Page 256 - Introduction to Continuum Mechanics

240 The Elastic Solid

so that the phase velocity CL is obtained to be

Thus, we see that with CL given by Eq. (5.8.3), the wave motion considered is a possible one.
Since for this motion, the components of the rotation tensor
/

are zero at all times, it is known as a plane irrotational wave. As a particle oscillates, its volume
also changes harmonically [the dilatation e = EH - e(2n/l)cos(2ji/l)(xi - c^ t)], the wave is
thus also known as a dilatational wave.
From Eq. (5.8.3), we see that for the plane wave discussed, the phase velocity CL depends
only on the material properties and not on the wave length /. Thus any disturbance represented
by the superposition of any number of one-dimensional plane irrotational wave trains of
different wavelengths propagates, without changing the form of the disturbance (no longer
sinusoidal), with the velocity equal to the phase velocity c^. In fact, it can be easily seen [from
Eq. (5.5.11)] that any irrotational disturbance given by

is a possible motion in the absence of body forces provided that u\ (*]_, t) is a solution of the
simple wave equation

It can be easily verified that Ui = /($), where s = xi ±CL t satisfies the above equation for any
function/, so that disturbances of any form given by/(s) propagate without changing its form
with wave speed c/,. In other words, the phase velocity is also the rate of advance of a finite
train of waves, or, of any arbitrary disturbance, into an undisturbed region.

Example 5.8.1
Consider a displacement field

for a material half-space that lies to the right of the plane xi = 0.