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.