Page 259 - Introduction to Continuum Mechanics

Plane Equivolumina! Wave 243

Here again the phase velocity cj is independent of the wavelength /, so that it again has the
additional significance of being the wave velocity of a finite train of equivoluminal waves, or
of any arbitrary equivoluminal disturbance into an undisturbed region.

The ratio of the two phase velocities CL and cj is

Since A = 2ft v/(l - 2v\ the ratio is found to depend only on v, in fact

For steel with v - 0.3 , CL/CJ = v% = 1.87. We note that since v<—, c^ is always greater
than c-p.

Example 5.9.1
Consider a displacement field

for a material half-space that lies to the right of the plane x\ = 0
(a) Determine a ,fi and / if the applied displacement onjcj = 0 is given by u = (b sin wffa
e
(b) Determine a, ft and / if the applied surface traction on xi = 0 is t = (dsin o>0 2
Solution. The problem is analogous to that of the previous example.
(a) Using HI (0,?) = bsin <o t, we have

and

(b) Using t = -72162 = (dsin a) i)*2 gi ves

and