Аналiз умов поширення хвиль у двошаровiй гiдродинамiчнiй системi з вiльною поверхнею

The study examines the problem of the propagation of internal and surface waves in a two-layer hydrodynamic system "a half-space - a layer - a layer with a free surface". A mathematical model in a linear approximation is presented. The research problem is formulated under the assumption that the fluids are ideal and incompressible. The mathematical formulation of the problem is given in a dimensionless form. Expressions for the deviation of the contact interface η1(x, t) and the free surface η2(x, t) in the form of traveling waves are found. Expressions for the potentials ϕ1(x, z, t) and ϕ2(x, z, t), whose gradients describe the propagation velocities in the layers Ω1 and Ω2 respectively, are obtained in an analytical form. A dispersion relation that connects the wave number and the wave propagation frequency is derived. The roots of the dispersion relation, which are the frequencies of wave propagation on the contact interface and on the free surface, are found. An analysis of the roots of the dispersion relation depending on the geometric and physical parameters of the system is carried out. In particular, the dependence of the wave propagation frequencies on the wave number without considering surface tension is analyzed.The conducted research indicates that in the absence of surface tension (T1 = T2 = 0), the density ratio ρ acts as a defining parameter that governs both the quantitative and qualitative characteristics of the wave modes in the considered system. A transition from the classical state of the system with clearly separated fast surface and slow internal modes to a regime of their inversion was identified, which is a significant result for a deeper understanding of the dynamics of strongly stratified fluids.The consideration of surface tension forces reveals a complex interaction between the effects of density stratification and capillarity. Capillary forces lead to a substantial increase in wave frequencies and can become a dominant factor for internal modes, effectively neutralizing the influence of density changes. At the same time, it has been established that the density ratio ρ retains its role as the key parameter that determines the qualitative structure of the modes, including the possibility of their complete inversion under conditions of strong fluid stratification.