book is suitable for self-study. The book is written at a level suitable for advanced undergraduates or beginning graduate students in solid mechanics, computational mechanics and fracture mechanics.

Elastic Waves in Layered Media

Author : William Maurice Ewing
Publisher :
ISBN 13 :
Total Pages : 402 pages
Book Rating : 4.:/5 (41 download)

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Book Synopsis Elastic Waves in Layered Media by : William Maurice Ewing

Download or read book Elastic Waves in Layered Media written by William Maurice Ewing and published by . This book was released on 1957 with total page 402 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Seismic Wave Propagation and Scattering in the Heterogeneous Earth : Second Edition

Author : Haruo Sato
Publisher : Springer Science & Business Media
ISBN 13 : 3642230288
Total Pages : 505 pages
Book Rating : 4.6/5 (422 download)

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Book Synopsis Seismic Wave Propagation and Scattering in the Heterogeneous Earth : Second Edition by : Haruo Sato

Download or read book Seismic Wave Propagation and Scattering in the Heterogeneous Earth : Second Edition written by Haruo Sato and published by Springer Science & Business Media. This book was released on 2012-03-08 with total page 505 pages. Available in PDF, EPUB and Kindle. Book excerpt: Seismic waves - generated both by natural earthquakes and by man-made sources - have produced an enormous amount of information about the Earth's interior. In classical seismology, the Earth is modeled as a sequence of uniform horizontal layers (or spherical shells) having different elastic properties and one determines these properties from travel times and dispersion of seismic waves. The Earth, however, is not made of horizontally uniform layers, and classic seismic methods can take large-scale inhomogeneities into account. Smaller-scale irregularities, on the other hand, require other methods. Observations of continuous wave trains that follow classic direct S waves, known as coda waves, have shown that there are heterogeneities of random size scattered randomly throughout the layers of the classic seismic model. This book focuses on recent developments in the area of seismic wave propagation and scattering through the randomly heterogeneous structure of the Earth, with emphasis on the lithosphere. The presentation combines information from many sources to present a coherent introduction to the theory of scattering in acoustic and elastic materials and includes analyses of observations using the theoretical methods developed. The second edition especially includes new observational facts such as the spatial variation of medium inhomogeneities and the temporal change in scattering characteristics and recent theoretical developments in the envelope synthesis in random media for the last ten years. Mathematics is thoroughly rewritten for improving the readability. Written for advanced undergraduates or beginning graduate students of geophysics or planetary sciences, this book should also be of interest to civil engineers, seismologists, acoustical engineers, and others interested in wave propagation through inhomogeneous elastic media.

Fundamentals of Seismic Wave Propagation

Author : Chris Chapman
Publisher : Cambridge University Press
ISBN 13 : 9781139451635
Total Pages : 646 pages
Book Rating : 4.4/5 (516 download)

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Book Synopsis Fundamentals of Seismic Wave Propagation by : Chris Chapman

Download or read book Fundamentals of Seismic Wave Propagation written by Chris Chapman and published by Cambridge University Press. This book was released on 2004-07-29 with total page 646 pages. Available in PDF, EPUB and Kindle. Book excerpt: Fundamentals of Seismic Wave Propagation, published in 2004, presents a comprehensive introduction to the propagation of high-frequency body-waves in elastodynamics. The theory of seismic wave propagation in acoustic, elastic and anisotropic media is developed to allow seismic waves to be modelled in complex, realistic three-dimensional Earth models. This book provides a consistent and thorough development of modelling methods widely used in elastic wave propagation ranging from the whole Earth, through regional and crustal seismology, exploration seismics to borehole seismics, sonics and ultrasonics. Particular emphasis is placed on developing a consistent notation and approach throughout, which highlights similarities and allows more complicated methods and extensions to be developed without difficulty. This book is intended as a text for graduate courses in theoretical seismology, and as a reference for all academic and industrial seismologists using numerical modelling methods. Exercises and suggestions for further reading are included in each chapter.

Seismic Waves in Laterally Inhomogeneous Media Part II

Author : Ivan Psencik
Publisher : Birkhäuser
ISBN 13 : 3034890494
Total Pages : 386 pages
Book Rating : 4.0/5 (348 download)

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Book Synopsis Seismic Waves in Laterally Inhomogeneous Media Part II by : Ivan Psencik

Download or read book Seismic Waves in Laterally Inhomogeneous Media Part II written by Ivan Psencik and published by Birkhäuser. This book was released on 2012-12-06 with total page 386 pages. Available in PDF, EPUB and Kindle. Book excerpt: The special issue contains contributions presented at the international workshop Seismic waves in laterally inhomogeneous media IV, which was held at the Castle of Trest, Czech Republic, May 22-27, 1995. The workshop, which was attended by about 100 seismologists from more than 10 countries, was devoted mainly to the current state of theoretical and computational means of study of seismic wave propagation in complex structures. The special issue can be of interest for theoretical, global and explorational seismologists. The first part contains papers dealing with the study and the use of various methods of solving forward and inverse problems in complicated structures. Among other methods, discrete-wave number method, the finite-difference method, the edge-wave supperposition method and the ray method are studied and used. Most papers contained in the second part are related to the ray method. The most important topics are two-point ray tracing, grid calculations of travel times and amplitudes and seismic wave propagation in anisotropic media.

Seismic Waves in Laterally Inhomogeneous Media

Author : Ivan Psencik
Publisher : Birkhäuser
ISBN 13 : 3034892136
Total Pages : 340 pages
Book Rating : 4.0/5 (348 download)

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Book Synopsis Seismic Waves in Laterally Inhomogeneous Media by : Ivan Psencik

Download or read book Seismic Waves in Laterally Inhomogeneous Media written by Ivan Psencik and published by Birkhäuser. This book was released on 2012-12-06 with total page 340 pages. Available in PDF, EPUB and Kindle. Book excerpt: The special issue contains contributions presented at the international workshop Seismic waves in laterally inhomo- geneous media IV, which was held at the Castle of Trest, Czech Republic, May 22-27, 1995. The workshop, which was attended by about 100 seismologists from more than 10 countries, was devoted mainly to the current state of theoretical and computational means of study of seismic wave propagation in complex structures. The special issue can be of interest for theoretical, global and explorational seismologists. The first part contains papers dealing with the study and the use of various methods of solving forward and inverse problems in complicated structures. Among other methods, discrete-wave number method, the finite-difference method, the edge-wave supperposition method and the ray method are studied and used. Most papers contained in the second part are related to the ray method. The most important topics are two-point ray tracing, grid calculations of travel times and amplitudes and seismic wave propagation in anisotropic media.

Seismic Wave Propagation in Stratified Media

Author : Brian Kennett
Publisher : ANU E Press
ISBN 13 : 192153673X
Total Pages : 298 pages
Book Rating : 4.9/5 (215 download)

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Book Synopsis Seismic Wave Propagation in Stratified Media by : Brian Kennett

Download or read book Seismic Wave Propagation in Stratified Media written by Brian Kennett and published by ANU E Press. This book was released on 2009-05-01 with total page 298 pages. Available in PDF, EPUB and Kindle. Book excerpt: Seismic Wave Propagation in Stratified Media presents a systematic treatment of the interaction of seismic waves with Earth structure. The theoretical development is physically based and is closely tied to the nature of the seismograms observed across a wide range of distance scales - from a few kilometres as in shallow reflection work for geophysical prospecting, to many thousands of kilometres for major earthquakes. A unified framework is presented for all classes of seismic phenomena, for both body waves and surface waves. Since its first publication in 1983 this book has been an important resource for understanding the way in which seismic waves can be understood in terms of reflection and transmission properties of Earth models, and how complete theoretical seismograms can be calculated. The methods allow the development of specific approximations that allow concentration on different seismic arrivals and hence provide a direct tie to seismic observations.

Waves And Rays In Seismology: Answers To Unasked Questions (Third Edition)

Author : Michael A Slawinski
Publisher : World Scientific
ISBN 13 : 9811226458
Total Pages : 654 pages
Book Rating : 4.8/5 (112 download)

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Book Synopsis Waves And Rays In Seismology: Answers To Unasked Questions (Third Edition) by : Michael A Slawinski

Download or read book Waves And Rays In Seismology: Answers To Unasked Questions (Third Edition) written by Michael A Slawinski and published by World Scientific. This book was released on 2020-11-20 with total page 654 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is a book on seismology dealing with advanced aspects of wave propagation in complex media. It can also be viewed as a book on mathematical modelling, wherein the accuracy of describing seismic phenomena exemplifies the modelling itself. The book gives an insight into the power of abstractness by applying the same mathematical methods and strategies to solve a variety of different physical problems. This book covers a broad range of topics in an advanced yet accessible manner. Each chapter is accompanied by a number of solved exercises, which render the book convenient for a lecturer and facilitate its use for an independent study. The details of mathematical methods are discussed in the appendices, which form a substantial portion of the book.

Seismic Wave Propagation and Scattering in the Heterogenous Earth

Author : Haruo Sato
Publisher : Springer Science & Business Media
ISBN 13 : 3540896236
Total Pages : 308 pages
Book Rating : 4.5/5 (48 download)

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Book Synopsis Seismic Wave Propagation and Scattering in the Heterogenous Earth by : Haruo Sato

Download or read book Seismic Wave Propagation and Scattering in the Heterogenous Earth written by Haruo Sato and published by Springer Science & Business Media. This book was released on 2008-12-17 with total page 308 pages. Available in PDF, EPUB and Kindle. Book excerpt: Seismic waves – generated both by natural earthquakes and by man-made sources – have produced an enormous amount of information about the Earth's interior. In classical seismology, the Earth is modeled as a sequence of uniform horizontal layers (or sperical shells) having different elastic properties and one determines these properties from travel times and dispersion of seismic waves. The Earth, however, is not made of horizontally uniform layers, and classic seismic methods can take large-scale inhomogeneities into account. Smaller-scale irregularities, on the other hand, require other methods. Observations of continuous wave trains that follow classic direct S waves, known as coda waves, have shown that there are heterogeneities of random size scattered randomly throughout the layers of the classic seismic model. This book focuses on recent developments in the area of seismic wave propagation and scattering through the randomly heterogeneous structure of the Earth, with emphasis on the lithosphere. The presentation combines information from many sources to present a coherent introduction to the theory of scattering in acoustic and elastic materials and includes analyses of observations using the theoretical methods developed.