Physical meaning of Laplace-Beltrami eigenfunctions?

The eigenfunctions of Laplace-Beltrami operator are often used as the basis of functions defined on some manifolds. It seems that there is some kind of connection between eigen analysis of Laplace-Beltrami operator and the natural vibration analysis of objects. I wonder, is my intuition true? What is the physical meaning of Laplace-Beltrami eigenfunctions?

For now, I only know that the eigenfunctions of the Laplace-Beltrami operator are real and orthogonal, thus they could be used as the basis of functions on the manifold where the functions are defined.

-
Actually, I posted the same question on the two sites because I'm not sure which category fits my question more. Thanks anyway for providing help, @John Rennie. –  Fei Zhu Dec 24 '12 at 11:12