3R11. Linear Elastic Waves. - JC Harris (Theor and Appl Mech Dept, Univ of Illinois, Urbana IL). Cambridge UP, New York. 2001. 162 pp. ISBN 0-521-64368-6. $69.95.

Reviewed by L Gaul (Inst A of Mech, Univ of Stuttgart, Pfaffenwaldring 9, Stuttgart, 70550, Germany) and S Hurlebaus (Inst A of Mech, Univ of Stuttgart, Allmandring 5 b, Stuttgart, 70550, Germany).

This work provides, in six chapters, a basic coverage of the science and technology of linear elastic waves. Each chapter contains its own summary, some problem statements, and a list of references.

Chapter 1, Simple Wave Solutions summarizes the basic equations of linear elasticity without any derivation. Then, the Laplace and Fourier transforms and their inverses are introduced. For distinguishing between propagation of a wave and vibration of a bounded medium the author introduced the Poisson summation equation. It is unusual to explain the dispersion feature by wave propagation in a one-dimensional discrete lattice. This would be more appropriate in a book on solid state physics.

Chapter 2, Kinematical Descriptions of Waves describes the kinematics of time-dependent and time-harmonic plane waves. The latter is also used for explaining the asymptotic ray expansion. The author constructs spherical and cylindrical waves from collections of homogeneous and inhomogeneous plane waves, as opposed to the direct derivation from the solution of the equations of motions in the corresponding coordinate system. From a didactic point of view, the readership, in this case the students, would surely prefer the other, the more obvious way. However, for people working in the area it is nice to obtain an additional view.

Chapter 3, entitled Reflection, Refraction and Interfacial Waves, deals with waves at an interface between two materials having different densities and wave velocities. Furthermore, the chapter describes waves that propagate along an interface, while decaying perpendicularly away from it. In this chapter, it becomes obvious that the author does not cover the subject in a complete manner. By considering the reflection, he deals only with an incident longitudinal plane wave, and the refraction is treated only for an incident shear SV plane wave. For a textbook as well as for a reference book, it would be advantageous to deal with the missing cases as well. Obviously, this chapter does not incorporate all possible cases, since the reader is expected to solve the reflection coefficients himself (the solution is given, but the derivation is left to the reader). Furthermore, it would be helpful for a better understanding by the students (the readership in the author’s opinion) to show plots of some reflection and refraction coefficients as a function of the angle of incidence. However, the explanation of the phase matching condition is excellent. As a consequence of dealing with reflections, the Rayleigh wave is introduced in this chapter as well. Again some plots presenting the decaying vertical and horizontal amplitudes with depth of the Rayleigh wave, as well as the orientation of the Rayleigh wave particle orbit would be helpful.

Chapter 4, Green’s Tensor and Integral Representation, discusses the formulation of the integral representations of solutions for rather general problems in elastic wave propagation. In this chapter, the reciprocity identity, the Green’s tensor for a full space, the principle of limiting absorption, the integral representation for a source and a scattering problem, and uniqueness of an unbounded region are introduced. The chapter closes with an example that uses the introduced ideas to derive an integral representation for the scattering of an acoustic wave by an elastic inclusion.

According to its title, Radiation and Diffraction, Chapter 5 deals with the basic propagation processes that are encountered when studying radiation or edge diffraction. The first problem under consideration consists of calculating the transient, antiplane radiation excited by a line source at the surface of a half-space using the Cagniard-deHoop method to invert the integral transformations. The second one consists of calculating the time-harmonic, inplane radiation from a two-dimensional center of compression buried in a half-space by using plane wave spectral techniques and the method of steepest descent. Finally, the last one treats the calculation of the diffraction of a time harmonic plane antiplane shear wave by a semi-infinite crack using the Wiener-Hopf method and by using matched asymptotic expansions. An appendix describing the relation between the diffraction integral and the Fresnel integral closes the chapter.

The last chapter, Guided Waves and Dispersion, treats antiplane shear problems. The guided waves are constructed by using partial waves, and their dispersions are calculated by using the transverse resonance principle. Both harmonic and transient excitations of a closed waveguide are examined by using a mode expansion. The harmonic excitation of an open waveguide by a line source is also studied by using both ray and mode representations. The last problem under consideration deals with the propagation in a closed waveguide with a slowly varying thickness using an asymptotic expansion that combines features of both rays and modes. The chapter ends by examining the propagation of information and energy with the group velocity.

All derivations are carefully developed, however, more illustrations would enhance the mathematical development and understanding. Plenty of textual explanation is provided to clarify the topic under consideration. The book has a detailed table of contents and a rich subject index. However, the reference list could be extended by adding some other basic wave propagation books such as Graff 1, Rose 2, Bedford and Drumheller 3, Doyle 4, Kolsky 5 etc. Due to the incomplete treatment of some problem areas and neglecting some basic topics, the book provides an initiative for the reader to extend the ideas and to solve problems which are not included.

In summary, Linear Elastic Waves is a useful work whose major contribution lies in its description and mathematical derivations. Readers may find, however, that the book owes a substantial debt to Achenbach’s classic treatment of the subject 6, since Professor Achenbach was the research advisor of the author. However, Achenbach’s book is out of print, and therefore, this book, which is only about half of the size of Achenbach’s book, would be a welcome replacement to graduate students having started in the subject of wave propagation.

Graff KF (1991), Wave Motion in Elastic Solids, Dover Publications, New York.
Rose JL (1999), Ultrasonic Waves in Solid Media, Cambridge Univ Press, Cambridge.
Bedford A and Drumheller DS (1996), Introduction to Elastic Wave Propagation, John Wiley & Son.
Doyle JF (1997), Wave Propagation in Structures, Springer, New York.
Kolsky H (1963), Stress Waves in Elastic Solids, Dover Publications, New York.
Achenbach JD (1993), Wave Propagation in Elastic Solids, Vol 16 of Applied Mathematics and Mechanics, North Holland, Amsterdam.