Matching Conditions on Capillary Ripples: Polarization

The matching conditions at the interface between two non-mixed fluids at rest are obtained directly using the equation of movement of the whole media. This is a non-usual point of view in hydrodynamics courses and our aim is to fix ideas about the intrinsic information contained in the matching conditions, on fluids in this case. Afterward, it is analyzed the polarization of the normal modes at the interface and it is shown that this information can be achieved through a physical analysis and reinforced later by the matching conditions. A detailed analysis of the matching conditions is given to understand the role that plays the continuity of the stress tensor through the interface on the physics of the surface particle movement. The main importance of the viscosity of each medium is deduced.