  We study ergodic properties of stochastic geometric wave equations on a
particular model with the target being the 2D sphere while considering the
space variable-independent solutions only. This simplification leads to a
degenerate stochastic equation in the tangent bundle of the 2D sphere. Studying
this equation, we prove existence and non-uniqueness of invariant probability
measures for the original problem and we obtain also results on attractivity
towards an invariant measure. We also present a suitable numerical scheme for
approximating the solutions subject to a sphere constraint.
