In this talk we consider the long-time behavior of solutions to the two dimensional non-homogeneous Euler equations under the Boussinesq approximation posed on a periodic channel. We prove inviscid damping  for the linearized equations around the stably stratified Couette flow using stationary-phase methods of oscillatory integrals. We discuss how these oscillatory integrals arise,  what are the main regularity requirements to carry out the stationary-phase arguments, and how to achieve such regularities.