  We compute the waves propagating on a compact 3-manifold of constant positive
curvature with a non trivial topology: the Poincar\'e dodecahedral space that
is a plausible model of multi-connected universe. We transform the Cauchy
problem to a mixed problem posed on a fundamental domain determined by the
quaternionic calculus. We adopt a variational approach using a space of finite
elements that is invariant under the action of the binary icosahedral group.
The computation of the transient waves is validated with their spectral
analysis by computing a lot of eigenvalues of the Laplace-Beltrami operator.
