Том 8

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Browsing Том 8 by Author "Demidov, Valerii"

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    Analysis of wave propagation conditions in a two-layer hydro-dynamic system with a free surface
    (2025) Naradovyi, Volodymyr; Huriev, Vasyl; Demidov, Valerii
    The study examines the problem of the propagation of internal and surface waves in a two-layerhydrodynamic system "a half-space - a layer - a layer with a free surface". A mathematical model ina linear approximation is presented. The research problem is formulated under the assumption thatthe fluids are ideal and incompressible. The mathematical formulation of the problem is given in adimensionless form. Expressions for the deviation of the contact interface η1(x,t) and the free surfaceη2(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 Ω1and Ω2respectively, are obtainedin an analytical form. A dispersion relation that connects the wave number and the wave propagationfrequency is derived. The roots of the dispersion relation, which are the frequencies of wave propagationon the contact interface and on the free surface, are found. An analysis of the roots of the dispersionrelation 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 surfacetension is analyzed.The conducted research indicates that in the absence of surface tension (T1= T2= 0), the densityratio ρ acts as a defining parameter that governs both the quantitative and qualitative characteristics ofthe wave modes in the considered system. A transition from the classical state of the system with clearlyseparated fast surface and slow internal modes to a regime of their inversion was identified, which is asignificant 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 densitystratification and capillarity. Capillary forces lead to a substantial increase in wave frequencies and canbecome 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 thatdetermines the qualitative structure of the modes, including the possibility of their complete inversionunder conditions of strong fluid stratification.