Book contents
- Frontmatter
- Contents
- Foreword
- 1 Linear elastic waves
- 2 Canonical acoustic-wave problems
- 3 Canonical elastic-wave problems
- 4 Radiation and impedance
- 5 Integral equations for crack scattering
- 6 Scanned acoustic imaging
- 7 Acoustic diffraction in viscous fluids
- 8 Near-cut-off behavior in waveguides
- Appendix A Asymptotic expansions
- Appendix B Some special functions
- References
- Index
1 - Linear elastic waves
Published online by Cambridge University Press: 05 October 2010
- Frontmatter
- Contents
- Foreword
- 1 Linear elastic waves
- 2 Canonical acoustic-wave problems
- 3 Canonical elastic-wave problems
- 4 Radiation and impedance
- 5 Integral equations for crack scattering
- 6 Scanned acoustic imaging
- 7 Acoustic diffraction in viscous fluids
- 8 Near-cut-off behavior in waveguides
- Appendix A Asymptotic expansions
- Appendix B Some special functions
- References
- Index
Summary
Chapter 1 provides the background, both the model equations and some of the mathematical transformations, needed to understand linear elastic waves. Only the basic equations are summarized, without derivation. Both Fourier and Laplace transforms and their inverses are introduced and important sign conventions settled. The Poisson summation formula is also introduced and used to distinguish between a propagating wave and vibration of a bounded body. A general survey of books and collections of papers that bear on the contents of the book are discussed at the end of the chapter.
A linear wave carries information at a particular velocity, the group velocity, which is characteristic of the propagation structure or environment. It is this transmission of information that gives linear waves their special importance. In order to introduce this aspect of wave propagation, we discuss propagation in one-dimensional periodic structures. Such structures are dispersive and therefore transmit information at a speed different from the wavespeed of their individual components.
Model equations
The equations of linear elasticity consist of:
(1) the conservation of linear and angular momentum; and
(2) a constitutive relation relating force and deformation.
In the linear approximation the density ρ is constant. The conservation of mechanical energy follows from (1) and (2). The most important feature of the model is that the force exerted across a surface, oriented by the unit normal nj, by one part of a material on the other is given by the traction ti = τjinj, where τji is the stress tensor.
- Type
- Chapter
- Information
- Elastic Waves at High FrequenciesTechniques for Radiation and Diffraction of Elastic and Surface Waves, pp. 1 - 22Publisher: Cambridge University PressPrint publication year: 2010