We study the modulational instability of a shallow water model, with and without surface tension, which generalizes the Whitham equation to include bi-directional propagation. Without surface tension, the small amplitude periodic traveling waves are modulationally unstable if their wave number is greater than a critical wave number predicting a Benjamin-Feir type instability and the result qual...

It is proven that small-amplitude steady periodic water waves with infinite depth are unstable respect to long-wave perturbations. This modulational instability was first observed more than half a century ago by Benjamin and Feir. has been rigorously only in the case of finite depth. We provide completely different self-contained approach prove spectral for both cases. © 2022 Courant Institute ...