# Acoustic and Elastic Wave Fields in Geophysics, Part I

**By**

- Avital Kaufman, Department of Geophysics, Colorado School of Mines, Golden, CO, USA
- A.L. Levshin, Department of Physics, University of Colorado, Boulder, CO 80390, USA

This book is dedicated to basic physical principles of the propagation of acoustic and elastic waves. It consists of two volumes. The first volume includes 8 chapters and extended Appendices explaining mathematical aspects of discussed problems. The first chapter is devoted to Newton's laws, which, along with Hooke's law, govern the behavior of acoustic and elastic waves. Basic concepts of mechanics are used in deriving equations which describe wave phenomena. The second and third chapters deal with free and forced vibrations as well as wave propagation in one dimension along the system of elementary masses and springs which emulates the simplest elastic medium.In addition, shear waves propagation along a finite and infinite string are discussed.

In chapter 4 the system of equations describing compressional waves is derived.The concepts of the density of the energy carried by waves, the energy flux, and the Poynting vector are introduced. Chapter 5 is dedicated to propagation ofspherical, cylindrical, and plane waves in homogeneous media, both in time andfrequency domains. Chapter 6 deals with interference and diffraction. Thetreatment is based on Helmholtz and Kirchhoff formulae. The detailed discussion of Fresnel's and Huygens's principles is presented. In Chapter 7 the effects of interference of waves with close wave numbers and frequencies are considered. Concepts such as the wave group, the group velocity, andthe stationary phase important for understanding propagation of dispersive waves are introduced. The final chapter of the first volume is devoted to the principles of geometrical acoustics in inhomogeneous media.

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### Book information

- Published: March 2000
- Imprint: ELSEVIER
- ISBN: 978-0-444-50336-7

### Reviews

...the authors have done a good job in presenting wave theory starting with very elementary matters and extending this to the detailed 'ramifications' of waves. This book should be profitably read by students and also by specialists, who do not always have the 'ramifications' at their disposal. I await volume 2 with anticipation.

P.G. Malischewsky, Friedrich-Schiller University Jena, Germany, Geophysical Journal International

...The book should be extremely valuable for all who need a rigorous physical and mathematical wave-propagation background which, as a rule, is omitted or drastically shortened (thus difficult to understand) in more specialized monographs. Moreover, most explanations include some innovative ideas, interesting analogies, and/or examples that make the text attractive for wave propagation specialists and experienced lecturers, too.J. Zahradnik and O. Novotny, Charles University, The Leading Edge

...should be useful to graduate students and researchers.H. Kirchner, Pure and Applied Geophysics

### Table of Contents

**Introduction. List of Symbols. Newton's laws and parlide motion.** Newton's laws. Motion of system of particles. **Free and forced vibrations.** Hooke's law of springs. Free vibrations of the system: mass-spring. Forced vibrations of the system: mass-spring. Principles of measuring vibrations. **Propagation.** Propagation of waves along a system of masses and springs. Solution of 1-D wave equation. Boundar conditions. Transversal waves in a spring. **Basic equations for dilatational waves.** Introduction. Wave phenomena in gas and fluid. Wave equation and boundary conditions for dilatational waves. The kinetic and potential energy of the wave flux of the energy Poynting vector. Boundary value problem. Theorem of uniqueness. Gravitational waves in a fluid. **Waves in homogeneous medium.** Spherical waves from an elementary source. Cylindrical waves from linear source in homogenous medium. Plane waves in homogeneous medium. **Interference and diffration.** Superposition of waves in an uniform medium, caused by a system of primary sources. Helmholtz formula. Kirchhoff diffration theory. Fraunhofer and Fresnel diffration. Helmholtz - Kirchhoff formula. Huygens - Fresnel principles. Relationship of potential with initial conditions. Poisson's formula. Transition and transportation equations. **Superposition of sinusoidal waves with different frequencies and wave lengths.** Wave group: Phase and group velocities. Superposition of sinussoidal waves and the method of stationary phase. **Principles of geometrical acoustics.** Introduction. Rays and their general features. Behaviour of rays when velocity is a function of one cartesian coordinate. Behaviour of rays when velocity is a function of one coordintae r. Rays near interfaces. Time fields. **Appendices.** Vector algebra. Scalar field and gradient. Vector fields. Complex numbers. Linear ordinary differential equations with constant coefficients. Fourier series. Fourier integral. Duhamel integral. **References.**