Waves on the Interface of Two-Liquid Layers in a Vertical Periodic Motion
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Two-liquid layers which are horizontally bounded by two rigid walls are vibrated vertically with constant frequency and amplitude. In such a case, the standing waves which are formed on the interface of the two-liquid layers are theoretically studied by a linear theory. The lower layer is assumed to be an incompressible ideal fluid and the upper one to be a viscous fluid. The equation governing the amplitude of the wave on the interface is reduced to an equation of the Mathieu type with viscous damping. The two-liquid layers can be made most unstable by the vertical oscillations if their frequency is twice the natural frequency of our system. However, parametric resonance is possible only when the amplitude of the vertical oscillation exceeds a threshold value. Its value depends mainly on the state of the upper layer. The properties of the standing waves on the interface is clarified through a stability diagram and the growth rate of the disturbances.
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