Internal waves are a ubiquitous and significant means of momentum and energy transport in the oceans, atmosphere, and astrophysical bodies. Here, we show that internal wave propagation in nonuniform density stratifications, which are prevalent throughout nature, has a direct mathematical analogy with the classical optical problem of a Fabry-Perot multiple-beam light interferometer. We rigorously establish this correspondence, and furthermore provide the first experimental demonstration of an internal wave interferometer, based on the theory of resonant transmission of internal waves. © 2010 The American Physical Society.