The problem of exactly determining the general linearly-elastic vibrations of even such a symmetric solid as the rectangular parallelepiped (cuboid) is surprisingly difficult. For stress-free surfaces, an infinite matrix system arises [1,2]. Physically [3], at the surfaces a coupling of transverse and longitudinal waves occurs, so individual component waves cannot in general be treated independently. In this note we restrict attention to the plane-strain case and seek analytic solutions for the natural oscillations of a cuboid with two pairs of opposite sides free.In standard notation (c.f. [4]), the state of plane strain is defined by displacements u, v as functions of only x, y and time t, with w zero, and thus all quantities are independent of z. The stresses rxz and r~z are zero, and
• ~z = v(~-~x +~) (1)where v is Poisson's ratio. (On the surfaces z = constant, the conditions of vanishing w and ~-~, ~-y~ are "mixed boundary conditions", and it is known ([5], p. 321) that such conditions on all surfaces of a cuboid lead to a simple but complete solution for natural oscillations.) We consider the rectangular cross-section 0 ~< x ~< a, 0 ~< y ~< b with stress-free sides and seek exact separable steady oscillating solutions. Thus we set u = f(x)g(y) exp (-ioot) (2) with ~o ~ 0, and a similar form for v. Use of the boundary conditions in the dynamical wave equation gives ~f(0) g"(y) = [0,o2f(0) + (X + 2ix)~"(0)]g(y)where 0 is the density and h, Ix are the Lam6 constants. The differential equation (3) is of second order, but the coefficient on the right *Permanent address: