  Biharmonic wave equations are of importance to various applications including
thin plate analyses. In this work, the numerical approximation of their
solutions by a $C^1$-conforming in space and time finite element approach is
proposed and analyzed. Therein, the smoothness properties of solutions to the
continuous evolution problem is embodied. High potential of the presented
approach for more sophisticated multi-physics and multi-scale systems is
expected. Time discretization is based on a combined Galerkin and collocation
technique. For space discretization the Bogner--Fox--Schmit element is applied.
Optimal order error estimates are proven. The convergence and performance
properties are illustrated with numerical experiments.
