Eigen 1-forms of the laplacian and riemannian submersions

Let $\pi : Z \longrightarrow Y$ be a fiber bundle where Y and Z are compact Riemannian manifolds without boundary. We are primarily interested in the case where $\pi$ is a Riemannian submersion with minimal fibers; this is the case, for example, where Z is the sphere bundle of some vector bundle over Y or where Z is a principal bundle over Y.