Mathematical study of the plenar oscillations of a heavy, almost homogeneous, incompressible, inviscid liquid partially filling a container

Abstract

In this paper, the authors, after showing that the problem of the small oscillations of a heavy heterogeneous liquid, which fills partially a container, is not a classical problem with discrete spectrum, study in details the two-dimensional problem in the particular case where the density of the liquid in the equilibrium position can be approximated by a linear function of the height of the particle, which differs very little from a constant, in the fluid domain. Then, the fluid is called “almost homogeneous in the fluid domain”. They prove that, in this case, the spectrum is real and decomposed in two parts: an essential spectrum which fills a bounded interval and a point spectrum formed by a sequence of eigenvalues tending to infinity, by means of the methods of the functional analysis. Finally, they explicit the spectrum in the particular case of a rectangular container.