The problem of normal oscillations of a viscous stratified fluid with an elastic membrane

Normal oscillations of a viscous stratified fluid partially filling an arbitrary vessel and bounded above by an elastic horizontal membrane are studied. In this case, we consider a scalar model problem that reflects the main features of the vector spatial problem. The characteristic equation for the eigenvalues of the model problem is obtained, the structure of the spectrum and the asymptotics of the branches of the eigenvalues are studied. Assumptions are made about the structure of the oscillation spectrum of a viscous stratified fluid bounded by an elastic membrane for an arbitrary vessel. It is proved that the spectrum of the problem is discrete, located in the right complex half-plane symmetrically with respect to the real axis, and has a single limit point $+\infty$. Moreover, the spectrum is localized in a certain way in the right half-plane, the location zone depends on the dynamic viscosity of the fluid.