Analysis of three-dimensional standing waves in a two-layer fluid

This article deals with the study of three-dimensional standing waves in a two-layer hydrodynamic system with a finite thickness of layers in a parallelepiped-shaped basin. The model of the problem is constructed for a non-viscous incompressible fluid under the influence of gravity and surface tension forces. The mathematical formulation of the problem involves the Laplace equations for the velocity potentials, two kinematic conditions and one dynamic condition at the interface between two fluids, as well as boundary conditions on the lateral walls, bottom and top. Introducing characteristic scales allows to identify a small parameter in the model equations – the nonlinearity coefficient. So, the mathematical model is reduced to a linear system of differential equations by the method of nondimensionalization. Solutions in the form of standing waves for deviation of the contact surface of two layers and velocity potentials of the upper and lower layers are obtained. It is shown that boundary conditions on the side walls of liquids limit the values of wave numbers to certain discrete values. The formula for the frequency of the standing internal wave depending on the physical parameters of the system is derived. It has been revealed that two natural numbers determine the possible frequencies of points at the contact surface between two fluid layers, which are known as modes of oscillation. Possible forms of standing waves containing nodal lines are graphically depicted, describing the simplest solutions for this problem. The general solution to any problem describing internal standing waves can be expressed as a linear combination of obtained elementary solutions, which are series represented as double sums