Page 257 - Introduction to Continuum Mechanics

Plane Irrotational Wave 241

(a) Determine a, /?, and / if the applied displacement on the plane jcj = 0 is given by
e
ii = (bsin ft>0 l
e
(b) Determine a ,{3, and / if the applied surface traction onjcj = 0 is given by t = (dsin ft>?) i-
Solution. The given displacement field is the superposition of two longitudinal elastic waves
having the same velocity of propagation CL in the positive x\ direction and is therefore a
possible elastic solution.
(a) To satisfy the displacement boundary condition, one simply sets
MI CO.M = ft sin w/ (ii)

or

Since this relation must be satisfied for all time t, we have

and the elastic wave has the foi m

Note that the wavelength is inversely proportional to the forcing frequency a). That is, the
higher the forcing frequency the smaller the wavelength of the elastic wave.
(b) To satisfy the traction boundary condition onxi = 0, one requires that

that is, at x\ = 0, TU = -d sin CD t, T 2i = T$i = 0. For the assumed displacement field

therefore,

i.e.,

To satisfy this relation for all time t, we have