A linearly elastic material occupies −h <= x3 <= h. The motion of every point of
the plane x3 = h is prescribed by the displacement components
u1 = b cos pt, u2 = 0, u3 = b sin pt,
where b and p are constants, and the plane x3 = −h is kept fixed.
Assume that the motion of the whole region may be expressed in the form
u1 = f(x3) cos pt, u2 = 0, u3 = g(x3) sin pt
and find differential equations for the functions f and g.
Solve these differential equations subject to appropriate boundary conditions and
hence find expressions for the displacements u1 and u3 in terms of x3 and t.