  It is proven that small-amplitude steady periodic water waves with infinite
depth are unstable with respect to long-wave perturbations. This modulational
instability was first observed more than half a century ago by Benjamin and
Feir. It has been proven rigorously only in the case of finite depth. We
provide a completely different and self-contained approach to prove the
spectral modulational instability for water waves in both the finite and
infinite depth cases.
