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1 - Linear elastic waves

Published online by Cambridge University Press:  05 October 2010

John G. Harris
Affiliation:
University of Illinois, Urbana-Champaign
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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. (1) the conservation of linear and angular momentum; and

  2. (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 Frequencies
Techniques for Radiation and Diffraction of Elastic and Surface Waves
, pp. 1 - 22
Publisher: Cambridge University Press
Print publication year: 2010

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  • Linear elastic waves
  • John G. Harris, University of Illinois, Urbana-Champaign
  • Book: Elastic Waves at High Frequencies
  • Online publication: 05 October 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511781094.002
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  • Linear elastic waves
  • John G. Harris, University of Illinois, Urbana-Champaign
  • Book: Elastic Waves at High Frequencies
  • Online publication: 05 October 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511781094.002
Available formats
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Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Linear elastic waves
  • John G. Harris, University of Illinois, Urbana-Champaign
  • Book: Elastic Waves at High Frequencies
  • Online publication: 05 October 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511781094.002
Available formats
×