  This work is devoted to the theoretical study of the stability of two
superposed horizontal liquid layers bounded by two solid planes and subjected
to a horizontal temperature gradient.
  The liquids are supposed to be immiscible with a nondeformable interface.
  The forces acting on the system are buoyancy and interfacial tension. Four
different flow patterns and temperature profiles are found for the basic state.
A linear perturbative analysis with respect to two and three dimensional
perturbations reveals the existence of three kind of patterns. Depending on the
relative height of both liquids several situations are predicted: either wave
propagation from cold to the hot regions, or waves propagating in the opposite
direction or still stationary longitudinal rolls. The behavior of three
different pairs of liquids which have been used in experiments on bilayers
under vertical gradient by other authors have been examined. The instability
mechanisms are discussed and a qualitative interpretation of the different
behaviors exhibited by the system is provided. In some configurations it is
possible to find a codimension-two point created by the interaction of two Hopf
modes with different frequencies and wavenumbers. These results suggest to
consider two liquid layers as an interesting prototype for the study of
propagation and interaction of waves in the context of the B\'enard-Marangoni
problem.
