Motivated by the analysis of the propagation of internal waves in a
stratified ocean, we consider in this article the incompressible Euler
equations with variable density in a flat strip, and we study the evolution of
perturbations of the hydrostatic equilibrium corresponding to a stable vertical
strati-fication of the density. We show the local well-posedness of the
equations in this configuration and provide a detailed study of their linear
approximation. Performing a modal decomposition according to a Sturm-Liouville
problem associated to the background stratification, we show that the linear
approximation can be described by a series of dispersive perturbations of
linear wave equations. When the so called Brunt-Vais{\"a}l{\"a} frequency is
not constant, we show that these equations are coupled, hereby exhibiting a
phenomenon of dispersive mixing. We then consider more specifically shallow
water configurations (when the horizontal scale is much larger than the depth);
under the Boussinesq approximation (i.e. neglecting the density variations in
the momentum equation), we provide a well-posedness theorem for which we are
able to control the existence time in terms of the relevant physical scales. We
can then extend the modal decomposition to the nonlinear case and exhibit a
non-linear mixing of different nature than the dispersive mixing mentioned
above. Finally, we discuss some perspectives such as the sharp stratification
limit that is expected to converge towards two-fluids systems