5 - Waves in Elastic Solids

Summary

The vibrations of panels in a car, the squeal of a train's brakes and the devastation left by an earthquake (see figure 5.1) are all examples of elastic wave propagation that are familiar to us. One of the features that these phenomena have in common is that the size of the elastic deformation is small compared to other length scales involved. The door panels of a car may only move by tenths of a millimetre to cause a noise, but the dimensions of the door itself may be of the order of a metre. A building need only be raised and tilted by a few tens of centimetres to cause considerable damage, but the length scale associated with the earth's crust is tens of kilometres. For this reason, most common elastic wave phenomena are approximately linear and we shall concentrate on these in this chapter.

Derivation of the Governing Equation

Consider the propagation of small amplitude waves in an ideal elastic body. By ideal, we mean that the stress–strain relationship is linear and isotropic and that the wave motion causes either no change in temperature or no heat flow within the body. Recall that the position vector, x = (x₁, x₂, x₃), of any point in the body after elastic deformation is related to its original position, X = (X₁, X₂, X₃), by a displacement vector, u, through the relationship, x = X + u. If we write u = (u₁, u₂, u₃) with u_i = u_i(x₁, x₂, x₃, t), we are working in an Eulerian frame.