Let $(M, g)$, $(N, h)$ be complete Riemannian manifolds (not necessarily compact). Let $f : M \rightarrow (0, \infty)$ be a smooth function, and finally let $$\overline{M} = M \times_f N$$ be the warped product of $M$ and $N$ with warping function $f$.
My question: is there a nice formula for $\Delta_{\overline{M}} : \Omega^p(\overline{M}) \rightarrow \Omega^p(\overline{M})$, the Laplacian operator of $\overline{M}$ on $p$-forms, in terms of $\Delta_M, \Delta_N$ and $f$?