Vibrations of Two-Layer Ideal Liquid in a Rigid Cylindrical Vessel with Elastic Bases

Abstract

We analyze the frequency equation of natural vibrations of a heavy ideal two-layer incompressible liquid placed in a rigid circular cylindrical vessel with elastic bases in the form of fixed thin circular plates. We consider the cases of axially symmetric (longitudinal) and asymmetric (transverse) vibrations of the liquid and plates and different limiting cases (degeneration of the plates into membranes, absolutely rigid plates, and the absence of the top plate). It is shown that the frequency spectrum of coupled asymmetric vibrations of the elastic bases and two-layer ideal liquid consists of three sets of frequencies corresponding to the vibrations of the top and bottom elastic bases and to the vibrations of the inner interface of liquids. The frequency spectrum of coupled axially symmetric vibrations consists of the same three sets as for the asymmetric vibrations and an additional frequency of vibration of the liquid column as a single whole. As an example, we perform the analytic and numerical investigations of the frequency spectrum of a homogeneous liquid with free surface and elastic bottom in the form of a membrane.