Capillary-gravity waves in a layered fluid

In this work, we analyze the stability of a gravity wave generated on the separation surface of two immiscible liquids inside a moving container and perturbed by a capillary wave. Such a phenomenon is experimentally observed when the amplitude and the frequency of the motion imposed to the container attain certain values. The evolution of the system is described by the variational principle. We assume that the motion of the system is decomposed into two modes: the gravity mode and the capillary mode. With suitable scaling assumptions, it is possible to show that the evolution of the gravity mode is determined by the forcing motion, while the capillary mode is excited by the nonlinear interactions between the capillary and gravity modes. Finally, an analytic dispersion relation is obtained for the pulsation of the capillary mode. This relation is a function of several quantities, all depending on the capillary wave number and the characteristics of the exciting motion.