Page 260 - Introduction to Continuum Mechanics

244 The Elastic Solid

Example 5.9.2

Consider the displacement field

(a) Show that this is an equivoluminal motion.
(b) From the equation of motion, determine the phase velocity c in terms of p, / ,p 0 and n
(assuming no body forces).
(c) This displacement field is used to describe a type of wave guide that is bounded by the plane
jc 2 = ±h. Find the phase velocity c if these planes are traction free.
Solution, (a) Since

thus, there is no change of volume at any time.

where k is known as the wave number and w is the circular frequency. The only nonzero
stresses are given by (note: MI = HI = 0)

The substitution of the stress components into the third equation of motion yields (the first
two equations are trivially satisfied)

*y
Therefore, with cj = H/p 0,