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The paper deals with studying trajectories of motion of individual liquid particles in a two-layer hydrodynamic system with a finite layer thickness as well as analyzing phase and group velocities of internal waves in the system. The problem is modeled for an inviscid incompressible fluid under action of the gravity and surface tension forces in a dimensionless form. Solutions of the problem are sought in the form of progressive waves using the multi-scale method. The solutions are expanded in terms of the nonlinearity coefficient. Dependence of the dispersion ratio of the wavenumber is investigated for different values of the surface tension coefficient and the ratio of the layer densities. Formulas are obtained for the group and phase velocities for internal gravity-capillary waves as well as in the limiting case for capillary waves. A comparison of the values of the phase and group velocities of internal waves for different values of the wave number is carried out. It is proved that with an increase in the wave number, the group velocity begins to outstrip the phase velocity, and their equality occurs at the minimum phase velocity. It is shown that the trajectories are ellipses in which the horizontal semi axes are larger than the vertical ones. Formulas are obtained for the semi axes of elliptic trajectories for each of the layers. The character of the change in the semi axes of elliptical trajectories is analyzed depending on the distance from the interface between two liquid layers as well as on the values of the wave number. It is proved that the semi axes of ellipses decrease unevenly with increasing distance from the boundary. The asymmetry of the particle trajectories of each of the layers is shown for the case when the thickness of the lower layer differs from the thickness of the lower layer. The study of the kinematic characteristics of the particle motion makes it possible to simulate real physical wave processes in the World Ocean. The results are also relevant for creating a theoretical basis for experiments.

The study was performed to analyze the flux of energy of internal gravitational-capillary waves in a two-layer hydrodynamic liquid system with finite layer thicknesses. The problem was considered for an ideal incompressible fluid in the field of gravity as well as taking into account the forces of surface tension. The problem was formulated in a dimensionless form for small values of the coefficient of nonlinearity. The dispersion of the gravitational-capillary progressive waves was studied in detail depending on the coefficient of surface tension and the ratio of layer densities. It was proved that with the increase in the wavenumber, the group velocity begins to pass ahead of the phase velocity and their equality occurs at the minimum of the phase velocity. Dependence of the total average energy flux on the wavenumber (wavelength) and thickness of the liquid layers was calculated and graphically analyzed for different values of physical quantities, in particular, density and the coefficient of surface tension. It follows from the analysis that the energy flux of gravitational internal waves increases to a certain maximum value with an increase in the thickness of the lower layer and then approaches a certain limit value. For capillary waves, the energy flux of internal waves is almost independent of the thickness of the lower layer. It was also shown that the average energy flux for gravitational waves at a stable amplitude is almost independent of the wavelength. On the contrary, for capillary waves, the energy flux increases sharply with an increase in the wavenumber. The results of the analysis of the energy flux of internal progressive waves make it possible to qualitatively assess physical characteristics in the development of environmental technologies that use internal undulatory motions in various aquatic environments as a source of energy

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