Page 261 - Introduction to Continuum Mechanics

Plane Equivoluminal Wave 245

Since k - 2ji/l, and a> = 2jic/l, therefore

(c) to satisfy the traction free boundary condition at*2 = ±h, we require that

=
therefore, T^lx =±h ~-upa sinph cos(kxi - a>i) - 0. In order for this relation to be
satisfied for all x\ and f, we must have

Thus,

Each value of n determines a possible displacement field, and the phase velocity c correspond-
ing to each mode is given by

This result indicates that the equivoluminal wave is propagating with a speed c greater than
the speed of a plane equivoluminal wave cj. Note that when/? = 0, c - cj as expected.

Example 5.9.3
An infinite train of harmonic plane waves propagates in the direction of the unit vector e n.
Express the displacement field in vector form for (a) a longitudinal wave, (b) a transverse wave.
Solution. Let x be the position vector of any point on a plane whose normal is e n and whose
distance from the origin is d (Fig. 5.3). Then x-e,, = d. Thus, in order that the particles on
the plane be at the same phase of the harmonic oscillation at any one time, the argument of