Synopsis

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 a vibration of a bounded body.

A linear wave carries information at a particular velocity, the group velocity, which is characteristic of the propagation structure or environment. It is this transmitting of information that gives linear waves their special importance. In order to introduce this aspect of wave propagation, propagation in one-dimensional periodic structures is discussed. 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 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.