Free oscillations of ideal stratified fluid in container

Oscillations of an ideal imcompressible fluid with a density which at standstill varies in the vertical direction are analyzed, assuming that the fluid container of arbitrary shape, has finite dimensions and is stationary. The problem is formulated for such a fluid completely filling a closed container with an initially stable stratification and for small movements under forces of gravity and buoyancy occurring within its volume. The corresponding equations for the velocity field, deviation of the pressure field from the static one at equilibrium, and deviation of the density field from the initial one are linearized in the Boussinesq approximation. A solenoidal displacement field is then introduced and the equation for the density field kinetics is integrated with respect to time. The resulting initial-value boundary-value problem is put in the form of an operator equation, upon introduction of both inherent and necessary Hilbert spaces. The oscillation spectrum at constant buoying frequency is analyzed next, including natural oscillations found from the corresponding eigenvalue problem, specifically for the case of exponential stratification in a cylindrical container.